Trajectory of the Universe - - A Mathematics, Physics and Philosophy NotebookThe whole purpose of physics is to find a number, with decimal points, etc! Otherwise you haven't done anything.

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A ''*-Algebra'' is an algebra equipped with an [[involution|Involution]] ''*''.

Links:
* [[WIKIPEDIA - *-Algebra|http://en.wikipedia.org/wiki/*-algebra]]

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A ''2-(31,15,7) Design'' is a [[Hadamard 2-Design|Hadamard Design]].

The lower bound for the number of non-isomorphic such designs is $22,478,260$.

Also see: [[2-(15,7,3) design|2-(15,7,3) Design]].

Papers:
* [[2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) Designs with Automorphisms of Odd Prime Order, and their Related Hadamard Matrices and Codes - I. Bouyukliev|http://caagt.ugent.be/preprints/DCChadamard-revised.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=11509957669821257584&hl=de&as_sdt=2000]]

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''AGL(4,2)'' is an [[affine general linear group|Affine General Linear Group]] which has order
\begin{eqnarray}
&&2^4 (2^4 - 1)(2^4 - 2^1)(2^4 - 2^2)(2^4 - 2^3) \\
&=&16(16 - 1)(16 - 2)(16 - 4)(16 - 8)  \\
&=& 16 \cdot 15 \cdot 14 \cdot 12 \cdot 8  = 16 \cdot 20.160 = 322.560
\end{eqnarray}

$AGL(4,2)$ is the [[automorphism group|Automorphism]] of the [[Reed-Muller code|Reed-Muller Code]] of length 16.

Links:
* [[Finite Relativity - S. H. Cullinane|http://finitegeometry.org/sc/16/finiterelat.html]]

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An ''Active Transformation'' transforms the basis elements of an algebra. See also [[passive transformations|Passive Transformation]].

> Polchinski admits that the condensed-matter sceptics have a point. "I don't think that string theorists have yet come up with anything that condensed-matter theorists don't already know," he says. The quantitative results tend to be re-derivations of answers that condensed-matter theorists had already calculated using more mundane methods.
> - [1] -

By means of ''\AdS/CFT Correspondence'' the mass-spectrum of a glueball, which is a bound state of [[gluons|Gluon]], has been calculated which is in perfect agreement with [[lattice QCD|Lattice QCD]]\-calculations.

See also:
* [[Anti de Sitter space|Anti De Sitter Space]]
* [[dS/CFT correspondence|dS/CFT Correspondence]]


Magazines:
* [[[1] String Theory Finds a Bench Mate (2011) - Z. Merali|http://www.nature.com/news/2011/111019/pdf/478302a.pdf]] [[local|magazines/478302a.pdf]]

Links:
* [[WIKIPEDIA - AdS/CFT Correspondence|http://en.wikipedia.org/wiki/AdS/CFT_correspondence]]

Videos:
* [[KITP Program: Holographic Duality and Condensed Matter Physics 2011|http://online.kitp.ucsb.edu/online/adscmt11/]]

''Adams' Theorem'' states:
If there exists a Hopf map $f: S^n \rightarrow S^{(n + 1)/2}$ with integer valued Hopf invariant $\gamma (f)$, then $n$ must equal to $1$, $3$, $7$ or $15$.

Given two elements $\mb A$ and $\mb X$ of an algebra $\mathcal A$, the ''Adjoint'' $ad_{\mb A}$ is defined as a linear map $ad_{\mb A}: \mathcal A \rightarrow \mathcal A$ given by the [[commutator|Commutator]] product:
\[
ad_{\mb A}(\mb X) = [\mb A, \mb X]
\]
If one replaces every element of an algebra by it's adjoint linear map one gets what is called the ''Adjoint Representation'' of the algebra. Commutation relations of the algebra are retained in the adjoint representation, i.e. given $[\mb A, \mb B] = \mb C$ it follows $[ad_{\mb A}, ad_{\mb B}] = ad_{\mb C}$.

Given elements $\mb A$, $\mb B$ and $\mb C$ of an algebra, they satisfy the ''Adjoint Properties'' if the following conditions are satisfied:
\begin{eqnarray}
\langle \mb A \mb B| \mb C \rangle& =& \langle \mb A | \mb C \mb B^* \rangle \\
\langle \mb A |\mb B \mb C \rangle& =& \langle   \mb B^* \mb A | \mb C \rangle
\end{eqnarray}

An ''Affine General Linear Group $AGL(n,\mathbb F)$'' is an extension of a [[general linear group|General Linear Group]] $GL(n,\mathbb F)$ and defined by
\[
AGL (n, \mathbb F) \equiv \{\gamma_{A,v}(u) = Au + v: A \in GL(n, \mathbb F) , v \in \mathbb F \}
\]
where $\gamma_{A,v}$ are so called ''Affine Linear Transformations'' which are maps $\gamma_{A,v} :\mathbb F \rightarrow \mathbb F$.
Thus $AGL(n,\mathbb F)$ combines linear maps with translations. $AGL(n,\mathbb F)$ contains the groups of these transformations as subgroups.

If the field $\mathbb F$ is finite of order $q$ one also writes:
\[
AGL(n,\mathbb F_q) \equiv AGL(n, q)
\]
In this case one has for the order
\[
ord (AGL(n,q)) = q^n ord (GL(n,q))
\]
!!!!Examples
* [[AGL(4,2)]]

Links:
* [[Binary Coordinate Systems - S. H. Cullinane|http://finitegeometry.org/sc/gen/coord.html]]
* [[Affine Groups and Small Binary Spaces - Expository Note - S. H. Cullinane|http://finitegeometry.org/sc/pg/dt/affinegps.html]]

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An ''Akivis Element'' in the [[free|Free Algebra]] [[nonassociative algebra|Nonassociative Algebra]] is a polynomial which can be expressed using only the [[commutator|Commutator]] and the [[associator|Associator]].

Every Akivis element is a [[primitive element|Primitive Element]], however the converse is not true.

!!!!Examples
In degree $4$ there are six Akivis elements, two involving commutators only,
\begin{eqnarray}
\mathfrak A_1 (\mb A, \mb B, \mb C, \mb D) &\equiv & [[[\mb A, \mb B], \mb C], \mb D] \\
\mathfrak A_2 (\mb A, \mb B, \mb C, \mb D)&\equiv & [[\mb A, \mb B] , [\mb C, \mb D]]
\end{eqnarray}
and the others being a combination of a commutator and an associator,
\begin{eqnarray}
\mathfrak A_3 (\mb A, \mb B, \mb C, \mb D)&\equiv & [[\mb A, \mb B, \mb C], \mb D] \\
\mathfrak A_4 (\mb A, \mb B, \mb C, \mb D)&\equiv & [\mb A, [\mb B, \mb C], \mb D]  \\
\mathfrak A_5 (\mb A, \mb B, \mb C, \mb D)&\equiv &  [[\mb A, \mb B], \mb C, \mb D]  \\
\mathfrak A_6(\mb A, \mb B, \mb C, \mb D) &\equiv & [[\mb A, \mb B, [\mb C, \mb D]]
\end{eqnarray}
For $\mathfrak A_3$ the order of the nesting of the commutator and the associator is converse to the one of $\mathfrak A_4$, $\mathfrak A_5$ and $\mathfrak A_6$.

In degree $4$ there are two primitive elements which are not Akivis elements, given by the [[quaternators|Quaternator]] $\mb p$ and $\mb q$.

Every primitive multilinear nonassociative polynomial of degree $4$ is a linear combination of permutations of these six Akivis elements and the two non-Akivis elements.

!!!!! Resolved form
Using results found under [[commutators of degree 4|Commutators of Degree 4]], we can write
\begin{eqnarray}
\mathfrak A_1 (\mb A, \mb B, \mb C, \mb D) &= & ((\mb{AB})\mb C)\mb D - ((\mb{BA})\mb C)\mb D - (\mb C(\mb{AB}))\mb D + (\mb C(\mb{BA}))\mb D - \mb D((\mb{AB})\mb C) + \mb D((\mb{BA})\mb C) + \mb D(\mb C(\mb{AB})) - \mb D(\mb C(\mb{BA})) \\
\mathfrak A_2 (\mb A, \mb B, \mb C, \mb D)& = &(\mb{AB})(\mb{CD}) - (\mb{AB})(\mb{DC}) - (\mb{BA})(\mb{CD}) + (\mb{BA})(\mb{DC}) - (\mb{CD})(\mb{AB}) + (\mb{DC})(\mb{AB}) + (\mb{CD})(\mb{BA}) - (\mb{DC})(\mb{BA})  \\
\end{eqnarray}
$\mathfrak A_1$ and $\mathfrak A_2$ contain all [[association types|Association Type]] possible in degree $4$.
Moreover
\begin{eqnarray}
\mathfrak A_3 (\mb A, \mb B, \mb C, \mb D)&= & [[\mb A, \mb B, \mb C], \mb D]  = ((\mb {AB} ) \mb C) \mb D - (\mb A (\mb {BC})) \mb D
- \mb D ((\mb {AB} ) \mb C) + \mb D (\mb A (\mb {BC})) \\
\mathfrak A_4 (\mb A, \mb B, \mb C, \mb D)&=& [\mb A, [\mb B, \mb C], \mb D]  = (\mb A (\mb {BC})) \mb D - \mb A ((\mb {BC})\mb D) -  (\mb A (\mb {CB})) \mb D + \mb A ((\mb {CB})\mb D) \\
\mathfrak A_5 (\mb A, \mb B, \mb C, \mb D)&= &  [[\mb A, \mb B], \mb C, \mb D] = ((\mb{AB})\mb C)\mb D - (\mb{AB})(\mb{CD}) - ((\mb{BA})\mb C)\mb D + (\mb{BA})(\mb{CD}) \\
\mathfrak A_6(\mb A, \mb B, \mb C, \mb D) &= & [[\mb A, \mb B, [\mb C, \mb D]] = (\mb{AB}) (\mb{CD}) - \mb A (\mb B (\mb{CD})) - (\mb{AB}) (\mb{DC}) + \mb A (\mb B (\mb{DC}))
\end{eqnarray}
Once again all association types are covered. One finds the interesting chain $2 \rightarrow 4, 4 \rightarrow 1, 1 \rightarrow 3, 3 \rightarrow 5$.

!!!!![[Tensorial|Tensor]] representation
\begin{eqnarray}
(\mathfrak A_1)_{\mu\nu\rho\sigma}^\tau &=& T_{\mu\nu}^\kappa T_{\kappa\rho}^\lambda T_{\lambda\sigma}^\tau \\
(\mathfrak A_2)_{\mu\nu\rho\sigma}^\tau&=&  T_{\mu\nu}^\kappa T_{\rho\sigma}^\lambda T_{\kappa\lambda}^\tau \\
(\mathfrak A_3)_{\mu\nu\rho\sigma}^\tau &=& A_{\mu\nu\rho}^\kappa T_{\kappa\sigma}^\tau \\
(\mathfrak A_4)_{\mu\nu\rho\sigma}^\tau &=& A_{\mu\kappa\sigma}^\tau T_{\nu\rho}^\kappa \\
(\mathfrak A_5)_{\mu\nu\rho\sigma}^\tau &=& T_{\mu\nu}^\kappa A_{\kappa\rho\sigma}^\tau \\
(\mathfrak A_6)_{\mu\nu\rho\sigma}^\tau&=& A_{\mu\nu\kappa}^\tau T_{\rho\sigma}^\kappa
\end{eqnarray}
where $T_{\mu\nu}^\rho$ is the [[torsion tensor|Torsion]] and $A_{\mu\nu\rho}^\sigma$ the [[nonassociativity tensor|Nonassociativity Tensor]]. (See also [[nested commutators and associators|Nested Commutators and Associators]]).


Papers:
* [[On Hopf Algebra Structures over Operads (2004) - R. Holtkamp|http://arxiv.org/abs/math/0407074v2]] [[local|papers/0407074v2.pdf]] [[pct. 27|http://scholar.google.com/scholar?hl=de&lr=&cites=10430974865186339281&um=1&ie=UTF-8&ei=Vxn4TuzLII_AtAab763nDw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCIQzgIwAA]]
* [[Spontaneous Compactification and Nonassociativity (2009) - E. K. Loginov|http://arxiv.org/pdf/0912.1729]] [[local|papers/0912.1729v1.pdf]] [[pct. 1|http://scholar.google.com/scholar?hl=de&lr=&cites=1394583167017500805&um=1&ie=UTF-8&ei=jiX4TpO6IIztsgagxaDpDw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCIQzgIwAA]].
* [[[1] Polynomial Identities for Tangent Algebras of Monoassociative Loops (2011) - M. R. Bremner, S. Madrigada|http://arxiv.org/abs/1111.6113v1]] [[local|papers/1111.6113v1.pdf]] pct. 0, prl. 10

Theses:
* [[Free Akivis Algebra (2005) - Ş. Findik|http://library.cu.edu.tr/tezler/5519.pdf]] [[local|theses/5519.pdf]] (Turkish)
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''Albert'' is a computer algebra system for doing calculus with [[nonassociative algebras|Nonassociative Algebra]].

Links:
* [[Albert website|http://www.cs.clemson.edu/~dpj/albertstuff/albert.html]]

Google books:
* [[Genius: In Their Own Words - The Intellectual Journeys of Seven Great 20th-century Thinkers - D. R. Steele, K. Mommer|http://books.google.de/books?hl=de&lr=&id=0mc4BKpAyr0C&oi=fnd&pg=PR7&ots=JuDZNqInRc&sig=u0vDtCSmEmyD9TNOV5p9GoGtIls]] [[local|google_books/GeniusInTheirOwnWord.pdf]] bct. 0

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Minimum encoding length principles, rooted in (algorithmic) information theory, quantify Ockham’s razor principle, and lead to a solid pragmatic foundation of inductive reasoning. Essentially, one can show that the more one can compress, the better one can predict, and vice versa.

Papers:
* [[A Formal Theory of Inductive Inference, Part I (1962) - R. Solomonoff|http://world.std.com/~rjs/1964pt1.pdf]] [[local|papers/1964pt1.pdf]] {{t1000Cite{[[pct. 1062|http://scholar.google.de/scholar?cites=14535061864587531020&as_sdt=2005&sciodt=2000&hl=de]]}}}
* [[A Formal Theory of Inductive Inference, Part II (1962) - R. Solomonoff Information and Control|http://world.std.com/~rjs/1964pt2.pdf]] [[local|papers/1964pt2.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=10530098178777296492&as_sdt=2005&sciodt=2000&hl=de]]
* [[The Discovery of Algorithmic Probability (1997) - R. J. Solomonoff|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.2763&rep=rep1&type=pdf]] [[local|papers/10.1.1.140.2763.pdf]] [[pct. 61|http://scholar.google.de/scholar?cites=2702713904196761945&as_sdt=2005&sciodt=2000&hl=de]]

Links:
* [[WIKIPEDIA - Algorithmic Probability|http://en.wikipedia.org/wiki/Algorithmic_probability]]
* [[WIKIPEDIA - Ray Solomonoff|http://en.wikipedia.org/wiki/Ray_Solomonoff]]
* [[SCHOLARPEDIA - Algorithmic probability|http://www.scholarpedia.org/article/Algorithmic_probability]]
* [[WIKIPEDIA - Inductive Inference|http://en.wikipedia.org/wiki/Inductive_inference]]
* [[WIKIPEDIA - Prior Probability|http://en.wikipedia.org/wiki/Diffuse_prior#Uninformative_priors]]
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An ''Alternating Group $A_n$'' of degree $n$ is defined by the even permutations of a set of $n$ elements with group operation the composition of even permutations.

$A_n$ is a subgroup of the symmetric group $S_n$.

Galois showed that for $n\ge 5$ $A_n$ is simple.

The order of of the alternating group is given by
\[
\operatorname{ord}(A_n) = \tfrac{n!}{2} = \frac {\operatorname{ord}(S_n)}{2}
\]
!!!!Isomorphisms
See: [[Projective general linear group|Projective General Linear Group]].

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> According to the weak anthropic principle, the conditional probability of finding yourself in a universe compatible with your existence equals 1.
> - Jürgen Schmidhuber - Algorithmic Theories of Everything -

According to the ''Antropic Principle'' the universe is the way it is because we exist.

The term "anthropic principle" was introduced by B. Carter in 1974 and defined in a non-controversial form: "What we can expect to observe must be restricted by the conditions necessary for our presence as observers". He calls this the ''Weak Anthropic Principle'' and defines the controversial ''Strong Anthropic Principle'' in the form: "The universe necessarily has the properties requisite for the existence of life at some stage in its history".

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Links:
* [[WIKIPEDIA - Apophis|http://en.wikipedia.org/wiki/99942_Apophis]]

Papers:
* [[Radial Motion into an Einstein-Rosen Bridge - N. J. Pop?awski|http://www.physics.indiana.edu/~nipoplaw/PLB_687_110.pdf]] [[local|papers/PLB_687_110.pdf]] pct. 0

Links:
* [[NATIONAL GEOGRAPHIC: Every Black Hole Contains Another Universe? - K. Than|http://news.nationalgeographic.com/news/2010/04/100409-black-holes-alternate-universe-multiverse-einstein-wormholes/]]
* [[Is the Big Bang a Black Hole? - P. Gibbs|http://www.desy.de/user/projects/Physics/Relativity/BlackHoles/universe.html]]
* [[The Universe is Not a Black Hole - S. Carroll|http://blogs.discovermagazine.com/cosmicvariance/2010/04/28/the-universe-is-not-a-black-hole/]]
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An ''Associahedron $\mathcal K(n)$'' is an $(n?2)$-dimensional solid [[polytope|Polytope]] (or polyhedron). There is exactly one associahedron of each dimension.

{{center{[img(450px+, )[images/associahedron.jpg]]}}}
{{center{[img(300px+, )[images/associahedron2.jpg]]}}}

In dimension $3$ the associahedron is known as [[Stasheff polytope|Stasheff Polytope]] $\mathcal K(5)$.

Papers:
* [[Root Systems and Generalized Associahedra - S. Fomin, N. Reading|http://arxiv.org/PS_cache/math/pdf/0505/0505518v3.pdf]] [[pct. 45|http://scholar.google.de/scholar?cites=6384921924290557765&hl=de]]
* [[The Diagonal of the Stasheff Polytope - J.-L. Loday|http://www-igm.univ-mlv.fr/~jyt/anr/articles/AA-infinity3.pdf]]
* [[Cluster Algebras: Notes for the CDM-03 Conference - S. Fomin, A. Zelevinsky|http://arxiv.org/PS_cache/math/pdf/0311/0311493v2.pdf]]
* [[The Multiple Facets of the Associahedron - J.-L. Loday|http://www.claymath.org/programs/outreach/academy/LectureNotes05/Lodaypaper.pdf]] pct. 0

Links:
* [[Strange Associations|http://www.ams.org/featurecolumn/archive/associahedra.html]]

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The ''Associator'' is defined as:
\[
[\mb A,\mb B, \mb C] \equiv (\mb{AB})\mb C - \mb A(\mb{BC}) \equiv \mb{AB} \cdot \mb C -\mb A \cdot \mb{BC}
\]
The latter notation is found frequently in literature.

A set of three elements $\mb A$, $\mb B$, $\mb C$ satisfying
\[
[\mb A, \mb B, \mb C] = 0
\]
will be called an ''Associative Triad''. (I.e. such elements lie in the [[nucleus|Nucleus]]).

The components of the associator form a tensor, which will be referred to as [[nonassociativity tensor|Nonassociativity Tensor]].

!!!!Properties
1. ''Linearity''
\[
[\sum_i \lambda_i \mb A_i,\sum_j  \mu_j \mb B_j,\sum_k  \nu_k \mb C_k ] = \sum_{i,j,k} \lambda_i  \mu_j  \nu_k \mb [ \mb A_i,\mb B_j, \mb C_k]
\]
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<<tiddler [[include_tiddlers/Astronomical Objects.html#"Astronomical Objects"]]>>

Links:
* [[Deep Sky Astrophotography by Jordi Gallego|http://astrosurf.com/jordigallego/index.html]]
* [[MIRROR IMAGE -  Peter Shah|http://www.astropix.co.uk/gallery.html]]
* [[Astroimages - Manfred Wasshuber|http://www.astroimages.at/gallery/gallery-nebel.htm]]
* [[Glen Youman's Astrophotos|http://www.astrophotos.net]]

See also:
* [[Astronomical objects|Astronomical Objects]]

Asymptotically free theories become weak at short distances, there is no Landau pole, and these [[quantum field theories|Quantum Field Theory]] are believed to be completely consistent down to any length scale.

Asymptotic freedom can be derived by calculating the beta-function describing the variation of the theory's [[coupling constant|Coupling Constant]] under the [[renormalization group|Renormalization Group]]. For small scales it has to be negative in this case.

For sufficiently short distances or large exchanges of momentum, an asymptotically free theory is amenable to perturbation theory calculations using [[Feynman diagrams|Feynman Diagram]]. Such situations are therefore more theoretically tractable than the long-distance, strong-coupling behaviour also often present in such theories, which is thought to produce confinement.

An example of an asymptotically free theory is [[QCD]]. The decrease of the strong coupling constant with energy has been dramatically confirmed with high precision, at DESY's electron-proton collider, HERA, and in studies at the mass scale of the [[Z boson|W and Z Bosons]] at CERN's Large Electron Positron (LEP) collider.

It has been suggested that the [[gravitational interactions|Gravitation]] could also be asymptotically free, a scenario known as [[asymptotically save gravity|Asymptotically Save Gravity]].


Links:
* [[WIKIPEDIA - Asymptotic Freedom|http://en.wikipedia.org/wiki/Asymptotic_freedom]]
* [[CERN Courier - Asymptotic Freedom wins Nobel (2004)|http://cerncourier.com/cws/article/cern/29178]]

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See [[Kalb-Ramond field|Kalb-Ramond Field]].

''BCH Code'' = ''Bose\-Chaudhuri\-Hocquenghem Code'' belong to a large class of cyclic [[blockcodes|Blockcode]].

Lectures:
* [[Notes on Coding Theory, The Definition of BCH and RS Codes - J. Beachy|http://www.math.niu.edu/~beachy/courses/523/08coding.pdf]]

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''Bach Brackets'' allow for a short notation for sums of tensor components that are a result of a symmetric or antisymmetric permutations of the indices of the tensor $T$.

Examples:
''Symmetrisation''
\begin{eqnarray}
T_{(ij)} &= &\frac1{2!}(T_{ij} +T_{ji}) \\
\\ \, \\
T_{(ijk)}& =& \frac1{3!}(T_{ijk} +T_{ikj}+T_{jki} +T_{jik}+T_{kij}+T_{kji})
\end{eqnarray}

''Anti\-Symmetrisation''
\begin{eqnarray}
T_{[ij]} &= & \frac{1}{2!}(T_{ij} -T_{ji}) \\
\\ \, \\
T_{[ijk]} &= &\frac{1}{3!}(T_{ijk} -T_{ikj}+T_{jki}-T_{jik}+T_{kij}-T_{kji}) \\
&= &\frac 13 (T_{i[jk]} + T_{j[ki]} + T_{k[ij]}) \\
&= &\frac 13 \sigma_{ijk} T_{i[jk]}
\end{eqnarray}
with the [[cyclic sum|Cyclic Sum]] $\sigma_{ijk}$.
An even permutation leads to a positive, a negative one to a negative sign.
Indices between the brackets not to be affected by the permutation are to be set between vertical lines:
\[
T_{(i|jk|l)} = \frac1{2!}(T_{il} +T_{jl})
\]

Relation to the [[commutator|Commutator]]:
\[
S_{[a}T_{b]} = \frac{1}{2}[S_{a} , T_{b}]
\]
!!!!Decomposition
Given a tensor $T_{ij\ldots}$, it can always decomposed into a symmetric and an antisymmetric part.
In terms of Bach brackets this can be expressed as follows:
\[
T_{ij\ldots} = \frac {n!}{2} \left (T_{(ij\ldots)}+ T_{[ij\ldots]} \right )
\]
with $n$ the number of tensor indices.
!!!!!Example
\[
T_{ij} = \left (T_{(ij)} + T_{[ij]} \right ) = \frac 12 (T_{ij} +  T_{ji} + T_{ij} - T_{ji})
\]

<<tiddler [[include_tiddlers/Background Independence.html#"Background Independence"]]>>

<<tiddler [[include_tiddlers/Baker-Campbell-Hausdorff Formula.html#"Baker-Campbell-Hausdorff Formula"]]>>

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<<tiddler [[include_tiddlers/Bargmann-Hall-Wightman Theorem.html#"Bargmann-Hall-Wightman Theorem"]]>>

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Links:
* [[Bel and Bel-Robinson Tensors|http://www.phy.olemiss.edu/~luca/Topics/b/bel.html]] [[local|html/bel.html]]

[[General Relativity|General Relativity]] describes macroscopic (spinless) matter. Therefore a symmetric momentum current appears in the [[Einstein equation|Einstein Field Equations]]. In order to deal with spinor matter, which in general has an asymmetric canonical momentum tensor, one has to execute a symmetrisation procedure. This procedure has to transform the canonical momentum and spin currents into new ones, which also fulfil the conservation laws of momentum and total angular momentum. Furthermore the new momentum current has to be symmetric.
 In general there are many possibilities for such a transformation. In GR, however, this transformation is further restricted by the fact, that there is no equation for the spin current. Therefore the new spin current must vanish. The ''Procedure'' which accomplishes this, is the one of ''Belinfante and Rosenfeld''. No such complicated operation is needed in the [[Einstein-Cartan theory|Einstein-Cartan Theory]]. From this point of view the EC theory is the natural extension of GR into microphysics.

<<tiddler [[include_tiddlers/Bell's Theorem.html#"Bell's Theorem"]]>>

<<tiddler [[include_tiddlers/Berry Phase.html#"Berry Phase"]]>>

Links:
* [[WIKIPEDIA - Bertrand Russell|http://de.wikipedia.org/wiki/Bertrand_Russell]]

Videos:
* [[Bertrand Russell - To our Descendants|http://www.youtube.com/watch?v=g3jnEqXhDNI&feature=related]]
* [[Bertrand Russell on God (1959)|http://www.youtube.com/watch?v=2aPOMUTr1qw&feature=related]]
* [[Bertrand Russell on Clarity and Exact Thinking|http://www.youtube.com/watch?v=mpJcn0Otk7I&feature=related]]

Links:
* [[WIKIPEDIA - Bessel Function|http://en.wikipedia.org/wiki/Bessel_function]]

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* Is there a principle - supposedly mathematical in nature - which allows one to derive all of physics and if so, what is it ?
* What is [[consciousness|Consciousness]] ?
* Can we be [[immortal|Immortality]] ? If we are not by nature, can we engineer immortality or at least (considerably) increase human lifespan ?

A ''Bilinear Covariant'' is a product of the form $\mb {\bar \Psi \Gamma \Psi}$, where $\mb {\Gamma}$ is a $4 \times 4$ complex matrix and $\bs \Psi$ and $\mb {\bar \Psi}$ are a [[Dirac spinor|Dirac Spinor]] and its adjoint respectively.
$16$ linearly independent $\mb {\bar \Psi \Gamma \Psi}$ are maximally possible (which actually span the underlying Clifford algebra [[Cl(1,3)]]). They are given by:
\begin{eqnarray}
S'(\mb x') &= &S(\mb x) = \mb {\bar \Psi}(\mb x) \bs {\Psi} (\mb x)\quad \text{ (1 scalar)} \\
&=& \vert \bs \psi_a \vert^2 +  \vert \bs \psi_b \vert^2 -  \vert \bs \psi_c \vert^2 - \vert \bs \psi_d \vert^2\\
&=& \psi_1^2 +  \psi_2^2 +  \psi_3^2 +  \psi_4^2 - \psi_5^2 -  \psi_6^2 -  \psi_7^2 -  \psi_8^2 \\
\mb V'^{\mu}(\mb x')& =& \mb{V^\mu}(\mb x) =  \Lambda^\mu_\nu \bar {\mb \Psi}(\mb x) \bs \gamma^\nu \bs {\Psi} (\mb x) \quad \text{ (4 vectors)} \\
\mb B'^{\mu\nu}(\mb x')& =& \mb B^{\mu\nu}(\mb x) =  \frac{i}{2} \Lambda^\mu_\rho \Lambda^\mu_\sigma \mb {\bar \Psi} (\mb x) [\bs \gamma^\rho,\bs \gamma^\sigma] \bs {\Psi} (\mb x) \quad \text{ (6 bivectors/antisymmetric tensors)} \\
\mb T'(\mb x') &= &\mb T(\mb x) = \det(\bs \Lambda) \Lambda^\mu_\nu \mb {\bar \Psi(\mb x)} \bs \gamma^\nu \bs \gamma^5 \bs {\Psi}(\mb x) \quad \text{ (4 pseudovectors)} \\
P'(\mb x') &= &P(\mb x) = \det(\bs \Lambda) \mb {\bar \Psi}(\mb x) \bs \gamma^5 \bs {\Psi} (\mb x) \quad \text{ (1 pseudoscalar)} \\
\end{eqnarray}
where the Dirac representation of the [[gamma matrices|Gamma Matrices]] was used.

These bilinear covariants play an important role in determining the possible Lorentz invariant couplings of the Dirac spinor field to other fields.

!!!!Examples
Coupling between the Dirac field and the
* electromagnetic field: $\mathcal L \propto \bar { \mb \Psi}(\mb x) \bs \gamma^\mu \bs {\Psi}(\mb x)  A_\mu(\mb x)   $
* pseudoscalar meson field: $  \mathcal L \propto \mb {\bar \Psi (\mb x)} \bs \gamma^5 \bs {\Psi} (\mb x) \Phi (\mb x)$

<html><center><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_60.html" width=51% height=86></iframe></center></html>Papers:
* [[Observables, Operators, and Complex Numbers in the Dirac Theory - D. Hestenes|http://www.intalek.com/Index/Projects/Research/Observ-opers.pdf]] {{t100Cite{[[pct. 100|http://scholar.google.de/scholar?cites=1346678541492404014&as_sdt=2005&sciodt=2000&hl=de]]}}}
* [[The Electromagnetic Form of the Dirac Electron Theory - A. G. Kyriakos|http://redshift.vif.com/JournalFiles/V11NO2PDF/V11N2KYR.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=5255194155955097275&hl=de&as_sdt=2000]]

Lectures:
* [[Spatial Reflection, Bilinear Covariants, Charge Conjugation, and Time Reversal|http://www.physics.buffalo.edu/phy511/Chapter%2012%20RQM.pdf]]

A ''Binary Code'' of length $n$ and dimension $k$ is a $k$?dimensional vector subspace of $\mathbb F^n_2$. The ''(Hamming) Weight'' of a vector of  $\mathbb F^n_2$ is the number of non-zero coordinates it contains.

<<tiddler [[include_tiddlers/Binary Icosahedral Group.html#"Binary Icosahedral Group"]]>>

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The ''Binomial Coefficient'' ${n \choose k}$ is defined by
\[
{n \choose k} = \frac{n \cdot (n-1) \cdots (n-k+1)}   {k \cdot (k-1) \cdots 1} = \frac{n!}{k!(n-k)!}\,,\,\ 0\leq k\leq n \qquad
\]
For other $n$ and $k$ it is $0$.
!!!!Properties
\[
\sum_{k=0}^{n} {n \choose k} = 2^n
\]
\[
 {n \choose k} = {n-1 \choose k-1} + {n-1 \choose k}
\]
\[
 {n \choose k} =  {n \choose n - k}
\]
!!!!Examples
\[
{7 \choose 3} = \frac{7!}{3!(7-3)!} = \frac{7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(3 \cdot 2 \cdot 1)(4 \cdot 3 \cdot 2 \cdot 1)}  = \frac{7\cdot 6 \cdot 5}{3\cdot 2\cdot 1} = \frac{210}{6} = 35
\]
\[
\sum_{k=0}^{7} {7 \choose k} = 2^7 = 128  = 1 + 7 + 21 + 35 + 35 + 21 + 7 + 1
\]
\begin{eqnarray}
\sum_{k=0}^{15} {15 \choose k} = 2^{15} = 32.768  & =& 1 + 15 + 105 + 455 + 1.365 + 3.003 + 5.005+ 6.435 +  \\
               &&5.005 + 3.003 + 1.365 + 455 + 105 + 15 + 1
\end{eqnarray}
Notice that the numbers correspond to rows in [[Pascal's triangle|Pascal's Triangle]].

Links:
* [[WIKIPEDIA - Binomial Coefficient|http://en.wikipedia.org/wiki/Binomial_coefficient]]
* [[Online Binomial Coefficient Calculator|http://www.ohrt.com/odds/binomial.php]]

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<<tiddler [[include_tiddlers/Biquaternion.html#"Biquaternion"]]>>

''Birefringence'' or ''Double Refraction'' is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through an anisotropic material.
For a medium having no birefringence means that it has a single lightcone.

<<tiddler [[include_tiddlers/Bisedenion.html#"Bisedenion"]]>>

Papers:
* [[On the Physical Interpretation and the Mathematical Structure of the Combinatorial Hierarchy - T. Bastin, H. P. Noyes, J Amson, CW Kilmister|http://www.slac.stanford.edu/pubs/slacpubs/2250/slac-pub-2304.pdf]]  [[pct. 32|http://scholar.google.de/scholar?cites=9803140199832021391&hl=de]]
* [[A Short Introduction to BIT-STRING PHYSICS - H. P. Noyes|http://arxiv.org/PS_cache/hep-th/pdf/9707/9707020v1.pdf]] [[pct. 17|http://scholar.google.de/scholar?cites=15737130025158251635&hl=de]]
* [[A Finite Particle Number Approach to Quantum Physics - H. P. Noyes|http://www.slac.stanford.edu/pubs/slacpubs/2750/slac-pub-2906.pdf]] [[pct. 7|http://scholar.google.de/scholar?cites=346310480967340547&hl=de]]
* [[Fractal Strings as the Basis of Cantorian-fractal Spacetime and the Fine Structure Constant - C. Castro|http://www.scribd.com/doc/13049520/Fractal-Strings-and-Cantorian-Spacetimes-]] pct. 0
* [[From Bit-Strings (part way) to Quaternions - H. P. Noyes|http://www.slac.stanford.edu/pubs/slacpubs/5250/slac-pub-5431.pdf]] pct. 0

Google books:
* [[The Theory of Indistinguishables - A. F. Parker-Rhodes|http://books.google.com/books?id=hHG0IuGm2V8C&dq=The+Theory+of+Indistinguishables&printsec=frontcover&source=bl&ots=LcSyuRupqm&sig=WyIUI6i63Y1Mxcx9o3Sjc7BZ9iQ&hl=de&ei=RZn0SbCJJoKO_Qal56jsCQ&sa=X&oi=book_result&ct=result&resnum=7#PPP1,M1]] [[local|google_books/TheTheoryOfIndistinguishables.pdf]] [[bct. 38|http://scholar.google.de/scholar?cites=5917646935173349348&hl=de]]

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A ''(Left) Bol Algebra'' is a [[Lie Triple system|Lie Triple System]], having in addition a binary operation $(\,,\,)$, satisfying
\begin{eqnarray}
(\mb A, \mb B) &=& - (\mb B, \mb A) \\
(\mb A, \mb B, (\mb C, \mb D)) &= & ((\mb A, \mb B, \mb C), \mb D) + (\mb C,(\mb A, \mb B, \mb D)) + (\mb C, \mb D,(\mb A, \mb B)) + ((\mb A,\mb B),(\mb C, \mb D)) \\
\end{eqnarray}
for any $\mb A$,$\mb B$,$\mb C$,$\mb D$ of its vector space.
(Right Bol algebras can be defined in a similar fashion and do not exhibit new mathematics. One can therefore restrict the treatment to either a left or a right Bol algebra. We'll consider the left case).

For any local analytic [[(left/right) Bol loop|Bol Loop]], a structure of a [[Bol algebra|Bol Algebra]] can be introduced on the [[tangent space|Tangent Algebra]] at unit in a canonical way and is called the (left/right) tangent Bol algebra.
''Theorem''
Any Bol algebra is isomorphic to a tangent Bol algebra, associated uniquely to some [[local analytic Bol loop|Bol Loop]].

Bol algebras generalise [[Lie algebras|Lie Algebra]] and [[Malcev algebras|Malcev Algebra]].

If the trilinear operation in the above definition vanishes identically then the definition becomes that of a Lie algebra.

In any Malcev algebra a ternary bracket can be de?ned by
\[
(\mb A, \mb B, \mb C) = ((\mb A, \mb  B), \mb C) - \frac13 \mb J(\mb A, \mb B, \mb C)
\]

Papers:
* [[The Representation of Bol Algebras (2003) - Ndoune, T. B. Bouetou|http://arxiv4.library.cornell.edu/PS_cache/math/pdf/0305/0305050v1.pdf]] [[local|papers/0305050v1.pdf]] pct. 0
* [[Sabinin's Method for Classification of Local Bol Loops (1999) - A. Vanžurová|http://dml.cz/bitstream/handle/10338.dmlcz/701638/WSGP_18-1998-1_19.pdf]] [[local|papers/WSGP_18-1998-1_19.pdf]] pct. 0
* [[On the Structure of Bol Algebras (2003) - T. B. Bouetou|http://arxiv.org/PS_cache/math/pdf/0310/0310096v1.pdf]] [[local|papers/0310096v1.pdf]] pct. 0

Documents:
* [[Proceedings of the Eighteenth Annual University-wide Seminar WORKSHOP 2009 at the Czech Technical University in Prague|http://www.cvut.cz/pracoviste/odbor-vedy-a-vyzkumu/stranky/konference-veletrhy-a-vystavy/ws2009.pdf]] [[local|documents/ws2009.pdf]]

<<tiddler [[include_tiddlers/Bol Loop.html#"Bol Loop"]]>>

> Over the course of eternity anything is possible. After some Big Bang in the far future, it’s possible that you yourself will re-emerge. But it’s more likely that you will be reincarnated as an isolated brain, without the baggage of stars and galaxies. In terms of probability, it’s "cheaper".
> - Andrei Linde -
<br>{{center{[img(242px+, )[images/BoltzmannBrain.jpg]]}}}

Links:
* [[WIKIPEDIA - Boltzmann Brain|http://en.wikipedia.org/wiki/Boltzmann_brain]]
* [[Richard Feynman on Boltzmann Brains - Sean Carroll|http://blogs.discovermagazine.com/cosmicvariance/2008/12/29/richard-feynman-on-boltzmann-brains/]]

<<tiddler [[include_tiddlers/Boltzmann Constant.html#"Boltzmann Constant"]]>>

Bore Hole experiments allow for testing possible violations of Newton's inverse-square law. Such violations have been reported and are referred to as ''Bore Hole Anomaly''.

Papers:
* [[Test of Newton's Inverse-Square Law in the Greenland Ice Cap - M. E. Ander, M. A. Zumberge, T. Lautzenhiser, R. L. Parker, C. L. V. Aiken, M. R. Gorman, M. M. Nieto, A. P. R. Cooper, J. F. Ferguson, E. Fisher, G. A. McMechan, G. Sasagawa, J. M. Stevenson, G. Backus, A. D. Chave, J. Greer, P. Hammer, B. L. Hansen, J. A. Hildebrand, J. R. Kelty, C. Sidles, J. Wirtz|http://www.whoi.edu/science/AOPE/people/achave/Site/Next_files/28.pdf]] [[pct. 40|http://scholar.google.com/scholar?hl=de&lr=&cites=10079090251362993710&um=1&ie=UTF-8&ei=57jBSqmvMZOe4QbnxMyLCA&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]

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<br><<tiddler [[include_tiddlers/Bra-Ket Notation.html#"Bra-Ket Notation"]]>>

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 In so called ''Brane World Scenarios'' which are cosmological models with extra dimensions it is assumed that ordinary matter is confined to a surface, called a brane, embedded in a higher dimensional spacetime.

These models are in contrast with [[Kaluza-Klein models|Kaluza-Klein Theory]] where matter fields also extend to the extra compact dimensions.

Example: [[Randall-Sundrum model|Randall-Sundrum Model]].

Papers:
* [[Einstein-Cartan Gravity Excludes Extra Dimensions - N. J. Poplawski|http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.4324v1.pdf]] pct. 0
* [[Gravity, Higher Dimensions, Nanotechnology and Particle Physics - M. Ito|http://www.iop.org/EJ/article/1742-6596/89/1/012019/jpconf7_89_012019.pdf?request-id=b3f418bd-8e76-40c4-b483-15282196284f]] pct. 0

Links:
* [[PHYSORG: Light Bending by a Black Hole may offer Proof of Extra Dimensions|http://www.physorg.com/news/2010-11-black-hole-proof-extra-dimensions.html]]

Videos:
* [[Detecting an Extra Dimension|http://www.youtube.com/watch?v=hpx9YklIrMQ&feature=channel]]

A ''(Left) Bruck Loop'' or ''K\-Loop'' is a [[(left) Bol loop|Bol Loop]], satisfying the ''Left Bruck Identity''
\[
(\mb A \mb B) (\mb A  \mb B) = \mb A (\mb B (\mb B \mb A))
\]
or equivalently the ''Automorphic Inverse Identity''
\[
(\mb A \mb B)^{-1}  = \mb A^{-1} \mb B^{-1}
\]
Left Bruck loops are equivalent to Ungar's [[gyrocommutative gyrogroups|Gyrogroup]].

A ''Burgers Vector'' characterizes a [[dislocation|Dislocation]].

{{center{[img(407px+, )[images/BurgersVector.jpg]]}}}
Links:
* [[Dislocations - Institut für Angewandte Physik der Technischen Universität Wien|http://www.iap.tuwien.ac.at/www/surface/stm_gallery/dislocations]]

A ''C''${}^*$''-Algebra'' is a [[Banach*-Algebra|Banach Algebra]] $\mathcal A$ over the field of complex numbers, satisfying the so called ''C${}^*$-Identity''
\[
\|\mb A^* \mb A\| =  \|\mb A\| \|\mb A\| = \|\mb A\|^2
\]
or equivalently
\[
\|\mb A \mb A^* \| = \|\mb A\|^2
\]
$\forall \mb A \in \mathcal A$.

Every $C^*$-algebra per definition is a Banach *-algebra, however the converse is not true in general.

Examples of C${}^*$-algebras are:
* Algebras $\mathcal A(H)$ of bounded linear operators on a [[Hilbert space|Hilbert Space]] $H$.
* Selfadjoint subalgebras $\mathcal A'(H)$ of $\mathcal A(H)$, closed in respect to a norm topology.
According to the [[Gelfand-Neumark-Segal theorem|Gelfand-Naimark Theorem]] any C${}^*$-algebra is isomorphic to an algebra $\mathcal A'(H)$.

A special class of C${}^*$-algebras are [[Von Neumann algebras|Von Neumann Algebra]].

!!!!Historical
In 1943 Gelfand und Neumark introduced the concept of a ''B${}^*$-Algebra''. Later on they could show that any B${}^*$-algebra is a C${}^*$-algebra, which makes the notion of a B${}^*$-algebra superfluous nowadays. Yet it prevails in the older literature.

Papers:
* [[Jordan C*-Algebras - J. D. M. Wright|http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.mmj/1029001946]] [[pct. 88|http://scholar.google.de/scholar?cites=11867268592626511517&as_sdt=2005&sciodt=2000&hl=de]]
* [[State Spaces of C*-Algebras - E. M. Alfsen, H. Hanche-Olsen, F. W. Shultz|http://www.kryakin.com/files/Acta_Mat_%282_55%29/acta150_107/144/144_8.pdf]] [[local|papers/144_8.pdf]] [[pct. 69|http://scholar.google.de/scholar?cites=3007707331766062373&as_sdt=2005&sciodt=2000&hl=de]]
* [[C*-algebras in Tensor Categories - P. Bouwknegt, K. Hannabuss, V. Mathai|http://arxiv.org/PS_cache/math/pdf/0702/0702802v2.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=11912668112600513410&as_sdt=2005&sciodt=2000&hl=de]]

Links:
* [[WIKIPEDIA - C*-Algebra|http://en.wikipedia.org/wiki/C*-algebra#Some_history:_B.2A-algebras_and_C.2A-algebras]]
* [[WIKIPEDIA - Von Neumann Algebra|http://en.wikipedia.org/wiki/Von_Neumann_algebra]]

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<<tiddler [[include_tiddlers/CPT Theorem.html#"CPT Theorem"]]>>

<html><center><img src="images/cpt.jpg" style="width: 230px; "/></center></html>

Links:
* [[Cages - A. E. Brouwer| http://www.win.tue.nl/~aeb/graphs/cages/cages.html]]

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<<tiddler [[include_tiddlers/Callias Index Theorem.html#"Callias Index Theorem"]]>>

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<<tiddler [[include_tiddlers/Canonical Expansion.html#"Canonical Expansion"]]>>

The idea of ''Canonical Polyvector Quantization'' is to lift a non-linear field theory to [[polyvector space|Polyvector Space]], casting it to a quasi-linear formulation. This should allow for applying the classical tools of canonical field quantization.
Therefore on the level of polyvector geometry a quantized polyvector field can be seen as represented by states of a collection of [[harmonic polyvector oscillators|Harmonic Oscillator]] which in fact can be (highly) [[anharmonic oscillators|Anharmonic Oscillator]] on the level of conventional field theory.

Seen more generally, due to the linearity of the description in a polyvecor tangent space, one can expect the axioms of [[quantum mechanics|Quantum Mechanics]] to go through. Therefore it should be possible to lift all the "tools of trade" of [[(relativistic) quantum field theory|Quantum Field Theory]] in a flat spacetime background to polyvector space, also based on a "flat" background.

!!!!Agenda
* One would expect a generalization of the canonical anti-commutation relations of the Dirac creation and anihilation field operators, which depend on the algebra of the respective polyvector space. That is, instead of quantizing the classical [[Dirac equation|Dirac Equation]] one starts out canonically quantizing the [[polyvector Dirac equation|Polyvector Dirac Equation]]. <br><br>
* One can check the formalism by calculating the [[vacuum|Vacuum]] energy. New terms should show up (which are due to nonlinearities in the classical setting) and if one is lucky enough they counter the "ugly" and infamous leading term derived via classial quantum field theory. (That is the hope is to fix the [[cosmological constant|Cosmological Constant]] problem this way).

// TODO to be worked out ... //

See also:
* [[Polyvector quantization|Polyvector Quantization]]

The ''Cantor Set'' is a perfect compromise between the discrete and the [[continuum|Continuum Hypothesis]]. It is a discrete structure, yet it has the same cardinality as the continuum.

A subset of a [[projective geometry|Projective Geometry]] $PG(n, q)$ or an [[affine geomety|Affine Geometry]] $AG(n, q)$ is called a ''Cap'' if no three of its points are collinear. In the case of a [[projective plane|Projective Plane]] a cap is also referred to as an ''Arc''.

A cap of cardinality $k$ is called a ''$k$-Cap''.

The largest $k$ for which a $k$-cap in $PG(n, 2)$ exists is equal to $2^n$ (complement of a hyperplane).

The ''Cardinality'' of a finite set $S$, denoted $\operatorname{card}(S)$ is equal to the number of elements of the set. E.g. $\operatorname {card}(\{2, 4, 6\}) = 3$.

<<tiddler [[include_tiddlers/Carl Gustav Jung.html#"Carl Gustav Jung"]]>>

<<tiddler [[include_tiddlers/Carlos Castro.html#"Carlos Castro"]]>>

The ''Cartan Matrix'' $g_{ij}$ of a rank $r$ [[root system|Root Lattice]] is a $r \times r$ matrix given by
\[
g_{ij} = 2\frac{\langle\mb e_i|\mb e_j\rangle}{\langle \mb e_i|\mb e_i \rangle}
\]
where $\mb e_i$ are the [[simple roots|Simple Root]]. The entries are independent of the choice of simple roots (up to ordering).

A Cartan matrix can also be interpreted as a [[metric tensor|Metric Tensor]].

Papers:
* [[Strings on Orbifolds: An Introduction - H.-P. Nilles|http://cdsweb.cern.ch/record/184239/files/198802208.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=16984624762816364962&hl=de&as_sdt=2000]]

Given a [[Lie Algebra|Lie Group]] with generators $\mb T_i$, a ''Cartan Subalgebra'' is generated by a maximal subset of $N$ commuting [[generators|Generator]] $\mb T_j$, i.e. for which
\[
[\mb T_j, \mb T_k] = 0\text{,}   \quad  \forall \, j,k \in \{1,\ldots N\}
\]
Every finite-dimensional Lie algebra contains at least one Cartan subalgebra. In general, a Lie algebra may have more than one Cartan subalgebra, but they all have the same dimension $N$, called the [[rank|Rank]] of the Lie algebra. The Cartan subalgebras of a [[semisimple|Simple Algebra]] Lie algebra are [[maximal|Subalgebra]] Abelian [[subalgebras|Subalgebra]]. However, the converse is not true. A maximal Abelian subalgebra of a semisimple Lie algebra need not be a Cartan Subalgebra (i.e. there could be several Abelian subalgebras of rank $N$, some of them being Cartan subalgebras, some not).

The ''Cartan Tensor'' $C^\rho_{\mu\nu}$ (a.k.a. ''Modified Torsion Tensor'') is defined by
\begin{equation}
C^\rho_{\mu\nu} = T^\rho_{\mu\nu} + T_\mu \delta_\nu^\rho + T_\nu \delta_\mu^\rho
\end{equation}
with $T^\rho_{\mu\nu}$ the uncontracted and $T_\mu = T^\nu_{\mu\nu}$ the contracted [[Cartan torsion tensor|Torsion]] (a.k.a ''Torsion Vector''). For the latter different normalisations are found in literature.

Papers:
* [[On a Completely Antisymmetric Cartan Torsion Tensor - L. Fabbri|http://arxiv.org/PS_cache/gr-qc/pdf/0608/0608090v2.pdf]] [[pct. 2|http://scholar.google.de/scholar?hl=de&lr=&cites=1653535039168438446]] - There exists yet another different version.

The ''Cartan\-Laptev Method'' is a reinterpretation/generalization of the methods of the mobile frame and of equivalence of Élie Cartan by the Russian geometer German Fedorovich Laptev. The method is used in [[web-theory|Web]].

Papers:
* [[The Cartan-Laptev Method in the Study of G-structures on Manifolds - N. M. Ostianu|http://83.149.209.141/php/getFT.phtml?jrnid=into&paperid=22&volume=30&year=2002&issue=&fpage=5&what=fullt&option_lang=eng]] pct. 0 - Russian

<<tiddler [[include_tiddlers/Casimir Effect.html#"Casimir Effect"]]>>

<<tiddler [[include_tiddlers/Casson Handle.html#"Casson Handle"]]>>

<<tiddler [[include_tiddlers/Catalan Number.html#"Catalan Number"]]>>

<<tiddler [[include_tiddlers/Catastrophe Theory.html#"Catastrophe Theory"]]>>

<<tiddler [[include_tiddlers/Category Theory.html#"Category Theory"]]>>

<<tiddler [[include_tiddlers/Cauchy-Riemann Operator.html#"Cauchy-Riemann Operator"]]>>

Simulations revealed that [[euclidean quantum gravity|Quantum Gravity]] is missing an important ingredient as nonperturbative superpositions of $4$-dimensional universes are inherently unstable.

The reason is that it does not implement the notion of causality and therefore space and time are treated equally.

The method of ''Causal Dynamical Triangulations'' (''CDT'') of spacetime fixes this problem.

It turned out that for the model to work, the [[cosmological constant|Cosmological Constant]] has to be included right from the outset and that is has to correspond with a [[de Sitter geometry|De Sitter Space]].

An important result of the simulations is that the number of spacetime dimensions depends on the scale. That is, the universe has something akin to a [[fractal (self similar) structure|Fractal]].
At short scales the number of [[spectral dimensions|Spectral Dimension]] drops from the classical $4$ to a value of about $2$.

Unlike other approaches to quantum gravity the recipe of causal dynamical triangulations is very robust. It is insensitive to a variety of small-scale details, a property known as universality. (A well-known phenomenon in statistical mechanics).

See also:
* [[Fractal spacetime|Fractal Spacetime]].
* [[Spin network|Spin Network]]
* [[Block universe|Block Universe]]
* [[Emergent spacetime|Emergent Spacetime]]
* [[Spacetime condensate|Spacetime Condensate]]
* [[Causal sets|Causal Sets]]

Papers:
* [[CDT - an Entropic Theory of Quantum Gravity (2010) - J. Ambjørn, A. Görlich, J. Jurkiewicz, R. Loll|http://arxiv.org/PS_cache/arxiv/pdf/1007/1007.2560v1.pdf]] [[local|papers/1007.2560v1.pdf]] pct. 0

Presentations:
* [[Numerical Simulations of Causal Dynamical Triangulations - J. Ambjørn, A. Görlich, J. Jurkiewicz, R. Loll|http://www.pact.cpes.sussex.ac.uk/~dl79/CLAQG/Jurkiewicz.pdf]] [[local|presentations/Jurkiewicz.pdf]]

Documents:
* [[Using Causality to Solve the Puzzle of Quantum Spacetime|http://www.stealthskater.com/Documents/Strings_09.pdf]] [[local|documents/Strings_09.pdf]] [[dct. 2|http://scholar.google.de/scholar?hl=de&lr=&cites=8265450132220242769&um=1&ie=UTF-8&ei=q8ItTZirPM31sgaw94zZBw&sa=X&oi=science_links&ct=sl-citedby&resnum=2&ved=0CCMQzgIwAQ]]

Links:
* [[WIKIPEDIA - Causal Dynamical Triangulation|http://en.wikipedia.org/wiki/Causal_dynamical_triangulation]]

Videos:
* [[Lectures given by Renate Loll at Perimeter Institute|http://pirsa.org/index.php?p=speaker&name=Renate_Loll]]

''Cayley's Theorem'' states that every group is isomorphic to a subgroup of the symmetric group.

Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups) as a permutation group of some underlying set. Thus, theorems which are true for permutation groups are true for groups in general.

<<tiddler [[include_tiddlers/Cayley-Dickson Algebra.html#"Cayley-Dickson Algebra"]]>>

<<tiddler [[include_tiddlers/Cayley-Dickson Doubling.html#"Cayley-Dickson Doubling"]]>>

<<tiddler [[include_tiddlers/Cellular Automaton.html#"Cellular Automaton"]]>>

<<tiddler [[include_tiddlers/Center.html#"Center"]]>>

<<tiddler [[include_tiddlers/Central Charge.html#"Central Charge"]]>>

<<tiddler [[include_tiddlers/Centralizer.html#"Centralizer"]]>>

<<tiddler [[include_tiddlers/Chameleon Particle.html#"Chameleon Particle"]]>>

<br><<tiddler [[include_tiddlers/Chaos.html#"Chaos"]]>>

<<tiddler [[include_tiddlers/Chaotic Quantization.html#"Chaotic Quantization"]]>>

Given a $n \times n$-matrix $\mb M$ the ''Characteristic Polynomial'' $p_{\mb M} (\lambda)$ is defined by
\begin{equation}
p_{\mb M} (\lambda) \equiv \det(\lambda \mb I_n- \mb M)
\end{equation}
It is the solution to the [[eigenvalue problem|Eigenvalue Theory]] $ \mb M \mb A = \lambda \mb A$.
The equation
\begin{equation}
p_{\mb M} (\lambda)=0
\end{equation}
is called the ''Characteristic Equation''.

!!!!Examples
$1 \times 1$''-matrix''
\begin{equation}
p_{m} (\lambda)= \lambda-m
\end{equation}
$2 \times 2$''-matrix''
\begin{eqnarray}
p_{\mb M} (\lambda) &=&  \lambda^2 ? \lambda (m_{11} + m_{22}) + (m_{11}m_{22} - m_{12}m_{21}) \\
& =& \lambda^2 ? \operatorname{Tr} (\mb M) \lambda + \det (\mb M)
\end{eqnarray}

<<tiddler [[include_tiddlers/Checkerboard Lattice.html#"Checkerboard Lattice"]]>>

The ''Chevalley Groups'' are the [[automorphism groups|Automorphism]] of the [[Lie algebras|Lie Algebra]] defined over the [[finite fields|Galois Field]].

<<tiddler [[include_tiddlers/Chiral Anomaly.html#"Chiral Anomaly"]]>>

<<tiddler [[include_tiddlers/Chirality.html#"Chirality"]]>>

The ''Christoffel Symbols'' describe the symmetric ([[Levi-Civita-part|Levi-Civita Connection]]) of a general [[gravitational connection|Connection]].

One distinguishes:
''Christoffel Symbols of first kind''
\[
\{\lambda,\mu,\nu\}
\]
''Christoffel Symbols of second kind''
\[
\Chr{\lambda}{\mu\nu} \equiv g^{\lambda\rho} \{\rho,\mu,\nu\}
\]
A definition of the latter is required as the Christoffel connection is not tensorial which would imply the possibility of raising and lowering indices.

<<tiddler [[include_tiddlers/Church-Turing Hypothesis.html#"Church-Turing Hypothesis"]]>>

<<tiddler [[include_tiddlers/Clifford Algebra.html#"Clifford Algebra"]]>>

<<tiddler [[include_tiddlers/Clifford Algebra Electroweak Theory.html#"Clifford Algebra Electroweak Theory"]]>>

<<tiddler [[include_tiddlers/Clifford Analysis.html#"Clifford Analysis"]]>>

<<tiddler [[include_tiddlers/Clifford Geometric Algebra.html#"Clifford Geometric Algebra"]]>>

<<tiddler [[include_tiddlers/Closed Timelike Curve.html#"Closed Timelike Curve"]]>>

<<tiddler [[include_tiddlers/Coding Theory.html#"Coding Theory"]]>>

<<tiddler [[include_tiddlers/Coherence Law.html#"Coherence Law"]]>>

<<tiddler [[include_tiddlers/Coherent State.html#"Coherent State"]]>>

<<tiddler [[include_tiddlers/Cold Fusion.html#"Cold Fusion"]]>>

<<tiddler [[include_tiddlers/Coleman-Mandula Theorem.html#"Coleman-Mandula Theorem"]]>>

<<tiddler [[include_tiddlers/Collapse of the Wavefunction.html#"Collapse of the Wavefunction"]]>>

<<tiddler [[include_tiddlers/Collective Unconscious.html#"Collective Unconscious"]]>>

A ''Collineation'' (or ''Projective Transformation'', ''Projectivity'') is a bijection between [[projective planes or spaces|Projective Space]], that maps straight lines to straight lines. Hence rectangles are mapped to rectangles (in particular squares are mapped to rectangles). Collineations do not preserve sizes or angles but do preserve coincidences and cross-ratios: two properties which are important in projective geometry. Collineations form a [[group|Group]].

<html><center><img src="images/colllineation.jpg" style="width: 380px; "/></center></html>

Papers:
* [[Combinatorics Entering the Third Millennium - P. J. Cameron|http://www.maths.qmul.ac.uk/~pjc/preprints/pfhist.pdf]] pct. 0

Your comments are very welcome. Please refer to a [[tiddler|What is a Tiddler]] or topic if possible. Thanx a lot !

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The ''Commutator'' (a.k.a. ''Lie Bracket'') of two elements $\mb A$, $\mb B$ of an algebra $\mathcal A$ is defined as:
\[
[\mb A,\mb  B] = \mb{AB} - \mb{BA}
\]
A product defined in this way is called ''Commutator Product'' or ''Lie Product''. An algebra that is obtained from an algebra $\mathcal A$ by replacing the product $\mb{AB}$ with the commutator $[\mb A, \mb B]$ is denoted $\mathcal A^?$.

!!!!Properties
1. ''Antisymmetry''
\[
[\mb A,\mb  B] = - [\mb A,\mb  B]
\]
2. ''Linearity''
\[
[\sum_i \mb \lambda_i \mb A_i,\sum_j  \mu_j\mb B_j] = \sum_{i,j} \lambda_i \mu_j [\mb A_i, \mb B_j]
\]

!!!!Identities
\[
[\mb  A,\mb  A] =0
\]
\[
[\mb  A^*,\mb  B] = - [\mb  A,\mb  B] = [\mb  A,\mb  B^*]
\]
\[
[\mb  A,[\mb  B,\mb  C]] = - [\mb  A,[\mb  C,\mb  B]] = [[\mb  C,\mb  B],\mb  A]  =  - [[\mb  B,\mb  C],\mb  A]
\]
\begin{eqnarray}
[[[\mb A, \mb B], \mb C],\mb D] = ((\mb{AB})\mb C)\mb D ? ((\mb{BA})\mb C)\mb D ? (\mb C(\mb{AB}))\mb D + (\mb C(\mb{BA}))\mb D ? \\\mb D((\mb{AB})\mb C) + \mb D((\mb{BA})\mb C) + \mb D(\mb C(\mb{AB})) ? \mb D(\mb C(\mb{BA}))
\end{eqnarray}
\begin{eqnarray}
[[\mb A, \mb C], [\mb B,\mb D]] = (\mb{AC})(\mb{BD}) ? (\mb{CA})(\mb{BD}) ? (\mb{AC})(\mb{DB}) + (\mb{CA})(\mb{DB}) ? \\(\mb{BD})(\mb{AC}) + (\mb{BD})(\mb{CA}) + (\mb{DB})(\mb{AC}) ? (\mb{DB})(\mb{CA})
\end{eqnarray}

<<tiddler [[include_tiddlers/Commutators of Degree 4.html#"Commutators of Degree 4"]]>>

<<tiddler [[include_tiddlers/Commutators of Degree 5.html#"Commutators of Degree 5"]]>>

<<tiddler [[include_tiddlers/Complex Action.html#"Complex Action"]]>>

<<tiddler [[include_tiddlers/Complex General Relativity.html#"Complex General Relativity"]]>>

<<tiddler [[include_tiddlers/Complex Hamiltonian.html#"Complex Hamiltonian"]]>>

<<tiddler [[include_tiddlers/Complex Spacetime.html#"Complex Spacetime"]]>>

A ''Composition Algebra'' (or [[normed algebra|Normed Algebra]]) is an algebra with a [[multiplicative norm|Norm]].

''Theorems''
* Every composition algebra over a field (of characteristic not equal to $2$) can be obtained by repeated application of the [[Cayley-Dickson construction|Cayley-Dickson Doubling]].
* As composition algebras are normed algebras the [[Hurwitz Theorem]] applies.
* Over any field there is (up to [[isomorphism|Homomorphism]]) exactly one [[Split Composition Algebra|Split Algebra]] of dimension $2$, $4$ and $8$.

A unital composition algebra is called a ''Hurwitz Algebra''.

Furthermore, all triple composition algebras have been determined, up to [[isotopy|Isotopy]], by \McCrimmon.

Papers:
* [[Composition Algebras and their Automorphisms - N. Jacobson|http://www.springerlink.com/content/x432872v0pt48081/fulltext.pdf]]  [[local|papers/CompositionAlgebrasAndTheirAutomorphisms.pdf]] {{t100Cite{[[pct. 136|http://scholar.google.de/scholar?cites=6291925051205774178&hl=de]]}}}

Google books:
* [[Octonions, Jordan Algebras, and Exceptional Groups - T. A. Springer, F. D. Veldkamp|http://books.google.com/books?id=UaeqA5tvSlAC&dq=veldkamp+octonions&printsec=frontcover&source=bl&ots=tbHZdFNhi5&sig=39Rh3jzn3czJgzgJv59gppsL-XI&hl=de&sa=X&oi=book_result&resnum=2&ct=result#PPA18,M1]] {{t100Cite{[[bct. 120|http://scholar.google.de/scholar?cites=910798344559818255&hl=de]]}}}

<<tiddler [[include_tiddlers/Compton Wavelength.html#"Compton Wavelength"]]>>

''Comtrans Algebras'' are [[ternary algebras|Ternary Algebra]] and are due to [[Jonathan D. H. Smith|http://orion.math.iastate.edu/jdhsmith/]].

Their introduction (around 1988) sprang from attempts to finding an algebraic construction similar to local [[Akivis algebras|Akivis Algebra]] for a [[three-web|3-Web]] in the tangent bundle of a coordinate [[n-ary loop|N-Quasigroup]] of a [[(n+1)-web|Web]].

The role played by comtrans algebras is analogous to the one played by [[Lie algebras|Lie Algebra]] of [[Lie groups|Lie Group]]. Furthermore they are analogues of [[Mal'cev|Malcev Algebra]] and Akivis algebras.

Per definitionem, a comtrans algebra satisfies the [[left alternative identity|Alternative Algebra]]
\[
[\mb A, \mb A, \mb B] = 0
\]
and consists of two ternary analogues of the binary commutator (basic trilinear operations), a ''Commutator'' $[\mb A, \mb B, \mb C]$ and a ''Translator'' $\langle \mb A, \mb B, \mb C \rangle$, the latter satisfying the [[Jacobi identity|Jacobian]]
\[
\langle \mb A, \mb B, \mb C \rangle +  \langle \mb B, \mb C, \mb A \rangle + \langle \mb C, \mb A, \mb B \rangle = 0
\]
such that together the commutator and translator obey the so called ''Comtrans Identity''
\[
[\mb A, \mb B, \mb C] = \langle \mb A, \mb B, \mb C \rangle
\]
@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/Akivis.jpg][Comments]]}}}&nbsp;@@

<<tiddler [[include_tiddlers/Conformal Anomaly.html#"Conformal Anomaly"]]>>

<<tiddler [[include_tiddlers/Conformal Cyclic Cosmology.html#"Conformal Cyclic Cosmology"]]>>

<<tiddler [[include_tiddlers/Conformal Field Theory.html#"Conformal Field Theory"]]>>

<<tiddler [[include_tiddlers/Conformal Transformation.html#"Conformal Transformation"]]>>

The ''Conformal Weyl Group'' is the $10$-parameter [[Poincaré group|Poincaré Transformation]] supplemented with a $1$-parameter group of scale transformations.
\[
x'_{\mu}  = e^{\theta} x_\mu\text{,} \quad \bs \psi' (\mb x') = e^{?k\theta} \bs \psi (\mb x)\text{;} \quad k,\theta= const.
\]

<<tiddler [[include_tiddlers/Conic Sedenion.html#"Conic Sedenion"]]>>

Links:
* [[Website Kevin Carmody - Hypernumbers|http://web.archive.org/web/20041204062721/kevincarmody.com/math/hypernumbers.html]]
* [[Tony Smith's Homepage - Zero Divisor Algebras|http://www.valdostamuseum.org/hamsmith/NDalg.html#rulebim]]

Papers:
* [[Circular and Hyperbolic Quaternions, Octonions, and Sedenions - K. Carmody|http://web.archive.org/web/20050130075442/kevincarmody.com/math/sedenions1.pdf]] [[local|papers/sedenions1.pdf]] [[pct. 18|http://scholar.google.de/scholar?cites=12950820281512531271&hl=de]]
* [[Circular and Hyperbolic Quaternions, Octonions, and Sedenions - Further Results - K. Carmody|http://web.archive.org/web/20050130102121/kevincarmody.com/math/sedenions2.pdf]] [[local|papers/sedenions2.pdf]] [[pct. 18|http://scholar.google.de/scholar?cites=10942444327132985935&hl=de]]

<<tiddler [[include_tiddlers/Conjugacy Class.html#"Conjugacy Class"]]>>

<<tiddler [[include_tiddlers/Consciousness.html#"Consciousness"]]>>

> I once was talking to a theologian and he said, "God is infinity." Well, I asked, which one?
> - Charles Musès -

The ''Continuum Hypothesis (CH)'' (advanced by Georg Cantor in 1877 and also known as ''Cantor’s Continuum Hypothesis'') states that if $X \subseteq \mathbb R$ is an uncountable set then there exists a bijection $\pi : X \rightarrow \mathbb R$.
Put it differently:
There are no cardinals strictly between $\aleph_0$ and $2^{\aleph_0}$. The latter cardinal number is also often denoted by $\mathfrak{c}$; it is the cardinality of the continuum (the set of real numbers). In this case $2^{\aleph_0} = \aleph_1$.
I.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers.

Establishing the truth or falsehood of the continuum hypothesis is the first of [[Hilbert's twenty-three problems|Hilbert's Problems]]. The hypothesis can neither be disproved nor be proved using the axioms of Zermelo\–Fraenkel set theory, provided set theory is consistent.

The ''Generalized Continuum Hypothesis (GCH)'' states that for every infinite set $X$ there are no cardinals strictly between $|X|$ and $|2^{X}|$ (the cardinality of the power set).
The extended continuum hypothesis is also independent of the usual axioms of set theory, the Zermelo\-Fraenkel axioms together with the axiom of choice (ZFC).
The GCH reproduces the CH in case that $X = \mathbb N\,$.


Links:
* [[WIKIPEDIA - Continuum Hypothesis|http://en.wikipedia.org/wiki/Continuum_hypothesis]]
* [[WIKIPEDIA - Aleph Number|http://en.wikipedia.org/wiki/Aleph_number]]

<<tiddler [[include_tiddlers/Contorsion.html#"Contorsion"]]>>

<<tiddler [[include_tiddlers/Conway Groups.html#"Conway Groups"]]>>

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!!!!!Revisions
<<<
2011.01.16 2.4.1 in init(), corrected double addition of CookieManager to backstage
2009.08.05 2.4.0 changed CookieJar output format to support odd symbols in option names (e.g. '@')
2008.09.14 2.3.2 fixed handling for blocked cookies (was still allowing some blocked cookies to be set)
2008.09.12 2.3.1 added blocked[] array and allowBrowserCookie() test function for selective blocking of changes to browser cookies based on cookie name
2008.09.08 2.3.0 extensive code cleanup: defined removeCookie(), renamed cookies, added 'button' param for stand-alone "bake cookies" button, improved init of shadow [[CookieManager]] and [[CookieJar]] tiddlers for compatibility with new [[CookieSaverPlugin]].
2008.07.11 2.2.1 fixed recursion error in hijack for saveOptionCookie()
2008.06.26 2.2.0 added chkCookieManagerNoNewCookies option
2008.06.05 2.1.3 replaced hard-coded definition for "CookieJar" title with option variable
2008.05.12 2.1.2 add "cookies" task to backstage (moved from BackstageTasks)
2008.04.09 2.1.0 added options: chkCookieManagerAddToAdvancedOptions
2008.04.08 2.0.1 automatically include CookieManager control panel in AdvancedOptions shadow tiddler
2007.08.02 2.0.0 converted from inline script
2007.04.29 1.0.0 initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.CookieManagerPlugin= {major: 2, minor: 4, revision: 1, date: new Date(2011,1,16)};
//}}}
//{{{
config.macros.cookieManager = {
	target:
		config.options.txtCookieJar||"CookieJar",
	blockedCookies:
		[],
	allowBrowserCookie: function(name) {
		return true;
	},
	displayStatus: function(msg) {
		if (config.options.chkMonitorBrowserCookies && !startingUp)
			displayMessage("CookieManager: "+msg);
	},
	init: function() {
		if (config.options.txtCookieJar===undefined)
			config.options.txtCookieJar=this.target;
		if (config.options.chkAllowBrowserCookies===undefined)
			config.options.chkAllowBrowserCookies=false;
		if (config.options.chkMonitorBrowserCookies===undefined)
			config.options.chkMonitorBrowserCookies=false;

		config.shadowTiddlers.CookieManager=
			 "/***\n"
			+"!!![[Browser cookies:|CookieManagerPlugin]] "
			+"{{fine{<<option chkAllowBrowserCookies>>enable <<option chkMonitorBrowserCookies>>monitor}}}\n"
			+"^^Review, modify, or delete browser cookies..."
			+"To block specific cookies, see [[CookieManagerPluginConfig]].^^\n"
			+"@@display:block;width:30em;{{smallform small{\n<<cookieManager>>}}}@@\n"
			+"***/\n";

		// add CookieManager to shadow CookieJar
		var h="/***\n<<tiddler CookieManager>>\n***/\n";
		var t=(config.shadowTiddlers[this.target]||"").replace(new RegExp(h.replace(/\*/g,'\\*'),''),'')
		config.shadowTiddlers[this.target]=h+t;

		if (config.options.chkCookieManagerAddToAdvancedOptions===undefined)
			config.options.chkCookieManagerAddToAdvancedOptions=true;
		if (config.options.chkCookieManagerAddToAdvancedOptions)
			config.shadowTiddlers.AdvancedOptions+="\n!!CookieManager\n><<tiddler CookieManager>>";

		// add "cookies" backstage task
		if (config.tasks && !config.tasks.cookies) { // for TW2.2b3 or above
			config.tasks.cookies = {
				text: "cookies",
				tooltip: "manage cookie-based option settings",
				content: "{{groupbox{<<tiddler [["+this.target+"]]>>}}}"
			}
			config.backstageTasks.push("cookies");
		}
	},
	handler: function(place,macroName,params,wikifier,paramString,tiddler) {
		var span=createTiddlyElement(place,"span");
		span.innerHTML=(params[0]&&params[0].toLowerCase()=="button")?this.button:this.panel;
		this.setList(span.firstChild.list);
	},
	panel: '<form style="display:inline;margin:0;padding:0" onsubmit="return false"><!--\
		--><select style="width:99%" name="list" \
			onchange="this.form.val.value=this.value.length?config.options[this.value]:\'\';"><!--\
		--></select><br>\
		<input type="text" style="width:98%;margin:0;" name="val" title="enter an option value"><br>\
		<input type="button" style="width:33%;margin:0;" value="get" title="refresh list" \
			onclick="config.macros.cookieManager.setList(this.form.list);"><!--\
		--><input type="button" style="width:33%;margin:0;" value="set" title="save cookie value" \
			onclick="var cmc=config.macros.cookieManager;\
				var opt=this.form.list.value; var v=this.form.val.value; \
				var msg=opt+\' is a blocked cookie.  OK to proceed?\';\
				if ((!cmc.blockedCookies.contains(opt) && cmc.allowBrowserCookie(opt))||confirm(msg)) {\
					config.options[opt]=opt.substr(0,3)==\'txt\'?v:(v.toLowerCase()==\'true\'); \
					saveOptionCookie(opt);config.macros.cookieManager.setList(this.form.list);\
				}"><!--\
		--><input type="button" style="width:33%;margin:0;" value="del" title="remove cookie" \
			onclick="var cmc=config.macros.cookieManager; var opt=this.form.list.value; \
				var msg=opt+\' is a blocked cookie.  OK to proceed?\';\
				if ((!cmc.blockedCookies.contains(opt) && cmc.allowBrowserCookie(opt))||confirm(msg)) {\
					removeCookie(this.form.list.value,true); \
					cmc.setList(this.form.list);\
				}"><br>\
		<input type="button" style="width:50%;margin:0;" value="bake cookies" \
			title="save current cookie-based option values into a tiddler" \
			onclick="return config.macros.cookieManager.bake(this,false)"><!--\
		--><input type="button" style="width:50%;margin:0;" value="bake all options" \
			title="save ALL option values (including NON-COOKIE values) into a tiddler" \
			onclick="return config.macros.cookieManager.bake(this,true)"><!--\
		--></form>\
	',
	button: '<form style="display:inline;margin:0;padding:0" onsubmit="return false"><!--\
		--><input type="button" style="margin:0;" value="bake cookies" \
			title="save current browser-based cookie values into a tiddler" \
			onclick="return config.macros.cookieManager.bake(this,false)"><!--\
		--></form>\
	',
	getCookieList: function() {
		var cookies = { };
		if (document.cookie != "") {
			var p = document.cookie.split("; ");
			for (var i=0; i < p.length; i++) {
				var pos=p[i].indexOf("=");
				if (pos==-1) cookies[p[i]]="";
				else cookies[p[i].substr(0,pos)]=unescape(p[i].slice(pos+1));
			}
		}
		var opt=new Array(); for (var i in config.options) if (cookies[i]) opt.push(i); opt.sort();
		return opt;
	},
	setList: function(list) {
		if (!list) return false;
		var opt=this.getCookieList();
		var sel=list.selectedIndex;
		while (list.options.length > 1) { list.options[1]=null; } // clear list (except for header item)
		list.options[0]=new Option("There are "+opt.length+" cookies...","",false,false);
		if (!opt.length) { list.form.val.value=""; return; } // no cookies
		var c=1;
		for(var i=0; i<opt.length; i++) {
			var txt="";
			if  (opt[i].substr(0,3)=="chk")
				txt+="["+(config.options[opt[i]]?"\u221A":"_")+"] ";
			txt+=opt[i];
			list.options[c++]=new Option(txt,opt[i],false,false);
		}
		list.selectedIndex=sel>0?sel:0;
		list.form.val.value=sel>0?config.options[list.options[sel].value]:"";
	},
	header:
		"/***\n"
		+"!!![[Baked cookies:|CookieManagerPlugin]]\n"
		+"^^Press {{smallform{<<cookieManager button>>}}} to save the current browser cookies... "
		+"then hand-edit this section to customize the results.^^\n"
		+"***/\n",
	format: function(name) {
		if (name.substr(0,3)=='chk')
			return '\tconfig.options["'+name+'"]='+(config.options[name]?'true;':'false;');
		return '\tconfig.options["'+name+'"]="'+config.options[name]+'";';
	},
	bake: function(here,all) {
		if (story.isDirty(this.target)) return false; // target is being hand-edited... do nothing
		var text=store.getTiddlerText(this.target);
		if (text.indexOf(this.header)==-1) {
			text+=this.header;
			displayMessage("CookieManager: added 'Baked Cookies' section to CookieJar");
		}
		var who=config.options.txtUserName;
		var when=new Date();
		var tags=['systemConfig'];
		var tid=store.getTiddler(this.target)||store.saveTiddler(this.target,this.target,text,who,when,tags,{});
		if (!tid) return false; // if no target... do nothing
		if (all) {
			var opts=new Array();
			for (var i in config.options) if (i.substr(0,3)=='chk'||i.substr(0,3)=='txt') opts.push(i);
			opts.sort();
		}
		else var opts=this.getCookieList();
		var t=tid.text;
		if (t.indexOf(this.header)==-1) t+=this.header;
		t+='\n// '+opts.length+(all?' options':' cookies')+' saved ';
		t+=when.formatString('on DDD, MMM DDth YYYY at 0hh:0mm:0ss');
		t+=' by '+who+'//\n';
		t+='//^^(edit/remove username check and/or individual option settings as desired)^^//\n';
		t+='//{{{\n';
		t+='if (config.options.txtUserName=="'+who+'") {\n';
		for (i=0; i<opts.length; i++) t+=config.macros.cookieManager.format(opts[i])+"\n";
		t+='}\n//}}}\n';
		store.saveTiddler(this.target,this.target,t,who,when,tags,tid?tid.fields:{});
		story.displayTiddler(story.findContainingTiddler(this),this.target);
		story.refreshTiddler(this.target,null,true);
		var msg=opts.length+(all?' options':' cookies')+' have been saved in '+this.target+'.  ';
		msg+='Please review all stored settings.';
		displayMessage(msg);
		return false;
	}
}
//}}}
//{{{
// Hijack saveOptionCookie() to add cookie blocking and monitoring messages
config.macros.cookieManager.saveOptionCookie=saveOptionCookie;
window.saveOptionCookie=function(name,force)
{
	var cmc=config.macros.cookieManager; // abbrev
	if (force || ((config.options.chkAllowBrowserCookies || name=="chkAllowBrowserCookies")
		&& !cmc.blockedCookies.contains(name) && cmc.allowBrowserCookie(name))) {
		cmc.saveOptionCookie.apply(this,arguments);
		cmc.displayStatus(name+"="+config.options[name]);
	} else cmc.displayStatus("setting of '"+name+"' is blocked");
}

// if removeCookie() function is not defined by TW core, define it here.
if (window.removeCookie===undefined) {
	window.removeCookie=function(name) {
		document.cookie = name+'=; expires=Thu, 01-Jan-1970 00:00:01 UTC; path=/;';
	}
}

// ... and then hijack it to add cookie blocking and monitoring messages
config.macros.cookieManager.removeCookie=removeCookie;
window.removeCookie=function(name,force)
{
	var cmc=config.macros.cookieManager; // abbrev
	if (!cmc.getCookieList().contains(name))
		return; // not a current cookie!
	if (force || ((config.options.chkAllowBrowserCookies || name=="chkAllowBrowserCookies")
		&& !cmc.blockedCookies.contains(name) && cmc.allowBrowserCookie(name))) {
		cmc.removeCookie.apply(this,arguments);
		cmc.displayStatus("deleted "+name);
	} else cmc.displayStatus("deletion of '"+name+"' is blocked");
}
//}}}

/***
|Name|CookieManagerPluginConfig|
|Source|http://www.TiddlyTools.com/#CookieManagerPluginConfig|
|Requires|CookieManagerPlugin|
|Description|custom settings for [[CookieManagerPlugin]]|
!!!!!Browser Cookie Configuration:
***/
// // <<option chkAllowBrowserCookies>> store ~TiddlyWiki option settings using private browser cookies
// // <<option chkMonitorBrowserCookies>> monitor browser cookie activity (shows a message whenever a cookie is updated)
//{{{
// default settings:
config.options.chkAllowBrowserCookies=false;	// if FALSE, this blocks *all* cookies
config.options.chkMonitorBrowserCookies=false;
//}}}

// // Individual cookie names can be prevented from being created, modified, or deleted in your browser's stored cookies by adding them to the {{{config.macros.cookieManager.blockedCookies}}} array:
//{{{
var bc=config.macros.cookieManager.blockedCookies; // abbreviation
bc.push("txtMainTab");			// TiddlyWiki:  SideBarTabs
bc.push("txtTOCSortBy");		// TiddlyTools: TableOfContentsPlugin
bc.push("txtCatalogTab");		// TiddlyTools: CatalogTabs
//}}}
// // You can also define a javascript test function that determines whether or not any particular cookie name should be blocked or allowed.  The following function should return FALSE if the browser cookie should be blocked, or TRUE if changes to the cookie should be allowed:
//{{{
config.macros.cookieManager.allowBrowserCookie=function(name) {
	// add tests based on specific cookie names and runtime conditions
	return true;
}
//}}}

The ''Copernican Principle'' states that the universe is homogenous - when viewed on a very large scale, different parts of the universe look essentially the same.

The Copernican principle is a built-in assumption of the current favoured solutions to [[Einstein's equations|Einstein Field Equations]], called the Friedmann\-Robertson\-Walker space-times.

<<tiddler [[include_tiddlers/Coquaternion.html#"Coquaternion"]]>>

/***
|Name|CoreTweaks|
|Source|http://www.TiddlyTools.com/#CoreTweaks|
|Version||
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.2.0|
|Type|plugin|
|Description|a small collection of overrides to TW core functions|
This tiddler contains small changes to TW core functions that correct or enhance standard features or behaviors.
***/
//{{{
// calculate TW version number - used to determine which tweaks should be applied
var ver=version.major+version.minor/10+version.revision/100;
//}}}
/***
----

***/
// // to be fixed in 2.6.0:
// // {{block{
/***
!!!1151 adjust popup placement when root element is in scrolled DIV
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/1151
When a popup link is placed inside a DIV with style "overflow:scroll" or "overflow:auto" and that DIV is then scrolled, the position of the resulting popup appears further down the page that intended, because it is not adjusting for the relative scroll offset of the containing DIV.  This tweak patches the Popup.place() function to calculate and subtract the current scroll offset from the computed popup position, so that it appears in the correct location on the page.

Test case: //(scroll to the bottom of this DIV and click on "test popup")//
{{groupbox{
 <<tiddler ScrollBox with: CoreTweaks##1151test 12em>>}}}/%
!1151test
<<tiddler About>>
<<showPopup tiddler:About label:"test popup" tip:About popupClass:sticky>>
!end
%/
***/
//{{{
window.findScrollOffsetX=function(obj) {
	var x=0;
	while(obj) {
		if (obj.scrollLeft && obj.nodeName!='HTML')
			x+=obj.scrollLeft;
		obj=obj.parentNode;
	}
	return -x;
}

window.findScrollOffsetY=function(obj) {
	var y=0;
	while(obj) {
		if (obj.scrollTop && obj.nodeName!='HTML')
			y+=obj.scrollTop;
		obj=obj.parentNode;
	}
	return -y;
}

var fn=Popup.place.toString();
if (fn.indexOf('findScrollOffsetX')==-1) { // only once
	fn=fn.replace(/var\s*rootLeft\s*=/,'var rootLeft = window.findScrollOffsetX(root) +');
	fn=fn.replace(/var\s*rootTop\s*=/,'var rootTop = window.findScrollOffsetY(root) +');
	eval('Popup.place='+fn);
}
//}}}
// // }}}}}}// // {{block{
/***
!!!1147 tiddler macro with params does not refresh
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/1147
when the {{{<<tiddler SomeTiddler>>}}} macro is handled, the resulting span has extra attributes: {{{refresh='content'}}} and {{{tiddler='SomeTiddler'}}}.  If SomeTiddler is changed, {{{store.notify('SomeTiddler')}}} triggers {{{refreshDisplay()}}}, which automatically re-renders transcluded content in any span that has these extra attributes.  However, when additional arguments are passed by using {{{<<tiddler SomeTiddler with: arg arg arg ...>>}}} then the resulting span does NOT get the extra attributes noted above and, as a consequence, the transcluded content is not being refreshed, even though the underlying tiddler has changed

To correct this, in {{{config.macros.tiddler.handler}}}:
*set the 'refresh' and 'tiddler' attributes even when arguments are present in the macro
*store the arguments themselves in an attribute (e.g, 'args'), using as a space-separated, bracketed list
Then, in {{{config.refreshers.content}}}:
*retrieve the stored arguments (if any) and the tiddler source
*substitute arguments into source and re-render the span with the updated content

***/
//{{{
config.refreshers.content=function(e,changeList) {
		var title = e.getAttribute("tiddler");
		var force = e.getAttribute("force");
		var args = e.getAttribute("args"); // ADDED
		if(force != null || changeList == null || changeList.indexOf(title) != -1) {
			removeChildren(e);
//			wikify(store.getTiddlerText(title,""),e,null,store.fetchTiddler(title)); // REMOVED
			config.macros.tiddler.transclude(e,title,args); // ADDED
			return true;
		} else
			return false;
};

config.macros.tiddler.handler=function(place,macroName,params,wikifier,paramString,tiddler) {
	params = paramString.parseParams("name",null,true,false,true);
	var names = params[0]["name"];
	var tiddlerName = names[0];
	var className = names[1] || null;
	var args = params[0]["with"];
	var wrapper = createTiddlyElement(place,"span",null,className);
//	if(!args) { // REMOVED
		wrapper.setAttribute("refresh","content");
		wrapper.setAttribute("tiddler",tiddlerName);
// 	} // REMOVED
	if(args!==undefined) wrapper.setAttribute("args",'[['+args.join(']] [[')+']]'); // ADDED
	this.transclude(wrapper,tiddlerName,args); // REFACTORED TO ...tiddler.transclude
}

// REFACTORED FROM ...tiddler.handler
config.macros.tiddler.transclude=function(wrapper,tiddlerName,args) {
	var text = store.getTiddlerText(tiddlerName); if (!text) return;
	var stack = config.macros.tiddler.tiddlerStack;
	if(stack.indexOf(tiddlerName) !== -1) return;
	stack.push(tiddlerName);
	try {
		if (typeof args == "string") args=args.readBracketedList(); // ADDED
		var n = args ? Math.min(args.length,9) : 0;
		for(var i=0; i<n; i++) {
			var placeholderRE = new RegExp("\\$" + (i + 1),"mg");
			text = text.replace(placeholderRE,args[i]);
		}
		config.macros.tiddler.renderText(wrapper,text,tiddlerName,null); // REMOVED UNUSED 'params'
	} finally {
		stack.pop();
	}
};
//}}}
// // }}}}}}// // {{block{
/***
!!!1134 allow leading whitespace in section headings / TBD handle shadow tiddler sections
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/1134
This tweak REPLACES and extends {{{store.getTiddlerText()}}} so it can return sections defined in shadow tiddlers as well as permitting use of leading whitespace in section headings.
***/
//{{{
TiddlyWiki.prototype.getTiddlerText = function(title,defaultText)
{
	if(!title) return defaultText;
	var parts = title.split(config.textPrimitives.sectionSeparator);
	var title = parts[0];
	var section = parts[1];
	var parts = title.split(config.textPrimitives.sliceSeparator);
	var title = parts[0];
	var slice = parts[1]?this.getTiddlerSlice(title,parts[1]):null;
	if(slice) return slice;
	var tiddler = this.fetchTiddler(title);
	var text = defaultText;
	if(this.isShadowTiddler(title))
		text = this.getShadowTiddlerText(title);
	if(tiddler)
		text = tiddler.text;
	if(!section) return text;
	var re = new RegExp("(^!{1,6}[ \t]*" + section.escapeRegExp() + "[ \t]*\n)","mg");
	re.lastIndex = 0;
	var match = re.exec(text);
	if(match) {
		var t = text.substr(match.index+match[1].length);
		var re2 = /^!/mg;
		re2.lastIndex = 0;
		match = re2.exec(t); //# search for the next heading
		if(match)
			t = t.substr(0,match.index-1);//# don't include final \n
		return t;
	}
	return defaultText;
};
//}}}
// // }}}}}}// // {{block{
/***
!!!824 ~WindowTitle - alternative to combined ~SiteTitle/~SiteSubtitle in window titlebar
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/824 - OPEN
This tweak allows definition of an optional [[WindowTitle]] tiddler that, when present, provides alternative text for display in the browser window's titlebar, instead of using the combined text content from [[SiteTitle]] and [[SiteSubtitle]] (which will still be displayed as usual in the TiddlyWiki document header area).

Note: this ticket replaces http://trac.tiddlywiki.org/ticket/401 (closed), which proposed using a custom [[PageTitle]] tiddler for this purpose.  ''If you were using the previous '401 ~PageTitle' tweak, you will need to rename [[PageTitle]] to [[WindowTitle]] to continue to use your custom window title text''
***/
//{{{
config.shadowTiddlers.WindowTitle='<<tiddler SiteTitle>> - <<tiddler SiteSubtitle>>';
window.getPageTitle=function() { return wikifyPlain('WindowTitle'); }
store.addNotification('WindowTitle',refreshPageTitle); // so title stays in sync with tiddler changes
//}}}
// // }}}}}}// // {{block{
/***
!!!471 'creator' field for new tiddlers
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/471 - OPEN
This tweak HIJACKS the core's saveTiddler() function to automatically add a 'creator' field to a tiddler when it is FIRST created. You can use """<<view creator>>""" (or """<<view creator wikified>>""" if you prefer) to show this value embedded directly within the tiddler content, or {{{<span macro="view creator"></span>}}} in the ViewTemplate and/or EditTemplate to display the creator value in each tiddler.
***/
//{{{
// hijack saveTiddler()
TiddlyWiki.prototype.CoreTweaks_creatorSaveTiddler=TiddlyWiki.prototype.saveTiddler;
TiddlyWiki.prototype.saveTiddler=function(title,newTitle,newBody,modifier,modified,tags,fields)
{
	var existing=store.tiddlerExists(title);
	var tiddler=this.CoreTweaks_creatorSaveTiddler.apply(this,arguments);
	if (!existing) store.setValue(title,'creator',config.options.txtUserName);
	return tiddler;
}
//}}}
// // }}}}}}
// // fixed in ~TW2.4.3
// // {{block{
/***
!!!444 'tiddler' and 'place' - global variables for use in computed macro parameters
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/444 - CLOSED:FIXED - TW2.4.3 - http://trac.tiddlywiki.org/changeset/8367
When invoking a macro, this tweak makes the current containing tiddler object and DOM rendering location available as global variables (window.tiddler and window.place, respectively).  These globals can then be used within //computed macro parameters// to retrieve tiddler-relative and/or DOM-relative values or perform tiddler-specific side-effect functionality.
***/
//{{{
if (ver<2.43) {
window.coreTweaks_invokeMacro = window.invokeMacro;
window.invokeMacro = function(place,macro,params,wikifier,tiddler) {
	var here=story.findContainingTiddler(place);
	window.tiddler=here?store.getTiddler(here.getAttribute('tiddler')):tiddler;
	window.place=place;
	window.coreTweaks_invokeMacro.apply(this,arguments);
}
}
//}}}
// // }}}}}}
// // fixed in ~TW2.4.2:
// // {{block{
/***
!!!823 apply option values via paramifiers (e.g. #chk...and #txt...)
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/823 - CLOSED:FIXED - TW2.4.2 http://trac.tiddlywiki.org/changeset/7988
This tweak extends and ''//replaces//'' the core {{{invokeParamifier()}}} function to support use of ''option paramifiers'' that set TiddlyWiki option values on-the-fly, directly from a document URL.

If a paramifier begins with 'chk' (checkbox) or 'txt' (text field), it's value will be automatically stored in {{{config.options.*}}}, adding to or overriding any existing 'chk' or 'txt' option values that may have already been loaded from browser cookies and/or assigned by the TW core or plugin initialization functions using hard-coded default values.  Note: option values that have been overriden by paramifiers are only applied during the current document session, and are not //automatically// retained.  However, if you edit an overridden option value during that session, then the modified value is, of course, saved in a browser cookie, as usual.
***/
//{{{
if (ver<2.42) {
function invokeParamifier(params,handler)
{
	if(!params || params.length == undefined || params.length <= 1)
		return;
	for(var t=1; t<params.length; t++) {
		var p = config.paramifiers[params[t].name];
		if(p && p[handler] instanceof Function)
			p[handler](params[t].value);
		else { // not a paramifier with handler()... check for an 'option' prefix
			var h=config.optionHandlers[params[t].name.substr(0,3)];
			if (h && h.set instanceof Function)
				h.set(params[t].name,params[t].value);
		}
	}
}
}
//}}}
// // }}}}}}
// // closed: won't fix //(leave as core tweaks)//
// // {{block{
/***
!!!637 TiddlyLink tooltip - custom formatting
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/637 - CLOSED: WON'T FIX
This tweak modifies the tooltip format that appears when you mouseover a link to a tiddler.  It adds an option to control the date format, as well as displaying the size of the tiddler (in bytes)

Tiddler link tooltip format:
{{stretch{<<option txtTiddlerLinkTootip>>}}}
^^where: %0=title, %1=username, %2=modification date, %3=size in bytes, %4=description slice^^
Tiddler link tooltip date format:
{{stretch{<<option txtTiddlerLinkTooltipDate>>}}}
***/
//{{{
config.messages.tiddlerLinkTooltip='%0 - %1, %2 (%3 bytes) - %4';
config.messages.tiddlerLinkTooltipDate='DDD, MMM DDth YYYY 0hh12:0mm AM';

config.options.txtTiddlerLinkTootip=
	config.options.txtTiddlerLinkTootip||config.messages.tiddlerLinkTooltip;
config.options.txtTiddlerLinkTooltipDate=
	config.options.txtTiddlerLinkTooltipDate||config.messages.tiddlerLinkTooltipDate;

Tiddler.prototype.getSubtitle = function() {
	var modifier = this.modifier;
	if(!modifier) modifier = config.messages.subtitleUnknown;
	var modified = this.modified;
	if(modified) modified = modified.formatString(config.options.txtTiddlerLinkTooltipDate);
	else modified = config.messages.subtitleUnknown;
	var descr=store.getTiddlerSlice(this.title,'Description')||'';
	return config.options.txtTiddlerLinkTootip.format([this.title,modifier,modified,this.text.length,descr]);
};
//}}}
// // }}}}}}// // {{block{
/***
!!!607 add HREF link on permaview command
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/607 - CLOSED: WON'T FIX
This tweak automatically sets the HREF for the 'permaview' sidebar command link so you can use the 'right click' context menu for faster, easier bookmarking.  Note that this does ''not'' automatically set the permaview in the browser's current location URL... it just sets the HREF on the command link.  You still have to click the link to apply the permaview.
***/
//{{{
config.macros.permaview.handler = function(place)
{
	var btn=createTiddlyButton(place,this.label,this.prompt,this.onClick);
	addEvent(btn,'mouseover',this.setHREF);
	addEvent(btn,'focus',this.setHREF);
};
config.macros.permaview.setHREF = function(event){
	var links = [];
	story.forEachTiddler(function(title,element) {
		links.push(String.encodeTiddlyLink(title));
	});
	var newURL=document.location.href;
	var hashPos=newURL.indexOf('#');
	if (hashPos!=-1) newURL=newURL.substr(0,hashPos);
	this.href=newURL+'#'+encodeURIComponent(links.join(' '));
}
//}}}
// // }}}}}}// // {{block{
/***
!!!458 add permalink-like HREFs on internal TiddlyLinks
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/458 - CLOSED: WON'T FIX
This tweak assigns a permalink-like HREF to internal Tiddler links (which normally do not have any HREF defined).  This permits the link's context menu (right-click) to include 'open link in another window/tab' command.  Based on a request from Dustin Spicuzza.
***/
//{{{
window.coreTweaks_createTiddlyLink=window.createTiddlyLink;
window.createTiddlyLink=function(place,title,includeText,theClass,isStatic,linkedFromTiddler,noToggle)
{
	// create the core button, then add the HREF (to internal links only)
	var link=window.coreTweaks_createTiddlyLink.apply(this,arguments);
	if (!isStatic)
		link.href=document.location.href.split('#')[0]+'#'+encodeURIComponent(String.encodeTiddlyLink(title));
	return link;
}
//}}}
// // }}}}}}
// // open tickets:
// // {{block{
/***
!!!608/609/610 toolbars - toggles, separators and transclusion
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/608 - OPEN (more/less toggle)
http://trac.tiddlywiki.org/ticket/609 - OPEN (separators)
http://trac.tiddlywiki.org/ticket/610 - OPEN (wikify tiddler/slice/section content)

This combination tweak extends the """<<toolbar>>""" macro to add use of '<' to insert a 'less' menu command (the opposite of '>' == 'more'), as well as use of '*' to insert linebreaks and "!" to insert a vertical line separator between toolbar items.  In addition, this tweak add the ability to use references to tiddlernames, slices, or sections and render their content inline within the toolbar, allowing easy creation of new toolbar commands using TW content (such as macros, links, inline scripts, etc.)

To produce a one-line style, with "less" at the end, use
| ViewToolbar| foo bar baz > yabba dabba doo < |
or to use a two-line style with more/less toggle:
| ViewToolbar| foo bar baz > < * yabba dabba doo |
***/
//{{{
merge(config.macros.toolbar,{
	moreLabel: 'more\u25BC',
	morePrompt: 'Show additional commands',
	lessLabel: '\u25C4less',
	lessPrompt: 'Hide additional commands',
	separator: '|'
});
config.macros.toolbar.onClickMore = function(ev) {
	var e = this.nextSibling;
	e.style.display = 'inline'; // show menu
	this.style.display = 'none'; // hide button
	return false;
};
config.macros.toolbar.onClickLess = function(ev) {
	var e = this.parentNode;
	var m = e.previousSibling;
	e.style.display = 'none'; // hide menu
	m.style.display = 'inline'; // show button
	return false;
};
config.macros.toolbar.handler = function(place,macroName,params,wikifier,paramString,tiddler) {
	for(var t=0; t<params.length; t++) {
		var c = params[t];
		switch(c) {
			case '!':  // ELS - SEPARATOR (added)
				createTiddlyText(place,this.separator);
				break;
			case '*':  // ELS - LINEBREAK (added)
				createTiddlyElement(place,'BR');
				break;
			case '<': // ELS - LESS COMMAND (added)
				var btn = createTiddlyButton(place,
					this.lessLabel,this.lessPrompt,config.macros.toolbar.onClickLess,'moreCommand');
				break;
			case '>':
				var btn = createTiddlyButton(place,
					this.moreLabel,this.morePrompt,config.macros.toolbar.onClickMore,'moreCommand');
				var e = createTiddlyElement(place,'span',null,'moreCommand');
				e.style.display = 'none';
				place = e;
				break;
			default:
				var theClass = '';
				switch(c.substr(0,1)) {
					case '+':
						theClass = 'defaultCommand';
						c = c.substr(1);
						break;
					case '-':
						theClass = 'cancelCommand';
						c = c.substr(1);
						break;
				}
				if(c in config.commands)

					this.createCommand(place,c,tiddler,theClass);
				else { // ELS - WIKIFY TIDDLER/SLICE/SECTION (added)
					if (c.substr(0,1)=='~') c=c.substr(1); // ignore leading ~
					var txt=store.getTiddlerText(c);
					if (txt) {
						// trim any leading/trailing newlines
						txt=txt.replace(/^\n*/,'').replace(/\n*$/,'');
						// trim PRE format wrapper if any
						txt=txt.replace(/^\{\{\{\n/,'').replace(/\n\}\}\}$/,'');
						// render content into toolbar
						wikify(txt,createTiddlyElement(place,'span'),null,tiddler);
					}
				} // ELS - end WIKIFY CONTENT
				break;
		}
	}
};
//}}}
// // }}}}}}// // {{block{
/***
!!!529 IE fixup - case-sensitive element lookup of tiddler elements
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/529 - OPEN
This tweak hijacks the standard browser function, document.getElementById(), to work-around the case-INsensitivity error in Internet Explorer (all versions up to and including IE7) //''Note: This tweak is only applied when using IE, and only for lookups of rendered tiddler elements within the containing 'tiddlerDisplay' element.''//
***/
//{{{
if (config.browser.isIE) {
document.coreTweaks_coreGetElementById=document.getElementById;
document.getElementById=function(id) {
	var e=document.coreTweaks_coreGetElementById(id);
	if (!e || !e.parentNode || e.parentNode.id!='tiddlerDisplay') return e;
	for (var i=0; i<e.parentNode.childNodes.length; i++)
		if (id==e.parentNode.childNodes[i].id) return e.parentNode.childNodes[i];
	return null;
};
}
//}}}
// // }}}}}}// // {{block{
/***
!!!890 add conditional test to """<<tiddler>>""" macro
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/890 - OPEN
This tweak extends the {{{<<tiddler>>}}} macro syntax so you can include a javascript-based //test expression// to determine if the tiddler transclusion should be performed:
{{{
<<tiddler TiddlerName if:{{...}} with: param param etc.>>
}}}
If the test is ''true'', then the tiddler is transcluded as usual.  If the test is ''false'', then the transclusion is skipped and //no output is produced//.
***/
//{{{
config.macros.tiddler.if_handler = config.macros.tiddler.handler;
config.macros.tiddler.handler = function(place,macroName,params,wikifier,paramString,tiddler)
{
	params = paramString.parseParams('name',null,true,false,true);
	if (!getParam(params,'if',true)) return;
	this.if_handler.apply(this,arguments);
};
//}}}
// // }}}}}}// // {{block{
/***
!!!831 backslash-quoting for embedding newlines in 'line-mode' formats
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/831 - OPEN
This tweak pre-processes source content to convert 'double-backslash-newline' into {{{<br>}}} before wikify(), so that literal newlines can be embedded in line-mode wiki syntax (e.g., tables, bullets, etc.)
***/
//{{{
window.coreWikify = wikify;
window.wikify = function(source,output,highlightRegExp,tiddler)
{
	if (source) arguments[0]=source.replace(/\\\\\n/mg,'<br>');
	coreWikify.apply(this,arguments);
}
//}}}
// // }}}}}}// // {{block{
/***
!!!683 FireFox3 Import bug: 'browse' button replacement
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/683 - OPEN
The web standard 'type=file' input control that has been used as a local path/file picker for TiddlyWiki no longer works as expected in FireFox3, which has, for security reasons, limited javascript access to this control so that *no* local filesystem path information can be revealed, even when it is intentional and necessary, as it is with TiddlyWiki.  This tweak provides alternative HTML source that patches the backstage import panel.  It replaces the 'type=file' input control with a text+button combination of controls that invokes a system-native secure 'file-chooser' dialog box to provide TiddlyWiki with access to a complete path+filename so that TW functions properly locate user-selected local files.
>Note: ''This tweak also requires http://trac.tiddlywiki.org/ticket/604 - cross-platform askForFilename()''
***/
//{{{
if (window.Components) {
	var fixhtml='<input name="txtBrowse" style="width:30em"><input type="button" value="..."'
		+' onClick="window.browseForFilename(this.previousSibling,true)">';
	var cmi=config.macros.importTiddlers;
	cmi.step1Html=cmi.step1Html.replace(/<input type='file' size=50 name='txtBrowse'>/,fixhtml);
}

merge(config.messages,{selectFile:'Please enter or select a file'}); // ready for I18N translation

window.browseForFilename=function(target,mustExist) { // note: both params are optional
	var msg=config.messages.selectFile;
	if (target && target.title) msg=target.title; // use target field tooltip (if any) as dialog prompt text
	// get local path for current document
	var path=getLocalPath(document.location.href);
	var p=path.lastIndexOf('/'); if (p==-1) p=path.lastIndexOf('\\'); // Unix or Windows
	if (p!=-1) path=path.substr(0,p+1); // remove filename, leave trailing slash
	var file=''
	var result=window.askForFilename(msg,path,file,mustExist); // requires #604
	if (target && result.length) // set target field and trigger handling
		{ target.value=result; target.onchange(); }
	return result;
}
//}}}
// // }}}}}}// // {{block{
/***
!!!604 cross-platform askForFilename()
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/604 - OPEN
invokes a system-native secure 'file-chooser' dialog box to provide TiddlyWiki with access to a complete path+filename so that TW functions properly locate user-selected local files.
***/
//{{{
window.askForFilename=function(msg,path,file,mustExist) {
	var r = window.mozAskForFilename(msg,path,file,mustExist);
	if(r===null || r===false)
		r = window.ieAskForFilename(msg,path,file,mustExist);
	if(r===null || r===false)
		r = window.javaAskForFilename(msg,path,file,mustExist);
	if(r===null || r===false)
		r = prompt(msg,path+file);
	return r||'';
}

window.mozAskForFilename=function(msg,path,file,mustExist) {
	if(!window.Components) return false;
	try {
		netscape.security.PrivilegeManager.enablePrivilege('UniversalXPConnect');
		var nsIFilePicker = window.Components.interfaces.nsIFilePicker;
		var picker = Components.classes['@mozilla.org/filepicker;1'].createInstance(nsIFilePicker);
		picker.init(window, msg, mustExist?nsIFilePicker.modeOpen:nsIFilePicker.modeSave);
		var thispath = Components.classes['@mozilla.org/file/local;1'].createInstance(Components.interfaces.nsILocalFile);
		thispath.initWithPath(path);
		picker.displayDirectory=thispath;
		picker.defaultExtension='html';
		picker.defaultString=file;
		picker.appendFilters(nsIFilePicker.filterAll|nsIFilePicker.filterText|nsIFilePicker.filterHTML);
		if (picker.show()!=nsIFilePicker.returnCancel)
			var result=picker.file.path;
	}
	catch(ex) { displayMessage(ex.toString()); }
	return result;
}

window.ieAskForFilename=function(msg,path,file,mustExist) {
	if(!config.browser.isIE) return false;
	try {
		var s = new ActiveXObject('UserAccounts.CommonDialog');
		s.Filter='All files|*.*|Text files|*.txt|HTML files|*.htm;*.html|';
		s.FilterIndex=3; // default to HTML files;
		s.InitialDir=path;
		s.FileName=file;
		return s.showOpen()?s.FileName:'';
	}
	catch(ex) { displayMessage(ex.toString()); }
	return result;
}

window.javaAskForFilename=function(msg,path,file,mustExist) {
	if(!document.applets['TiddlySaver']) return false;
	// TBD: implement java-based askFile(...) function
	try { return document.applets['TiddlySaver'].askFile(msg,path,file,mustExist); }
	catch(ex) { displayMessage(ex.toString()); }
}
//}}}
// // }}}}}}// // {{block{
/***
!!!657 wrap tabs onto multiple lines
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/657 - OPEN
This tweak inserts an extra space element following each tab, allowing them to wrap onto multiple lines if needed.
***/
//{{{
config.macros.tabs.handler = function(place,macroName,params)
{
	var cookie = params[0];
	var numTabs = (params.length-1)/3;
	var wrapper = createTiddlyElement(null,'div',null,'tabsetWrapper ' + cookie);
	var tabset = createTiddlyElement(wrapper,'div',null,'tabset');
	tabset.setAttribute('cookie',cookie);
	var validTab = false;
	for(var t=0; t<numTabs; t++) {
		var label = params[t*3+1];
		var prompt = params[t*3+2];
		var content = params[t*3+3];
		var tab = createTiddlyButton(tabset,label,prompt,this.onClickTab,'tab tabUnselected');
		createTiddlyElement(tab,'span',null,null,' ',{style:'font-size:0pt;line-height:0px'}); // ELS
		tab.setAttribute('tab',label);
		tab.setAttribute('content',content);
		tab.title = prompt;
		if(config.options[cookie] == label)
			validTab = true;
	}
	if(!validTab)
		config.options[cookie] = params[1];
	place.appendChild(wrapper);
	this.switchTab(tabset,config.options[cookie]);
};
//}}}
// // }}}}}}// // {{block{
/***
!!!628 hide 'no such macro' errors
***/
// // {{groupbox small{
/***
http://trac.tiddlywiki.org/ticket/628 - OPEN
When invoking a macro that is not defined, this tweak prevents the display of the 'error in macro... no such macro' message.  This is useful when rendering tiddler content or templates that reference macros that are defined by //optional// plugins that have not been installed in the current document.

<<option chkHideMissingMacros>> hide 'no such macro' error messages
***/
//{{{
if (config.options.chkHideMissingMacros===undefined)
	config.options.chkHideMissingMacros=false;

window.coreTweaks_missingMacro_invokeMacro = window.invokeMacro;
window.invokeMacro = function(place,macro,params,wikifier,tiddler) {
	if (!config.macros[macro] || !config.macros[macro].handler)
		if (config.options.chkHideMissingMacros) return;
	window.coreTweaks_missingMacro_invokeMacro.apply(this,arguments);
}
//}}}
// // }}}}}}
// // <<foldHeadings>>

<<tiddler [[include_tiddlers/Correspondence Principle.html#"Correspondence Principle"]]>>

<<tiddler [[include_tiddlers/Cosmic Coincidences.html#"Cosmic Coincidences"]]>>

<<tiddler [[include_tiddlers/Cosmic Creation and God.html#"Cosmic Creation and God"]]>>

<<tiddler [[include_tiddlers/Cosmic Horizon.html#"Cosmic Horizon"]]>>

<<tiddler [[include_tiddlers/Cosmic Microwave Background.html#"Cosmic Microwave Background"]]>>

<<tiddler [[include_tiddlers/Cosmic String.html#"Cosmic String"]]>>

<<tiddler [[include_tiddlers/Cosmological Constant.html#"Cosmological Constant"]]>>

<<tiddler [[include_tiddlers/Cosmological Equation of State.html#"Cosmological Equation of State"]]>>

<br><<tiddler [[include_tiddlers/Cosmological Natural Selection.html#"Cosmological Natural Selection"]]>>

<<tiddler [[include_tiddlers/Cosmology.html#"Cosmology"]]>>

<<tiddler [[include_tiddlers/Coupled Map Lattice.html#"Coupled Map Lattice"]]>>

<<tiddler [[include_tiddlers/Covariant Derivative.html#"Covariant Derivative"]]>>

<<tiddler [[include_tiddlers/Covariant Entropy Bound.html#"Covariant Entropy Bound"]]>>

A ''Coxeter Lattice'' $\mathbb A_n$ is defined by
\[
\mathbb A_n \equiv \{x \in \mathbb Z^{n+1} : \sum_{i= 1}^{n+1} x_i = 0 \}
\]

The ''Coxeter\-Todd Lattice $K_{12}$'' is a $12$-dimensional [[even integral lattice|Lattice]], having [[kissing number|Kissing Number]] $756$. It is the only extremal $3$-modular lattice in $12$ dimensions and its vectors have minimal norm of $4$. It is a sublattice of the [[Leech lattice|Leech Lattice]].

Papers:
* [[The Genus of the Coxeter-Todd Lattice - R. Scharlau, B. B. Venkov|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.46.3615&rep=rep1&type=pdf]] [[pct. 4|http://scholar.google.com/scholar?hl=de&lr=&cites=12773864681937385350&um=1&ie=UTF-8&ei=ylUfS_zXJNCLsAbdqZmuCw&sa=X&oi=science_links&resnum=1&ct=sl-citedby&ved=0CBUQzgIwAA]]

>Imagine you have never ever met any other being on planet earth and come up with the idea that the creation of you has been initiated by somebody like you. Wouldn't that be a crazy idea ?
> - [[Markus's wisdom|Markus's Wisdom - Mathematics and Physics]] -

> ... our universe ... possibly ... a by-product of a search for something else ...
> - Jürgen Schmidhuber - Algorithmic Theories of Everything -

Is it conceivable that the universe we life in has been created by some kind of intelligence in a parent (predecessor) universe ?
Some questions in this respect:
* What does it take to create a universe ? (If some preceding intelligence has done it, it seems likely that we are pretty much like them and we could do it as well one day). Hence to understand how to build a universe might allow us to better understand the origin of our own universe. <br> <br>
* Why would some kind of intelligence in a universe be interested in sprouting a child universe ?

Some thoughts, ideas and speculations:
* A complex system doesn't appear "out of nothing" or out of randomness, rather it should have a long evolutionary history. As our universe, in particular as it harbours intelligent life, is a pretty complex system, it may well be that the few billion years since the big bang are not enough to explain __all__ the necessary steps evolution had to take to end up with a universe the way we see it today. Therefore, introducing an ancestry of universes gives evolution more time, as now the time available to produce our universe as an output is a few billion years times the number of predecessor universes.<br><br>
* The big bang singularity has a ridiculously [[low entropy|Entropy of the Universe]] whereas the entropy of a black hole is high. (An oddity [[Roger Penrose|Roger Penrose]] over and over again has pointed to). If one assumes that a black hole naturally gives rise to a baby universe, it seems quite mysterious what effect it would take to convert the initial high entropy state to a low entropy one. An alternative is "intelligence". I.e. only such universes are low entropy universes that have been created due to the intervention of intelligence. The natural creation may well exist alongside, but these offspring universes most probably are low in complexity (as high in entropy) and therefore will not give rise to intelligence.  <br> <br>
* The explanation of the emergence of consciousness and intelligence could be that it guarantees the replication of the universe, a necessary step in an evolutionary process. I.e. consciousness is one possible trait of a universe guaranteeing survival in the evolutionary process. (Expressed in a more colloquial way: Consciousness is the "sexuality of the universe", destined to reproduce it and to contribute to the evolution of its kind). If this it true, evolution also is taking place on a higher level than what we are accustomed to.<br><br>
* If our universe is "programmed" for life, this means that the [[strong anthropic principle|Anthropic Principle]] holds and the scenario alluded to here is a concrete realisation of it. This implies that the universe may not have completely unfolded yet and it offers an explanation as to why we see an ongoing progress, characterized by a directionality towards more complexity and organisation which is contrary to what one would naively expect due to the second law of thermodynamics. (For an extreme conclusion from this fact, see [[omega point theory|Omega Point]]). Furthermore this makes it quite likely that our planet is not the only place in the universe harbouring life (i.e. "we are not alone").<br> <br>
* In this respect I came up with a totally strange Gedankenexperiment: Suppose we create matter and anti-matter by means of pair production. From the anti-matter we form a black hole whereas the matter we assemble in a low entropy state. (Maybe one could even create a conscious brain out of the matter part. Curiously enough, experts say that to create a universe in the laboratory it only takes a few pounds of matter, incidentally, just about the mass of a brain of a highly developed intelligent species). If the anti-matter black hole gives rise to a daughter universe then it seems that the whole of this universe is EPR\-correlated with this complex structure in our universe. (And if this complex structure is conscious, this consciousness is EPR\-correlated with the whole of a universe). We can turn this argument upside down and wonder if the whole of our universe is EPR\-correlated with some preceding intelligence. So then, could our consciousness be correlated with some maybe further developed intelligence in another universe ? Could this explain how ideas come into the world ? I.e. could that mean that seemingly new things we come up with are just inherited from some foregoing intelligence and are not really new ? <br> <br>
* Furthermore this model could serve to explain the [[fine tuning|Fine Tuning]] of our universe.<br><br>
* Why does evolution bring about species like us, having brains, capable of doing abstract thinking like in mathematics, which appears to be totally redundant when it comes to mere survival and reproduction. (I.e. the "game" of evolution) ? The answer could be this: we are more or less "like them", and they had to have such brains to be smart enough to create our universe (maybe in a sophisticated laboratory setup).

{{center{[img(449px+, )[images/ancient_maths.jpg]]}}}
* One does not face the [[Boltzmann brain paradox|Boltzmann Brain Paradox]] resulting from the assumption that the universe and life therein is the result of an initial appropriate quantum fluctuation.<br><br>
* A quite compelling explanation for why our universe was intelligently created, is that it is a [["simulation"|The Simulation Argument]] (or "program") running on a "computer" of an advanced civilisation. (Although I have heard of this argument several times before, I was never much convinced of it. Yet in this context it is way more plausible and interpretable to me). If one makes the weak assumption that an advanced civilisation is doing information processing and like us tries to built ever more potent information processing machines, consequently the structures of the processing units have to get smaller and smaller. The ultimate limit - as far as we know today - is the Planck scale. If it were possible to read and write units on this fundamental scale, this would be the ultimate computer. But this means manipulating the very fabric of spacetime itself. To do manipulations on these small scales, probably enormous energies are required. Hence one is acting in an energy regimen not so far away from that where the creation of small black holes (or baby universes) takes place. Yet contrary to blindly smashing matter into one another to create a black hole, probably having high entropy, here one is doing it in a very ordered way. Therefore the creation of a low entropy black hole (and hence a descending universe) could be explained  by intelligence (or "consciousness" ?), i.e. being the by-product of an advanced civilisation having the sheer desire to do ever better information processing, a trend also observable in our civilisation, so maybe one day ... In this scenario our universe could be a "Game of Life" (or more generally a [[cellular automaton|Cellular Automaton]]) simulation (having universal computing power), running on a sheet of spacetime instead on a silicon wafer, initiated by a "Super Conway" in a predecessor universe. The replacement of silicon as the fundamental substrate by spacetime is what may help make the whole scenario appear more plausible. (Every physical process is a computation and every computation can only be done harnessing a physical process). Therefore the days of silicon based computers seem to be counted and our "primitive" notion of a computer may change over time). See also: [[digital physics|Digital Physics]].<br><br>
* If this scenario is true, we could speak of [["intelligent design"|Intelligent Design]] of our universe. Yet this does not necessarily imply the existence of a monotheistic God. Rather the existence of beings that are not more of a God than what we would be one day if we were to create a universe in the laboratory having the potential to bring about intelligent/conscious life would be enough of an explanation. Besides God and the multiverse (or landscape) this offers a "third way" (one "in between") for making plausible why the [[world around us is so special and unlikely|Speciality of the World]].
© by Markus Maute, 2010

See also:
* [[Organic universe|Organic Universe]]
* [[Are we sitting inside a black hole ?|Are we sitting inside a Black Hole ?]]
* [[Cosmic creation and God|Cosmic Creation and God]]
* [[Cosmological natural selection|Cosmological Natural Selection]]

Papers:
* [[The Natural Selection of Universes Containing Intelligent Life (1995) - E. R. Harrison|http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1995QJRAS..36..193H&amp;data_type=PDF_HIGH&amp;whole_paper=YES&amp;type=PRINTER&amp;filetype=.pdf]] [[local|papers/natural_selection.pdf]] [[pct. 50|http://scholar.google.de/scholar?hl=de&lr=&cites=6461940979141576906&um=1&ie=UTF-8&ei=xgb6Tc_sBIrMswaI3-0F&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCQQzgIwAA]] - Stunningly [[Edward R. Harrison|http://en.wikipedia.org/wiki/Edward_Robert_Harrison]] came up with nearly exactly the same idea. I only came across his paper after having written most of the things above.
* [[Possible Implications of the Quantum Theory of Gravity (1994) - L. Crane|http://arxiv.org/PS_cache/hep-th/pdf/9402/9402104v1.pdf]] [[local|papers/9402104v1.pdf]] [[pct. 16|http://scholar.google.de/scholar?hl=de&lr=&cites=8606615866263834534&um=1&ie=UTF-8&ei=3gb6TbK1EI3dsgbW1_nODw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCQQzgIwAA]]
* [[Message in the Sky (2005) - S. Hsu, A. Zee|http://arxiv.org/pdf/physics/0510102v3.pdf]] [[local|papers/0511135v1.pdf]] [[pct. 11|http://scholar.google.de/scholar?cites=13115471811645741443&as_sdt=2005&sciodt=0,5&hl=de]]
* [[Universes out of Almost Empty Space (2007) - S. Ansoldi, E. I. Guendelman|http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.1233v3.pdf]] [[local|papers/0706.1233v3.pdf]] [[pct. 8|http://scholar.google.de/scholar?hl=de&lr=&cites=5473380222756217248&um=1&ie=UTF-8&ei=4wX6Tcu-L8nBswaW2InwDw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB0QzgIwAA]]
* [[Child Universes in the Laboratory (2006) - S. Ansoldi, E. I. Guendelman|http://arxiv.org/PS_cache/gr-qc/pdf/0611/0611034v1.pdf]] [[local|papers/0611034v1.pdf]] [[pct. 8|http://scholar.google.de/scholar?cites=6340669719597987427&as_sdt=2005&sciodt=0,5&hl=de]]
* [[The Universe out of a Monopole in the Laboratory? (2006) - N. Sakai, K. Nakao, H. Ishihara, M. Kobayashi|http://arxiv.org/PS_cache/gr-qc/pdf/0602/0602084v3.pdf]] [[local|papers/0602084v3.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=8198911644294153817&as_sdt=2005&sciodt=0,5&hl=de]]
* [[The Thermodynamic Arrow: Puzzles and Pseudo-puzzles (2003) - H. Price|http://www.usyd.edu.au/time/price/preprints/Price2.pdf]] [[local|papers/Price2.pdf]] [[pct. 6|http://scholar.google.de/scholar?cites=5212559579570928960&as_sdt=2005&sciodt=0,5&hl=de]]
* [[Life, the Universe, and almost Everything: Signs of Cosmic Design? (2009) - R. Vaas|http://arxiv.org/ftp/arxiv/papers/0910/0910.5579.pdf]] [[local|papers/0910.5579.pdf]] [[pct. 5|http://scholar.google.de/scholar?hl=de&lr=&cites=8978909965771824602&um=1&ie=UTF-8&ei=-Af6TeC5MISPswaJspzjDw&sa=X&oi=science_links&ct=sl-citedby&resnum=2&ved=0CCIQzgIwAQ]]
* [[The Real Message in the Sky (2005) - D. Scott, J. P. Zibin|http://arxiv.org/pdf/physics/0510102v3]] [[local|papers/0511135v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?hl=de&lr=&cites=9384517530783217412&um=1&ie=UTF-8&ei=Hgj6TePQO8TvsgbI4-zzDw&sa=X&oi=science_links&ct=sl-citedby&resnum=2&ved=0CCMQzgIwAQ]]
* [[How to Create a Universe (2007) - G. McCabe|http://philsci-archive.pitt.edu/archive/00003196/01/Spec.pdf]] [[local|papers/Spec.pdf]] pct. 0

Documents:
* [[The Evolution and Developement of the Universe (2007) - C. Vidal|http://arxiv.org/pdf/0912.5508v2]] [[local|documents/0912.pdf]] [[dct. 3|http://scholar.google.de/scholar?hl=de&lr=&cites=9135411750829929976&um=1&ie=UTF-8&ei=1Qj6TcGyKsbzsgaf7qXfDw&sa=X&oi=science_links&ct=sl-citedby&resnum=3&ved=0CDgQzgIwAg]]
* [[Evo Devo Universe? A Framework for Speculations on Cosmic Culture - J. M. Smart|http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20100003004_2010003041.pdf]] [[local|documents/20100003004_2010003041.pdf]] [[dct. 1|http://scholar.google.de/scholar?hl=de&lr=&cites=4736085125113793802&um=1&ie=UTF-8&ei=CAn6TdiKC87LtAb9kagI&sa=X&oi=science_links&ct=sl-citedby&resnum=2&ved=0CCYQzgIwAQ]]

Links:
* [[Scholarpedia - Time's Arrow and Boltzmann's Entropy|http://www.scholarpedia.org/article/Time%27s_arrow_and_Boltzmann%27s_entropy]]
* [[Evo Devo Universe|http://evodevouniverse.com]]
* [[The Role of Life in the Cosmological Replication Cycle - B. A. Balázs|http://astro.elte.hu/~bab/Role_Life_Univp.htm]]

Audio:
* [[Build Your Own Universe|http://www.npr.org/templates/story/story.php?storyId=6545246]] - Highly recommended in this context.

Journals:
* Is it Possible to Create a Universe in the Laboratory by Quantum Tunneling? (1990) - E. Farhi, A. H. Guth, J. Guven {{t100Cite{[[jct. 217|http://scholar.google.de/scholar?hl=de&lr=&cites=15044497668404537883&um=1&ie=UTF-8&ei=_Qn6TayZL43Nsgbj0uzvDw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CBoQzgIwAA]]}}}
* An Obstacle to Creating a Universe in the Laboratory (1987) - E. Farhi, A. H. Guth [[jct. 136|http://scholar.google.de/scholar?hl=de&lr=&cites=16616542130931513973&um=1&ie=UTF-8&ei=QQr6TemdFtHIswaZ5IUB&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB0QzgIwAA]]

<<tiddler [[include_tiddlers/Cybernetics.html#"Cybernetics"]]>>

In group theory the ''Cycle Notation'' is used to describe permutations of elements of a set $\Omega$ in terms of cycles constituting it.
A cycle of $\Omega$ is a permutation of its elements which maps the elements of some subset $S \subset \Omega$ to each other in a cyclic fashion, while fixing (i.e., mapping to themselves) all other elements (i.e. those of $\bar S$). The set $S$ is called the [[orbit|Orbit]] of the cycle.

!!!!Examples
\begin{eqnarray}
\begin{pmatrix} 1 & 2 & 3  \\ 3 & 2 & 1 \end{pmatrix} &\equiv& (1\ 3) \\
\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 1 & 5 & 4\end{pmatrix} &\equiv& (1\ 3)(4\ 5) \\
\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1\end{pmatrix} &\equiv& (1\ 2\ 5)(3\ 4)
\end{eqnarray}
Links:
* [[WIKIPEDIA - Cycle Notation|http://en.wikipedia.org/wiki/Cycle_notation]]

A ''Cyclic Group'' $C_n$ of [[order|Order]] $n$ is a [[group|Group]] that is generated by a single elements, say $g$. The set of elements $G$ consists of all powers of this  generator:
\[
G = \lbrace g^n \mid n \in \mathbb{Z} \rbrace
\]
The only subgroups of a cyclic group are the group itself and the identity.

Cyclic groups are the simplest groups and they are completely classified: For every $n \in \mathbb N$ there exists a cyclic group $C_n$ with exactly $n$ elements.
Furthermore there exists an ''Infinite Cyclic Group'', the additive group over $\mathbb{Z}$.
Every other cyclic group is isomorphic to one of the aforementioned ones.

Links:
* [[WIKIPEDIA - Cyclic Group|http://en.wikipedia.org/wiki/Cyclic_group]]

<<tiddler [[include_tiddlers/D'Alembert Equation.html#"D'Alembert Equation"]]>>

The ''D'Alembert Operator'' (a.k.a. ''d'Alembertian'' or ''Wave Operator'') generalizes the [[Laplace operator|Laplace Equation]] $\Delta$ to Minkowski-space and is given by

\begin{eqnarray}
\square & \equiv & \partial_\mu \partial^\mu \\
&= & \eta_{\mu\nu} \partial^\nu \partial^\mu \\
& =& \frac{\partial^2}{c^2\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2} \\
&=& {\partial^2 \over c^2\partial t^2} - \Delta
\end{eqnarray}

Links:
* [[WIKIPEDIA - D'Alembert Operator|http://en.wikipedia.org/wiki/D'Alembert_operator]]

<<tiddler [[include_tiddlers/D-Brane.html#"D-Brane"]]>>

<<tiddler [[include_tiddlers/Dark Energy.html#"Dark Energy"]]>>

<<tiddler [[include_tiddlers/Dark Energy for Dummies.html#"Dark Energy for Dummies"]]>>

<<tiddler [[include_tiddlers/Dark Flow.html#"Dark Flow"]]>>

<html><center><a href="http://apod.nasa.gov/apod/ap070516.html"><img src="images/DarkMatter.jpg" style="width: 482px; "/></a></center></html>
''Dark Matter'' was introduces to "explain", based on [[Einsteins field equations|Einstein Field Equations]],
* the gravitational field needed for the galactic rotation curves,
* gravitational lensing of galaxies,
* the formation of structures in the universe.
It also appears in the spectral decomposition of the cosmic microwave background radiation.  However, there is no single observational hint at particles which could make up this dark matter. As a consequence, there are attempts to describe the same effects by a modification of the gravitational field equations, e.g. of Yukawa form, or by a modification of the dynamics of particles, like the [[MOND]] ansatz. Due to the lack of direct detection of dark matter particles, all those attempts are on the same footing.

Videos:
* [[Hubblecast EPISODE 05: Hubble Finds Ring of Dark Matter|http://www.space.com/php/video/player.php?video_id=150407Dark_matter]]

Links:
* [[Galaxy Cluster Cl 0024+17 (ZwCl 0024+1652)|http://imgsrc.hubblesite.org/hu/db/images/hs-2007-17-b-print.jpg]]

<<tiddler [[include_tiddlers/David Bohm.html#"David Bohm"]]>>

David Hilbert hielt am 8. September 1930 in Königsberg eine Rede unter dem Titel "Naturerkennen und Logik". Ein [[Ausschnitt|http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.mp3]] von vier Minuten wurde für das Radio aufgezeichnet und ist bis heute erhalten geblieben.

<<tiddler [[include_tiddlers/De Finetti Theorem.html#"De Finetti Theorem"]]>>

<<tiddler [[include_tiddlers/De Sitter Relativity.html#"De Sitter Relativity"]]>>

<<tiddler [[include_tiddlers/De Sitter Space.html#"De Sitter Space"]]>>

<<tiddler [[include_tiddlers/Deconfinement.html#"Deconfinement"]]>>

[[Welcome]]

<<tiddler [[include_tiddlers/Defect.html#"Defect"]]>>

<<tiddler [[include_tiddlers/Deformation Quantization.html#"Deformation Quantization"]]>>

<<tiddler [[include_tiddlers/Degree 4 Association Type Expansions.html#"Degree 4 Association Type Expansions"]]>>

<<tiddler [[include_tiddlers/Degree 5 Association Type Expansions.html#"Degree 5 Association Type Expansions"]]>>

<<tiddler [[include_tiddlers/Desargues' Theorem.html#"Desargues' Theorem"]]>>

<<tiddler [[include_tiddlers/Design.html#"Design"]]>>

<<tiddler [[include_tiddlers/Diffeomorphism.html#"Diffeomorphism"]]>>

>Enter cellular automata. Like partial differential equations, they have space and time built-in but on a discrete grid, not on a continuum. They have state variables at each site but only a few bits' worth, not an infinite information storage (in a single real number, you can encode the Library of Congress with plenty of room to spare). Two decades ago, the difficulties of modelling physics in this way appeared insurmountable. Today, it is clear that we can do all that differential equations can do, and more, because ''it is differential equations that are the poor man's cellular automata not the other way around!'' This development, of course, parallels an evolution in mathematical thought, certainly stimulated by our communion with digital computers: combinatorics, once relegated to a Cinderella role, has replaced the calculus as the queen of mathematics.
> - Tommaso Toffoli - Occam, Turing, von Neumann, Jaynes: How much can you get for how little? (A conceptual introduction to cellular automata) -

See also: [[Discrete differential geometry|Discrete Differential Geometry]].

Theses:
* [[The Stationary Einstein-Maxwell Equations (2001) - S. Tieu|http://www.collectionscanada.gc.ca/obj/s4/f2/dsk3/ftp04/MQ61610.pdf]] [[local|theses/MQ61610.pdf]]

Google books:
* [[Conformal Differential Geometry and its Generalizations (1996) - M. A. Akivis, V. V. Goldberg|http://books.google.com/books?id=7LXqI-Vp0bkC&printsec=frontcover&dq=conformal+differential+geometry+akivis&source=bl&ots=HPDhw3yqfl&sig=Zs6dxhfKt3K0NyoIX7AX0eaF3u4&hl=de&ei=MfC5S7_HOIutOLeE9aAL&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAkQ6AEwAA]] [[local|google_books/ConformalDifferentialGeometry.pdf]] [[bct. 90|http://scholar.google.de/scholar?cites=10064986817017083809&hl=de&as_sdt=2000]]
* [[Élie Cartan (1869 - 1951) (1993) - M. A. Akivis, B. A. Rosenfeld|http://books.google.com/books?id=WV3kzdYZdXIC&printsec=frontcover&dq=elie+cartan+akivis&source=bl&ots=gLp9cSXaJB&sig=bvTnFdcKwVaAATR5-QT2xbovAhw&hl=de&ei=rR6PTKPFFMvOswaB5NH1Cw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBgQ6AEwAA]] [[local|google_books/ElieCartan.pdf]] [[bct. 8|http://scholar.google.de/scholar?cites=12253586960719631095&hl=de&as_sdt=2000]]

<<tiddler [[include_tiddlers/Diffusion Equation.html#"Diffusion Equation"]]>>

> You insist that there is something a machine cannot do. If you will tell me precisely what it is that a machine cannot do, then I can always make a machine which will do just that!
> - John von Neumann -

There are many variants of ''Digital Physics'' (also referred to as ''Digital Philosophy''), but most of them have in common that physical reality and mental activity is viewed as digitized information processing.

Digital philosophy can be regarded as a modern reinterpretation of Gottfried Leibniz's monist metaphysics, one that replaces Leibniz's monads with aspects of the theory of cellular automata, assuming that the universe is a gigantic Turing-complete cellular automaton.
So far there is no unambiguous physical evidence against the possibility that "everything is just a computation".

Some people that are regarded as adherers to the concept of digital philosophy are: Gottfried Wilhelm Leibniz, Konrad Zuse, Edward Fredkin, Stephen Wolfram, [[Gregory Chaitin]], Jürgen Schmidhuber and Seth Lloyd.

Jürgen Schmidhuber pointed out that the simplest explanation of the universe would be a very simple Turing machine programmed to systematically execute all possible programs computing all possible histories for all types of computable physical laws. Furthermore there is an optimally efficient way of computing all computable universes based on Leonid Levin's universal search algorithm. He expanded this work by combining Ray Solomonoff's theory of [[inductive inference|Algorithmic Probability]] with the assumption that quickly computable universes are more likely than others.

The idea of a fundamental discrete entity being the building block of physical reality has appeared over and over again in history in many different guises, as for example:
* [[Planck units|Planck Units]]
* Monads (Leibnitz)
* Urs (Weizäcker)
* Bits (Wheeler)
* Metrons ([[Heim|Heim Theory]])
* Ons (Goertzel)

See also:
* [[Cellular automaton|Cellular Automaton]]
* [[Process physics|Process Physics]]
* [[Discrete spacetime|Discrete Spacetime]]
* [[Spacetime condensate|Spacetime Condensate]]
* [[Spin networks|Spin Network]]
* [[World crystal|World Crystal]]
* [[Ultrafinitism]]

Links:
* [[WIKIPEDIA - Digital Physics|http://en.wikipedia.org/wiki/Digital_physics]]
* [[Zuse's Thesis: The Universe is a Computer - Jürgen Schmidhuber|http://www.idsia.ch/~juergen/digitalphysics.html]]
* [[Digital Philosophy.org|http://www.digitalphilosophy.org/]]
* [[Theory of Universal Learning Machines & Universal AI|http://www.idsia.ch/~juergen/unilearn.html]]

Papers:
* [[A Computer Scientist’s View of Life, the Universe, and Everything (1999) - J. Schmidhuber|http://arxiv.org/PS_cache/quant-ph/pdf/9904/9904050v1.pdf]] [[local|papers/9904050v1.pdf]] [[pct. 63|http://scholar.google.de/scholar?cites=5213277605102533365&as_sdt=2005&sciodt=2000&hl=de]]
* [[Algorithmic Theories of Everything (2000) - J. Schmidhuber|http://arxiv.org/PS_cache/quant-ph/pdf/0011/0011122v2.pdf]] [[local|papers/0011122v2.pdf]] [[pct. 46|http://scholar.google.de/scholar?cites=7282820845356865291&hl=de]]

Magazines:
* [[SPEKTRUM DER WISSENSCHAFT · SPEZIAL 3/07 - Alle Berechenbaren Universen (2007) - Von J. Schmidhuber|http://www.idsia.ch/~juergen/Spektrum2007.pdf]] [[local|magazines/Spektrum2007.pdf]]

Videos:
* [[The Computational Universe - S. Lloyd|http://www.edge.org/3rd_culture/lloyd2/lloyd2_p2.html]]

A ''Dihedral group'' $\mathcal D_n$ is the group of symmetries of a regular polygon, including both rotations and reflections.  They are finite groups.
For $n > 2$ dihedral groups are non-Abelian permutation groups.

<<tiddler [[include_tiddlers/Dilaton.html#"Dilaton"]]>>

<<tiddler [[include_tiddlers/Dimensionality of the World.html#"Dimensionality of the World"]]>>

<<tiddler [[include_tiddlers/Dirac Algebra.html#"Dirac Algebra"]]>>

<<tiddler [[include_tiddlers/Dirac Equation.html#"Dirac Equation"]]>>

<<tiddler [[include_tiddlers/Dirac Equation in Curved Spacetime.html#"Dirac Equation in Curved Spacetime"]]>>

<<tiddler [[include_tiddlers/Dirac Operator.html#"Dirac Operator"]]>>

<<tiddler [[include_tiddlers/Dirac-Hestenes Equation.html#"Dirac-Hestenes Equation"]]>>

<<tiddler [[include_tiddlers/Dirac-Nambu-Goto Action.html#"Dirac-Nambu-Goto Action"]]>>

<<tiddler [[include_tiddlers/Disclination.html#"Disclination"]]>>

Many sophisticated properties of differential-geometric objects find their simple explanation within the ''Discrete Differential Geometry''.

Papers:
* [[Discrete Differential Calculus, Graphs, Topologies and Gauge Theory (1994) - A. Dimakis, F. Müller-Hoissen|http://arxiv.org/PS_cache/hep-th/pdf/9404/9404112v2.pdf]] [[local|papers/9404112v2.pdf]] [[pct. 72|http://scholar.google.de/scholar?cites=15140035106911831270&hl=de&as_sdt=2000]]
* [[Discrete Riemannian Geometry (1998) - A. Dimakis, F. Müller-Hoissen|http://arxiv.org/PS_cache/gr-qc/pdf/9808/9808023v1.pdf]] [[local|papers/9808023v1.pdf]] [[pct. 53|http://scholar.google.de/scholar?cites=15200090245903738717&hl=de&as_sdt=2000]]
* [[Discrete Differential Geometry. Consistency as Integrability (2005) - A. I. Bobenko, Y. B. Suris|http://arxiv.org/PS_cache/math/pdf/0504/0504358v1.pdf]] [[local|papers/0504358v1.pdf]] [[pct. 42|http://scholar.google.com/scholar?hl=de&lr=&cites=10389079491011402303&um=1&ie=UTF-8&ei=X_HgTP-vPMrGswbB1YGODA&sa=X&oi=science_links&ct=sl-citedby&resnum=3&ved=0CC4QzgIwAg]]

Lectures:
* [[Discrete Differential Geometry: An Applied Introduction|http://mesh.brown.edu/3DPGP-2007/pdfs/sg06-course01.pdf]] [[local|lectures/sg06-course01.pdf]]

>Having finite time jumps clearly indicates in what direction we should search for a satisfactory quantum model (of gravity): Schrödinger's equation will be a finite difference equation in the time direction. Take that as a modified picture for the small-distance structure of the theory!
> - Gerard t'Hooft -

See also:
* [[Quasicrystals|Quasicrystal]]

Papers:
* [[The Spectrum of a Quasiperiodic Schrödinger Operator (1987) - A. Sütő|http://www.springerlink.com/content/j823w7373574766g/fulltext.pdf]] [[local|papers/QuasiperiodicSchroedingerOperator.pdf]] [[pct. 86|http://scholar.google.com/scholar?cites=5173763421814546962&as_sdt=2005&sciodt=2000&hl=de]]
* [[Uniform Spectral Properties of One-dimensional Quasicrystals, I. Absence of Eigenvalues (1999) - D. Damanik, D. Lenz|http://arxiv.org/PS_cache/math-ph/pdf/9903/9903011v1.pdf]] [[local|papers/9903011v1.pdf]] [[pct. 51|http://scholar.google.de/scholar?cites=1845280361973064339&as_sdt=2005&sciodt=2000&hl=de]]
* [[A Discrete Schrödinger Spectral Problem and Associated Evolution Equations (2002) - M. Boiti, M. Bruschi, F. Pempinelli, B. Prinari|http://arxiv.org/PS_cache/nlin/pdf/0206/0206012v1.pdf]] [[local|papers/0206012v1.pdf]] [[pct.14|http://scholar.google.de/scholar?hl=de&lr=&cites=10517076482321512711&um=1&ie=UTF-8&ei=hBZ-TdC_DM7esga7xNHtBw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCEQzgIwAA]]
* [[The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian (2007) - D. Damanik, M. Embree, A. Gorodetski, S. Tcheremchantsev|http://www.ruf.rice.edu/~dtd3/DEGT-FD.pdf]] [[local|papers/DEGT-FD.pdf]] [[pct. 12|http://scholar.google.de/scholar?cites=17303396921058694404&as_sdt=2005&sciodt=2000&hl=de]]
* [[Relics of the Primordial Origin of Space and Time in the Low Energy World (1996) - C. Wolf|http://www.springerlink.com/content/m175322034390r46/fulltext.pdf]] [[local|papers/primordial_origin.pdf]] [[pct. 1|http://scholar.google.de/scholar?hl=de&lr=&cites=2887684162962929052&um=1&ie=UTF-8&ei=TBZ-TZ7WCIrAswb38cDyBw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CBwQzgIwAA]]

<<tiddler [[include_tiddlers/Discrete Spacetime.html#"Discrete Spacetime"]]>>

''Dislocations'' (or ''Line Defects'') are lines along which whole rows of atoms in a solid are arranged anomalously. The resulting irregularity in spacing is most severe along a line called the line of dislocation. Line defects can weaken or strengthen solids.

One distinguishes two primary types of dislocations: ''Edge Dislocations'' and ''Screw Dislocations''. Mixed dislocations are intermediate between these.

In principle, point defects make a crystal [[viscoelastic|Elasticity]], whereas ''Dislocations'' cause [[plasticity|Plasticity]].


<html><center><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_60.html" width=51% height=86></iframe></center></html>Papers:
* [[An Elastoplastic Theory of Dislocations as a Physical Field Theory with Torsion (2001) - M. Lazar|http://arxiv.org/PS_cache/cond-mat/pdf/0105/0105270v3.pdf]] [[local|papers/0105270v3.pdf]] [[pct. 31|http://scholar.google.de/scholar?cites=4328843154662662595&as_sdt=2005&sciodt=2000&hl=de]]
* [[Effective Dislocation Lines in Continuously Dislocated Crystals I. Material Anholonomity (2007) - A.Trz?sowski|http://arxiv.org/ftp/arxiv/papers/0709/0709.1793.pdf]] [[local|papers/0709.1793.pdf]] pct. 0
* [[Orowan's Formula, Differential Geometry and Four-dimensional Material Space (2002) - K. Yamasaki|http://struct.geosociety.jp/trgj/46/4604.pdf]] [[local|papers/4604.pdf]] pct. 0

Links:
* [[WIKIPEDIA - Disclocation|http://en.wikipedia.org/wiki/Dislocation]]
* [[NDT Ressource Center - Linear Defects - Dislocations|http://www.ndt-ed.org/EducationResources/CommunityCollege/Materials/Structure/linear_defects.htm]]

<<tiddler [[include_tiddlers/Dissipative System.html#"Dissipative System"]]>>

An algebra $\mathcal{A}$ is called a ''Division algebra'' if it possesses no [[zero-divisors|Zero Divisor]]. I.e. for any element $\mb A \in \mathcal A$ and any non-zero element $ \mb B \in \mathcal A$ there exists exactly one element $\mb X \in \mathcal A$ and  $\mb Y \in \mathcal A$ respectively, such that $\mb A = \mb{BX}$ and  $\mb A = \mb{YB}$.

Division algebras are also referred to as ''compact'' algebras.

''Theorem (M. Kervaire, J. Milnor)''
Any finite-dimensional real division algebra must be of dimension $1$, $2$, $4$, or $8$.

However over the [[p-adic numbers|P-adic Number]] there are an infinite number of division algebras.

An example of a division algebra of order $16$ over the rational numbers is described in [1] and [2]. It is based on a modified [[Cayley-Dickson doubling process|Cayley-Dickson Doubling]], yet it doesn't yield [[alternative algebras|Alternative Algebra]] if applied to the [[complex numbers|Complex Number]] or the real [[quaternion algebra|Quaternion]].

Papers:
* [[[1] On a Construction for Division Algebras of Order 16 (1945) - R. D. Schafer|http://www.ams.org/journals/bull/1945-51-08/S0002-9904-1945-08385-2/S0002-9904-1945-08385-2.pdf]] [[local|papers/S0002-9904-1945-08385-2.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=8310201224475751139&as_sdt=2005&sciodt=2000&hl=de]]
* [[[2] Equivalence in a Class of Division Algebras of Order 16 (1946) - R. D. Schafer|http://www.ams.org/journals/bull/1946-52-10/S0002-9904-1946-08665-6/S0002-9904-1946-08665-6.pdf]] [[local|papers/S0002-9904-1946-08665-6.pdf]] pct. 0

The ''Dixon\-Souriau Equations'' are a generalization of the [[Mathisson-Papapetrou equations|Mathisson-Papapetrou Equations]] in that an additional electromagnetic field is assumed.
In the absence of [[torsion|Torsion]] the equations are given by:
\begin{eqnarray}
\frac{D\tilde p^\mu}{D\tau} & = &  -\frac{1}{2} {R^\mu}_{\nu\lambda\sigma} S^{\nu\lambda} u^\sigma + eF^\mu{}_\nu u^\nu  -\frac\lambda2 S^{\nu\sigma}
\partial^\mu F_{\nu\sigma} \\
 \frac{DS^{\mu\nu}}{D\tau}& = &\tilde p^\mu u^\nu- \tilde p^\nu
 u^\mu +\lambda [S^{\mu\sigma}F_\sigma^\nu - S^{\nu \sigma}F_\sigma^\mu]
\end{eqnarray}
with
\[
\tilde{p}^{\mu} \equiv p^\mu - \frac{DS^{\mu\nu}}{D\tau}u_\nu
\]
In  addition to the Mathisson\-Papapetrou equations the equations contain the [[electromagnetic field strength tensor|Field Strength Tensor]] $F^{\mu\nu}$ and $\lambda$, which is an electromagnetic coupling scalar.

!!!!Special Cases
The Dixon\-Souriau equations reduce to the Van Holten equations whenever the particle’s four-momentum and four-velocity become co-linear. It has also been shown that the equations reduce to the well known Bargmann\-Michel\-Telegdi equations in the limit of the weak and homogeneous external field.

Papers:
* [[On the Electrodynamics of Spinning Particles - J. W. Van Holten|http://www.nikhef.nl/pub/services/biblio/preprints/h90-22.pdf]] [[local|papers/h90-22.pdf]] [[pct. 36|http://scholar.google.de/scholar?cites=5311923282338670619&hl=de&as_sdt=2000]]
* [[Modèle de Particule à Spin Dans le Champ Electromagnétique et Gravitationnel - J. M. Souriau|http://www.jmsouriau.com/Publications/JMSouriau-ModPartSpin1974.pdf]] [[local|papers/JMSouriau-ModPartSpin1974.pdf]] [[pct. 20|http://scholar.google.de/scholar?cites=4757212981966671457&hl=de&as_sdt=2000]] - One of the original papers.
* [[Charged Particles with Spin in a Gravitational Wave and a Uniform Magnetic Field - M. Mohseni|http://arxiv.org/PS_cache/gr-qc/pdf/0510/0510094v2.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=445315149535384639&hl=de&as_sdt=2000]] - With excellent literature review on the topic.
* [[Spin-Rotation Couplings: Spinning Test Particles and Dirac Field - D. Bini, Luca Lusanna|http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.0791v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=7777017156572952812&hl=de]]
* [[Spinning Particles in General Relativity - F. Cianfrani, G. Montani|http://arxiv.org/PS_cache/gr-qc/pdf/0701/0701080v1.pdf]] pct. 0

Links:
* [[Site Officiel de Jean-Marie Souriau|http://www.jmsouriau.com/]]

Journals:
* [[Spinning Particles in Schwarzschild Spacetime - R. H. Rietdijk, J. W. Van Holten|journals/SpinningParticleSchwarzschildMetric.djvu]] [[jct. 36|http://scholar.google.de/scholar?cites=15970824269076798034&hl=de&as_sdt=2000]]

<<tiddler [[include_tiddlers/Dold-Thom Theorem.html#"Dold-Thom Theorem"]]>>

<<tiddler [[include_tiddlers/Domain Wall.html#"Domain Wall"]]>>

The ''Double Factorial $n!!$'' is the product of all positive integers less or equal to $n$, having the same parity as $n$:
\[
n!! = n (n-2) (n-4)\cdots
\]
Note that $ n!!$ is not the same as $ (n!)!$.
!!!!Examples
$10!! = 10\cdot 8\cdot 6\cdot 4\cdot 2 = 3.840 $
$7!! = 7 \cdot 5 \cdot 3 \cdot 1 = 105 $

!!!!Properties
* $(2n)!! = 2^n n!$
* $ (2n+1)!! = \frac{(2n+1)!}{2^n n!}$

<<tiddler [[include_tiddlers/Draft.html#"Draft"]]>>

<<tiddler [[include_tiddlers/Dual Number.html#"Dual Number"]]>>

<<tiddler [[include_tiddlers/Dual Quaternion.html#"Dual Quaternion"]]>>

<<tiddler [[include_tiddlers/Duality Involution.html#"Duality Involution"]]>>

The [[energy-momentum tensor|Stress Energy Tensor]] does not change if an [[electromagnetic field|Electrodynamics]] is transformed by a so called ''Duality Rotation'':
\[
F'_{\mu\nu} = F_{\mu\nu} cos (\delta) + \tilde F_{\mu\nu} sin (\delta)
\]
Consequently, although a given electromagnetic tensor uniquely defines the electromagnetic energy-momentum tensor $T_{\mu\nu}$, the converse is not true. Given $T_{\mu\nu}$, $F_{\mu\nu}$ is defined only up to duality rotations.
Furthermore the currents transform according to
\[
j'_{\mu} = j_{\mu} cos (\delta) + \tilde j_{\mu} sin (\delta)
\]

More explicitely one has
\begin{eqnarray}
\vec E' & =& \vec E \cos (\alpha) + \vec B \sin (\alpha)  \\
\vec B'& = &\vec B\cos (\alpha) - \vec E \sin (\alpha)
\end{eqnarray}
and Gauß's law becomes
\[
\vec \nabla \times \vec E'  + \frac{\partial \vec B'}{\partial t} = \rho'
\]

If one assumes $\alpha = \pi/2$, one gets
\begin{eqnarray}
\vec E' & =&\vec B \\
\vec B'& = &-\vec E
\end{eqnarray}
which defines a [[duality involution|Duality Involution]] ${}^\sim$.
The associated transition $F \rightarrow \tilde F$ corresponds to the [[duality|Duality]] of electric and magnetic fields, i.e. the map:
\[
\vec E \rightarrow -\vec B, \quad  \vec B \rightarrow \vec E
\]
Lectures:
* [[Problems and Solutions - G. Mammadov|http://gmammado.mysite.syr.edu/notes/Electromagnetic_Field_Strength_Tensor.pdf]]

Google books:
* [[Modern Nonlinear Optics, Part 2 - M. W. Evans|http://books.google.com/books?id=9p0kK6IG94gC&pg=PA333&lpg=PA333&dq=%22Larmor%22+%22Rainich+group%22&source=bl&ots=tR3pyIOp_a&sig=e79EOZT3diri9gmkjGtNZhh9s5A&hl=de&sa=X&oi=book_result&resnum=1&ct=result#PPA332,M1]] [[bct. 89|http://scholar.google.de/scholar?cites=16148624411202458834&hl=de&as_sdt=2000]]

<<tiddler [[include_tiddlers/Dynamical Casimir Effect.html#"Dynamical Casimir Effect"]]>>

In 1947, Eugene Dynkin simplified the process of classifying complex semi-simple [[Lie algebras|Lie Algebra]] by using what became known as ''Dynkin Diagrams''.
Roughly speaking a Dynkin diagram records the configuration of an algebra’s [[simple roots|Root Vector]].

To construct a Dynkin diagram one uses the facts that:
* Every root in a rank $l$ algebra can be expressed as an integer sum or difference of $l$ simple roots.
* The relative lengths and interior angle between pairs of simple roots fits one of four cases.

Each node in a Dynkin diagram represents one of the algebra’s simple roots. It is represented by a circle. (Sometimes the circle is made black if the root is a short one). Two nodes are connected by zero, one, two or three lines  depending on the angle between them, which can be $ \frac\pi 2$ ,$ \frac {2\pi} 3$,  $ \frac {3\pi} 4$,  $ \frac {5\pi} 6$.
If a pair of roots has different length an arrow is used to point towards the shorter one.

!!!!Examples
<html><center><img src="images/roots.jpg" style="width: 603px; "/></center></html>
<html><center><img src="images/SO(2n)_Dynkin.jpg" style="width: 250px; "/></center></html>
In the case of simply laced groups, i.e. groups where all simple roots have the same length, only the first two cases occur, i.e. $\langle r_i|r_j\rangle = 0$ or $\langle r_i|r_j\rangle = -1$.

A ''Dyon'' is a particle that carries electric and magnetic charges.

<<tiddler [[include_tiddlers/E-Infinity Theory.html#"E-Infinity Theory"]]>>

''E6'' is the third largest of the 5 exceptional [[Lie groups|Lie Group]]. It is 78 dimensional, having 72 roots.
Geometrically the group is related to the tetrahedron.
E6 is the group of [[isometries|Isometry]] of the [[bioctonionic|Bioctonion]] projective plane.

Papers:
* [[The Structure of E6 - A. D. Wangberg|http://ir.library.oregonstate.edu/dspace/bitstream/1957/7446/1/thesis.pdf]] [[local|papers/thesis.pdf]]
* [[An Investigation of E6 Grand Unified Model - W. Lin|http://140.122.100.145/ntnuj/j35/j35-15.pdf]] [[local|papers/j35/j35-15.pdf]]

''$E_7$'' is the second largest of the 5 exceptional [[Lie groups|Lie Group]]. It is 133 dimensional, having 126 roots.
Geometrically the group is related to the octahedron.
$E_7$ is the group of [[isometries|Isometry]] of the [[quaterooctonionic|Quaterooctonions]] projective plane.

Papers:
* [[The Chevalley group G2(2) of order 12096 and the octonionic root system of E7 - M. Koca, R. Koc, N. O. Koca|http://arxiv.org/PS_cache/hep-th/pdf/0509/0509189v2.pdf]]

<<tiddler [[include_tiddlers/E8.html#"E8"]]>>

<<tiddler [[include_tiddlers/E8 Lattice.html#"E8 Lattice"]]>>

$E_{8\left(8\right)}$ is the real split form of the [[Lie group|Lie Group]] [[E8]].
For $E_{8\left(8\right)}$  there exist $453.060$ different irreducible representations.
!!!!Historical
In 2007 the character table for $E_8$ was calculated. Conceptualising, designing and running the calculations took a team of 19 mathematicians four years. The final computation took more than three days of solid processing time on a Sage supercomputer.
What came out was a $453.060 \times 453.060$-matrix which contains over $60$ GB of data which is more than $60$ times as much data as the human genome sequence.


Papers:
* [[The Minimal Unitary Representation of E8(8) (2002) - M. Günaydin, K. Koepsell, H. Nicolai|http://arxiv.org/PS_cache/hep-th/pdf/0109/0109005v2.pdf]] [[local|papers/0109005v2.pdf]] [[pct. 42|http://scholar.google.de/scholar?cites=13037309818640150601&hl=de&as_sdt=2000]]
* [[An Exceptional Geometry for d = 11 Supergravity? (2000) - K. Koepsell, H. Nicolai, H. Samtleben|http://arxiv.org/PS_cache/hep-th/pdf/0006/0006034v1.pdf]] [[local|papers/0006034v1.pdf]] [[pct. 29|http://scholar.google.de/scholar?cites=6784179047447449427&hl=de&as_sdt=2000]]

Presentations:
* [[The Character Table for E8 or how we wrote down a 453060 x 453060 Matrix and Found Happiness - D. Vogan|http://www-math.mit.edu/~dav/E8TALK.pdf]] [[local|lectures/E8TALK.pdf]]

<<tiddler [[include_tiddlers/E8(C).html#"E8(C)"]]>>

The ''ECE (Einstein\-Cartan\-Evans) Theory'' was developed by Myron Evans.

One of the paradigms of the theory is that the unification of quantum mechanics and general relativity occurs by accepting objectivity and causality and rejecting indeterminacy.

Papers:
* [[The Bianchi Identity of Differential Geometry M. W. Evans, H. Eckardt|http://aias.us/documents/uft/paper88.pdf]] pct.0

Links:
* [[Evans on Torsion|http://www.americanantigravity.com/documents/Myron-Evans-Interview.pdf]]
@@display:block;text-align:center;[img[My comments ...|images/comment.gif][Comments]]&nbsp;@@

''EVP'' = ''Electric Voice Phenomenon''.

Links:
* [[WIKIPEDIA - Electronic Voice Phenomenon|http://en.wikipedia.org/wiki/Electronic_voice_phenomena]]
* [[Stimmen aus einer anderen Welt - Chronik und Technik der Tonbandstimmenforschung - von Hildegard Schäfer|http://www.rodiehr.de/a_27_s_stimmen_inhalt.htm#I N H A L T]]

<<tiddler [[include_tiddlers/Eddington Number.html#"Eddington Number"]]>>

<<tiddler [[include_tiddlers/Edward Witten.html#"Edward Witten"]]>>

Papers:
* [[Special-Relativistic Resolution of Ehrenfest's Paradox: Comments on Some Recent Statements by T. E. Phipps, Jr. -O. Gron|http://128.112.100.2/~mcdonald/examples/mechanics/gron_fp_11_623_81.pdf]]

<<tiddler [[include_tiddlers/Einstein Field Equations.html#"Einstein Field Equations"]]>>

<<tiddler [[include_tiddlers/Einstein Space.html#"Einstein Space"]]>>

<<tiddler [[include_tiddlers/Einstein-Cartan Theory.html#"Einstein-Cartan Theory"]]>>

<<tiddler [[include_tiddlers/Einstein-Hilbert Action.html#"Einstein-Hilbert Action"]]>>

<<tiddler [[include_tiddlers/Einstein-Podolsky-Rosen Paradox.html#"Einstein-Podolsky-Rosen Paradox"]]>>

<<tiddler [[include_tiddlers/Elasticity.html#"Elasticity"]]>>

<<tiddler [[include_tiddlers/Electron Volt.html#"Electron Volt"]]>>

The ''Electroweak Gauge Potential'' is given by:
\[
\mb{W}_{\mu} = (W^0_\mu, W^1_\mu, W^2_\mu, W^3_\mu) \equiv (B_\mu, W^1_\mu, W^2_\mu, W^3_\mu)
\]
One therefore has $16$ field components for the electroweak field.

The ''Electro\-Weak [[Field Strength Tensor|Field Strength Tensor]]'' derived from it is
\[
\mb{W}_{\mu\nu}  = \partial_{\mu} \mb{W}_{\nu} - \partial_{\nu} \mb{W}_{\mu} + g \mb{W}_{\mu} \times \mb{W}_{\nu}
\]
which is antisymmetric.

In component notation this reads
\[
W_{\mu\nu}^{a}  = \partial_{\mu} W^{a}_{\nu} - \partial_{\nu} W^{a}_{\mu} + g \epsilon_{abc}W^{b}_{\mu}W^{c}_{\nu}
\]

''Electroweak Currents'' are in general decomposed into a vector current $V^\mu_a$ and an axial-vector current $A^\mu_a$.
The vector parts of the charge changing current and the isovector piece of the electromagnetic current are three components of a vector in isospace. All $3$ components are conserved.
The axial current is not conserved, even in the chiral limit.

Books:
* [[Electroweak Theory - E. A. Paschos|books/electroweak_theory_emmanuel_paschos.pdf]] [[bct. 6|http://scholar.google.de/scholar?cites=2474110183633045376&hl=de&as_sdt=2000]]

<<tiddler [[include_tiddlers/Emergent Gravity.html#"Emergent Gravity"]]>>

In ''Emergent Relativity'', [[special relativity|Special Relativity]] is regarded as a theory statistically emerging from a deeper (essentially non-relativistic) level of dynamics. It dates back to works of David Bohm in the early 50s, but it received a real boost with the advancement of [[quantum-gravity|Quantum Gravity]] approaches.

In recent years it has appeared under various disguises in quantum-gravity models based on spacetime foam pictures, in [[loop quantum gravity|Loop Quantum Gravity]] models, in [[non-commutative geometry|Noncommutative Geometry]] models or in [[black-hole|Black Hole]] physics.
At a strictly phenomenological level, one can understand fluctuations of the Newtonian mass as originating from the idea that the medium in which propagation occurs ("spacetime") involves some sort of "granularity" (usually considered in quantum gravity models).
On the basis of experience with condensed-matter systems, one can expect that granularity of the medium might lead to corrections in the local dispersion relation and hence to modifications in local effective mass of a particle.

<<tiddler [[include_tiddlers/Emergent Spacetime.html#"Emergent Spacetime"]]>>

<<tiddler [[include_tiddlers/Entropy.html#"Entropy"]]>>

<<tiddler [[include_tiddlers/Entropy of the Universe.html#"Entropy of the Universe"]]>>

>I was sitting on a chair in my patent office in Bern. Suddenly a thought struck me: If a man falls freely, he would not feel his weight. I was taken aback. The simple thought experiment made a deep impression on me. It was what led me to the theory of gravity.
>- Albert Einstein (1922) -

The ''Weak Equivalence Principle'' states that all particles follow the same path in a gravitational field independent of their mass. This fact is also known as "equality of inertial and gravitational masses".

It has been argued [1] that the inertial mass is not well defined in general relativity. (It depends on the spatial coordinates and therefore has no physical meaning). Indeed, Denisov and Solov'ov have found an explicit change of variables for the [[Schwarzschild metric|Schwarzschild Metric]] such that the mass in the new coordinates has a different value.
In fact the notion of mass is intimately related to the concept of flat metric.

Notice, that general relativity can equivalently be expressed by means of [[teleparallel gravity|Teleparallel Gravity]] which can do without the weak equivalence principle. (I.e. it seems that the conventional formulation of general relativity contains a redundancy).

The ''Strong Equivalence Principle'' states that an accelerated reference frame is equivalent to gravitation, or that mass curves spacetime, and accelerated motion is due to the curvature. Technically speaking this means that all physical laws that hold in flat Minkowski space (i.e. "special relativity") continue to hold in every reference frame provided one replaces derivatives by covariant derivatives.
Due to this principle, among all the possible physical fields, only gravity displays the special feature of being negligible at least in a given point of a given system of reference. For a symmetric metric connection this is assured by Weyl's theorem, which states, that the connection has the feature of being "removable" at least in a given point with a given choice of the system of coordinates. (This boils down to the existence of [[Riemann normal coordinates|Riemann Normal Coordinates]]).

The most general connection, still permitting the symmetric part of it to represent gravity as required by Weyl's theorem, realizing the physical meaning of the strong equivalence principle, is given by the inclusion of a [[totally antisymmetric Cartan torsion (axial vector torsion)|Torsion]] term in the connection. (In other words, one does not spoil the property of particles following geodesic paths).

!!!!Equivalence principles and quantum mechanics
It is well known that quantum mechanics and GR clash, i.e. they are incompatible (which is at the heart of the desire to further [[unify|Unification]] physics). The equivalence principles may be a good place to see that happen.

Several arguments have been put forward as to why the equivalence principles (at least in their classical formulation) do not exists in quantum mechanics [2]:
* Because the phase shift of a neutron interference experiment in a gravitational field depends on the mass, the weak EP is regarded as not being valid for quantum phenomena.
* The influence of a nonlinear Newtonian potential (or of curved space-time) cannot be transformed away using accelerated frames.
* The wave function solving the Schrödinger equation in a homogeneous gravitational field or in an accelerated frame depends on the mass. This is true in the relativistic as well as in the non-relativistic domain.
* The path of particles with spin may depend on the spin direction.
* In the context of a causal interpretation of quantum mechanics, it has been shown that quantum theory does not obey the EP.

!!!!An oddity
In the classical formulation of the theory of relativity, the strong equivalence principle is equivalent to saying that locally (in exactly one spacetime point, to be precise) the connection vanishes, provided one "picks" the appropriate reference frame. Yet the [[Riemann tensor|Riemann Tensor]] in general does not vanish in this frame (i.e. in the spacetime point in regards). This is quite strange a situation, because the very idea of General Relativity is that gravity is spacetime, possessing Riemannian curvature. The strong equivalence principle usually is sold by saying that gravity can be transformed away locally (which is probably based on the assumption that it is coded in the connection), but what about the non-vanishing of Riemannian curvature in the "distinguished" frame ?
The problem can be overcome by using the teleparallel formulation of gravity instead, which can completely reproduce GR without invoking Riemannian curvature. (Einstein seems to have had the right hunch, having spent so much of his lifetime with teleparallel gravity).

Therefore my conclusion: ''There is no (true) strong equivalence principle in General Relativity''. Gravity being Riemannian curvature is an illusion and a lucky accident, it being able to authentically describe gravity. If it turns out that teleparallel gravity is the correct description of gravity - and I am quite convinced of that -, many books have to be rewritten, claiming that gravity is (Riemannian) curvature. Note, that this does by no means mean that general relativity is wrong, but it is interpreted in the wrong way, leading to a roadblock, which may explain why there was no success in bringing it together with quantum mechanics, one of the outstanding problems in theoretical physics of the last 100 years ("más ó menos"). There is just no way to further push the envelope with this wrong interpretation.


{{center{[img(309px+, )[images/einstein-crack.jpg]]}}}
!!!!Generalisations
An idea by the author of this WIKI Blog is to generalise the classical strong equivalence principle in that one not only removes gravity locally, but all gauge fields. (In other words, the classical equivalence principle of GR and the gauge principle of Yang\-Mills theory should be two aspects of one and the same more general principle). This may be achieved by introducing [[canonical coordinates|Canonical Coordinates]]. Yet this implies that one has to face the calamities of a description in terms of a  more "weird" spacetime geometry (certainly not a [[Riemannian geometry|Riemann Space]] any more). A geometrical framework to achieve this goal is [[polyvector geometry|Polyvector Space]] and the generalised equivalence principle thus will be referred to as [[polyvecoctor equivalence principle|Polyvector Equivalence Principle]].
This new principle may also help to shed new light on the problems encountered in the context with quantum mechanics (see above).

<html><center><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_90.html" width=71% height=110>
</iframe></center></html>
Papers:
* [[[2] On the Equivalence Principle in Quantum Theory (1995) - C. Lämmerzahl|http://arxiv.org/PS_cache/gr-qc/pdf/9605/9605065v1.pdf]] [[local|papers/9605065v1.pdf]] [[pct. 46|http://scholar.google.de/scholar?cites=2734963173128124719&as_sdt=2005&sciodt=2000&hl=de]]
* [[Questioning the Equivalence Principle (2001) - T. Damour|http://arxiv.org/PS_cache/gr-qc/pdf/0109/0109063v1.pdf]] [[local|papers/0109063v1.pdf]] [[pct. 32|http://scholar.google.de/scholar?cites=8944760755624431639&as_sdt=2005&sciodt=2000&hl=de]]
* [[Einstein's Apple His First Principle of Equivalence - E. Schucking|http://arxiv.org/PS_cache/gr-qc/pdf/0703/0703149v1.pdf]] [[local|papers/0703149v1.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=15065206914202310161&as_sdt=2005&sciodt=2000&hl=de]]
* [[Equivalence Principle and Electromagnetic Field: No Birefringence, no Dilaton, and no Axion (2007) - F. W. Hehl, Y. N. Obukhov|http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.3422v1.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=10650799015431134281&as_sdt=2005&sciodt=2000&hl=de]]
* [[On the Principle of Equivalence (2009) - L. Fabbri|http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.2541v2.pdf]] [[local|papers/0905.2541v2.pdf]] pct. 0

Journals:
* [1] Mass and Energy in General Relativity (1995) - Y. Bozhkov, Waldyr A. Rodrigues
@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/albert_einstein.jpg][Comments]]}}}&nbsp;@@

''Equations of Motion'' are [[differential equations|Differential Equation]] which remain valid if transformed differentially to new coordinates, even if the transformation is not integrable in the Schwarz sense.

<<tiddler [[include_tiddlers/Equivalence Principles.html#"Equivalence Principles"]]>>

<<tiddler [[include_tiddlers/Euler Characteristic.html#"Euler Characteristic"]]>>

<<tiddler [[include_tiddlers/Event Horizon.html#"Event Horizon"]]>>

<<tiddler [[include_tiddlers/Exact Sequence.html#"Exact Sequence"]]>>

Videos:
* [[The Quest for a Living World|http://discovermagazine.com/video/science-videos/quest-for-a-living-world]]

The algebra $C^\infty (\mathcal M)$ of smooth real functions on $\mathcal M$ determines $\mathcal M$ up to a [[diffeomophism|Diffeomorphism]].

The four-dimensional Euclidean space $\mathbb R^4$ can be given infinitely many nondiffeomorphic (exotic) differential structures.

The 28 differential structures on [[S7|7-Sphere]] and some [[homeomorphic|Homeomorphism]] [[homogeneous spaces|Homogeneous Space]] can be distinguished by their spectra provided an appropriate [[metric|Metric Tensor]] is chosen.

Papers:
* [[Fifty Years Ago: Topology of Manifolds in the 50's and 60's - J. Milnor|http://www.math.sunysb.edu/~jack/PREPRINTS/pcity-lec.pdf]]
* [[Exotic Smoothness and Physics - C. H. Brans|http://arxiv.org/PS_cache/gr-qc/pdf/9405/9405010v1.pdf]]
* [[Differential Structures Geometrization of Quantum Mechanics - T. Asselmeyer-Maluga, H. Rosé|http://arxiv.org/PS_cache/gr-qc/pdf/0511/0511089v3.pdf]]
* [[Exotic Spheres and Curvature - M. Joachim, D. J. Wraith|http://www.ams.org/bull/2008-45-04/S0273-0979-08-01213-5/S0273-0979-08-01213-5.pdf]]
* [[Exotic Smoothness and Particle Physics - J. Sladkowski|http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3Ahep-th%2F9604137]]
* [[Spacetime Models, Fundamental Interactions and Noncommutative Geometry - J. S Ladkowski|http://arxiv.org/PS_cache/hep-th/pdf/9610/9610093v1.pdf]]

Papers:
* [[The Exponential Map on the Cayley-Dickson Algebras - G. Moreno|http://arxiv.org/PS_cache/math/pdf/0405/0405424v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=16528539938669895237&hl=de]]

/***
|Name|ExportTiddlersPlugin|
|Source|http://www.TiddlyTools.com/#ExportTiddlersPlugin|
|Documentation|http://www.TiddlyTools.com/#ExportTiddlersPluginInfo|
|Version|2.9.6|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|interactively select/export tiddlers to a separate file|
!!!!!Documentation
>see [[ExportTiddlersPluginInfo]]
!!!!!Inline control panel (live):
><<exportTiddlers inline>>
!!!!!Revisions
<<<
2011.02.14 2.9.6 fix OSX error: use picker.file.path
2010.02.25 2.9.5 added merge checkbox option and improved 'merge' status message
|please see [[ExportTiddlersPluginInfo]] for additional revision details|
2005.10.09 0.0.0 development started
<<<
!!!!!Code
***/
//{{{
// version
version.extensions.ExportTiddlersPlugin= {major: 2, minor: 9, revision: 6, date: new Date(2011,2,14)};

// default shadow definition
config.shadowTiddlers.ExportTiddlers='<<exportTiddlers inline>>';

// add 'export' backstage task (following built-in import task)
if (config.tasks) { // TW2.2 or above
	config.tasks.exportTask = {
		text:'export',
		tooltip:'Export selected tiddlers to another file',
		content:'<<exportTiddlers inline>>'
	}
	config.backstageTasks.splice(config.backstageTasks.indexOf('importTask')+1,0,'exportTask');
}

config.macros.exportTiddlers = {
	$: function(id) { return document.getElementById(id); }, // abbreviation
	label: 'export tiddlers',
	prompt: 'Copy selected tiddlers to an export document',
	okmsg: '%0 tiddler%1 written to %2',
	failmsg: 'An error occurred while creating %1',
	overwriteprompt: '%0\ncontains %1 tiddler%2 that will be discarded or replaced',
	mergestatus: '%0 tiddler%1 added, %2 tiddler%3 updated, %4 tiddler%5 unchanged',
	statusmsg: '%0 tiddler%1 - %2 selected for export',
	newdefault: 'export.html',
	datetimefmt: '0MM/0DD/YYYY 0hh:0mm:0ss',  // for 'filter date/time' edit fields
	type_TW: "tw", type_PS: "ps", type_TX: "tx", type_CS: "cs", type_NF: "nf", // file type tokens
	type_map: { // maps type param to token values
		tiddlywiki:"tw", tw:"tw", wiki: "tw",
		purestore: "ps", ps:"ps", store:"ps",
		plaintext: "tx", tx:"tx", text: "tx",
		comma:     "cs", cs:"cs", csv:  "cs",
		newsfeed:  "nf", nf:"nf", xml:  "nf", rss:"nf"
	},
	handler: function(place,macroName,params) {
		if (params[0]!='inline')
			{ createTiddlyButton(place,this.label,this.prompt,this.togglePanel); return; }
		var panel=this.createPanel(place);
		panel.style.position='static';
		panel.style.display='block';
	},
	createPanel: function(place) {
		var panel=this.$('exportPanel');
		if (panel) { panel.parentNode.removeChild(panel); }
		setStylesheet(store.getTiddlerText('ExportTiddlersPlugin##css',''),'exportTiddlers');
		panel=createTiddlyElement(place,'span','exportPanel',null,null)
		panel.innerHTML=store.getTiddlerText('ExportTiddlersPlugin##html','');
		this.initFilter();
		this.refreshList(0);
		var fn=this.$('exportFilename');
		if (window.location.protocol=='file:' && !fn.value.length) {
			// get new target path/filename
			var newPath=getLocalPath(window.location.href);
			var slashpos=newPath.lastIndexOf('/'); if (slashpos==-1) slashpos=newPath.lastIndexOf('\\');
			if (slashpos!=-1) newPath=newPath.substr(0,slashpos+1); // trim filename
			fn.value=newPath+this.newdefault;
		}
		return panel;
	},
	togglePanel: function(e) { var e=e||window.event;
		var cme=config.macros.exportTiddlers; // abbrev
		var parent=resolveTarget(e).parentNode;
		var panel=cme.$('exportPanel');
		if (panel==undefined || panel.parentNode!=parent)
			panel=cme.createPanel(parent);
		var isOpen=panel.style.display=='block';
		if(config.options.chkAnimate)
			anim.startAnimating(new Slider(panel,!isOpen,e.shiftKey || e.altKey,'none'));
		else
			panel.style.display=isOpen?'none':'block' ;
		if (panel.style.display!='none') {
			cme.refreshList(0);
			cme.$('exportFilename').focus();
			cme.$('exportFilename').select();
		}
		e.cancelBubble = true; if (e.stopPropagation) e.stopPropagation(); return(false);
	},
	process: function(which) { // process panel control interactions
		var theList=this.$('exportList'); if (!theList) return false;
		var count = 0;
		var total = store.getTiddlers('title').length;
		switch (which.id) {
			case 'exportFilter':
				count=this.filterExportList();
				var panel=this.$('exportFilterPanel');
				if (count==-1) { panel.style.display='block'; break; }
				this.$('exportStart').disabled=(count==0);
				this.$('exportDelete').disabled=(count==0);
				this.displayStatus(count,total);
				if (count==0) { alert('No tiddlers were selected'); panel.style.display='block'; }
				break;
			case 'exportStart':
				this.go();
				break;
			case 'exportDelete':
				this.deleteTiddlers();
				break;
			case 'exportHideFilter':
			case 'exportToggleFilter':
				var panel=this.$('exportFilterPanel')
				panel.style.display=(panel.style.display=='block')?'none':'block';
				break;
			case 'exportSelectChanges':
				var lastmod=new Date(document.lastModified);
				for (var t = 0; t < theList.options.length; t++) {
					if (theList.options[t].value=='') continue;
					var tiddler=store.getTiddler(theList.options[t].value); if (!tiddler) continue;
					theList.options[t].selected=(tiddler.modified>lastmod);
					count += (tiddler.modified>lastmod)?1:0;
				}
				this.$('exportStart').disabled=(count==0);
				this.$('exportDelete').disabled=(count==0);
				this.displayStatus(count,total);
				if (count==0) alert('There are no unsaved changes');
				break;
			case 'exportSelectAll':
				for (var t = 0; t < theList.options.length; t++) {
					if (theList.options[t].value=='') continue;
					theList.options[t].selected=true;
					count += 1;
				}
				this.$('exportStart').disabled=(count==0);
				this.$('exportDelete').disabled=(count==0);
				this.displayStatus(count,count);
				break;
			case 'exportSelectOpened':
				for (var t=0; t<theList.options.length; t++) theList.options[t].selected=false;
				var tiddlerDisplay=this.$('tiddlerDisplay');
				for (var t=0; t<tiddlerDisplay.childNodes.length;t++) {
					var tiddler=tiddlerDisplay.childNodes[t].id.substr(7);
					for (var i=0; i<theList.options.length; i++) {
						if (theList.options[i].value!=tiddler) continue;
						theList.options[i].selected=true; count++; break;
					}
				}
				this.$('exportStart').disabled=(count==0);
				this.$('exportDelete').disabled=(count==0);
				this.displayStatus(count,total);
				if (count==0) alert('There are no tiddlers currently opened');
				break;
			case 'exportSelectRelated':
				// recursively build list of related tiddlers
				function getRelatedTiddlers(tid,tids) {
					var t=store.getTiddler(tid); if (!t || tids.contains(tid)) return tids;
					tids.push(t.title);
					if (!t.linksUpdated) t.changed();
					for (var i=0; i<t.links.length; i++)
						if (t.links[i]!=tid) tids=getRelatedTiddlers(t.links[i],tids);
					return tids;
				}
				// for all currently selected tiddlers, gather up the related tiddlers (including self) and select them as well
				var tids=[];
				for (var i=0; i<theList.options.length; i++)
					if (theList.options[i].selected) tids=getRelatedTiddlers(theList.options[i].value,tids);
				// select related tiddlers (includes original selected tiddlers)
				for (var i=0; i<theList.options.length; i++)
					theList.options[i].selected=tids.contains(theList.options[i].value);
				this.displayStatus(tids.length,total);
				break;
			case 'exportListSmaller':	// decrease current listbox size
				var min=5;
				theList.size-=(theList.size>min)?1:0;
				break;
			case 'exportListLarger':	// increase current listbox size
				var max=(theList.options.length>25)?theList.options.length:25;
				theList.size+=(theList.size<max)?1:0;
				break;
			case 'exportClose':
				this.$('exportPanel').style.display='none';
				break;
		}
		return false;
	},
	displayStatus: function(count,total) {
		var txt=this.statusmsg.format([total,total!=1?'s':'',!count?'none':count==total?'all':count]);
		clearMessage();	displayMessage(txt);
		return txt;
	},
	refreshList: function(selectedIndex) {
		var theList = this.$('exportList'); if (!theList) return;
		// get the sort order
		var sort;
		if (!selectedIndex)   selectedIndex=0;
		if (selectedIndex==0) sort='modified';
		if (selectedIndex==1) sort='title';
		if (selectedIndex==2) sort='modified';
		if (selectedIndex==3) sort='modifier';
		if (selectedIndex==4) sort='tags';

		// unselect headings and count number of tiddlers actually selected
		var count=0;
		for (var t=5; t < theList.options.length; t++) {
			if (!theList.options[t].selected) continue;
			if (theList.options[t].value!='')
				count++;
			else { // if heading is selected, deselect it, and then select and count all in section
				theList.options[t].selected=false;
				for ( t++; t<theList.options.length && theList.options[t].value!=''; t++) {
					theList.options[t].selected=true;
					count++;
				}
			}
		}

		// disable 'export' and 'delete' buttons if no tiddlers selected
		this.$('exportStart').disabled=(count==0);
		this.$('exportDelete').disabled=(count==0);

		// show selection count
		var tiddlers = store.getTiddlers('title');
		if (theList.options.length) this.displayStatus(count,tiddlers.length);

		// if a [command] item, reload list... otherwise, no further refresh needed
		if (selectedIndex>4) return;

		// clear current list contents
		while (theList.length > 0) { theList.options[0] = null; }
		// add heading and control items to list
		var i=0;
		var indent=String.fromCharCode(160)+String.fromCharCode(160);
		theList.options[i++]=
			new Option(tiddlers.length+' tiddlers in document', '',false,false);
		theList.options[i++]=
			new Option(((sort=='title'   )?'>':indent)+' [by title]', '',false,false);
		theList.options[i++]=
			new Option(((sort=='modified')?'>':indent)+' [by date]', '',false,false);
		theList.options[i++]=
			new Option(((sort=='modifier')?'>':indent)+' [by author]', '',false,false);
		theList.options[i++]=
			new Option(((sort=='tags'    )?'>':indent)+' [by tags]', '',false,false);

		// output the tiddler list
		switch(sort) {
			case 'title':
				for(var t = 0; t < tiddlers.length; t++)
					theList.options[i++] = new Option(tiddlers[t].title,tiddlers[t].title,false,false);
				break;
			case 'modifier':
			case 'modified':
				var tiddlers = store.getTiddlers(sort);
				// sort descending for newest date first
				tiddlers.sort(function (a,b) {if(a[sort] == b[sort]) return(0); else return (a[sort] > b[sort]) ? -1 : +1; });
				var lastSection = '';
				for(var t = 0; t < tiddlers.length; t++) {
					var tiddler = tiddlers[t];
					var theSection = '';
					if (sort=='modified') theSection=tiddler.modified.toLocaleDateString();
					if (sort=='modifier') theSection=tiddler.modifier;
					if (theSection != lastSection) {
						theList.options[i++] = new Option(theSection,'',false,false);
						lastSection = theSection;
					}
					theList.options[i++] = new Option(indent+indent+tiddler.title,tiddler.title,false,false);
				}
				break;
			case 'tags':
				var theTitles = {}; // all tiddler titles, hash indexed by tag value
				var theTags = new Array();
				for(var t=0; t<tiddlers.length; t++) {
					var title=tiddlers[t].title;
					var tags=tiddlers[t].tags;
					if (!tags || !tags.length) {
						if (theTitles['untagged']==undefined) { theTags.push('untagged'); theTitles['untagged']=new Array(); }
						theTitles['untagged'].push(title);
					}
					else for(var s=0; s<tags.length; s++) {
						if (theTitles[tags[s]]==undefined) { theTags.push(tags[s]); theTitles[tags[s]]=new Array(); }
						theTitles[tags[s]].push(title);
					}
				}
				theTags.sort();
				for(var tagindex=0; tagindex<theTags.length; tagindex++) {
					var theTag=theTags[tagindex];
					theList.options[i++]=new Option(theTag,'',false,false);
					for(var t=0; t<theTitles[theTag].length; t++)
						theList.options[i++]=new Option(indent+indent+theTitles[theTag][t],theTitles[theTag][t],false,false);
				}
				break;
			}
		theList.selectedIndex=selectedIndex; // select current control item
		this.$('exportStart').disabled=true;
		this.$('exportDelete').disabled=true;
		this.displayStatus(0,tiddlers.length);
	},
	askForFilename: function(here) {
		var msg=here.title; // use tooltip as dialog box message
		var path=getLocalPath(document.location.href);
		var slashpos=path.lastIndexOf('/'); if (slashpos==-1) slashpos=path.lastIndexOf('\\');
		if (slashpos!=-1) path = path.substr(0,slashpos+1); // remove filename from path, leave the trailing slash
		var filetype=this.$('exportFormat').value.toLowerCase();
		var defext='html';
		if (filetype==this.type_TX) defext='txt';
		if (filetype==this.type_CS) defext='csv';
		if (filetype==this.type_NF) defext='xml';
		var file=this.newdefault.replace(/html$/,defext);
		var result='';
		if(window.Components) { // moz
			try {
				netscape.security.PrivilegeManager.enablePrivilege('UniversalXPConnect');
				var nsIFilePicker = window.Components.interfaces.nsIFilePicker;
				var picker = Components.classes['@mozilla.org/filepicker;1'].createInstance(nsIFilePicker);
				picker.init(window, msg, nsIFilePicker.modeSave);
				var thispath = Components.classes['@mozilla.org/file/local;1'].createInstance(Components.interfaces.nsILocalFile);
				thispath.initWithPath(path);
				picker.displayDirectory=thispath;
				picker.defaultExtension=defext;
				picker.defaultString=file;
				picker.appendFilters(nsIFilePicker.filterAll|nsIFilePicker.filterText|nsIFilePicker.filterHTML);
				if (picker.show()!=nsIFilePicker.returnCancel) var result=picker.file.path;
			}
			catch(e) { alert('error during local file access: '+e.toString()) }
		}
		else { // IE
			try { // XPSP2 IE only
				var s = new ActiveXObject('UserAccounts.CommonDialog');
				s.Filter='All files|*.*|Text files|*.txt|HTML files|*.htm;*.html|XML files|*.xml|';
				s.FilterIndex=defext=='txt'?2:'html'?3:'xml'?4:1;
				s.InitialDir=path;
				s.FileName=file;
				if (s.showOpen()) var result=s.FileName;
			}
			catch(e) {  // fallback
				var result=prompt(msg,path+file);
			}
		}
		return result;
	},
	initFilter: function() {
		this.$('exportFilterStart').checked=false; this.$('exportStartDate').value='';
		this.$('exportFilterEnd').checked=false;  this.$('exportEndDate').value='';
		this.$('exportFilterTags').checked=false; this.$('exportTags').value='';
		this.$('exportFilterText').checked=false; this.$('exportText').value='';
		this.showFilterFields();
	},
	showFilterFields: function(which) {
		var show=this.$('exportFilterStart').checked;
		this.$('exportFilterStartBy').style.display=show?'block':'none';
		this.$('exportStartDate').style.display=show?'block':'none';
		var val=this.$('exportFilterStartBy').value;
		this.$('exportStartDate').value
			=this.getFilterDate(val,'exportStartDate').formatString(this.datetimefmt);
		if (which && (which.id=='exportFilterStartBy') && (val=='other'))
			this.$('exportStartDate').focus();

		var show=this.$('exportFilterEnd').checked;
		this.$('exportFilterEndBy').style.display=show?'block':'none';
		this.$('exportEndDate').style.display=show?'block':'none';
		var val=this.$('exportFilterEndBy').value;
		this.$('exportEndDate').value
			=this.getFilterDate(val,'exportEndDate').formatString(this.datetimefmt);
		 if (which && (which.id=='exportFilterEndBy') && (val=='other'))
			this.$('exportEndDate').focus();

		var show=this.$('exportFilterTags').checked;
		this.$('exportTags').style.display=show?'block':'none';

		var show=this.$('exportFilterText').checked;
		this.$('exportText').style.display=show?'block':'none';
	},
	getFilterDate: function(val,id) {
		var result=0;
		switch (val) {
			case 'file':
				result=new Date(document.lastModified);
				break;
			case 'other':
				result=new Date(this.$(id).value);
				break;
			default: // today=0, yesterday=1, one week=7, two weeks=14, a month=31
				var now=new Date(); var tz=now.getTimezoneOffset()*60000; now-=tz;
				var oneday=86400000;
				if (id=='exportStartDate')
					result=new Date((Math.floor(now/oneday)-val)*oneday+tz);
				else
					result=new Date((Math.floor(now/oneday)-val+1)*oneday+tz-1);
				break;
		}
		return result;
	},
	filterExportList: function() {
		var theList  = this.$('exportList'); if (!theList) return -1;
		var filterStart=this.$('exportFilterStart').checked;
		var val=this.$('exportFilterStartBy').value;
		var startDate=config.macros.exportTiddlers.getFilterDate(val,'exportStartDate');
		var filterEnd=this.$('exportFilterEnd').checked;
		var val=this.$('exportFilterEndBy').value;
		var endDate=config.macros.exportTiddlers.getFilterDate(val,'exportEndDate');
		var filterTags=this.$('exportFilterTags').checked;
		var tags=this.$('exportTags').value;
		var filterText=this.$('exportFilterText').checked;
		var text=this.$('exportText').value;
		if (!(filterStart||filterEnd||filterTags||filterText)) {
			alert('Please set the selection filter');
			this.$('exportFilterPanel').style.display='block';
			return -1;
		}
		if (filterStart&&filterEnd&&(startDate>endDate)) {
			var msg='starting date/time:\n'
			msg+=startDate.toLocaleString()+'\n';
			msg+='is later than ending date/time:\n'
			msg+=endDate.toLocaleString()
			alert(msg);
			return -1;
		}
		// if filter by tags, get list of matching tiddlers
		// use getMatchingTiddlers() (if MatchTagsPlugin is installed) for full boolean expressions
		// otherwise use getTaggedTiddlers() for simple tag matching
		if (filterTags) {
			var fn=store.getMatchingTiddlers||store.getTaggedTiddlers;
			var t=fn.apply(store,[tags]);
			var tagged=[];
			for (var i=0; i<t.length; i++) tagged.push(t[i].title);
		}
		// scan list and select tiddlers that match all applicable criteria
		var total=0;
		var count=0;
		for (var i=0; i<theList.options.length; i++) {
			// get item, skip non-tiddler list items (section headings)
			var opt=theList.options[i]; if (opt.value=='') continue;
			// get tiddler, skip missing tiddlers (this should NOT happen)
			var tiddler=store.getTiddler(opt.value); if (!tiddler) continue;
			var sel=true;
			if ( (filterStart && tiddler.modified<startDate)
			|| (filterEnd && tiddler.modified>endDate)
			|| (filterTags && !tagged.contains(tiddler.title))
			|| (filterText && (tiddler.text.indexOf(text)==-1) && (tiddler.title.indexOf(text)==-1)))
				sel=false;
			opt.selected=sel;
			count+=sel?1:0;
			total++;
		}
		return count;
	},
	deleteTiddlers: function() {
		var list=this.$('exportList'); if (!list) return;
		var tids=[];
		for (i=0;i<list.length;i++)
			if (list.options[i].selected && list.options[i].value.length)
				tids.push(list.options[i].value);
		if (!confirm('Are you sure you want to delete these tiddlers:\n\n'+tids.join(', '))) return;
		store.suspendNotifications();
		for (t=0;t<tids.length;t++) {
			var tid=store.getTiddler(tids[t]); if (!tid) continue;
			var msg="'"+tid.title+"' is tagged with 'systemConfig'.\n\n";
			msg+='Removing this tiddler may cause unexpected results.  Are you sure?'
			if (tid.tags.contains('systemConfig') && !confirm(msg)) continue;
			store.removeTiddler(tid.title);
			story.closeTiddler(tid.title);
		}
		store.resumeNotifications();
		alert(tids.length+' tiddlers deleted');
		this.refreshList(0); // reload listbox
		store.notifyAll(); // update page display
	},
	go: function() {
		if (window.location.protocol!='file:') // make sure we are local
			{ displayMessage(config.messages.notFileUrlError); return; }
		// get selected tidders, target filename, target type, and notes
		var list=this.$('exportList'); if (!list) return;
		var tids=[]; for (var i=0; i<list.options.length; i++) {
			var opt=list.options[i]; if (!opt.selected||!opt.value.length) continue;
			var tid=store.getTiddler(opt.value); if (!tid) continue;
			tids.push(tid);
		}
		if (!tids.length) return; // no tiddlers selected
		var target=this.$('exportFilename').value.trim();
		if (!target.length) {
			displayMessage('A local target path/filename is required',target);
			return;
		}
		var merge=this.$('exportMerge').checked;
		var filetype=this.$('exportFormat').value.toLowerCase();
		var notes=this.$('exportNotes').value.replace(/\n/g,'<br>');
		var total={val:0};
		var out=this.assembleFile(target,filetype,tids,notes,total,merge);
		if (!total.val) return; // cancelled file overwrite
		var link='file:///'+target.replace(/\\/g,'/');
		var samefile=link==decodeURIComponent(window.location.href);
		var p=getLocalPath(document.location.href);
		if (samefile) {
			if (config.options.chkSaveBackups) { var t=loadOriginal(p);if(t)saveBackup(p,t); }
			if (config.options.chkGenerateAnRssFeed && saveRss instanceof Function) saveRss(p);
		}
		var ok=saveFile(target,out);
		displayMessage((ok?this.okmsg:this.failmsg).format([total.val,total.val!=1?'s':'',target]),link);
	},
	plainTextHeader:
		 'Source:\n\t%0\n'
		+'Title:\n\t%1\n'
		+'Subtitle:\n\t%2\n'
		+'Created:\n\t%3 by %4\n'
		+'Application:\n\tTiddlyWiki %5 / %6 %7\n\n',
	plainTextTiddler:
		'- - - - - - - - - - - - - - -\n'
		+'|     title: %0\n'
		+'|   created: %1\n'
		+'|  modified: %2\n'
		+'| edited by: %3\n'
		+'|      tags: %4\n'
		+'- - - - - - - - - - - - - - -\n'
		+'%5\n',
	plainTextFooter:
		'',
	newsFeedHeader:
		 '<'+'?xml version="1.0"?'+'>\n'
		+'<rss version="2.0">\n'
		+'<channel>\n'
		+'<title>%1</title>\n'
		+'<link>%0</link>\n'
		+'<description>%2</description>\n'
		+'<language>en-us</language>\n'
		+'<copyright>Copyright '+(new Date().getFullYear())+' %4</copyright>\n'
		+'<pubDate>%3</pubDate>\n'
		+'<lastBuildDate>%3</lastBuildDate>\n'
		+'<docs>http://blogs.law.harvard.edu/tech/rss</docs>\n'
		+'<generator>TiddlyWiki %5 / %6 %7</generator>\n',
	newsFeedTiddler:
		'\n%0\n',
	newsFeedFooter:
		'</channel></rss>',
	pureStoreHeader:
		 '<html><body>'
		+'<style type="text/css">'
		+'	#storeArea {display:block;margin:1em;}'
		+'	#storeArea div {padding:0.5em;margin:1em;border:2px solid black;height:10em;overflow:auto;}'
		+'	#pureStoreHeading {width:100%;text-align:left;background-color:#eeeeee;padding:1em;}'
		+'</style>'
		+'<div id="pureStoreHeading">'
		+'	TiddlyWiki "PureStore" export file<br>'
		+'	Source'+': <b>%0</b><br>'
		+'	Title: <b>%1</b><br>'
		+'	Subtitle: <b>%2</b><br>'
		+'	Created: <b>%3</b> by <b>%4</b><br>'
		+'	TiddlyWiki %5 / %6 %7<br>'
		+'	Notes:<hr><pre>%8</pre>'
		+'</div>'
		+'<div id="storeArea">',
	pureStoreTiddler:
		'%0\n%1',
	pureStoreFooter:
		'</div><!--POST-BODY-START-->\n<!--POST-BODY-END--></body></html>',
	assembleFile: function(target,filetype,tids,notes,total,merge) {
		var revised='';
		var now = new Date().toLocaleString();
		var src=convertUnicodeToUTF8(document.location.href);
		var title = convertUnicodeToUTF8(wikifyPlain('SiteTitle').htmlEncode());
		var subtitle = convertUnicodeToUTF8(wikifyPlain('SiteSubtitle').htmlEncode());
		var user = convertUnicodeToUTF8(config.options.txtUserName.htmlEncode());
		var twver = version.major+'.'+version.minor+'.'+version.revision;
		var v=version.extensions.ExportTiddlersPlugin; var pver = v.major+'.'+v.minor+'.'+v.revision;
		var headerargs=[src,title,subtitle,now,user,twver,'ExportTiddlersPlugin',pver,notes];
		switch (filetype) {
			case this.type_TX: // plain text
				var header=this.plainTextHeader.format(headerargs);
				var footer=this.plainTextFooter;
				break;
			case this.type_CS: // comma-separated
				var fields={};
				for (var i=0; i<tids.length; i++) for (var f in tids[i].fields) fields[f]=f;
				var names=['title','created','modified','modifier','tags','text'];
				for (var f in fields) names.push(f);
				var header=names.join(',')+'\n';
				var footer='';
				break;
			case this.type_NF: // news feed (XML)
				headerargs[0]=store.getTiddlerText('SiteUrl','');
				var header=this.newsFeedHeader.format(headerargs);
				var footer=this.newsFeedFooter;
				break;
			case this.type_PS: // PureStore (no code)
				var header=this.pureStoreHeader.format(headerargs);
				var footer=this.pureStoreFooter;
				break;
			case this.type_TW: // full TiddlyWiki
			default:
				var currPath=getLocalPath(window.location.href);
				var original=loadFile(currPath);
				if (!original) { displayMessage(config.messages.cantSaveError); return; }
				var posDiv = locateStoreArea(original);
				if (!posDiv) { displayMessage(config.messages.invalidFileError.format([currPath])); return; }
				var header = original.substr(0,posDiv[0]+startSaveArea.length)+'\n';
				var footer = '\n'+original.substr(posDiv[1]);
				break;
		}
		var out=this.getData(target,filetype,tids,fields,merge);
		var revised = header+convertUnicodeToUTF8(out.join('\n'))+footer;
		// if full TW, insert page title and language attr, and reset all MARKUP blocks...
		if (filetype==this.type_TW) {
			var newSiteTitle=convertUnicodeToUTF8(getPageTitle()).htmlEncode();
			revised=revised.replaceChunk('<title'+'>','</title'+'>',' ' + newSiteTitle + ' ');
			revised=updateLanguageAttribute(revised);
			var titles=[]; for (var i=0; i<tids.length; i++) titles.push(tids[i].title);
			revised=updateMarkupBlock(revised,'PRE-HEAD',
				titles.contains('MarkupPreHead')? 'MarkupPreHead' :null);
			revised=updateMarkupBlock(revised,'POST-HEAD',
				titles.contains('MarkupPostHead')?'MarkupPostHead':null);
			revised=updateMarkupBlock(revised,'PRE-BODY',
				titles.contains('MarkupPreBody')? 'MarkupPreBody' :null);
			revised=updateMarkupBlock(revised,'POST-SCRIPT',
				titles.contains('MarkupPostBody')?'MarkupPostBody':null);
		}
		total.val=out.length;
		return revised;
	},
	getData: function(target,filetype,tids,fields,merge) {
		// output selected tiddlers and gather list of titles (for use with merge)
		var out=[]; var titles=[];
		var url=store.getTiddlerText('SiteUrl','');
		for (var i=0; i<tids.length; i++) {
			out.push(this.formatItem(store,filetype,tids[i],url,fields));
			titles.push(tids[i].title);
		}
		// if TW or PureStore format, ask to merge with existing tiddlers (if any)
		if (filetype==this.type_TW || filetype==this.type_PS) {
			var txt=loadFile(target);
			if (txt && txt.length) {
				var remoteStore=new TiddlyWiki();
				if (version.major+version.minor*.1+version.revision*.01<2.52) txt=convertUTF8ToUnicode(txt);
				if (remoteStore.importTiddlyWiki(txt)) {
					var existing=remoteStore.getTiddlers('title');
					var msg=this.overwriteprompt.format([target,existing.length,existing.length!=1?'s':'']);
					if (merge) {
						var added=titles.length; var updated=0; var kept=0;
						for (var i=0; i<existing.length; i++)
							if (titles.contains(existing[i].title)) {
								added--; updated++;
							} else {
								out.push(this.formatItem(remoteStore,filetype,existing[i],url));
								kept++;
							}
						displayMessage(this.mergestatus.format(
							[added,added!=1?'s':'',updated,updated!=1?'s':'',kept,kept!=1?'s':'',]));
					}
					else if (!confirm(msg)) out=[]; // empty the list = don't write file
				}
			}
		}
		return out;
	},
	formatItem: function(s,f,t,u,fields) {
		if (f==this.type_TW)
			var r=s.getSaver().externalizeTiddler(s,t);
		if (f==this.type_PS)
			var r=this.pureStoreTiddler.format([t.title,s.getSaver().externalizeTiddler(s,t)]);
		if (f==this.type_NF)
			var r=this.newsFeedTiddler.format([t.saveToRss(u)]);
		if (f==this.type_TX)
			var r=this.plainTextTiddler.format([t.title, t.created.toLocaleString(), t.modified.toLocaleString(),
				t.modifier, String.encodeTiddlyLinkList(t.tags), t.text]);
		if (f==this.type_CS) {
			function toCSV(t) { return '"'+t.replace(/"/g,'""')+'"'; } // always encode CSV
			var out=[ toCSV(t.title), toCSV(t.created.toLocaleString()), toCSV(t.modified.toLocaleString()),
				toCSV(t.modifier), toCSV(String.encodeTiddlyLinkList(t.tags)), toCSV(t.text) ];
			for (var f in fields) out.push(toCSV(t.fields[f]||''));
			var r=out.join(',');
		}
		return r||"";
	}
}
//}}}
/***
!!!Control panel CSS
//{{{
!css
#exportPanel {
	display: none; position:absolute; z-index:12; width:35em; right:105%; top:6em;
	background-color: #eee; color:#000; font-size: 8pt; line-height:110%;
	border:1px solid black; border-bottom-width: 3px; border-right-width: 3px;
	padding: 0.5em; margin:0em; -moz-border-radius:1em;-webkit-border-radius:1em;
}
#exportPanel a, #exportPanel td a { color:#009; display:inline; margin:0px; padding:1px; }
#exportPanel table {
	width:100%; border:0px; padding:0px; margin:0px;
	font-size:8pt; line-height:110%; background:transparent;
}
#exportPanel tr { border:0px;padding:0px;margin:0px; background:transparent; }
#exportPanel td { color:#000; border:0px;padding:0px;margin:0px; background:transparent; }
#exportPanel select { width:98%;margin:0px;font-size:8pt;line-height:110%;}
#exportPanel input  { width:98%;padding:0px;margin:0px;font-size:8pt;line-height:110%; }
#exportPanel textarea  { width:98%;padding:0px;margin:0px;overflow:auto;font-size:8pt; }
#exportPanel .box {
	border:1px solid black; padding:3px; margin-bottom:5px;
	background:#f8f8f8; -moz-border-radius:5px;-webkit-border-radius:5px; }
#exportPanel .topline { border-top:2px solid black; padding-top:3px; margin-bottom:5px; }
#exportPanel .rad { width:auto;border:0 }
#exportPanel .chk { width:auto;border:0 }
#exportPanel .btn { width:auto; }
#exportPanel .btn1 { width:98%; }
#exportPanel .btn2 { width:48%; }
#exportPanel .btn3 { width:32%; }
#exportPanel .btn4 { width:24%; }
#exportPanel .btn5 { width:19%; }
!end
//}}}
!!!Control panel HTML
//{{{
!html
<!-- target path/file  -->
<div>
<div style="float:right;padding-right:.5em">
<input type="checkbox" style="width:auto" id="exportMerge" CHECKED
	title="combine selected tiddlers with existing tiddlers (if any) in export file"> merge
</div>
export to:<br>
<input type="text" id="exportFilename" size=40 style="width:93%"><input
	type="button" id="exportBrowse" value="..." title="select or enter a local folder/file..." style="width:5%"
	onclick="var fn=config.macros.exportTiddlers.askForFilename(this); if (fn.length) this.previousSibling.value=fn; ">
</div>

<!-- output format -->
<div>
format:
<select id="exportFormat" size=1>
	<option value="TW">TiddlyWiki HTML document (includes core code)</option>
	<option value="PS">TiddlyWiki "PureStore" HTML file (tiddler data only)</option>
	<option value="TX">TiddlyWiki plain text TXT file (tiddler source listing)</option>
	<option value="CS">Comma-Separated Value (CSV) data file</option>
	<option value="NF">RSS NewsFeed XML file</option>
</select>
</div>

<!-- notes -->
<div>
notes:<br>
<textarea id="exportNotes" rows=3 cols=40 style="height:4em;margin-bottom:5px;" onfocus="this.select()"></textarea>
</div>

<!-- list of tiddlers -->
<table><tr align="left"><td>
	select:
	<a href="JavaScript:;" id="exportSelectAll"
		onclick="return config.macros.exportTiddlers.process(this)" title="select all tiddlers">
		&nbsp;all&nbsp;</a>
	<a href="JavaScript:;" id="exportSelectChanges"
		onclick="return config.macros.exportTiddlers.process(this)" title="select tiddlers changed since last save">
		&nbsp;changes&nbsp;</a>
	<a href="JavaScript:;" id="exportSelectOpened"
		onclick="return config.macros.exportTiddlers.process(this)" title="select tiddlers currently being displayed">
		&nbsp;opened&nbsp;</a>
	<a href="JavaScript:;" id="exportSelectRelated"
		onclick="return config.macros.exportTiddlers.process(this)" title="select tiddlers related to the currently selected tiddlers">
		&nbsp;related&nbsp;</a>
	<a href="JavaScript:;" id="exportToggleFilter"
		onclick="return config.macros.exportTiddlers.process(this)" title="show/hide selection filter">
		&nbsp;filter&nbsp;</a>
</td><td align="right">
	<a href="JavaScript:;" id="exportListSmaller"
		onclick="return config.macros.exportTiddlers.process(this)" title="reduce list size">
		&nbsp;&#150;&nbsp;</a>
	<a href="JavaScript:;" id="exportListLarger"
		onclick="return config.macros.exportTiddlers.process(this)" title="increase list size">
		&nbsp;+&nbsp;</a>
</td></tr></table>
<select id="exportList" multiple size="10" style="margin-bottom:5px;"
	onchange="config.macros.exportTiddlers.refreshList(this.selectedIndex)">
</select><br>

<!-- selection filter -->
<div id="exportFilterPanel" style="display:none">
<table><tr align="left"><td>
	selection filter
</td><td align="right">
	<a href="JavaScript:;" id="exportHideFilter"
		onclick="return config.macros.exportTiddlers.process(this)" title="hide selection filter">hide</a>
</td></tr></table>
<div class="box">

<input type="checkbox" class="chk" id="exportFilterStart" value="1"
	onclick="config.macros.exportTiddlers.showFilterFields(this)"> starting date/time<br>
<table cellpadding="0" cellspacing="0"><tr valign="center"><td width="50%">
	<select size=1 id="exportFilterStartBy"
		onchange="config.macros.exportTiddlers.showFilterFields(this);">
		<option value="0">today</option>
		<option value="1">yesterday</option>
		<option value="7">a week ago</option>
		<option value="30">a month ago</option>
		<option value="file">file date</option>
		<option value="other">other (mm/dd/yyyy hh:mm)</option>
	</select>
</td><td width="50%">
	<input type="text" id="exportStartDate" onfocus="this.select()"
		onchange="config.macros.exportTiddlers.$('exportFilterStartBy').value='other';">
</td></tr></table>

<input type="checkbox" class="chk" id="exportFilterEnd" value="1"
	onclick="config.macros.exportTiddlers.showFilterFields(this)"> ending date/time<br>
<table cellpadding="0" cellspacing="0"><tr valign="center"><td width="50%">
	<select size=1 id="exportFilterEndBy"
		onchange="config.macros.exportTiddlers.showFilterFields(this);">
		<option value="0">today</option>
		<option value="1">yesterday</option>
		<option value="7">a week ago</option>
		<option value="30">a month ago</option>
		<option value="file">file date</option>
		<option value="other">other (mm/dd/yyyy hh:mm)</option>
	</select>
</td><td width="50%">
	<input type="text" id="exportEndDate" onfocus="this.select()"
		onchange="config.macros.exportTiddlers.$('exportFilterEndBy').value='other';">
</td></tr></table>

<input type="checkbox" class="chk" id=exportFilterTags value="1"
	onclick="config.macros.exportTiddlers.showFilterFields(this)"> match tags<br>
<input type="text" id="exportTags" onfocus="this.select()">

<input type="checkbox" class="chk" id=exportFilterText value="1"
	onclick="config.macros.exportTiddlers.showFilterFields(this)"> match titles/tiddler text<br>
<input type="text" id="exportText" onfocus="this.select()">

</div> <!--box-->
</div> <!--panel-->

<!-- action buttons -->
<div style="text-align:center">
<input type=button class="btn4" onclick="config.macros.exportTiddlers.process(this)"
	id="exportFilter" value="apply filter">
<input type=button class="btn4" onclick="config.macros.exportTiddlers.process(this)"
	id="exportStart" value="export tiddlers">
<input type=button class="btn4" onclick="config.macros.exportTiddlers.process(this)"
	id="exportDelete" value="delete tiddlers">
<input type=button class="btn4" onclick="config.macros.exportTiddlers.process(this)"
	id="exportClose" value="close">
</div><!--center-->
!end
//}}}
***/

<<tiddler [[include_tiddlers/Exterior Derivative.html#"Exterior Derivative"]]>>

/***
|Name|ExternalTiddlersPlugin|
|Source|http://www.TiddlyTools.com/#ExternalTiddlersPlugin|
|Documentation|http://www.TiddlyTools.com/#ExternalTiddlersPluginInfo|
|Version|1.3.3|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Requires|TemporaryTiddlersPlugin, SectionLinksPlugin (optional, recommended)|
|Description|retrieve and wikify content from external files or remote URLs|
This plugin extends the {{{<<tiddler>>}}} macro syntax so you can retrieve and wikify content directly from external files or remote URLs.  You can also define alternative "fallback" sources to provide basic "import on demand" handling by automatically creating/importing tiddler content from external sources when the specified ~TiddlerName does not already exist in your document.
!!!!!Documentation
>see [[ExternalTiddlersPluginInfo]]
!!!!!Configuration
<<<
<<option chkExternalTiddlersImport>> automatically create/import tiddlers when using external fallback references
{{{usage: <<option chkExternalTiddlersImport>>}}}
<<option chkExternalTiddlersQuiet>> don't display messages when adding tiddlers ("quiet mode")
{{{usage: <<option chkExternalTiddlersQuiet>>}}}
<<option chkExternalTiddlersTemporary>> tag retrieved tiddlers as 'temporary'(requires [[TemporaryTiddlersPlugin]])
{{{usage: <<option chkExternalTiddlersTemporary>>}}}
tag retrieved tiddlers with: <<option txtExternalTiddlersTags>>
{{{usage: <<option txtExternalTiddlersTags>>}}}

__password-protected server settings //(optional, if needed)//:__
>username: <<option txtRemoteUsername>> password: <<option txtRemotePassword>>
>{{{usage: <<option txtRemoteUsername>> <<option txtRemotePassword>>}}}
>''note: these settings are also used by [[LoadTiddlersPlugin]] and [[ImportTiddlersPlugin]]''
<<<
!!!!!Revisions
<<<
2011.04.27 1.3.3 merge/clone defaultCustomFields for saving in TiddlySpace
|please see [[ExternalTiddlersPluginInfo]] for additional revision details|
2007.11.25 1.0.0 initial release - moved from CoreTweaks
<<<
!!!!!Code
***/
//{{{
version.extensions.ExternalTiddlersPlugin= {major: 1, minor: 3, revision: 3, date: new Date(2011,4,26)};

// optional automatic import/create for missing tiddlers
if (config.options.chkExternalTiddlersImport==undefined) config.options.chkExternalTiddlersImport=true;
if (config.options.chkExternalTiddlersTemporary==undefined) config.options.chkExternalTiddlersTemporary=true;
if (config.options.chkExternalTiddlersQuiet==undefined) config.options.chkExternalTiddlersQuiet=false;
if (config.options.txtExternalTiddlersTags==undefined) config.options.txtExternalTiddlersTags="external";
if (config.options.txtRemoteUsername==undefined) config.options.txtRemoteUsername="";
if (config.options.txtRemotePassword==undefined) config.options.txtRemotePassword="";

config.macros.tiddler.externalTiddlers_handler = config.macros.tiddler.handler;
config.macros.tiddler.handler = function(place,macroName,params,wikifier,paramString,tiddler)
{
	params = paramString.parseParams("name",null,true,false,true);
	var names = params[0]["name"];
	var list = names[0];
	var items = list.split("|");
	var className = names[1] ? names[1] : null;
	var args = params[0]["with"];

	// UTILITY FUNCTIONS
	function extract(text,tids) { // get tiddler source content from plain text or TW doc
		if (!text || !tids || !tids.length) return text; // no text or no tiddler list... return text as-is
		var remoteStore=new TiddlyWiki();
		if (!remoteStore.importTiddlyWiki(text)) return text; // not a TW document... return text as-is
		var out=[]; for (var t=0;t<tids.length;t++)
			{ var txt=remoteStore.getTiddlerText(tids[t]); if (txt) out.push(txt); }
		return out.join("\n");
	}
	function substitute(text,args) { // replace "substitution markers" ($1-$9) with macro param values (if any)
		if (!text || !args || !args.length) return text;
		var n=args.length; if (n>9) n=9;
		for(var i=0; i<n; i++) { var re=new RegExp("\\$" + (i + 1),"mg"); text=text.replace(re,args[i]); }
		return text;
	}
	function addTiddler(src,text,tids) { // extract tiddler(s) from text and create local copy
		if (!config.options.chkExternalTiddlersImport) return; // not enabled... do nothing
		if (!text || !tids || !tids.length) return; // no text or no tiddler list... do nothing
		var remoteStore=new TiddlyWiki();
		if (!remoteStore.importTiddlyWiki(text)) // not a TW document... create a single tiddler from text
			makeTiddler(src,text,tids[0]);
		else // TW document with "permaview-like" suffix... copy tiddler(s) from remote store
			for (var t=0;t<tids.length;t++)
				insertTiddler(src,remoteStore.getTiddler(tids[t]));
		return;
	}
	function makeTiddler(src,text,title) { // create a new tiddler object from text
		var who=config.options.txtUserName; var when=new Date();
		var msg="/%\n\nThis tiddler was automatically created using ExternalTiddlersPlugin\n";
		msg+="by %0 on %1\nsource: %2\n\n%/";
		var tags=config.options.txtExternalTiddlersTags.readBracketedList();
		if (config.options.chkExternalTiddlersTemporary) tags.pushUnique(config.options.txtTemporaryTag);
		var fields=merge({},config.defaultCustomFields,true)
		store.saveTiddler(null,title,msg.format([who,when,src])+text,who,when,tags,fields);
		if (!config.options.chkExternalTiddlersQuiet) displayMessage("Created new tiddler '"+title+"' from text file "+src);
	}
	function insertTiddler(src,t) { // import a single tiddler object into the current document store
		if (!t) return;
		var who=config.options.txtUserName; var when=new Date();
		var msg="/%\n\nThis tiddler was automatically imported using ExternalTiddlersPlugin\n";
		msg+="by %0 on %1\nsource: %2\n\n%/";
		var newtags=new Array().concat(t.tags,config.options.txtExternalTiddlersTags.readBracketedList());
		if (config.options.chkExternalTiddlersTemporary) newtags.push(config.options.txtTemporaryTag);
		var fields=merge(t.fields,config.defaultCustomFields,true)
		store.saveTiddler(null,t.title,msg.format([who,when,src])+t.text,t.modifier,t.modified,newtags,fields);
		if (!config.options.chkExternalTiddlersQuiet) displayMessage("Imported tiddler '"+t.title+"' from "+src);
	}
	function getGUID()  // create a Globally Unique ID (for async reference to DOM elements)
		 { return new Date().getTime()+Math.random().toString(); }

	// loop through "|"-separated list of alternative tiddler/file/URL references until successful
	var fallback="";
	for (var i=0; i<items.length; i++) { var src=items[i];
		// if tiddler (or shadow) exists, replace reference list with current source name and apply core handler
		if (store.getTiddlerText(src)) {
			arguments[2][0]=src; // params[] array
			var p=arguments[4].split(list); arguments[4]=p[0]+src+p[1]; // paramString
			this.externalTiddlers_handler.apply(this,arguments);
			break; // stop processing alternatives
		}

		// tiddler doesn't exist, and not an external file/URL reference... skip it
		if (!config.formatterHelpers.isExternalLink(src)) {
			if (!fallback.length) fallback=src; // title to use when importing external tiddler
			continue;
		}
		// separate 'permaview' list of tiddlers (if any) from file/URL (i.e., '#name name name..." suffix)
		var p=src.split("#"); src=p.shift(); var tids=p.join('#').readBracketedList(false);
		// if reference is to a remotely hosted document or the current document is remotely hosted...
		if (src.substr(0,4)=="http" || document.location.protocol.substr(0,4)=="http") {
			if (src.substr(0,4)!="http") // fixup URL for relative remote references
				{ var h=document.location.href; src=h.substr(0,h.lastIndexOf("/")+1)+src; }
			var wrapper = createTiddlyElement(place,"span",getGUID(),className); // create placeholder for async rendering
			var callback=function(success,params,text,src,xhr) { // ASYNC CALLBACK
				if (!success) { displayMessage(xhr.status); return; } // couldn't read remote file... report the error
				if (params.fallback.length)
					addTiddler(params.url,text,params.tids.length?params.tids:[params.fallback]); // import tiddler
				var wrapper=document.getElementById(params.id); if (!wrapper) return;
				wikify(substitute(extract(text,params.tids),params.args),wrapper); // ASYNC RENDER
			};
			var callbackparams={ url:src, id:wrapper.id, args:args, tids:tids, fallback:fallback }  // ASYNC PARAMS
			var name=config.options.txtRemoteUsername; // optional value
			var pass=config.options.txtRemotePassword; // optional value
			var x=doHttp("GET",src,null,null,name,pass,callback,callbackparams,null)
			if (typeof(x)=="string") // couldn't start XMLHttpRequest... report error
				{ displayMessage("error: cannot access "+src); displayMessage(x); }
			break; // can't tell if async read will succeed.... stop processing alternatives anyway.
		}
		else { // read file from local filesystem
			var text=loadFile(getLocalPath(src));
			if (!text) { // couldn't load file... fixup path for relative reference and retry...
				var h=document.location.href;
				var text=loadFile(getLocalPath(decodeURIComponent(h.substr(0,h.lastIndexOf("/")+1)))+src);
			}
			if (text) { // test it again... if file was loaded OK, render it in a class wrapper
				if (fallback.length) // create new tiddler using primary source name (if any)
					addTiddler(src,text,tids.length?tids:[fallback]);
				var wrapper=createTiddlyElement(place,"span",null,className);
				wikify(substitute(extract(text,tids),args),wrapper); // render
				break; // stop processing alternatives
			}
		}
	}
};
//}}}

An ''Extraspecial Group $2^{1+2d}$'' is a subgroup of $GL(2d, \mathbb F)$, for a field $\mathbb F$ of characteristic $0$.

Papers:
* [[Extremal Lattices - R. Scharlau, R. Schulze-Pillot|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.46.1119&rep=rep1&type=ps]] [[pct. 30|http://scholar.google.de/scholar?cites=16121390943809304545&hl=de&as_sdt=2000]]

<<tiddler [[include_tiddlers/F4.html#"F4"]]>>

<<tiddler [[include_tiddlers/Fano Plane.html#"Fano Plane"]]>>

<<tiddler [[include_tiddlers/Fano Planes - Classification.html#"Fano Planes - Classification"]]>>

In the following a description if what will be called ''Fano Spaces'' in terms of hypercomplex numbers is given:
!!!!Fano Point
[[Complex Numbers|Complex Number]]: The Fano point is defined by the imaginary unit $i$.

!!!!Fano Line
[[Quaternions|Quaternion]]: The Fano line is defined by the three imaginary units (Fano points). This relates to the fact that quaternions contain $3$ complex subalgebras.
<html><center><img src="images/Fano_line.jpg" style="width: 180px; "/></center></html>
!!!![[Fano Plane]]
[[Octonions|Octonion]]: The Fano plane is defined by the seven imaginary units (Fano points) and seven Fano lines. This relates to the fact that the octonions contain $7$ complex and $7$ quaternionic subalgebras.

!!!!Fano Tetrahedron
The Fano tetrahedron represents the projective space [[PG(3,2)]].

{{center{[img(429px+, )[images/fano_tetrahedron.jpg]]}}}
[[Sedenions|Sedenion]]: The Fano tetrahedron is built out of $15$ Fano planes. For this it is required that any $2$ of them have exactly one Fano line in common. This way the set of [[30 different Fano planes|Fano Planes - Classification]] splits up into two subsets with $15$ planes each. Out of both of them one can construct a Fano tetrahedron.
Every one of the $15$ Fano planes has $7$ Fano lines, summing up to $105$. As however every Fano line of a Fano tetrahedron is found in exactly $3$ Fano planes constituting it one has $35$ different Fano lines altogether.
These facts relate to the fact that the sedenions have $15$ octonionic or octonion-like, $35$ quaternionic and $15$ complex [[subalgebras|Sedenion Subalgebras]].

''4-D Fano Tetrahedron''
The 4-D Fano tetrahedron represents the projective space [[PG(4,2)]].

Links:
* [[A Finite Projective Space - D. A. Richter|http://homepages.wmich.edu/~drichter/projectivespace.htm]]
* [[Burkard Polster's Page|http://web.maths.monash.edu.au/~bpolster/]]

Google books:
* [[A Geometrical Picture Book - B. Polster|http://books.google.com/books?id=2PtPG4qjfZAC&printsec=frontcover&dq=intitle:A+intitle:Geometrical+intitle:Picture+intitle:Book&lr=&num=100&as_brr=0&as_pt=ALLTYPES&ei=l7VQSee3DYTMlQSRwpToBg#PPR15,M1]] [[local|google_books/AGeometricalPictureBook.pdf]] [[bct. 18|http://scholar.google.de/scholar?cites=7536479039420982593&hl=de]] brl. 10 - "A picture is worth a thousand formula".

<<tiddler [[include_tiddlers/Fermi Gamma-ray Space Telescope.html#"Fermi Gamma-ray Space Telescope"]]>>

The ''Fermi Paradox'' is the apparent contradiction between high estimates of the probability of the existence of extraterrestrial civilizations and the lack of evidence for such civilizations and contact with them.

Links:
* [[WIKIPEDIA - Fermi Paradox|http://en.wikipedia.org/wiki/Fermi_paradox]]

<<tiddler [[include_tiddlers/Fermi's Golden Rule.html#"Fermi's Golden Rule"]]>>

<<tiddler [[include_tiddlers/Fermionic Path Integral.html#"Fermionic Path Integral"]]>>

* [[A Possible Mechanism for Evading Temperature Quantum Decoherence in Living Matter by Feshbach Resonance (2009) - N. Poccia, A. Ricci, D. Innocenti, A. Bianconi|http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2695269/pdf/ijms-10-02084.pdf]] [[local|papers/ijms-10-02084.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=1650174702112680860&hl=de&as_sdt=2000]]

Lectures:
* [[Bose-Einstein Condensation in a Dilute Gas; the First 70 Years and some Recent Experiments - E. A. Cornell, E. Wieman|http://nobelprize.org/nobel_prizes/physics/laureates/2001/cornellwieman-lecture.pdf]] [[local|lectures/cornellwieman-lecture.pdf]]

The ''Feshbach\-Villars Representation'' casts the [[Klein-Gordon equation|Klein-Gordon Equation]] into two equations, both of which are first order in time.

<br><<tiddler [[include_tiddlers/Feynman Checkerboard.html#"Feynman Checkerboard"]]>>

<<tiddler [[include_tiddlers/Fiber Bundle.html#"Fiber Bundle"]]>>

<<tiddler [[include_tiddlers/Field.html#"Field"]]>>

!!!!Electrodynamics
The field strength tensor $F^{\mu\nu} $ is defined by:
\[
F_{\mu\nu}  = \partial_{\mu}A_{\nu} - \partial_{\nu} A_{\mu}
\]
Written out explicitely one has
\[
F_{\mu\nu} \equiv
\left(\begin{matrix}
0  &  E_x &  E_y & E_z \\
-E_x &   0  &  -B_z & B_y \\
-E_y & B_z &   0  &  -B_x \\
-E_z &  -B_y & B_x &   0  \\
\end{matrix}\right)
\]
Its [[dual|Duality Rotation]] $\tilde F^{\mu\nu} $ is defined by:
\[
\tilde{F}^{\mu\nu} \equiv \frac{1}{2}\, \varepsilon^{\mu\nu\alpha\beta}\,F_{\alpha\beta}  =
\begin{pmatrix}
0  & -B_x & -B_y & -B_z \\
B_x &   0  &  E_z & -E_y \\
B_y & -E_z &   0  &  E_x\\
B_z &  E_y & -E_x &   0 \\
\end{pmatrix}
\]
!!!!!Properties
* Antisymmetry: $F_{\mu\nu} = ? F_{\nu\mu}$
* Tracelesness: $F_{\mu\mu} = 0$
* $6$ independent components

Links:
* [[Les Médailles Fields|http://serge.mehl.free.fr/anx/med_fields.html]]

<<tiddler [[include_tiddlers/Fifth Force.html#"Fifth Force"]]>>

<<tiddler [[include_tiddlers/Fine Structure Constant.html#"Fine Structure Constant"]]>>

<<tiddler [[include_tiddlers/Fine Tuning.html#"Fine Tuning"]]>>

A ''Finite Geometry'' is any geometric system that has only a finite number of points. A finite geometry can have any (finite) number of dimensions. Euclidean geometry, for example, is not a finite geometry, as it is based on the real numbers.

Objects of investigation of finite geometry are finite [[incidence structures|Incidence Structure]]. Finite geometries are therefore also called ''Incidence Geometries''.

Finite geometries serve as an interface between geometry and discrete mathematics (in particular combinatorics).

Links:
* [[Elements of Finite Geometry - S. H. Cullinane|http://finitegeometry.org/]]
* [[Finite Geomtetries and Axiomatic Systems - B. Eastman|http://www.beva.org/math323/asgn5/nov5.htm]]

Essentially, a ''Finsler Manifold'' $\mathcal M$ is manifold where each tangent space $T\mathcal{M}$ is equipped with a Minkowski norm. This norm induces a canonical [[inner product|Scalar Product]].
However, in sharp contrast to the Riemannian case, Finsler inner products are not parametrized by points of $\mathcal M$, but by directions in $T\mathcal{M}$. Thus one can think of a Finsler manifold as a space where the inner product does not only depend on where you are, but also in which direction you are looking.
Still Finsler geometry contains many analogues of [[Riemannian geometry|Riemann Space]] such as lengths, geodesics, curvature, connections, covariant derivatives. Structure equations also hold. However, [[normal coordinates|Normal Coordinates]] do not generalize to the Finslerian case.
Finsler geometry is not a generalization of Riemannian geometry. It is better understood as Riemannian geometry without the quadratic restriction $F^2(x^1,\ldots,x^n; dx^1,\ldots,dx^n) = g_{\mu\nu}(x^1,\ldots,x^n) dx^\mu dx^\nu$.

Papers:
* [[Finsler Geometry is just Riemannian Geometry without the Quadratic Restriction (1996) - S.-S. Chern|http://www.ams.org/notices/199609/chern.pdf]]
@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/feynman_ico.jpeg][Comments]]}}}&nbsp;@@

<<tiddler [[include_tiddlers/First Bianchi Identity.html#"First Bianchi Identity"]]>>

The ''First Fundamental Form'' of a manifold is given by:
\[
|d\mb s|^2 = g_{\mu\nu}(\mb x) dx^\mu dx^\nu
\]
The properties of a manifold that can be described by means of the first fundamental form are part of the inner geometry of the manifold.

''2 dimensions'':
One usually defines $g_{11} = E$, $g_{12} = F$ and $g_{22} = G$.
Therefore:
\begin{eqnarray}
|d\mb s|^2 &= & E dx_1^2 + 2 F dx_1 dx_2 + G dx_2^2
\end{eqnarray}

__Derivation:__
(Maybe not perfect as we do an embedding in an ambient Euclidean space).

Let's regard a $m$-dimensional manifold $\mathcal M \in \mathbb R^n$ $m \le n$ parametrized by $\mb X(\mb x)$.
Furthermore let $\mb C(\tau)$ be a $1$-parameter curve in the manifold.
We have:
\[
\frac{d\mb C(\tau)}{d\tau} = \sum_{i=1}^m \frac{\partial \mb X (\mb x)}{\partial x_i}  \frac{d x_i}{d\tau} = \sum_{i=1}^m \mb e_i (\mb x)  \frac{d x_i}{d\tau}
\]
where the $\mb e_i(\mb x)$ define a local basis in the manifold.
The path length $l_{\tau_0}(\tau_1)$  of the curve is given by:
\[
l_{\tau_0}(\tau_1) = \int_{\tau_0}^{\tau_1} {\sqrt{\left \langle \frac{d\mb C(\tau)}{d\tau} | \frac{d\mb C(\tau)}{d\tau} \right \rangle} d\tau}
\]
and therefore
\begin{eqnarray}
l_{\tau_0}(\tau_1) & = & \int_{\tau_0}^{\tau_1} {\sqrt{\sum_{i,j} \langle \mb e_i (\mb x) | \mb e_j (\mb x) \rangle dx_i dx_j} d\tau} \\
   & = & \int_{\tau_0}^{\tau_1} {\sqrt{\sum_{i,j} g_{ij}(\mb x) dx_i dx_j} d\tau} \\
   & = & \int_{\tau_0}^{\tau_1} {\sqrt{|d\mb s|^2} d\tau} \\
\end{eqnarray}
with $g_{ij} (\mb x)$ the [[induced metric|Induced Metric]] of $\mathcal M$.

Given a [[projective geometry|Projective Geometry]] $PG(n,q)$, a ''Flat'' is a subspace of dimension $k?1$.

More generally, for $1 \le k \le n$, a subset $K \subseteq PG(n,q)$ is a ''$k$-Flat'' if $K$ is isomorphic to $PG(k,q)$.
E.g., a line is a $1$-flat, a plane is a $2$-flat and a solid is a $3$-flat.

The complement of a $(n-1)$-flat in $PG(n, q)$ is isomorphic to $AG(n, q)$.

<<tiddler [[include_tiddlers/Flexible Algebra.html#"Flexible Algebra"]]>>

The ''Floor Function'' maps a real number $x$ to the next smallest integer, i.e.
\[
\operatorname {floor} (x) \equiv \lfloor x \rfloor \equiv \max\, \{n\in\mathbb{Z}\mid n\le x\}
\]
Links:
* [[WIKIPEDIA - Floor Function|http://en.wikipedia.org/wiki/Floor_and_ceiling_functions]]

<<tiddler [[include_tiddlers/Fokker-Planck Equation.html#"Fokker-Planck Equation"]]>>

!!!!Clifford 3-form/volume-form
\begin{eqnarray}
dV = && \frac{1}{3!}  (dx_1\wedge dx_2 \wedge dx_3 + dx_2 \wedge dx_3 \wedge dx_1 +  dx_3 \wedge dx_1 \wedge dx_2 \\
&&- dx_2\wedge dx_1 \wedge dx_3 - dx_1 \wedge dx_3 \wedge dx_1 -  dx_3\wedge dx_2 \wedge dx_1)
\end{eqnarray}
In case that the coordinates are commutative the form equals zero.

!!!!Cayley\-Dickson 3-form/volume\-form
\begin{eqnarray}

2 \cdot 3! \cdot dV &=& [dx_1,[dx_2,dx_3]] + [dx_2,[dx_3,dx_1]] + [dx_3,[dx_1,dx_2]] \\
&=& [dx_1, (dx_2dx_3 - dx_3dx_2)] + [dx_2, (dx_3dx_1 - dx_1dx_3)] + [dx_3, (dx_1dx_2 - dx_2dx_1)]\\
&=& dx_1(dx_2dx_3 - dx_3dx_2) -  (dx_2dx_3 - dx_3dx_2) dx1 + \\
&&  dx_2(dx_3dx_1 - dx_1dx_3) -  (dx_3dx_1 - dx_1dx_3) dx_2 + \\
&&  dx_3(dx_1dx_2 - dx_2dx_1) -  (dx_1dx_2 - dx_2dx_1) dx_3  \\
&=& dx_1(dx_2dx_3)  -dx_1(dx_3dx_2) -  (dx_2dx_3)dx_1 + (dx_3dx_2)dx_1 + \\
&& dx_2(dx_3dx_1)  -dx_2(dx_1dx_3) -  (dx_3dx_1)dx_2 + (dx_1dx_3)dx_2 + \\
&& dx_3(dx_1dx_2)  -dx_3(dx_2dx_1) -  (dx_1dx_2)dx_3 + (dx_2dx_1)dx_3 \\
\end{eqnarray}
In case that the coordinates are associative the form equals zero.

Papers:
* [[Quantum Interpretations of the Four Color Theorem - P. C. Kainen|http://www9.georgetown.edu/faculty/kainen/qtm4ct.pdf]]

<<tiddler [[include_tiddlers/Fourth Order Bol Identities Expansions.html#"Fourth Order Bol Identities Expansions"]]>>

<html><center><img src="images/p_mannheim.jpg" style="width: 680px; "/></center></html>$\quad\quad\quad\quad$ - Philip Mannheim -

Papers:
* [[Living with Ghosts - S. W. Hawking, T. Hertog|http://arxiv.org/PS_cache/hep-th/pdf/0107/0107088v2.pdf]] [[pct. 68|http://scholar.google.de/scholar?hl=de&lr=&cites=5775590995509111619]]
* [[On the History of Fourth Order Metric Theories of Gravitation - R. Schimming, H. J. Schmidt|http://arxiv.org/PS_cache/gr-qc/pdf/0412/0412038v1.pdf]] [[pct. 11|http://scholar.google.de/scholar?hl=de&lr=&cites=11897331899145799901]]

<<tiddler [[include_tiddlers/Fractal.html#"Fractal"]]>>

Papers:
* [[Fractal Properties of Quantum Spacetime (2009) - D. Benedetti|http://arxiv.org/PS_cache/arxiv/pdf/0811/0811.1396v2.pdf]] [[local|papers/0811.1396v2.pdf]] [[pct. 9|http://scholar.google.com/scholar?hl=de&lr=&cites=15178166396834749168&um=1&ie=UTF-8&ei=oyfmTN3ZMY3HswaaqoWgCw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCQQzgIwAA]]

Links:
* [[PHYSORG,  Spacetime may have Fractal Properties on a Quantum Scale|http://www.physorg.com/news157203574.html]]
* [[WIKIPEDIA - Fractal Cosmology|http://en.wikipedia.org/wiki/Fractal_cosmology]]

<<tiddler [[include_tiddlers/Free Algebra.html#"Free Algebra"]]>>

<<tiddler [[include_tiddlers/Free Parameters of the Standard Model.html#"Free Parameters of the Standard Model"]]>>

<br><<tiddler [[include_tiddlers/Freeman Dyson.html#"Freeman Dyson"]]>>

<<tiddler [[include_tiddlers/Freudental's Magic Square.html#"Freudental's Magic Square"]]>>

The two independent ''Friedmann Equations'' are derived from the [[Einstein equations|Einstein Field Equations]] and describe a homogeneous, isotropic universe.
They read
\begin{eqnarray}
H^2 = &\left(\frac{\dot{a}}{a}\right)^2 & = &\frac{8 \pi G} 3 \rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}3 \\
\dot H + H^2 = &\frac{\ddot{a}} a & = & -\frac{4 \pi G} 3 \left(\rho+\frac{3p}{c^2}\right) + \frac{\Lambda c^2}3
\end{eqnarray}
where $\rho$ is the total energy density of the universe (sum of matter, radiation, [[dark energy|Dark Energy]]), and $p$ is the total pressure (sum of pressures of each component). $\Lambda$ is the [[cosmological constant|Cosmological Constant]].
$k$ is the curvature of $3$-dimensional space: $k = 0$ corresponds to a spatially flat, Euclidean Universe, $k > 0$ to positive curvature ($3$-sphere), and $k < 0$ to negative curvature (saddle).

The successful predictions of the radiation dominated era of cosmology, e.g., big bang nucleosynthesis and the formation of CMB anisotropies, provide evidence for the $\rho+\frac{3p}{c^2}$-term.

See also:
* [[Copernican principle|Copernican Principle]]


<html><center><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_60.html" width=51% height=86></iframe></center></html>
Links:
* [[WIKIPEDIA - Friedmann Equations|http://en.wikipedia.org/wiki/Friedmann_equations]]

According to the ''Frobenius Theorem'' the only finite dimensional associative division algebras over the real numbers are the real numbers, the complex numbers and the [[quaternions|Quaternion]].
An extended version states that every alternative [[division algebra|Division Algebra]] is isomorphic to one of the following: the algebra of real numbers, the algebra of complex numbers, the [[quaternions|Quaternion]] and the [[Cayley numbers|Octonion]].

<<tiddler [[include_tiddlers/Fubini-Study Metric.html#"Fubini-Study Metric"]]>>

<<tiddler [[include_tiddlers/Fun Stuff.html#"Fun Stuff"]]>>

The notion of a ''(Closed) $G$\-Structure'' of order $k$ referred to here was introduced by Akivis in 1975.

It is defined by a formally completely integrable system of exterior differential equations.
A solution of this system exists and depends on $N$ arbitrary constants, where $N$ is the number of linearly independent [[Pfaffian equations|Pfaffian Equation]] contained in the system.

A $G$-structure on a smooth manifold is said to be closed if it is completely defined by a finite number of [[structure constants|Structure Constants]].
In terms of a tensorial representation of the $G$-structure this means, that any tensor of rank $r > k+1$ is a concomitant of the tensors of rank  $r \le k+1$ characterising the structure.

The order $k$ of a closed $G$-structure is a measure of how close is is to [[Lie groups|Lie Group]] for which $k = 2$.

Examples of closed $G$-structures are:

!!!!!Order 1
[[Parallelizable|Parallelizability]] (or locally flat) $G$-structures, having vanishing [[torsion|Torsion]] and [[curvature|Nonassociativity Tensor]].

!!!!! Order 2
[[Lie groups|Lie Group]] which have non-trivial torsion, but vanishing curvature.
An $n$-element [[Lie group|Lie Group]] is defined by its structure constants $c^i_{jk}$ with $c^i_{jk} = -c^i_{kj}$, satisfying the [[Jacobi identities|Jacobian]].
The number of these constants is less than $\frac12 n^2(n - 1)$,  as the number of $n^3$ possible structure constants is reduced by the $n\left (\frac{n^2}{2} - \frac{n}{2} \right )$ relations.
In physics this condition is used quite frequently in that one requires that the [[commutators|Commutator]] of the elements of an algebra form a closed system.

!!!!! Order 3
Locally symmetric [[Riemann spaces|Riemann Space]].

!!!!! Order 4
[[Hexagonal 3-webs|Hexagonal 3-Web]].

<<tiddler [[include_tiddlers/G2.html#"G2"]]>>

The [[Chevalley group|Chevalley Group]] ''$G2(2)$'' is the [[automorphism group|Automorphism]] of the [[Lie algebra|Lie Algebra]] $\mathfrak g$${}_2$ defined over the [[finite field|Galois Field]] $\mathbb F_2$. It is one of the finite subgroups of the [[Lie group|Lie Group]] [[G2]]. $G2(2)$ is the automorphism group of the octonionic root system of the exceptional Lie group [[E7]]. It has the simple group $U_3(3)$ as a subgroup.
$G2(2)$ is imbeded in the [[projective geometry|Projective Geometry]] [[PG(6,2)]].

Papers:
* [[The Chevalley group G2(2) of order 12096 and the octonionic root system of E7 - M. Koca, R. Koc, N. O. Koca|http://arxiv.org/PS_cache/hep-th/pdf/0509/0509189v2.pdf]] [[local|papers/0509189v2.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=4528943419224839982&hl=de]]
* [[G 2(2) as the Automorphism Group of the Octonionic Root System of E 7 (1990) - F. Karsch, M. Koca||http://repositories.ub.uni-bielefeld.de/biprints/volltexte/2010/3948/pdf/ka_29.pdf]] [[local|papers/ka_29.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=7511449424108132970&as_sdt=2005&sciodt=2000&hl=de]] TRD
* [[Exceptional Groups, Symmetric Spaces and Applications - S. L. Cacciatori, B. L. Cerchiai|http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.0121v1.pdf]] [[local|papers/0906.0121v1.pdf]] pct. 0

<<tiddler [[include_tiddlers/G2(C).html#"G2(C)"]]>>

<<tiddler [[include_tiddlers/G2-Manifold.html#"G2-Manifold"]]>>

* [[GAP Online Manuals|http://www.gap-system.org/Doc/manuals.html]]
** [[GAP Release 4.4.12 Reference Manual|http://www.gap-system.org/Manuals/doc/ref/manual.pdf]] [[local|documents/GAPReferenceManual.pdf]]
** [[GUAVA - A GAP4 Package for Computing with Error-correcting Codes|http://www.gap-system.org/Manuals/pkg/guava3.10/doc/manual.pdf]] [[Html-version|http://www-history.mcs.st-and.ac.uk/~gap/Manuals/pkg/guava3.10/htm/chap0.html]] [[local|documents/GUAVAManual.pdf]]
** [[Loops Package|GAP Loops Package]]
** [[AtlasRep|http://www.math.rwth-aachen.de/~Thomas.Breuer/atlasrep/index.html]] - An interface between GAP and the Atlas of Group Representations, a database that comprises representations of many almost simple groups and information about their maximal subgroups.

Examples:
* [[Applied Abstract Algebra - D. Joyner, R. Kreminski, J. Turisco|http://www.usna.edu/Users/math/wdj/book/book.html]]

''$GL(4,2)$'' or ''$GL_2(4)$''  is the [[general linear group|General Linear Group]] of $4 \times 4$-matrices over a finite field with 2 elements.
It has order $2^6\cdot 3^2\cdot 5\cdot 7 = 20.160$ and is [[isomorphic|Isomorphism]] to the groups $A_8$ ([[alternating group|Alternating Group]]), $PGL(4,2)$ ([[projective general linear group|Projective General Linear Group]]), $PSL(4,2) = L_4(2)$ ([[projective special linear group|Projective General Linear Group]]) and $SL(4,2)$.

Although $PSL(3,4)$ happens to have the same order as $PSL(4,2)$, the groups are not isomorphic.

Papers:
* [[The Alternating Group A8 and the General Linear Group GL4(2) - J. Murray|http://www.emis.de/journals/MPRIA/1999/PA99I2/pdf/99201ai.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=11875994002033190097&hl=de]]

''GTG'' is an acronym for ''Gauge Theory of Gravity'' which describes gravitational fields by means of the [[spacetime algebra|Spacetime Algebra]] and [[Clifford geometric calculus|Clifford Geometric Algebra]]. It is based on a different approach than are [[gauge theories of gravity|Gauge Theory of Gravity]] of the [[Poincaré group|Poincaré Transformation]].


Papers:
* [[Gravity, Gauge Theories and Geometric Algebra (1998) - A. Lasenby, C. Doran, S. Gull|http://www.mrao.cam.ac.uk/~clifford/publications/ps/gravity.pdf]] [[local|papers/gravity.pdf]] {{t100Cite{[[pct. 106|http://scholar.google.de/scholar?cites=15459370966736119609&hl=de&as_sdt=2000]]}}}
* [[Gauge Theory Gravity with Geometric Calculus (2005) - D. Hestenes|http://geocalc.clas.asu.edu/pdf/GTG.w.GC.FP.pdf]] [[local|papers/GTG.w.GC.FP.pdf]] [[pct. 13|http://scholar.google.de/scholar?cites=1192000172061832273&as_sdt=2005&sciodt=2000&hl=de]]
* [[Spacetime Geometry with Geometric Calculus (2008) - D. Hestenes|http://geocalc.clas.asu.edu/pdf/SpacetimeGeometry.w.GC.proc.pdf]] [[local|papers/SpacetimeGeometry.w.GC.proc.pdf]] [[pct. 1|http://scholar.google.com/scholar?hl=de&lr=&cites=9273923840321077641&um=1&ie=UTF-8&ei=MfOsS9L3IJagsQb-tpGcAw&sa=X&oi=science_links&resnum=2&ct=sl-citedby&ved=0CBUQzgIwAQ]] prl. 9 - To be able to better compare the notation in the paper with the notation used in this WIKI, the following correspondences may be helpful: $L_{\mu\nu}^\lambda \sim\Gamma_{\mu\nu}^\lambda$, $g_\mu  \sim \mb e_\mu (\mb x)$,  $\gamma_a  \sim \mb e_a$.

Links:
* [[Cambridge University Geometric Algebra Research Group Home Page|http://www.mrao.cam.ac.uk/~clifford/index.html]] - Web site of the "inventors" of the theory.

<<tiddler [[include_tiddlers/GUT.html#"GUT"]]>>

A ''Galois Field'', denoted $GF(m)$, is a finite field with $m$ elements.

''Theorem'' (Galois, about 1830):
Up to isomorphism there exists a field with $m$ elements if and only if $m$ is a prime power, i.e. $m = p^n$ for some prime $p$.

If $m$ itself is prime, then $GF(m) = \mathbb Z_m$.

(Galois) fields do not have [[zero divisors|Zero Divisor]].

Finite fields are important in number theory, [[Lie group|Lie Group]] theory, algebraic geometry, Galois theory, cryptography and [[coding theory|Coding Theory]].

Papers:
* [[Division Algebras, Galois Fields, Quadratic Residues - G. Dixon|http://xxx.lanl.gov/PS_cache/hep-th/pdf/9302/9302113v1.pdf]] [[pct. 8|http://scholar.google.de/scholar?cites=4801232640434799846&hl=de]]

Lectures:
* [[Finite Fields - D. Mayhew|http://msor.victoria.ac.nz/twiki/pub/Courses/MATH324_2009T2/WebHome/notes1.pdf]]

<<tiddler [[include_tiddlers/Gamma 16 Lattice.html#"Gamma 16 Lattice"]]>>

<<tiddler [[include_tiddlers/Gamma Matrices.html#"Gamma Matrices"]]>>

<<tiddler [[include_tiddlers/Gamma Ray Burst.html#"Gamma Ray Burst"]]>>

<<tiddler [[include_tiddlers/Gamow State.html#"Gamow State"]]>>

<<tiddler [[include_tiddlers/Gauge Theory.html#"Gauge Theory"]]>>

<<tiddler [[include_tiddlers/Gauge Theory of Gravity.html#"Gauge Theory of Gravity"]]>>

The ''(q-ary) Gaussian Binomial Coefficient'' (a.k.a. ''q-binomial Coefficient'', ''Gaussian Number'' or ''Gaussian Polynomial'') is defined by:
\[
\begin{bmatrix} m \\ r \end{bmatrix}_q \equiv \frac{(1-q^m)(1-q^{m-1})\dots(1-q^{m-r+1})}{(1-q)(1-q^2)\dots(1-q^r)} = \prod_{i=0}^{r-1} \frac {1-q^{m-i}} {1-q^{i+1}}
\]
For $q = 1$ it coincides with the classical [[binomial coefficient|Binomial Coefficient]], hence it is a generalization thereof.

Papers:
* [[An Algebraic Interpretation of the q-Binomial Coefficients (2009) - M. Braun|http://www.ieja.net/papers/2009/V6/2-V6-2009.pdf]] [[local|papers/2-V6-2009.pdf]] pct. 0

Links:
* [[WIKIPEDIA - Gaussian Binomial|http://en.wikipedia.org/wiki/Gaussian_binomial]]
* [[WolframMathWorld - q-Binomial Coefficient|http://mathworld.wolfram.com/q-BinomialCoefficient.html]]

A ''Gaussian Integer'' is a complex number whose real and imaginary part are both integers. The set $\mathbb{Z}[i]$ of all Gaussian integers is given by
\[
\mathbb{Z}[i]=\{a+bi \mid a,b\in \mathbb{Z} \}
\]
The integers form an [[integer lattice|Integer Lattice]], the $\mathbb Z^2$-lattice (with [[kissing number|Kissing Number]] $4$).

<html><center><img src="images/Z2_lattice.jpg" style="width: 220px; "/></center></html>
Gaussian integers are [[integral elements|Integral Elements]] and can be regarded as a generalization of the integers $\mathbb Z \subset \mathbb R$ to the case of the complex plane $\mathbb C \cong \mathbb R^2$.

The rescaled Gaussian integers $\sqrt{2} \{\pm 1, \pm i \}$ are the non-zero [[roots|Root Vector]] of [[SO(4)]] $\cong$ [[SU(2)xSU(2)|SU(2)]] as they are orthogonal to one another.

See also:
* [[Hurwitz integers|Hurwitz Integer]] (quaternionic integers)
* [[Integral octonions|Integral Octonion]] (octonionic integers)

Links:
* [[On Quaternions and Octonions - J. H. Conway, D. A. Smith|books/QuaternionsAndOctonions.djvu]] [[bct. 53|http://scholar.google.de/scholar?cites=3990102742662413626&hl=de]]

<<tiddler [[include_tiddlers/Gelfand-Naimark Theorem.html#"Gelfand-Naimark Theorem"]]>>

Links:
* [[WIKIPEDIA - Geomerical Frustration|http://en.wikipedia.org/wiki/Geometrical_frustration]]
* [[Geomerical Frustration - Physics Today 02/2006|http://www.physics.rutgers.edu/grad/681/GFrustration_physics.today.pdf]]

>Einstein's "general relativity", ... has two central ideas: (1) Spacetime geometry "tells" mass-energy how to move; and (2) mass-energy "tells" spacetime geometry how to curve. ... the way spacetime tells mass-energy how to move is automatically obtained from the Einstein field equation by using the identity of Riemannian geometry, known as the Bianchi identity, which tells us that the covariant divergence of the Einstein tensor is zero.
> - I. Ciufolini, J. A. Wheeler - Gravitation and Inertia

The [[Riemannian geometry|Riemann Space]] underlying Einstein's theory can be formulated either in terms of the [[metric|Metric Tensor]] $g_{\mu\nu}$ or a frame field ([[vielbein|Tetrad]]) ${h_\mu}^a$.


Papers:
* [[Catalogue of Spacetimes - T. Muller, F. Grave|http://wwwvis.informatik.uni-stuttgart.de/~muelleta/CoS/catalogue_2010-04-01.pdf]] [[local|papers/catalogue_2010-04-01.pdf]] pct. 0

Links:
* [[Institut für Visualisierung und Interaktive Systeme (VIS) - Thomas Müller|http://www.vis.uni-stuttgart.de/~muelleta/]] lrl. 9 - GR and visualisation.

Videos:
* [[Einstein's Theory (lecture 1 - 12) - L. Susskind|http://www.youtube.com/view_play_list?p=6C8BDEEBA6BDC78D]]
* [[Lectures on General Relativity and Cosmology - T. Padmanabhan|http://gr-lectures-paddy.blogspot.com/]]
* [[Advanced Topics in General Relativity - T. Padmanabhan|http://pcc2341f.unige.ch/videos/VideosPadmanabhan.htm]]
* [[Caltech's Physics: Gravitational Waves - A Web-Based Course|http://elmer.tapir.caltech.edu/ph237/]]
* [[50 Years of the Cauchy Problem in General Relativity|http://fanfreluche.math.univ-tours.fr/Cauchy2.html]]

<<tiddler [[include_tiddlers/Generalized Second Law of Thermodynamics.html#"Generalized Second Law of Thermodynamics"]]>>

!!!!Rotation group:
The number $n$ of generators of $SO(p,q)$ with $p+q = N$ is given by:
\[
n = \frac{N(N-1)}{2}
\]
This number is equal to the number of (maximally) different off-diagonal elements of a symmetric $N\times N$-matrix.

!!!!!Examples
* $SO(3)$: The classical $3$ Euler angles.
* [[SO(4)]]: $n = 6$, i.e. the classical $3$ Euler angles + $3$ additional angles
* $SO(3,1)$: $n = 6$, i.e. the classical $3$ Euler angles and due to the Minkowski metric $3$ "imaginary angles" which correspond to Lorentz boosts.
* [[SO(7)]]: $n = 21$
* [[SO(8)]]: $n = 28$
* $SO(15)$: $n = 105$
* [[SO(16)]]: $n = 120$

!!!![[Codes|Blockcode]]
The ''Generator Matrix $G$'' of a code is a matrix with code words in its rows such that all linear combinations of the rows generate the whole of a linear code $[n,k,d]$ (i.e. all of its $2^k$ words).
$G$ is therefore a $k \times n$-matrix.
!!!!!Example
$[8,4]$-code:
\[
\mb{G} := \begin{pmatrix} 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1\\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1\\ 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \end{pmatrix}
\]
!!!![[Lattices|Lattice]]
Given a lattice $L$ with basis vectors $\{\mb e_1, \ldots, \mb e_n$\}, its ''Generator Matrix'' (or ''Basis Matrix'') ''$B$'' is defined by $B_{ij} \equiv (\mb e_i)_j$. I.e. it is a matrix with the rows holding the components of the basis vectors of the lattice.

By means of $B$, $L$ can be represented as follows:
\[
L = \{\vec x' : \vec x' = B \vec x = \sum_{i=1}^n x_i\mb e_i, \, x_i \in \mathbb Z\}
\]

<<tiddler [[include_tiddlers/Genome.html#"Genome"]]>>

<<tiddler [[include_tiddlers/Geodesic Equation.html#"Geodesic Equation"]]>>

<<tiddler [[include_tiddlers/Geodesic Loop.html#"Geodesic Loop"]]>>

<<tiddler [[include_tiddlers/Geometric Algebra.html#"Geometric Algebra"]]>>

<<tiddler [[include_tiddlers/Geometric Product.html#"Geometric Product"]]>>

''Geometrodynamics'' is the study of curved empty space and the evolution of this geometry with time according to [[Einstein’s equations of motion|Einstein Field Equations]].
The sources of curvature are conceived however differently in geometrodynamics and in the usual theory of relativity. In the latter any warping of the Riemannian space-time manifold is due to masses and fields of non-geometric origin. In geometrodynamics by contrast only those masses and fields are considered which can be built out of geometry itself.

Links:
* [[WIKIPEDIA - Gerard 't Hooft|http://en.wikipedia.org/wiki/Gerard_'t_Hooft]]

<<tiddler [[include_tiddlers/Geroch's Theorem.html#"Geroch's Theorem"]]>>

<<tiddler [[include_tiddlers/Ghost Condensation.html#"Ghost Condensation"]]>>

<<tiddler [[include_tiddlers/Ghost Field.html#"Ghost Field"]]>>

<<tiddler [[include_tiddlers/Gibbs State.html#"Gibbs State"]]>>

''Gleason’s Theorem'', which might be regarded as the most fundamental theorem of algebraic coding theory states, that every even, self-dual error correcting code can be generated by the [[Hamming code|Hamming Code]] and the [[Golay code|Golay Code]].

<<tiddler [[include_tiddlers/Gluon.html#"Gluon"]]>>

<<tiddler [[include_tiddlers/Golay Code.html#"Golay Code"]]>>

<<tiddler [[include_tiddlers/Goldstone Boson.html#"Goldstone Boson"]]>>

/***
|Name|GotoPlugin|
|Source|http://www.TiddlyTools.com/#GotoPlugin|
|Documentation|http://www.TiddlyTools.com/#GotoPluginInfo|
|Version|1.9.1|
|Author|Eric Shulman - ELS Design Studios|
|License|http://www.TiddlyTools.com/#LegalStatements <br>and [[Creative Commons Attribution-ShareAlike 2.5 License|http://creativecommons.org/licenses/by-sa/2.5/]]|
|~CoreVersion|2.1|
|Type|plugin|
|Requires||
|Overrides||
|Description|view any tiddler by entering it's title - displays list of possible matches|
''View a tiddler by typing its title and pressing //enter//.''  As you type, a list of possible matches is displayed.  You can scroll-and-click (or use arrows+enter) to select/view a tiddler, or press //escape// to close the listbox to resume typing.  When the listbox is ''//not//'' being displayed, press //escape// to clear the current text input and start over.
!!!Documentation
>see [[GotoPluginInfo]]
!!!Configuration
<<<
*Match titles only after {{twochar{<<option txtIncrementalSearchMin>>}}} or more characters are entered.<br>Use down-arrow to start matching with shorter input.  //Note: This option value is also set/used by [[SearchOptionsPlugin]]//.
*To set the maximum height of the listbox, you can create a tiddler tagged with <<tag systemConfig>>, containing:
//{{{
config.macros.gotoTiddler.listMaxSize=10;  // change this number
//}}}
<<<
!!!Revisions
<<<
2009.04.12 [1.9.1] support multiple instances with different filters by using per-element tiddler cache instead of shared static cache
|please see [[GotoPluginInfo]] for additional revision details|
2006.05.05 [0.0.0] started
<<<
!!!Code
***/
//{{{
version.extensions.GotoPlugin= {major: 1, minor: 9, revision: 1, date: new Date(2009,4,12)};

// automatically tweak shadow SideBarOptions to add <<gotoTiddler>> macro above <<search>>
config.shadowTiddlers.SideBarOptions=config.shadowTiddlers.SideBarOptions.replace(/<<search>>/,"&nbsp;{{button{ goto}}}\n<<gotoTiddler>><<search>>");

if (config.options.txtIncrementalSearchMin===undefined) config.options.txtIncrementalSearchMin=3;

config.macros.gotoTiddler= {
	listMaxSize: 10,
	listHeading: 'Found %0 matching title%1...',
	searchItem: "Search for '%0'...",
	handler:
	function(place,macroName,params,wikifier,paramString,tiddler) {
		var quiet	=params.contains("quiet");
		var showlist	=params.contains("showlist");
		var search	=params.contains("search");
		params = paramString.parseParams("anon",null,true,false,false);
		var instyle	=getParam(params,"inputstyle","");
		var liststyle	=getParam(params,"liststyle","");
		var filter	=getParam(params,"filter","");
		var html=this.html;
		var keyevent=window.event?"onkeydown":"onkeypress"; // IE event fixup for ESC handling
		html=html.replace(/%keyevent%/g,keyevent);
		html=html.replace(/%search%/g,search);
		html=html.replace(/%quiet%/g,quiet);
		html=html.replace(/%showlist%/g,showlist);
		html=html.replace(/%display%/g,showlist?'block':'none');
		html=html.replace(/%position%/g,showlist?'static':'absolute');
		html=html.replace(/%instyle%/g,instyle);
		html=html.replace(/%liststyle%/g,liststyle);
		html=html.replace(/%filter%/g,filter);
		if (config.browser.isIE) html=this.IEtableFixup.format([html]);
		var span=createTiddlyElement(place,'span');
		span.innerHTML=html; var form=span.getElementsByTagName("form")[0];
		if (showlist) this.fillList(form.list,'',filter,search,0);
	},
	html:
	'<form onsubmit="return false" style="display:inline;margin:0;padding:0">\
		<input name=gotoTiddler type=text autocomplete="off" accesskey="G" style="%instyle%"\
			title="Enter title text... ENTER=goto, SHIFT-ENTER=search for text, DOWN=select from list"\
			onfocus="this.select(); this.setAttribute(\'accesskey\',\'G\');"\
			%keyevent%="return config.macros.gotoTiddler.inputEscKeyHandler(event,this,this.form.list,%search%,%showlist%);"\
			onkeyup="return config.macros.gotoTiddler.inputKeyHandler(event,this,%quiet%,%search%,%showlist%);">\
		<select name=list style="display:%display%;position:%position%;%liststyle%"\
			onchange="if (!this.selectedIndex) this.selectedIndex=1;"\
			onblur="this.style.display=%showlist%?\'block\':\'none\';"\
			%keyevent%="return config.macros.gotoTiddler.selectKeyHandler(event,this,this.form.gotoTiddler,%showlist%);"\
			onclick="return config.macros.gotoTiddler.processItem(this.value,this.form.gotoTiddler,this,%showlist%);">\
		</select><input name="filter" type="hidden" value="%filter%">\
	</form>',
	IEtableFixup:
	"<table style='width:100%;display:inline;padding:0;margin:0;border:0;'>\
		<tr style='padding:0;margin:0;border:0;'><td style='padding:0;margin:0;border:0;'>\
		%0</td></tr></table>",
	getItems:
	function(list,val,filter) {
		if (!list.cache || !list.cache.length || val.length<=config.options.txtIncrementalSearchMin) {
			// starting new search, fetch and cache list of tiddlers/shadows/tags
			list.cache=new Array();
			if (filter.length) {
				var fn=store.getMatchingTiddlers||store.getTaggedTiddlers;
				var tiddlers=store.sortTiddlers(fn.apply(store,[filter]),'title');
			} else
				var tiddlers=store.getTiddlers("title","excludeLists");
			for(var t=0; t<tiddlers.length; t++) list.cache.push(tiddlers[t].title);
			if (!filter.length) {
				for (var t in config.shadowTiddlers) list.cache.pushUnique(t);
				var tags=store.getTags();
				for(var t=0; t<tags.length; t++) list.cache.pushUnique(tags[t][0]);
			}
		}
		var found = [];
		var match=val.toLowerCase();
		for(var i=0; i<list.cache.length; i++)
			if (list.cache[i].toLowerCase().indexOf(match)!=-1) found.push(list.cache[i]);
		return found;
	},
	getItemSuffix:
	function(t) {
		if (store.tiddlerExists(t)) return "";  // tiddler
		if (store.isShadowTiddler(t)) return " (shadow)"; // shadow
		return " (tag)"; // tag
	},
	fillList:
	function(list,val,filter,search,key) {
		if (list.style.display=="none") return; // not visible... do nothing!
		var indent='\xa0\xa0\xa0';
		var found = this.getItems(list,val,filter); // find matching items...
		found.sort(); // alpha by title
		while (list.length > 0) list.options[0]=null; // clear list
		var hdr=this.listHeading.format([found.length,found.length==1?"":"s"]);
		list.options[0]=new Option(hdr,"",false,false);
		for (var t=0; t<found.length; t++) list.options[list.length]=
			new Option(indent+found[t]+this.getItemSuffix(found[t]),found[t],false,false);
		if (search)
			list.options[list.length]=new Option(this.searchItem.format([val]),"*",false,false);
		list.size=(list.length<this.listMaxSize?list.length:this.listMaxSize); // resize list...
		list.selectedIndex=key==38?list.length-1:key==40?1:0;
	},
	keyProcessed:
	function(ev) { // utility function
		ev.cancelBubble=true; // IE4+
		try{event.keyCode=0;}catch(e){}; // IE5
		if (window.event) ev.returnValue=false; // IE6
		if (ev.preventDefault) ev.preventDefault(); // moz/opera/konqueror
		if (ev.stopPropagation) ev.stopPropagation(); // all
		return false;
	},
	inputEscKeyHandler:
	function(event,here,list,search,showlist) {
		if (event.keyCode==27) {
			if (showlist) { // clear input, reset list
				here.value=here.defaultValue;
				this.fillList(list,'',here.form.filter.value,search,0);
			}
			else if (list.style.display=="none") // clear input
				here.value=here.defaultValue;
			else list.style.display="none"; // hide list
			return this.keyProcessed(event);
		}
		return true; // key bubbles up
	},
	inputKeyHandler:
	function(event,here,quiet,search,showlist) {
		var key=event.keyCode;
		var list=here.form.list;
		var filter=here.form.filter;
		// non-printing chars bubble up, except for a few:
		if (key<48) switch(key) {
			// backspace=8, enter=13, space=32, up=38, down=40, delete=46
			case 8: case 13: case 32: case 38: case 40: case 46: break; default: return true;
		}
		// blank input... if down/enter... fall through (list all)... else, and hide or reset list
		if (!here.value.length && !(key==40 || key==13)) {
			if (showlist) this.fillList(here.form.list,'',here.form.filter.value,search,0);
			else list.style.display="none";
			return this.keyProcessed(event);
		}
		// hide list if quiet, or below input minimum (and not showlist)
		list.style.display=(!showlist&&(quiet||here.value.length<config.options.txtIncrementalSearchMin))?'none':'block';
		// non-blank input... enter=show/create tiddler, SHIFT-enter=search for text
		if (key==13 && here.value.length) return this.processItem(event.shiftKey?'*':here.value,here,list,showlist);
		// up or down key, or enter with blank input... shows and moves to list...
		if (key==38 || key==40 || key==13) { list.style.display="block"; list.focus(); }
		this.fillList(list,here.value,filter.value,search,key);
		return true; // key bubbles up
	},
	selectKeyHandler:
	function(event,list,editfield,showlist) {
		if (event.keyCode==27) // escape... hide list, move to edit field
			{ editfield.focus(); list.style.display=showlist?'block':'none'; return this.keyProcessed(event); }
		if (event.keyCode==13 && list.value.length) // enter... view selected item
			{ this.processItem(list.value,editfield,list,showlist); return this.keyProcessed(event); }
		return true; // key bubbles up
	},
	processItem:
	function(title,here,list,showlist) {
		if (!title.length) return;
		list.style.display=showlist?'block':'none';
		if (title=="*")	{ story.search(here.value); return false; } // do full-text search
		if (!showlist) here.value=title;
		story.displayTiddler(null,title); // show selected tiddler
		return false;
	}
}
//}}}

A ''Graded Lie Algebra'' is a [[Lie algebra|Lie Algebra]] endowed with a gradation which is compatible with the Lie bracket. A graded Lie algebra is a [[nonassociative graded algebra|Nonassociative Algebra]] under the bracket operation.

Papers:
* [[Graded Lie Algebras and q-commutative and r-associative Parameters - L. A. Wills-Toro, J. D. Vaelez, T. Craven|http://www.math.hawaii.edu/~tom/mathfiles/WillsSLAAlg.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=14651537263041308412&hl=de]]

Given a vector $\vec {\mb X} = \{\mb A_1, \ldots, \mb A_N\}$ of elements of an algebra, having a [[inner product|Scalar Product]] $<.|.>$, the ''Gram Matrix'' is defined as
\[
\mb M(\mb A_1, \ldots,\mb A_N) \equiv \langle \vec {\mb X}| \vec {\mb X}^t \rangle = \left ( \begin{matrix}  \langle \mb  A_1,\mb  A_1\rangle & \langle \mb  A_1, \mb  A_2\rangle & \ldots & \langle \mb  A_1,\mb  A_N\rangle \\  \langle \mb  A_2, \mb  A_1\rangle & \langle \mb  A_2, \mb  A_2\rangle & \ldots & \langle \mb  A_2, \mb  A_N\rangle \\  \ldots & \ldots & \ldots & \ldots \\  \langle \mb  A_N,\mb  A_1\rangle & \langle \mb  A_N,\mb  A_2\rangle & \ldots & \langle \mb  A_N, \mb A_N\rangle \\  \end{matrix} \right )
\]
or in component form
\[
M_{ij} = \langle \mb A_i|\mb A_j \rangle
\]
!!!!Properties
* Any Gram matrix is symmetric, since inner products are symmetric.
* Given a Gram matrix the vectors $\vec {\mb X}$ are determined up to [[isometry|Isometry]].
* Given a real symmetric positive semidefinite $N \times N$-matrix $A$, then $A$ is a Gram matrix. I.e. Gram matrices provide a concrete realization of all positive semidefinite matrices.

Links:
* [[WolframMathWorld - Distance-Transitive Graph|http://mathworld.wolfram.com/Distance-TransitiveGraph.html]]

<<tiddler [[include_tiddlers/Gravastar.html#"Gravastar"]]>>

<<tiddler [[include_tiddlers/Gravitation.html#"Gravitation"]]>>

<<tiddler [[include_tiddlers/Gravitational Constant.html#"Gravitational Constant"]]>>

<<tiddler [[include_tiddlers/Gravitationally Induced State Reduction.html#"Gravitationally Induced State Reduction"]]>>

The ''Gravitino'' is the conjectured [[supersymmetric|Supersymmetry]] partner of the graviton.
Its action is given by
\begin{equation}
\mathcal L= ? \frac{i}{2}  \Psi_\mu^* \gamma^{[\mu} \gamma^\nu \gamma^{\lambda]} \partial_\nu \Psi_\lambda
\end{equation}

Papers:
* [[Gravi-Weak Unification - F. Nestia, R. Percacci|http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.3307v2.pdf]]

Links:
* [[Prefixes & Words Based On Greek Number Names|http://home.comcast.net/~igpl/NWG.html]]

<<tiddler [[include_tiddlers/Green's Function.html#"Green's Function"]]>>

>... the further you go, the more creativity, the more ingenuity is required. To continue making progress, you will eventually need to come up with more and more complicated mathematical principles, novel principles that are not consequences of our current mathematical knowledge.
> - in "Thinking about Gödel and Turing..."

> Understanding is compression.

> I would claim that you understand something only if you can program it.

!!!!Applications
* [[Process Physics]]

Links:
* [[Website|http://www.umcs.maine.edu/~chaitin/]]

Papers:
* [[On Computable Numbers, with an Application to the Entscheidungsproblem - A. M. Turing|http://www.math.uic.edu/~vladot/mcs441/turing36.pdf]] [[local|papers/turing36.pdf]] {{t1000Cite{[[pct. 3502|http://scholar.google.de/scholar?cites=761850432140269779&hl=de]]}}}

Videos:
* [[Lectures on YOUTUBE|http://www.youtube.com/results?search_query=Gregory+Chaitin+Lecture+&search_type=&aq=f]]
* [[The Search for the Perfect Language (Lecture given at Perimeter Institute)|http://streamer.perimeterinstitute.ca/mediasite/viewer/NoPopupRedirector.aspx?peid=4ad2723d-ff8d-4a6f-8888-456572c6eb64&shouldResize=False#]]
* [[Leibniz, Complexity and Incompleteness|http://videolectures.net/ephdcs08_chaitin_lcai/]]

The ''Griess Algebra'' is the weight-$2$ subspace of the [[Moonshine VOA|Monstrous Moonshine]] $V^\natural$. It is a non-associative but commutative algebra of dimension $196.884 =196.883+1$ with a positive definite invariant bilinear form.

It has $48$-dimensional associative subalgebras.

Since Griess's construction of the [[Monster simple group|Monster Group]] as the [[automorphism group|Automorphism]] of this algebra, many attempts have been made in order to better understand its nature.

[[Conway|John Conway]] constructed a slightly modified version of it, called the ''Conway\-Griess Algebra''.

Links:
* [[WIKIPEDIA - Grigori Perelman|http://en.wikipedia.org/wiki/Grigori_Perelman]]

Videos:
* [[SWR2 Wissen 31.03.2008: Die Perelman-Vermutung|http://www.ardmediathek.de/ard/servlet/content/3517136?documentId=3230166]]

The ''Gronwall Conjecture'' (1912) is is related to the theory of [[3-webs|3-Web]] and states:

If a [[non-parallelizable|Parallelizability]] $3$-web $W$ in the (real or complex) plane is [[linearizable|Linearizability]], then, up to a [[projective transformation|Collineation]], there is a unique [[diffeomorphism|Diffeomorphism]] which maps $W$ into a linear $3$-web.

Papers:
* [[On the Linearizability of 3-webs - J. Grifone, Z. Muzsnay, J. Saab|http://www.math.klte.hu/~muzsnay/Pdf/Papers/web.pdf]] [[pct. 7|http://scholar.google.de/scholar?cites=1316333701315286845&hl=de]]

<<tiddler [[include_tiddlers/Group.html#"Group"]]>>

<<tiddler [[include_tiddlers/Groupoid.html#"Groupoid"]]>>

The composition of relativistic velocities can be described by algebraic structures called a ''Gyrogroups'' which were introduced by A. A. Ungar.
Gyrogroups are noncommutative and nonassociative which is related to [[Thomas precession|Thomas Precession]] in special theory of relativity.

Papers:
* [[Gyrogroups and Homogeneous Loops (1998) - A. N. Issa|http://streaming.ictp.trieste.it/preprints/P/98/051.pdf]] [[local|papers/051.pdf]] [[pct. 11|http://scholar.google.de/scholar?cites=1364092523502009345&as_sdt=2005&sciodt=2000&hl=de]]
* [[Left Distributive Quasigroups and Gyrogroups (2001) - A. N. Issa|http://journal.ms.u-tokyo.ac.jp/pdf/jms080101.pdf]] [[local|papers/jms080101.pdf]] [[pct. 3|http://scholar.google.com/scholar?hl=de&lr=&cites=18389812285134192099&um=1&ie=UTF-8&ei=hybwS5mfOMSD-Qbes-yzBw&sa=X&oi=science_links&resnum=1&ct=sl-citedby&ved=0CCAQzgIwAA]]
* [[Gyrogroups and Left Gyrogroups as Transversals of a Special Kind (2003) - E. Kuznetsov|http://adm.lnpu.edu.ua/downloads/issues/2003/N3/adm-n3-4.pdf]] [[local|papers/adm-n3-4.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=9130784278213815873&as_sdt=2005&sciodt=2000&hl=de]]

Links:
* [[WIKIPEDIA - Gyrovector Space|http://en.wikipedia.org/wiki/Gyrovector_space]]

Journals:
* [[Pacific Journal of Mathematics - Volume 193, No. 1, March 2000|http://pjm.berkeley.edu/pjm/2000/193-1/pjm-v193-n1-s.pdf]] [[local|journals/pjm-v193-n1-s.pdf]]

<<tiddler [[include_tiddlers/Gâteaux Derivative.html#"Gâteaux Derivative"]]>>

An algebra $\mathcal A$ is called a ''H*-Algebra'' (or ''Hilbert Algebra'') if it is a [[Banach algebra|Banach Algebra]] with an involution as well as a [[Hilbert space|Hilbert Space]]. Furthermore it has to satisfy conditions, relating the involution with the Hilbert space structure.

The exact definition is as follows:
# $\mathcal A$ is a symmetric Banach algebra,
# $\mathcal A$ is a Hilbert space,
# the [[norm|Norm]] in $\mathcal A$ coincides with the norm in the Hilbert space,
# $\langle \mb{AB}, \mb C \rangle = \langle \mb B, \mb A^* \mb C \rangle, \;\; \forall \mb A,  \mb B, \mb C \in \mathcal A$,
# $\mb A^* \mb A \ne 0, \;\;  \forall \mb A \ne 0$.

A H*-algebra is not necessarily [[associative|Nonassociative Algebra]] (e.g. Malcev H* algebras).

Papers:
* [[Nonassociative Real H*-algebras - M. Cabrera, J. Martínez, A. Rodríguez|http://www.raco.cat/index.php/PublicacionsMatematiques/article/viewFile/37564/37438]] [[pct. 7|http://scholar.google.de/scholar?cites=7420421535667737973&as_sdt=2005&sciodt=2000&hl=de]]
* [[Malcev H*-algebras - M. Cabrera, J. Martínez, A. Rodríguez|http://dmle.cindoc.csic.es/pdf/EXTRACTAMATHEMATICAE_1986_01_03_08.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=16085502481159523129&as_sdt=2005&sciodt=2000&hl=de]]
* [[Moufang H*-algebras - J. A. C. Mira|http://www.unex.es/extracta/Vol-17-2/17j2cuen.pdf]] pct. 0
* [[On Lie Derivations of 3-Graded Algebras A. J. C. Martín, C. M. González|http://www.maths.tcd.ie/pub/ims/bull48/R4802.pdf]] pct. 0

Links:
* [[WIKIPEDIA - H*-Algebra|http://de.wikipedia.org/wiki/H*-Algebra]]

<<tiddler [[include_tiddlers/H-Space.html#"H-Space"]]>>

> Most practitioners of QFT appear to ignore the implications of Haag's theorem entirely and prefer to go ahead producing numbers.
> - WIKIPEDIA -

''Haag's Theorem'' states that if a field at a certain time is related to a free one by a unitary transformation, as is the case in the interaction picture, then the field is inevitably free.

Links:
* [[WIKIPEDIA - Haag's Theorem|http://en.wikipedia.org/wiki/Haag%27s_theorem]]
@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/Haag.jpg][Comments]]}}}&nbsp; @@

In 1975, Rudolf Haag, Jan T. Łopuszański and Martin Sohnius published a proof (''Haag-Łopuszański-Sohnius Theorem'') which shows, that by weakening the assumptions of the [[Coleman-Mandula theorem|Coleman-Mandula Theorem]] allowing both commuting and anticommuting symmetry generators, there is a nontrivial extension of the [[Poincaré algebra|Poincaré Transformation]], namely the [[supersymmetry algebra|Supersymmetry]].

Journals:
* [[All Possible Generators of Supersymmetries of the S Matrix (1975) - R. Haag, J. T. Łopuszański, M. Sohnius|journals/SupersymmetryHaag.djvu]] {{t500Cite{[[jct. 959|http://scholar.google.de/scholar?hl=de&lr=&cites=7651531531072320161&um=1&ie=UTF-8&ei=JGB3TpbeE-Hm4QTapeC8DQ&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB8QzgIwAA]]}}}

A ''Hadamard Code $\operatorname{Had}(m)$'' is a (binary) $[2^m, m + 1, 2^{m?1}]$ - [[linear error-correcting code|Linear Blockcode]] which is equivalent to a [[first order Reed-Muller code|Reed-Muller Code]] $\operatorname{RM}(1,m)$. (It is a special Reed\-Muller code, having an equal number of "1"'s and "0"'s).

The dual code of a Hadamard code is an [[extended Hamming code|Hamming Code]].

Especially for large $m$ it has a poor error-correcting rate but it is capable of correcting many errors.
Hadamard codes may be described by a $(m + 1) \times 2^m$ generator matrix $G_m$.

!!!![[SAGE|http://www.sagenb.org/]]^^[[Help|Sage]]^^ examples
{{{
Ham = gap.HadamardCode(16)
N = gap.Elements(Ham)
gap.Size(N)
Aut = Ham.AutomorphismGroup()
gap.Size(Aut)
gap.Elements(Ham)
gap.WeightDistribution(Ham)
}}}

Papers:
* [[Z4-linear Hadamard and Extended Perfect Codes -D. S.Krotov|http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.0199v1.pdf]] [[pct. 51|http://scholar.google.de/scholar?cites=10926960367645449155&hl=de&as_sdt=2000]]

<<tiddler [[include_tiddlers/Hadamard Design.html#"Hadamard Design"]]>>

<<tiddler [[include_tiddlers/Hadamard Matrix.html#"Hadamard Matrix"]]>>

<<tiddler [[include_tiddlers/Hadamard Matrix - Examples.html#"Hadamard Matrix - Examples"]]>>

<<tiddler [[include_tiddlers/Hagedorn Temperature.html#"Hagedorn Temperature"]]>>

Von Neumann's proof of the ''Halting problem'' is equivalent to [[Cantor's diagonal argument|Cantors Diagonal Argument]]. Turing machines are based on a machine language, whereas the [[Gödel's theorems|Gödel's Theorems]] are based on LISP.

A ''Hamiltonian Group'' is a non-abelian ''Dedekind Group''. The latter is defined as a group for which every subgroup is [[normal|Normal Subgroup]]. All abelian groups are Dedekind groups.

The smallest example of a Hamiltonian group is the [[quaternion group|Quaternion Group]] $\mathcal Q_8$.

Any Hamiltonian group $\mathcal H$ is the direct product of quaternion groups, the direct sum of [[cyclic groups|Cyclic Group]] $\oplus_i\mathcal C_2$ and a periodic abelian group $\mathcal A$ all of whose elements have odd order, i.e.
\begin{equation}
\mathcal H = \mathcal Q_8 \times \oplus_i\ \mathcal C_2 \times \mathcal A
\end{equation}
!!!! Generalizations:
* See [[Hamilton Loop]].

<<tiddler [[include_tiddlers/Hamilton Loop.html#"Hamilton Loop"]]>>

<<tiddler [[include_tiddlers/Hamming Code.html#"Hamming Code"]]>>

<<tiddler [[include_tiddlers/Hamming Distance.html#"Hamming Distance"]]>>

The ''Hamming\-Weight'' (or ''Hamming\-Norm'') ''$w(x)$'' of a word of a [[code|Code]] $C$ is equal to the number of its letters not equal to "zero".

Formally:
\[w (c) \equiv \operatorname{ord}(\{c \in C : c \ne 0 \}
\]

The ''Minimum Weight''  $w_{min}$ of a code $C$ is defined as the weight of the lowest-weight code word.

For an [[orthogonal code|Dual Code]] one has: $w(x) \in 2\mathbb Z$.
For a [[selfdual code|Dual Code]]: $w(x) \in 4\mathbb Z$.

<<tiddler [[include_tiddlers/Harmonic Oscillator.html#"Harmonic Oscillator"]]>>

<<tiddler [[include_tiddlers/Hawking Radiation.html#"Hawking Radiation"]]>>

<<tiddler [[include_tiddlers/Heat Kernel.html#"Heat Kernel"]]>>

<<tiddler [[include_tiddlers/Heat Kernel Expansion.html#"Heat Kernel Expansion"]]>>

<<tiddler [[include_tiddlers/Heim Theory.html#"Heim Theory"]]>>

<<tiddler [[include_tiddlers/Heisenberg Algebra.html#"Heisenberg Algebra"]]>>

<<tiddler [[include_tiddlers/Helmholtz Conditions.html#"Helmholtz Conditions"]]>>

<<tiddler [[include_tiddlers/Hentzel-Peresi Identity.html#"Hentzel-Peresi Identity"]]>>

>For many years whenever I got into a different topic I found out who was behind the scene, and sure enough, it was Hermann Weyl.
> - Michael Atiyah [1] -

Links:
* [[[1] An Interview with Michael Atiyah|http://kryakin.com/files/Atiyah.pdf]]
* [[WIKIPEDIA - Hermann Weyl|http://en.wikipedia.org/wiki/Hermann_Weyl]]
* [[Weylmann.com|http://www.weylmann.com/]]

<<tiddler [[include_tiddlers/Hermitian Conjugate.html#"Hermitian Conjugate"]]>>

<<tiddler [[include_tiddlers/Hermitian Manifold.html#"Hermitian Manifold"]]>>

Given the real-valued function $f(x_1, x_2, \dots, x_n)$ for which it is assumed that all second partial derivatives exist, the ''Hessian Matrix'' of $f$ is defined as:
\[
(\mb H_f)_{ij}(\mb{x}) = \frac{\partial^2 f(\mb{x})}{\partial x_i\partial x_j} = \partial_i \partial_j f(\mb{x})\,\!
\]
Written out explicitly it is:
\[
\mb H_f(\mb{x}) = \begin{pmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1\,\partial x_n} \\  \\ \frac{\partial^2 f}{\partial x_2\,\partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2\,\partial x_n} \\  \\ \vdots & \vdots & \ddots & \vdots \\  \\ \frac{\partial^2 f}{\partial x_n\,\partial x_1} & \frac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{pmatrix}
\]
The ''Hessian Matrix'' describes the second order change of the function $f$. It appears as the term in the Taylor series expansion of $f$ which corresponds to this change:
\[
\Delta f(\mb{x}) =f(\mb{x}+\Delta\mb{x})\approx f(\mb{x}) + \mb J_f(\mb{x})\Delta \mb{x} +\frac{1}{2} \Delta\mb{x}^\mathrm{T} \mb H_f(\mb{x}) \Delta\mb{x}
\]
The first order change of $f$ is described by the [[Jacobian matrix|Jacobi Matrix]] $\mb J_f$.

If the second derivatives of $f$ are all continuous in a neighbourhood of $\mb x$ then $\mb H_f (\mb x)$ is symmetric in $\mb x$.

Links:
* [[The Curvature of a Hessian Metric - B. Totaro|http://arxiv.org/PS_cache/math/pdf/0401/0401381v2.pdf]] [[pct. 7|http://scholar.google.de/scholar?cites=3803753193891904130&hl=de&as_sdt=2000]]

The ''Heterotic String Theory'' is the only [[string theory|Superstring Theory]] with solely closed strings. In ten-dimensional space-time it is equipped with $\mathcal N = 1$ [[supersymmetry|Supersymmetry]] and an [[E8]]$\times$[[E8]] or [[SO(32)]] gauge group. Only for these two gauge groups one gets a cancellation of [[anomalies|Anomaly]].

The heterotic string is derived from the 26-dimensional bosonic string in that its excitations are split up into ''left-movers'' and ''right-movers'':

Left movers:
26-dimensional, bosonic, 16 dimensions compactified.
480 generators of [[E8]]$\times$[[E8]] or [[SO(32)]].

Right movers:
10 dimensional superstring with bosonic and fermionic degrees of freedom related by $\mathcal N = 1$ (local) supersymmetry.

In the low energy limit one gets the following effective action which modifies Einstein gravity:
\[
S = \int dx^4 \sqrt{-g} \; e^{-2\Phi} (R + 12\partial_\mu \Phi \partial^\mu \Phi ?  \frac{1}{2\cdot 3!} H_{\mu\nu\sigma}H^{\mu\nu\sigma})
\]
with $H_{\mu\nu\sigma}$ the antisymmetric [[Kalb-Ramond|Kalb-Ramond Field]]- or axion-field which can be decomposed according to:
\[
H_{\mu\nu\sigma} = \partial_{[\mu} B_{\nu\sigma]} + (\Omega_Y)_{\mu\nu\sigma} + (\Omega_L)_{\mu\nu\sigma}
\]
$\Omega_Y$ and $\Omega_L$ are Yang\-Mills- and Lorentz\-Chern Simons terms respectively.

Papers:
* [[Fermionic Subspaces of the Bosonic String - A. Chattaraputi, F. Englert, L. Houart, A. Taorminak|http://arxiv.org/PS_cache/hep-th/pdf/0212/0212085v1.pdf]] [[local|papers/0212085v1.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=3936724868441634729&hl=de]]
* [[Grand Unification in the Heterotic Brane World - P. K. S. Vaudrevange|http://arxiv.org/PS_cache/arxiv/pdf/0812/0812.3503v1.pdf]] pct. 0

<<tiddler [[include_tiddlers/Hexagonal 3-Web.html#"Hexagonal 3-Web"]]>>

<<tiddler [[include_tiddlers/Hierarchy Problem.html#"Hierarchy Problem"]]>>

<<tiddler [[include_tiddlers/Higgs Mechanism.html#"Higgs Mechanism"]]>>

This graph made its official appearance in the context of the construction of the [[sporadic simple group|Sporadic Group]] $HS$ which is a subgroup in the [[automorphism group|Automorphism]] of the graph.

{{center{[img(431px+, )[images/Higman_Sims_Graph2.jpg]]}}}
There are $704$ [[Hoffman-Singleton|Hoffman-Singleton Graph]] subgraphs in the Higman\-Sims graph.

The total number of automorphisms of the graph is $88.704.000= 352\cdot 252.000$, since there are $352$ ways of splitting the Higman\-Sims graph into a pair of Hoffman\-Singleton graphs.

Papers:
* [[On the Graphs of Hoffman-Singleton and Higman-Sims (2004) - P. R. Hafner|http://www.emis.de/journals/EJC/Volume_11/PDF/v11i1r77.pdf]] [[local|papers/v11i1r77.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=3772605880650005811&hl=de&as_sdt=2000]]

Videos:
* [[Projections of the Higman-Sims Graph from the Leech Lattice|http://www.youtube.com/watch?v=neUd794Gbg0]] [[local|videos/Projections of the Higman Sims graph from the Leech lattice.wmv]]

<<tiddler [[include_tiddlers/Hilbert Space.html#"Hilbert Space"]]>>

''Hilbert's Problems'' are a list of ''twenty-three problems in mathematics'' published by David Hilbert during 1900. The problems were all unsolved at the time.
!!!! Status of Resulution
* Problems 3, 7, 10, 11, 13, 14, 17, 19, 20 and 21 have a resolution that is accepted by consensus.
* Problems 1, 2, 5, 9, 12, 15, 18 and 22 have solutions that have partial acceptance, but there exists some controversy as to whether it resolves the problem.
* Problems 8 (the Riemann hypothesis) and 12 are unresolved, both being part of number theory.
* Problems 4, 6, 16 and 23 are regarded as too vague to be ever described as solved. The same applies for a 24$^{th}$ problem added later and withdrawn again.

Links:
* [[WIKIPEDIA - Hilbert's Problems|http://en.wikipedia.org/wiki/Hilbert%27s_problems]]

<<tiddler [[include_tiddlers/Hoffman-Singleton Graph.html#"Hoffman-Singleton Graph"]]>>

Links:
* [[Stanford Encyclopedia of Philosophy|http://plato.stanford.edu/entries/spacetime-holearg/]]

<<tiddler [[include_tiddlers/Holographic Principle.html#"Holographic Principle"]]>>

In analysis a ''Holomorphic Function'' (a.k.a. ''Regular Function'') is a function that is analytic and single-valued in a given region.

A function $f: X \rightarrow Y$ between two topological spaces $X$ and $Y$ is called a ''Homeomorphism'' if it has the following properties:
* $f$ is a bijection.
* $f$ is continuous.
* the inverse function $f ^{-1}$ is continuous.
The homeomorphisms form an equivalence relation on the class of all topological spaces, called ''Homeomorphism Classes''.

A ''Homogeneous Space'' is manifold or topological space on which a [[group|Group]] acts continuously by symmetry in a [[transitive|Transitivity]] way.

A homogeneous space of dimension $N$ admits a set of $\frac {N(N -1)}{2}$ Killing vectors.

The notion of homogeneous space has been coined by Élie Cartan although it is much older.

<<tiddler [[include_tiddlers/Homomorphism.html#"Homomorphism"]]>>

In topology, two continuous functions from a topological space to another are called ''homotopic'' if one can continuously deform the one into the other. Being homotopic is an equivalence relation on the set of all continuous functions between the two spaces.

Classically the equivalence classes induced by homotopy form a group, called ''Homotopy Group''.
The associated spaces are also [[isotopic|Isotopy]] spaces, the converse however is not generally true. Therefore homotopy is "blind" when it comes to distinguishing certain structures (e.g. [[loop- and quasigroup manifolds|Quasigroup Manifold]]).
A generalization of homotopy which "fixes" this problem is called ''H\-Homotopy'' and is related to the concept of [[H-spaces|H-Space]].

Papers:
* [[Origins and Breadth of the Theory of Higher Homotopies - J. Huebschmann|http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.2645v1.pdf]] [[pct. 6|http://scholar.google.de/scholar?cites=16196161903623431081&hl=de&as_sdt=2000]]

<<tiddler [[include_tiddlers/Hopf Algebra.html#"Hopf Algebra"]]>>

<<tiddler [[include_tiddlers/How to Quantize a Universe.html#"How to Quantize a Universe"]]>>

<<tiddler [[include_tiddlers/How to do Nonassociative Quantum Mechanics.html#"How to do Nonassociative Quantum Mechanics"]]>>

The ''Howe–Tucker String Action'' is equivalent to the [[Nambu Goto action|Dirac-Nambu-Goto Action]]. It is invariant under Weyl rescaling of the world metric and as a consequence, the string classical energy–momentum tensor has vanishing trace.

<<tiddler [[include_tiddlers/Hubble Constant.html#"Hubble Constant"]]>>

<<tiddler [[include_tiddlers/Hurwitz Integer.html#"Hurwitz Integer"]]>>

The ''Hurwitz Theorem'' states that every [[normed algebra|Normed Algebra]] over the real numbers with an identity element is isomorphic to either $\mathbb R $, $\mathbb C $, $\mathbb H $ or $\mathbb O $, i.e. the real numbers, the [[complex numbers|Complex Number]], the [[quaternions|Quaternion]] or the [[octonions|Octonion]]. The latter three can also be of [[split|Split Algebra]]-form. The split algebras possess [[zero divisors|Zero Divisor]].

Papers:
* [[Hurwitz Theorem and Parallelizable Spheres from Tensor Analysis (2001) - J. A. Nieto, L. N. Alejo-Armenta|http://arxiv.org/PS_cache/hep-th/pdf/0005/0005184v2.pdf]] [[local|papers/0005184v2.pdf]] [[pct. 13|http://scholar.google.de/scholar?hl=de&lr=&cites=5968398703903081476]] prl. 9

<<tiddler [[include_tiddlers/Hydrino.html#"Hydrino"]]>>

<<tiddler [[include_tiddlers/Hyperbolic Quaternion.html#"Hyperbolic Quaternion"]]>>

<<tiddler [[include_tiddlers/Hyperboloid.html#"Hyperboloid"]]>>

<<tiddler [[include_tiddlers/Hypercomplex Analysis.html#"Hypercomplex Analysis"]]>>

<<tiddler [[include_tiddlers/IKKT Model.html#"IKKT Model"]]>>

Given an algebra $\mathcal A$, an ''Ideal'' is a special kind of subalgebra $\mathcal A'$ of $\mathcal A$ with the property, that for any $\mb A' \in \mathcal A'$ and $\mb A \in \mathcal A$, $\mb {AA'} \in \mathcal A'$.
Expressed in a more sloppy manner: An element of an algebra cannot kick out an element of an ideal of it.

Example:
The set of even integers is an ideal in the ring of integers $\mathbb{Z}$.

''Idempotency'' is the property of an operation yielding the same result irrespective of it being applied once or several times.

!!!! Examples
* [[Projection]] operators:  P = PP = PPP ...
* Identity function: x = f(x) = f(f(x)) = f(f(f(x))) ...

!!!!Properties
* Every idempotent which is not zero and not the identity $\mb e$ is also a [[zero divisor|Zero Divisor]] as $\mb A^2 = \mb A$ implies $\mb A (\mb A - \mb e) = 0$.

<<tiddler [[include_tiddlers/Ideomotor Effect.html#"Ideomotor Effect"]]>>

/***
|Name|ImageSizePlugin|
|Source|http://www.TiddlyTools.com/#ImageSizePlugin|
|Version|1.2.1|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|adds support for resizing images|
This plugin adds optional syntax to scale an image to a specified width and height and/or interactively resize the image with the mouse.
!!!!!Usage
<<<
The extended image syntax is:
{{{
[img(w+,h+)[...][...]]
}}}
where ''(w,h)'' indicates the desired width and height (in CSS units, e.g., px, em, cm, in, or %). Use ''auto'' (or a blank value) for either dimension to scale that dimension proportionally (i.e., maintain the aspect ratio). You can also calculate a CSS value 'on-the-fly' by using a //javascript expression// enclosed between """{{""" and """}}""". Appending a plus sign (+) to a dimension enables interactive resizing in that dimension (by dragging the mouse inside the image). Use ~SHIFT-click to show the full-sized (un-scaled) image. Use ~CTRL-click to restore the starting size (either scaled or full-sized).
<<<
!!!!!Examples
<<<
{{{
[img(100px+,75px+)[images/meow2.jpg]]
}}}
[img(100px+,75px+)[images/meow2.jpg]]
{{{
[<img(34%+,+)[images/meow.gif]]
[<img(21% ,+)[images/meow.gif]]
[<img(13%+, )[images/meow.gif]]
[<img( 8%+, )[images/meow.gif]]
[<img( 5% , )[images/meow.gif]]
[<img( 3% , )[images/meow.gif]]
[<img( 2% , )[images/meow.gif]]
[img(  1%+,+)[images/meow.gif]]
}}}
[<img(34%+,+)[images/meow.gif]]
[<img(21% ,+)[images/meow.gif]]
[<img(13%+, )[images/meow.gif]]
[<img( 8%+, )[images/meow.gif]]
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[<img( 2% , )[images/meow.gif]]
[img(  1%+,+)[images/meow.gif]]
{{tagClear{
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<<<
!!!!!Revisions
<<<
2009.02.24 [1.2.1] cleanup width/height regexp, use '+' suffix for resizing
2009.02.22 [1.2.0] added stretchable images
2008.01.19 [1.1.0] added evaluated width/height values
2008.01.18 [1.0.1] regexp for "(width,height)" now passes all CSS values to browser for validation
2008.01.17 [1.0.0] initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.ImageSizePlugin= {major: 1, minor: 2, revision: 1, date: new Date(2009,2,24)};
//}}}
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var f=config.formatters[config.formatters.findByField("name","image")];
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<<tiddler [[include_tiddlers/Immortality.html#"Immortality"]]>>

<<tiddler [[include_tiddlers/Incidence Matrix.html#"Incidence Matrix"]]>>

An ''Incidence Structure'' is a triple $(P,B,I)$ with $P$ a set of ''Points'' (a.k.a. ''Variety''), $B$ a system of subsets of $V$, called ''Blocks'', and $I$ a so called ''Incidence Relation'' which describes the relationship between the points and the blocks of the incidence structure.

A point $p_i \in V$ is said to be incident with a block $b_j\in B$ if $p_i \in b_j$. An incident pair $(p_i,b_j)$ is called a ''Flag'', a non-incident pair an ''Anti\-Flag''.
An incidence relation can be represented by means of a [[incidence matrix|Incidence Matrix]].

Every incidence structure can be represented as a [[binary code|Blockcode]]. Such representations are unique up to isomorphisms.

''Index of a Subgroup''
The index of a subgroup $H$ of a group $G$ (usually denoted $|G:H|$ or $[G:H]$) is the “relative size” of $H$ in respect to $G$.
If $G$ and $H$ are finite, the index is simply the quotient of the [[orders|Order]] of $G$ and $H$. By Lagrange's theorem, this number is always a positive integer.
If $G$ and $H$ are infinite, the index is defined as the number of cosets of $H$ in $G$. If $H$ is a [[normal subgroup|Normal Subgroup]] of $G$, then the index is equal to the order of the [[quotient group|Quotient Group]] $G/H$.
!!!!Example
The special [[orthogonal group|Orthogonal Group]] $SO(n)$ has index 2 in respect to the orthogonal group $O(n)$.

<<tiddler [[include_tiddlers/Index Theorem.html#"Index Theorem"]]>>

The ''Induced Metric'' (or [[first fundamental form|First Fundamental Form]]) of a manifold $\mathcal M$ is the assignment of an [[inner product|Scalar Product]] to each point in the manifold:
\begin{equation}
\langle \; , \; \rangle: T\mathcal M \times T\mathcal M \rightarrow \mathbb R
\end{equation}
I.e. the induced metric is the scalar product restricted to the tangent spaces of $\mathcal M$.

<<tiddler [[include_tiddlers/Inflation.html#"Inflation"]]>>

<<tiddler [[include_tiddlers/Information Loss Paradox.html#"Information Loss Paradox"]]>>

/***
|Name|InlineJavascriptPlugin|
|Source|http://www.TiddlyTools.com/#InlineJavascriptPlugin|
|Documentation|http://www.TiddlyTools.com/#InlineJavascriptPluginInfo|
|Version|1.9.5|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|Insert Javascript executable code directly into your tiddler content.|
''Call directly into TW core utility routines, define new functions, calculate values, add dynamically-generated TiddlyWiki-formatted output'' into tiddler content, or perform any other programmatic actions each time the tiddler is rendered.
!!!!!Documentation
>see [[InlineJavascriptPluginInfo]]
!!!!!Revisions
<<<
2009.04.11 [1.9.5] pass current tiddler object into wrapper code so it can be referenced from within 'onclick' scripts
2009.02.26 [1.9.4] in $(), handle leading '#' on ID for compatibility with JQuery syntax
|please see [[InlineJavascriptPluginInfo]] for additional revision details|
2005.11.08 [1.0.0] initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.InlineJavascriptPlugin= {major: 1, minor: 9, revision: 5, date: new Date(2009,4,11)};

config.formatters.push( {
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}
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// // GLOBAL FUNCTION: $(...) -- 'shorthand' convenience syntax for document.getElementById()
//{{{
if (typeof($)=='undefined') { function $(id) { return document.getElementById(id.replace(/^#/,'')); } }
//}}}

<<tiddler [[include_tiddlers/Instanton.html#"Instanton"]]>>

An ''$n$-dimensional Integer Lattice $\mathbb Z^n$'' (not to be confused with an [[integral lattice|Lattice]]), a.k.a ''$n$-dimensional Cubic Lattice'', is defined as
\[
\mathbb Z^n \equiv  \{(x_1,x_2,\ldots,x_i, \ldots, x_n) : x_i \in \mathbb Z\}
\]
$\mathbb Z^n$ is [[self-dual|Lattice]] and its [[kissing number|Kissing Number]] is $2n$.

The [[automorphism group|Automorphism]] $Aut(\mathbb Z^n)$ consists of all sign changes of the $n$ coordinates ($= 2^n$) and all permutations ($= n!$). Hence $N(n) \equiv \operatorname{ord} (Aut (\mathbb Z^n)) = 2^n n! = (2n)!!$. (The latter is the [[Double factorial|Double Factorial]]).
Examples:
* $N(2) = 8$
* $N(4) = 384$
* $N(8) = 10.321.920$
* $N(16) = 1.371.195.958.099.968.000$
See also: [[Sloane's A000165|http://www.research.att.com/~njas/sequences/A000165]].

<<tiddler [[include_tiddlers/Integral Bioctonion.html#"Integral Bioctonion"]]>>

>Die ganze Zahl schuf der liebe Gott, alles Übrige ist Menschenwerk.
>- Leopold Kronecker
A set of elements selected from an algebra is called a set of ''Integer Elements'' if it satisfies the following four conditions:
# For each element, the coefficients of the [[characteristic equation|Characteristic Polynomial]] (rank equation) are integers.
# The set is closed under subtraction and multiplication.
# The set contains $1$.
# The set is not a subset of a larger set satisfying conditions 1, 2 and 3.

The unit norm ''Integral Elements'' of [[complex numbers|Complex Number]], [[quaternions|Quaternion]] and [[octonions|Octonion]] can be constructed recursively by the [[Cayley-Dickson procedure|Cayley-Dickson Doubling]] of pairing, starting with $\pm 1$, i.e. the integral elements of real numbers of unit norm which are the non-zero roots of [[SU(2)]] and continuing with adding $\pm \frac 12$, the weights of the spinor representation of $SU(2)$.
For further details see the following table:
<html><center><img src="images/IntegralElements.jpg" style="width: 640px;"/></center></html>

A remark:
Due to the relationship of the integral elements with the characteristic equation, they appear to be very interesting in respect to their applications in quantum mechanics (i.e. for quantizing systems).

Papers:
* [[Division Algebras with Integral Elements - M. Koca, N. Ozdes|http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?200035098]] [[local|papers/IntegralElements.pdf]] [[pct. 10|http://scholar.google.de/scholar?cites=10351995558760038720&hl=de]] prl. 10
* [[Octonions and Exceptional Groups? - W.-l. Lin|http://psroc.phys.ntu.edu.tw/cjp/v30/579.pdf]] [[local|papers/579.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=10586251098270676143&hl=de]] prl. 9

<<tiddler [[include_tiddlers/Integral Octonion.html#"Integral Octonion"]]>>

<<tiddler [[include_tiddlers/Intelligent Design.html#"Intelligent Design"]]>>

<<tiddler [[include_tiddlers/Interpretation of Quantum Mechanics.html#"Interpretation of Quantum Mechanics"]]>>

Links:
* [[WIKIPEDIA - Indra's Net|http://en.wikipedia.org/wiki/Indra's_net]]

<<tiddler [[include_tiddlers/Invariant Mass.html#"Invariant Mass"]]>>

<<tiddler [[include_tiddlers/Involution.html#"Involution"]]>>

<<tiddler [[include_tiddlers/Isometry.html#"Isometry"]]>>

<<tiddler [[include_tiddlers/Isomorphism.html#"Isomorphism"]]>>

<<tiddler [[include_tiddlers/Isospin.html#"Isospin"]]>>

<<tiddler [[include_tiddlers/Isotopy.html#"Isotopy"]]>>

!!!![[Quadratic forms|Quadratic Form]]
A quadratic form $q$ is said to be ''Isotropic'' if and only if there exists a non-zero vector $v$ such that $q(v)=0$.
Else $q$ is called ''Anisotropic''.
$q$ is anisotropic if and only if $q$ is a definite form, that is $q$ is either positive definite, i.e. $q(v) > 0, \; \forall \, v$ or $q$ is negative definite, i.e. $q(v) < 0, \; \forall \,v$.

[[JHyperComplex|http://www.jhypercomplex.com]] is a Java API for doing hypercomplex computations (both numerical and algebraic) being developed by the author of this Wiki.

It is the result of realizing that when doing calculations with hypercomplex numbers (e.g. quaternions, octonions) classically with [[paper and pencil|Paper and Pencil Physics and Mathematics]] one often runs into the the same stupid, mechanistic, boring and hence error-prone calculations.
Furthermore there are things one cannot do this way due to them being too complex. Furthermore for larger algebras (which are very interesting in respect their applications to physics !), playing around and experimenting is not feasible any more.

In the meantime \JHyperComplex has become quite a potent research tool (unique of its kind, I think) and has yielded quite a few interesting results.

This WIKI in parts is a byproduct of the development of this software and contains some results obtained with it.

Why would one require yet another computer algebra software ? First of all, \JHyperComplex is written in Java and is based on an object oriented design which makes it in principle expandable at will "without causing much pain".

I very much appreciate other computer algebra systems like [[Sage]] or [[MAGMA]] and use them frequently (many examples in this WIKI are based on such systems), however I have come to the conclusion that many problems require considering which system is best suited to solve it or can do it at all.
As some systems are "hard wired" they either allow one for a solution of a problem or (practically) they don't. In case of JHyperComplex there is always a solution to a given problem, given one is willing to code Java and extend the API accordingly.

There's a lot more that can be said about \JHyperComplex. If you have questions or are interested in purchasing an "as it is version" (which is all I can offer at the moment due to time constraints), please [[contact me here|Welcome]].
I should mention that JHyperComplex has already a quite good (\JUnit) test-coverage.

If you don't believe that hypercomplex numbers are interesting, you should check out another piece of software I have written (with the help of JHyperComplex), namely  [[HyperFract|http://www.HyperFract.com]].

{{center{[img(485px+, )[images/JacobianDeterminant.gif]]}}}
The ''Jacobi Determinant'' is the determinant of a square [[Jacobi matrix|Jacobi Matrix]]:
\[
\det (\mb J_{\mb f}(\mb{x})) = \begin{vmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\  \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_2} \\  \\ \vdots & \vdots & \ddots & \vdots \\  \\ \frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_n} \end{vmatrix}
\]
Its value gives the following information about the behavior of the function $\mb f$ in the point $\mb x$:
*  $> 0$  orientation preserving
* $< 0$ orientation reversing
* $= 0$ not invertible
* $= 1$ volume preserving

''General Relativity''
For a coordinate transformation $x_\mu \mapsto x'_\nu(\mb x) $ in general relativity, the Jacobi determinant $\det (\mb J_{\mb{x'}}(\mb{x}))$ can be written as
\[
\det (J_{\mb{\mb x'}}(\mb{x})) = \epsilon_{\mu\nu\rho\sigma} \frac{\partial x'^\mu}{\partial x^0} \frac{\partial x'^\nu}{\partial x^1} \frac{\partial x'^\rho}{\partial x^2} \frac{\partial x'^\sigma}{\partial x^3}
\]
Links:
* [[The Jacobian Determinant - Jeff Knisley|http://math.etsu.edu/MultiCalc/Chap3/Chap3-5/index.htm]] - Doesn't work well with firefox, better use other browser.

Given the real-valued function $f_i(x_1, x_2, \dots, x_n),\, i = 1,...,m,$ for which it is assumed that all partial derivatives exist, then the ''Jacobi Matrix'' (or short ''Jacobian'') of $\mb f$ is defined by:
\[
(\mb J_{\mb f})_{ij}(\mb{x}) = \frac{\partial f_i(\mb{x})}{\partial x_j} = \partial_j f_i(\mb{x})\,\!
\]
Written out explicitely it is:
\[
\mb J_{\mb{f}}(\mb{x}) = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\  \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_2} \\  \\ \vdots & \vdots & \ddots & \vdots \\  \\ \frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_n} \end{pmatrix}
\]
The ''Jacobian Matrix'' describes the first order change of the function or, put it differently, its tangent.
It appears as the term in the Taylor series expansion of $f_i$ which corresponds to the first order change:
\[
\Delta f_i(\mb{x}) = f_i(\mb{x}+\Delta\mb{x})\approx f_i(\mb{x}) + (\mb J_{\mb{f}})_i(\mb{x})\Delta \mb{x} + \frac{1}{2} \Delta\mb{x}^\mathrm{T} \mb H_{f_i}(\mb{x}) \Delta\mb{x}
\]
The second order change of $f_i$ is described by the [[Hessian matrix|Hessian Matrix]] $H_{f_i}$.

!!!! Applications
The Jacobian can be used to describe coordinate transformations $ \mb x = (x_1, x_2, \ldots, x_n) \to \mb x' = (x'_1(\mb x), x'_2 (\mb x), \ldots, x'_n (\mb x))$.
One has, up to first order:
\[
x'^i (\mb x + d\mb x) \mb{e}_i  = x'^i (\mb x) \mb{e}_i + \frac{\partial x'^i (\mb x)}{\partial x^j} dx^j  \mb{e}_i = x'^i (\mb x) \mb{e}_j + J_{\mb {x'}}(\mb {x})_{ij}  \mb{e}_j
\]

<<tiddler [[include_tiddlers/Jacobian.html#"Jacobian"]]>>

<<tiddler [[include_tiddlers/John Conway.html#"John Conway"]]>>

<<tiddler [[include_tiddlers/Jordan Algebra.html#"Jordan Algebra"]]>>

<<tiddler [[include_tiddlers/Jordan Identity.html#"Jordan Identity"]]>>

The the ''Jordan Triple Product'' $\{\mb A, \mb B,\mb C\}_J$ is defined as
\begin{equation}
\{\mb A, \mb B, \mb C \}_J =  (\mb A \mb B^*) \mb C + (\mb C \mb B^*) \mb A ? (\mb A \mb C) \mb B^*
\end{equation}

<<tiddler [[include_tiddlers/Josephson Junction.html#"Josephson Junction"]]>>

<<tiddler [[include_tiddlers/KO-Theory.html#"KO-Theory"]]>>

The ''Kadomtsev\-Petviashvili (KP) Equation'' is an extension of the [[Kortweg-De Vries Equation|Kortweg-De Vries Equation]] to $2+1$ dimensions.
The KP equation and its hierarchy is deeply related to the theory of [[Riemann surfaces|Riemann Space]] (Riemann\-Schottky problem).

The KP hierarchy makes its appearance in many areas of mathematics (in particular differential and algebraic geometry) and physics (from hydrodynamics to [[string theory|Superstring Theory]]).

!!!!Applications
The KP equation describes nonlinear fluid surface waves in a certain approximation and explains to some extent the formation of network patterns formed by line wave segments on a water surface.


Papers:
* [[Weakly Nonassociative Algebras, Riccati and KP Hierarchies (2008) - A. Dimakisa, F. Müller-Hoissen|http://arxiv.org/PS_cache/nlin/pdf/0701/0701010v4.pdf]] [[local|papers/0701010v4.pdf]] [[pct. 6|http://scholar.google.com/scholar?hl=de&lr=&cites=7729812748852738279&um=1&ie=UTF-8&sa=X&ei=QaE9TNHcDZySOLHDpb0P&ved=0CCMQzgIwAA]]

<<tiddler [[include_tiddlers/Kalb-Ramond Field.html#"Kalb-Ramond Field"]]>>

The ''Kemmer Equation'' describes a massive particle with spin 1 and was first derived in 1931 by Kemmer.
Its is a Dirac type equation but involves matrices obeying a different scheme of commutation rules. The theory can be developed in strikingly close correspondence to Dirac’s electron theory; practically all the definitions of physical quantities like spin, magnetic moment etc. have their exact counterpart.

A ''Kikkawa Space'' is a manifolds with affine connection such that all [[geodesic loops|Geodesic Loop]] of some neighborhood (at some point) are [[right-monoalternative|Alternative Algebra]].

Papers:
* [[On Kikkawa Spaces - L. Sabinina|http://www.iop.org/EJ/article/0036-0279/58/4/L13/RMS_58_4_L13.pdf?request-id=79c53b95-8005-40d6-b9d1-08c30b8dab36]]

A ''Kirkman Triple System'' of order $v$ (shortly denoted by $KTS(v)$) is a [[resolvable|Resolvable Design]] [[Steiner triple system|Steiner Triple System]] $STS(v)$. The case $v = 15$ became known as [[Kirkman's schoolgirl problem|Kirkman's Schoolgirl Problem]].

In 1971 D. K. Ray\-Choudhury and R. M. Wilson proved that at least one Kirkman triple system for every (non-negative) order exists, provided a STS exists for that order.

The smallest possibility has $v = 3$ with exactly one block and one parallel class, hence it is trivial.
For $v=9$ (which is related to the $3 \times 3$ magic square [1]) there is a single unique (up to an isomorphism) solution, while there are $7$ different systems for $v=15$.

Links:
* [[[1] WIKIPEDIA - Magic Square|http://en.wikipedia.org/wiki/Magic_square]]

<<tiddler [[include_tiddlers/Kirkman's Schoolgirl Problem.html#"Kirkman's Schoolgirl Problem"]]>>

<<tiddler [[include_tiddlers/Kirlian Photography.html#"Kirlian Photography"]]>>

<<tiddler [[include_tiddlers/Kissing Number.html#"Kissing Number"]]>>

<<tiddler [[include_tiddlers/Klein Four-group.html#"Klein Four-group"]]>>

Papers:
* [[History and Physics of the Klein Paradox (1999) - A. Calogeracos, N. Dombey|http://arxiv.org/PS_cache/quant-ph/pdf/9905/9905076v1.pdf]] [[local|papers/9905076v1.pdf]] [[pct. 43|http://scholar.google.de/scholar?cites=5733264893586937786&as_sdt=2005&sciodt=2000&hl=de]]

<<tiddler [[include_tiddlers/Klein-Gordon Equation.html#"Klein-Gordon Equation"]]>>

<<tiddler [[include_tiddlers/Kleinfeld Function.html#"Kleinfeld Function"]]>>

<<tiddler [[include_tiddlers/Kleinfeld Identities.html#"Kleinfeld Identities"]]>>

<<tiddler [[include_tiddlers/Kochen-Specker Theorem.html#"Kochen-Specker Theorem"]]>>

The ''Kolmogorov Complexity'' (a.k.a. ''Descriptive Complexity'', ''Kolmogorov\-Chaitin Complexity'', ''Stochastic Complexity'', ''Algorithmic Entropy'' or ''Program\-Size Complexity'') of an object is a measure of the minimal computational resources that are required to specify it in some fixed universal description language.
It can be shown that the Kolmogorov complexity of any string cannot be too much larger than the length of the string itself.

The notion of Kolmogorov complexity is quite deep and can be used to state and prove impossibility results akin to [[Gödel's incompleteness theorem|Gödel's Theorems]] and [[Turing's halting problem|Halting Problem]].

Kolmogorov complexity ignores runtime though.

Papers:
* [[Occam’s Razor as a Formal Basis for a Physical Theory (2002) - A. N. Soklakov|http://arxiv.org/PS_cache/math-ph/pdf/0009/0009007v3.pdf]] [[local|papers/0009007v3.pdf]] [[pct. 9|http://scholar.google.de/scholar?cites=12698824753203673377&as_sdt=2005&sciodt=2000&hl=de]]

Links:
* [[WIKIPEDIA - Kolmogorov Complexity|http://en.wikipedia.org/wiki/Kolmogorov_complexity]]

The ''Korteweg-de Vries Equation'' (''\KdV Equation'' for short) is a nonlinear partial differential equation of the form
\[
u_t - 6 u u_{xx} + u_{xxx} = 0
\]
The equation was first written down by Korteweg and de Vries in 1895 in connection with the evolution of long water waves down canals of rectangular cross section. One solution of the equation leads to a mathematical representation of [[solitons|Soliton]], which were observed for the first time in 1834 in water canals by John Scott Russell.
The \KdV-equation also arises in plasma physics, in the study of an harmonic lattices, and in the propagation of waves in elastic rods.

Links:
* [[WIKIPEDIA - Korteweg–de Vries Equation|http://en.wikipedia.org/wiki/Korteweg%E2%80%93de_Vries_equation]]

<<tiddler [[include_tiddlers/Krein Space.html#"Krein Space"]]>>

The ''Kretschmann Scalar K'' for an n-dimensional Riemann manifold is given by
\begin{equation}
K = R_{\mu\nu\rho\sigma} \, R^{\mu\nu\rho\sigma} = C_{\mu\nu\rho\sigma} \, C^{\mu\nu\rho\sigma} +\frac{4}{d-2} R_{\mu\nu}\, R^{\mu\nu} - \frac{2}{(n-1)(n-2)}R^2
\end{equation}
with $ C_{\mu\nu\rho\sigma}$ the [[Weyl tensor|Weyl Tensor]], $R_{\mu\nu}$ the [[Ricci tensor|Ricci Tensor]] and $R$ the [[Ricci scalar|Ricci Scalar]].
In 4 dimensions one has:
\begin{equation}
K = C_{\mu\nu\rho\sigma} \, C^{\mu\nu\rho\sigma} +\frac{1}{2} R_{\mu\nu}\, R^{\mu\nu} - \frac{1}{3}R^2
\end{equation}

<<tiddler [[include_tiddlers/Kähler Manifold.html#"Kähler Manifold"]]>>

<<tiddler [[include_tiddlers/LOFAR.html#"LOFAR"]]>>

If $\mathcal H$ is a subgroup of a finite group $\mathcal G$, then the [[order|Order]] of $\mathcal H$ divides the order of $\mathcal G$.

<<tiddler [[include_tiddlers/Lambda 16 Lattice.html#"Lambda 16 Lattice"]]>>

<<tiddler [[include_tiddlers/Landau Ghost.html#"Landau Ghost"]]>>

<<tiddler [[include_tiddlers/Landauer's Principle.html#"Landauer's Principle"]]>>

Links:
* [[WIKIPEDIA - Langlands Program|http://en.wikipedia.org/wiki/Langlands_program]]

<<tiddler [[include_tiddlers/Laplace Equation.html#"Laplace Equation"]]>>

The ''Laplace\-Beltrami Operator'' $\square$ is a generalisation of the [[Laplace operator|Laplace Equation]] for [[Riemannian|Riemann Space]] and [[pseudo-Riemannian manifolds|Pseudo-Riemannian Space]].
It is given by:
\[
 \square_g \equiv \Delta_g \equiv \ \frac{1} {\sqrt{g}} \ \partial_\mu \left ( \sqrt{g} g^{\mu\nu} \partial_{\nu}  \right )
\]

<<tiddler [[include_tiddlers/Large Hadron Collider.html#"Large Hadron Collider"]]>>

<<tiddler [[include_tiddlers/Large Number Hypothesis.html#"Large Number Hypothesis"]]>>

<<tiddler [[include_tiddlers/Large Numbers and the Wavefunction Collapse.html#"Large Numbers and the Wavefunction Collapse"]]>>

A ''Lattice'' is an algebra $\mathcal A$ with two operations ''$\wedge$'' (called ''Meet'' or ''And'') and ''$\vee$'' (called ''Join'' or ''Or'') for which, $\forall \mb A, \mb B, \mb C \in \mathcal A$, the following relations hold:
|!Relations|!Laws|
|$\mb A \wedge \mb A = \mb A$; $\;\mb A \vee \mb A = \mb A\quad$|[[Idempotency]]|
|$\mb A  \wedge \mb B = \mb B  \wedge \mb A$; $\;\mb A \vee \mb B = \mb B \vee \mb A\quad$ |''Commutativity''|
|$(\mb A  \wedge \mb B)  \wedge \mb C = \mb A  \wedge (\mb B  \wedge \mb C)$; $\;(\mb A \vee \mb B) \vee \mb C = \mb A \vee (\mb B \vee \mb C)$|''Associativity''|
|$\mb A \vee (\mb A \wedge \mb B) = \mb A$; $\;\mb A \wedge (\mb A \vee \mb B) = \mb A\quad$|''Absorption''|

<<tiddler [[include_tiddlers/Lattice Gas Cellular Automaton.html#"Lattice Gas Cellular Automaton"]]>>

<<tiddler [[include_tiddlers/Lattice QCD.html#"Lattice QCD"]]>>

/***
|''Name:''|LaunchApplicationPlugin|
|''Author:''|Lyall Pearce|
|''Source:''|http://www.Remotely-Helpful.com/TiddlyWiki/LaunchApplication.html|
|''License:''|[[Creative Commons Attribution-Share Alike 3.0 License|http://creativecommons.org/licenses/by-sa/3.0/]]|
|''Version:''|1.4.0|
|''~CoreVersion:''|2.3.0|
|''Requires:''| |
|''Overrides:''| |
|''Description:''|Launch an application from within TiddlyWiki using a button|
!!!!!Usage
<<<
{{{<<LaunchApplication "buttonLabel" "tooltip" "application" ["arguments" ...]>>}}}
{{{<<LaunchApplicationButton "buttonLabel" "tooltip" "application" ["arguments" ...]>>}}}
{{{<<LaunchApplicationLink "buttonLabel" "tooltip" "application" ["arguments" ...]>>}}}
* buttonLabel is anything you like
* tooltip is anything you like
* application is a path to the executable (which is Operating System dependant)
* arguments is any command line arguments the application requires.
* You must supply relative path from the location of the TiddlyWiki OR a fully qualified path
* Forward slashes works fine for Windows

{{{<<LaunchApplication...>>}}} functions the same as {{{<<LaunchApplicationButton...>>}}}

eg.

{{{
<<LaunchApplicationButton "Emacs" "Linux Emacs" "file:///usr/bin/emacs">>
}}}
<<LaunchApplicationButton "Emacs" "Linux Emacs" "file:///usr/bin/emacs">>

{{{
<<LaunchApplicationLink "LocalProgram" "Program relative to Tiddly html file" "localDir/bin/emacs">>
}}}
<<LaunchApplicationLink "LocalProgram" "Program relative to Tiddly html file" "localDir/bin/emacs">>

{{{
<<LaunchApplicationButton "Open Notepad" "Text Editing" "file:///e:/Windows/notepad.exe">>
}}}
<<LaunchApplicationButton "Open Notepad" "Text Editing" "file:///e:/Windows/notepad.exe">>

{{{
<<LaunchApplicationLink "C Drive" "Folder" "file:///c:/">>
}}}
<<LaunchApplicationLink "C Drive" "Folder" "file:///c:/">>


!!!!!Revision History
* 1.1.0 - leveraged some tweaks from from Bradly Meck's version (http://bradleymeck.tiddlyspot.com/#LaunchApplicationPlugin) and the example text.
* 1.2.0 - Make launching work in Linux too and use displayMessage() to give diagnostics/status info.
* 1.3.0 - execute programs relative to TiddlyWiki html file plus fix to args for firefox.
* 1.3.1 - parameters to the macro are properly parsed, allowing dynamic paramters using {{{ {{javascript}} }}} notation.
* 1.4.0 - updated core version and fixed empty tooltip and added launch link capability

<<<
***/
//{{{
version.extensions.LaunchApplication = {major: 1, minor: 4, revision: 0, date: new Date(2007,12,29)};
config.macros.LaunchApplication = {};
config.macros.LaunchApplicationButton = {};
config.macros.LaunchApplicationLink = {};

function LaunchApplication(appToLaunch,appParams) {
    if(! appToLaunch)
	return;
    var tiddlyBaseDir = self.location.pathname.substring(0,self.location.pathname.lastIndexOf("\\")+1);
    if(!tiddlyBaseDir || tiddlyBaseDir == "") {
	tiddlyBaseDir = self.location.pathname.substring(0,self.location.pathname.lastIndexOf("/")+1);
    }
    // if Returns with a leading slash, we don't want that.
    if(tiddlyBaseDir.substring(0,1) == "/") {
	tiddlyBaseDir = tiddlyBaseDir.substring(1);
    }
    if(appToLaunch.indexOf("file:///") == 0) // windows would have C:\ as the resulting file
    {
	tiddlyBaseDir = "";
	appToLaunch = appToLaunch.substring(8);
    }

    if (config.browser.isIE) {
	// want where the tiddly is actually located, excluding tiddly html file

	var theShell = new ActiveXObject("WScript.Shell");
	if(theShell) {
            // the app name may have a directory component, need that too
	    // as we want to start with current working dir as the location
	    // of the app.
	    var appDir = appToLaunch.substring(0, appToLaunch.lastIndexOf("\\"));
	    if(! appDir || appDir == "") {
		appDir = appToLaunch.substring(0, appToLaunch.lastIndexOf("/"));
	    }
	    appParams = appParams.length > 0 ? " \""+appParams.join("\" \"")+"\"" : "";
	    try {
		theShell.CurrentDirectory = decodeURI(tiddlyBaseDir + appDir);
		var commandString = ('"' +decodeURI(tiddlyBaseDir+appToLaunch) + '" ' + appParams);
		pluginInfo.log.push(commandString);
	        theShell.run(commandString);
	    } catch (e) {
		displayMessage("LaunchApplication cannot locate/execute file '"+tiddlyBaseDir+appToLaunch+"'");
		return;
	    }
	} else {
	    displayMessage("LaunchApplication failed to create ActiveX component WScript.Shell");
	}
    } else { // Not IE
	// want where the tiddly is actually located, excluding tiddly html file
	netscape.security.PrivilegeManager.enablePrivilege("UniversalXPConnect");
        var file = Components.classes["@mozilla.org/file/local;1"].createInstance(Components.interfaces.nsILocalFile);
        var launchString;
	try { // try linux/unix format
            launchString = decodeURI(tiddlyBaseDir+appToLaunch);
	    file.initWithPath(launchString);
	} catch (e) {
	    try { // leading slash on tiddlyBaseDir
                launchString = decodeURI("/"+tiddlyBaseDir+appToLaunch);
		file.initWithPath(launchString);
	    } catch (e) {
		try { // try windows format
		    launchString = decodeURI(appToLaunch).replace(/\//g,"\\");
		    file.initWithPath(launchString);
		} catch (e) {
		    try { // try windows format
			launchString = decodeURI(tiddlyBaseDir+appToLaunch).replace(/\//g,"\\");
			file.initWithPath(launchString);
		    } catch (e) {
			displayMessage("LaunchApplication cannot locate file '"+launchString+"' : "+e);
			return;
		    } // try windows mode
		} // try windows mode
	    }; // try with leading slash in tiddlyBaseDir
	}; // try linux/unix mode
	try {
	    if (file.isFile() && file.isExecutable()) {
		displayMessage("LaunchApplication executing '"+launchString+"' "+appParams.join(" "));
		var process = Components.classes['@mozilla.org/process/util;1'].createInstance(Components.interfaces.nsIProcess);
		process.init(file);
		process.run(false, appParams, appParams.length);
	    }
	    else
	    {
		displayMessage("LaunchApplication launching '"+launchString+"' "+appParams.join(" "));
		file.launch(); // No args available with this option
	    }
	} catch (e) {
	    displayMessage("LaunchApplication cannot execute/launch file '"+launchString+"'");
	}
    }
};

config.macros.LaunchApplication.handler = function (place,macroName,params,wikifier,paramString,tiddler) {
    // 0=ButtonText, 1=toolTip, 2=AppToLaunch, 3...AppParameters
    if (params[0] && (params[1] || params[1] == "") && params[2]) {
        var theButton = createTiddlyButton(place, getParam(params,"buttonText",params[0]), getParam(params,"toolTip",params[1]), onClickLaunchApplication);
        theButton.setAttribute("appToLaunch", getParam(params,"appToLaunch",params[2]));
        params.splice(0,3);
        theButton.setAttribute("appParameters", params.join(" "));
        return;
    }
}
config.macros.LaunchApplicationButton.handler = function (place,macroName,params,wikifier,paramString,tiddler) {
    config.macros.LaunchApplication.handler (place,macroName,params,wikifier,paramString,tiddler);
}

config.macros.LaunchApplicationLink.handler = function (place,macroName,params,wikifier,paramString,tiddler) {
    // 0=ButtonText, 1=toolTip, 2=AppToLaunch, 3...AppParameters
    if (params[0] && (params[1] || params[1] == "") && params[2]) {
        //var theLink = createExternalLink(place, getParam(params,"buttonText",params[0]));
        var theLink = createTiddlyButton(place, getParam(params,"buttonText",params[0]), getParam(params,"toolTip",params[1]), onClickLaunchApplication,"link");
        theLink.setAttribute("appToLaunch", getParam(params,"appToLaunch",params[2]));
        params.splice(0,3);
        theLink.setAttribute("appParameters", params.join(" "));
        return;
    }
}

function onClickLaunchApplication(e) {
	var theAppToLaunch = this.getAttribute("appToLaunch");
	var theAppParams = this.getAttribute("appParameters").readMacroParams();
	LaunchApplication(theAppToLaunch,theAppParams);
}

//}}}

<<tiddler [[include_tiddlers/Leech Lattice.html#"Leech Lattice"]]>>´

Given a [[loop|Loop]] $\mathcal L$, a ''Left Translation'' $L_{\mb A}: \mathcal L \rightarrow \mathcal L$  is defined as
\[
L_{\mb A} (\mb X) = \mb {AX}
\]
Similarly a ''Right Translation'' $R_{\mb A}: \mathcal L \rightarrow \mathcal L$ is defined as
\[
R_{\mb B} (\mb X) = \mb {XB}
\]

The composition of two right- (left-) translations is not necessarily a right- (left-) translation.

The set of left- and right-translations $\{L_{\mb A}, R_{\mb A} : \mb A \in \mathcal L\}$ generates a [[group|Group]], which is a permutation group acting on $\mathcal L$. It is known as the ''Multiplication Group'' of $\mathcal L$ and denoted $Mlt(\mathcal L)$. (Some authors prefer the notation $M(\mathcal L)$).

<<tiddler [[include_tiddlers/Leggett-Garg Inequality.html#"Leggett-Garg Inequality"]]>>

The $6$ elementary particles electron, electron-neutrino, muon, muon-neutrino, tauon and tauon-neutrino are called ''Leptons''. Leptons are subject to the [[electroweak|Electroweak Interactions]] and [[gravitational|Gravitation]] force.

The fundamental theorem of [[Riemannian geometry|Riemann Space]] states: On a Riemannian manifold there is a unique [[connection|Connection]] which is [[torsion-free|Torsion]] and [[compatible with the metric|Metric Compatibility]].

This connection is called the ''Levi\-Civita Connection'' (a.k.a. ''Riemannian\-'' or ''Christoffel Connection''). The connection coefficients are at times expressed by means of the [[Christoffel symbols|Christoffel Symbols]].

The relation with the [[metric tensor|Metric Tensor]] is given by:
\[
\Chr{\lambda}{\mu\nu} = \frac{1}{2} g^{\lambda \rho} (\partial_\mu g_{\rho \nu}  + \partial_\nu g_{\rho\mu}  - \partial_\rho g_{\mu \nu}  )
\]
which means that the metric tensor completely determines the Christoffel coefficients and vice versa which is characteristic of a [[Riemann manifold|Riemann Space]].

The Levi\-Civita connection is a symmetric connection, therefore
\[
\Chr{\lambda}{\mu\nu} = \Chr{\lambda}{\nu\mu}
\]
Due to its symmetry, it consists of $40 = 10\cdot 4$ independent components.


<html><center><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_60.html" width=51% height=86></iframe></center></html>

The ''Lewis\-Tolman Lever Paradox'' (or ''Right\-Angle Lever Paradox'') is one of the first paradoxes of special relativity proposed in 1909.

Papers:
* [[Right Angle Lever Paradox - J. C. Nickerson, R. T. McAdory|http://polaris.deas.harvard.edu/galileo/images/material/1469/351/reltorque.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=14416921559306520612&hl=de]]
* [[The Lewis-Tolman Lever Paradox - J. W. Butler|http://www.physics.princeton.edu/~mcdonald/examples/mechanics/butler_ajp_38_360_70.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=7742882603248677411&hl=de]]
* [[Covariant Formulation of Hooke's Law - O. Gron|http://www.physics.princeton.edu/~mcdonald/examples/mechanics/gron_ajp_49_28_81.pdf]] pct. 0
* [[The Lack of Rotation in a Moving Right Angle Lever - J. Franklin|http://arxiv.org/PS_cache/arxiv/pdf/0805/0805.1196v2.pdf]] pct. 0
* [[Relativistic Angular Momentum - N. Menicucci|http://panda.unm.edu/Courses/Finley/P495/TermPapers/relangmom.pdf]] pct. 0

<<tiddler [[include_tiddlers/Lie Algebra.html#"Lie Algebra"]]>>

The ''Lie Derivative'' $\mathcal L_V$ in respect to two vectors $W$ and $V$ is defined by:
\[
\mathcal{L}_V(W)_\mu = V^\nu D_\nu W_\mu ? W^\nu D_\nu V_\mu
\]
with $D$ the [[covariant derivative|Covariant Derivative]].
The Lie derivative can be generalized involving tensors.

<<tiddler [[include_tiddlers/Lie Group.html#"Lie Group"]]>>

<<tiddler [[include_tiddlers/Lie Triple System.html#"Lie Triple System"]]>>

>It is Lie's most remarkable insight that the bracket is determined by the degree two terms in the Taylor expansion of the product, and that is suffices as a basis for the entire local theory.
> - K. H. Hofmann, K. Strambach -

''I. Theorem''
Each local real analytic [[Lie group|Lie Group]] determines a [[Lie algebra|Lie Algebra]] in its [[tangent space|Tangent Algebra]] at the identity element.

''III. Theorem (also called: "Converse Lie Theorem")''
Any finite-dimensional Lie algebra over the real numbers is the Lie algebra associated to some (unique) Lie group.

''Generalisation'' of Lie's 3-rd theorem for [[quasigroups|Quasigroup]]:
In general an [[Akivis algebra|Akivis Algebra]] does not determine a local differentiable [[quasigroup|Quasigroup]] uniquely.
However for [[Malcev-|Malcev Algebra]] and [[Bol- algebras|Bol Algebra]] (which are particular cases of Akivis algebras) local Moufang and Bol quasigroups are determined in a unique way respectively.
As for [[monoassociative|Monoassociativity]] quasigroups, their local Akivis algebras do not determine them uniquely. However, a prolonged Akivis algebra defined in a fourth-order differential neighborhood determines a monoassociative quasigroup uniquely. Note that besides the operations of commutation and association, a prolonged Akivis algebra has two quaternary operations called quaternators. It is therefore a so called binary-ternary-quaternary algebra.
A key ingedient in the proof of Lie's Third Theorem is the [[(generalized) Baker Campbell Hausdorff formula|Baker Campbell Hausdorff Formula (BCH)]].

<<tiddler [[include_tiddlers/Lindblad Equation.html#"Lindblad Equation"]]>>

<<tiddler [[include_tiddlers/Linear Blockcode.html#"Linear Blockcode"]]>>

!!!!Web-theory
A [[n-Web|Web]] is said to be ''linearizable'' (''rectifiable'') if it is equivalent to a linear $n$-web, i.e. a $n$-web formed by $n$ one-parameter foliations of straight lines on a [[projective plane|Projective Space]]. (A stronger condition than linearizability is the notion of [[parallelizability|Parallelizability]]).

<<tiddler [[include_tiddlers/Liquid Crystal.html#"Liquid Crystal"]]>>

<<tiddler [[include_tiddlers/Lisi's E8 Model.html#"Lisi's E8 Model"]]>>

<<tiddler [[include_tiddlers/Logic.html#"Logic"]]>>

{{center{[img(670px+, )[images/MoufangBoolLoops.jpg]]}}}
A ''Loop'' is a group except that one allows the multiplication to be non-associative. It furthermore is a [[quasigroup|Quasigroup]] with a unit element. Many results in loop theory may by regarded as a generalization of results about [[groups|Group]].
Another way to see it: If one starts with an abelian group with its axioms and refrains from commutativity one is lead to non-abelian groups. If on the other hand one  gives up associativity one is lead to loops (which consequently could also be calld non-associative groups).

Loops which have an alternative, but not associative loop ring, have been completely characterized.

Although the loop-product is in general not associative, i.e.
\[
(\mb{AB})\mb C \ne \mb A(\mb{BC})
\]
it is associative up to [[homotopy|Homotopy]], i.e.
\[
(\mb{AB})\mb C \sim \mb A(\mb{BC})
\]

Some prominent loops are:
* [[Moufang loops|Moufang Loop]]
* [[Bol loops|Bol Loop]]
{{center{[img(524px+, )[images/LoopProperties.jpg]]}}}
Papers:
* [[Quasigroups, Loops, and Associative Laws - K. Kunen |http://www.math.wisc.edu/~kunen/quasi.ps]] [[pct. 32|http://scholar.google.com/scholar?hl=de&lr=&cites=10828658052078342113&um=1&ie=UTF-8&ei=gPRNSvPgBpi6sAbj-tXxBw&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]
*[[Historical Notes on Loop Theory - H. A. Pflugfelder | http://www.emis.de/journals/CMUC/pdf/cmuc0002/pflug.pdf]] [[pct.9|http://scholar.google.de/scholar?hl=de&lr=&cites=18435977413609797309]]
*[[Smooth Quasigroups and Loops: Forty-five Years of Incredible Growth - L. V. Sabinin |http://www.emis.de/journals/CMUC/pdf/cmuc0002/sabinin.pdf]] [[pct. 2|http://scholar.google.de/scholar?hl=de&lr=&cites=18081906196983711621]] - With lots of references to literature.
* [[Introduction to: Non-Associative Finite Invertible Loops - R. E. Cawagas|http://arxiv.org/PS_cache/arxiv/pdf/0907/0907.5059v1.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=2000212841965417426&hl=de&as_sdt=2000]]
* [[Automated Theorem Proving in Loop Theory - J. D. Phillips, D. Stanovsky | http://ftp.informatik.rwth-aachen.de/Publications/CEUR-WS/Vol-378/paper3.pdf]] pct. 0

Google Books:
* [[Loops in Group and Lie Theory - P. T. Nagy, K. Strambach|http://books.google.com/books?hl=de&lr=&id=V9m8lFyQhtAC&oi=fnd&pg=PR5&ots=H0AdJQEEHc&sig=qOyO3HIbryZigsCp3rvFIEujn_Q]]  [[bct. 22|http://scholar.google.de/scholar?cites=15403789945460484989&hl=de]]

<<tiddler [[include_tiddlers/Loop Quantum Gravity.html#"Loop Quantum Gravity"]]>>

<<tiddler [[include_tiddlers/Lorentz Group.html#"Lorentz Group"]]>>

<<tiddler [[include_tiddlers/Lorentz-Dirac Equation.html#"Lorentz-Dirac Equation"]]>>

''Lovelock Theory of Gravity'' (short ''Lovelock Gravity'') represents a unique class of higher curvature gravity theories with field equations that do not involve derivatives of the [[Riemann curvature tensor|Riemann Tensor]].

It was introduced by D. Lovelock in 1971 and can be regarded as a generalization of [[Einstein's theory of general relativity|General Relativity]] to higher dimensions. In dimension $D= 3$ and $ D= 4$ Lovelock's theory coincides with Einstein's theory, but in higher dimension both theories are different.
For $D > 4$ Einstein gravity can be thought of as a particular case of Lovelock gravity since the [[Einstein-Hilbert action|Einstein-Hilbert Action]] is one of several terms that constitute the Lovelock action.

Links:
* [[WIKIPEDIA - Lovelock Theory of Gravity|http://en.wikipedia.org/wiki/Lovelock_theory_of_gravity]]

A ''Low-density Parity-check Code'' (''LDPC Code'' or ''Gallagar Code'') is a [[linear error-correcting code|Linear Blockcode]] which was introduced in 1960 by Robert G. Gallager.

LDPC codes have parity-check matrices with a low density of "1's" (i.e. they are "sparse"), which renders low complexity decoding and leads to simple implementations.

It has been shown that these codes can achieve a good error performance that is very close to Shannon limit.

A special class are [[finite geometry|Finite Geometry]] LDPC codes, based on [[euclidean|Affine Geometry]] and [[projective geometries|Projective Space]].
One distinguishes four classes of such codes:
# Type\-I Euclidean geometry (EG)\-LDPC codes,
# type\-II EG\-LDPC codes,
# type\-I projective geometry (PG)\-LDPC codes,
# type\-II PG\-LDPC codes.

Papers:
* [[Low Density Parity Check Codes Based on Finite Geometries: A Rediscovery and New Results - Y. Kou, S. Lin, M. P.C. Fossorier|http://www.stanford.edu/class/ee379b/class_reader/ucd1.pdf]] [[local|papers/ucd1.pdf]] {{t500Cite{[[pct. 598|http://scholar.google.de/scholar?cites=10652403661149396541&hl=de]]}}}
* [[Structured Low-Density Parity-Check Codes - J. M. F. Moura, J. Lu, H. Zhang|http://www.ece.cmu.edu/~moura/papers/spm-jan04-moura-lu-zhang-ieeeexplore.pdf]] [[local|papers/spm-jan04-moura-lu-zhang-ieeeexplore.pdf]] [[pct. 35|http://scholar.google.de/scholar?cites=12666190617458772897&hl=de]]

<<tiddler [[include_tiddlers/Lucas' Problem.html#"Lucas' Problem"]]>>

<<tiddler [[include_tiddlers/M-Algebra.html#"M-Algebra"]]>>

<<tiddler [[include_tiddlers/M-Theory.html#"M-Theory"]]>>

!!!![[MAGMA|http://magma.maths.usyd.edu.au/calc/]]^^[[Help|MAGMA]]^^
One if the outstanding features of MAGMA is that it allows for the generation of [[lattices|Lattice]], a feature that is often missing in other computer algebra systems.

Links:
* [[MAGMA Computational Algebra System Home Page|http://magma.maths.usyd.edu.au/]]
* [[MAGMA Online Calculator|http://magma.maths.usyd.edu.au/calc/]]
* [[WIKIPEDIA - MAGMA Computer Algebra System|http://en.wikipedia.org/wiki/Magma_computer_algebra_system]]
* [[Solving Problems with MAGMA - W. Bosma, J. Cannon, C. Playoust, A. Steel|http://www.dms.auburn.edu/research/manuals/magma2.6/examples.pdf]]  [[local|lectures/SolvingProblemsWithMAGMA.pdf]] [[lct. 10|http://scholar.google.com/scholar?hl=de&lr=&cites=9486123372688473527&um=1&ie=UTF-8&ei=YuE2S46dNKfesAbnzbHSCA&sa=X&oi=science_links&resnum=10&ct=sl-citedby&ved=0CDkQzgIwCTgK]]
* [[Handbook of MAGMA Functions|http://www.msri.org/about/computing/docs/magma/]] [[local|documents/MAGMA]]
** [[Lattices|http://www.msri.org/about/computing/docs/magma/html/text826.htm]] [[local|documents/MAGMA/html/text826.htm]]
** [[Coding Theory|http://www.msri.org/about/computing/docs/magma/html/part16.htm]] [[local|documents/MAGMA/html/part16.htm]]
** [[Hadamard Matrices|http://www.msri.org/about/computing/docs/magma/html/text1517.htm]] [[local|file:///E:/Trajectory/documents/MAGMA/html/text1517.htm]]
** [[Incidence Structures and Designs|http://www.msri.org/about/computing/docs/magma/html/text1502.htm]] [[local|file:///E:/Trajectory/documents/MAGMA/html/text1502.htm]]

Examples:
* [[Applied Abstract Algebra - D. Joyner, R. Kreminski, J. Turisco|http://www.usna.edu/Users/math/wdj/book/book.html]]

Links:
* [[Maxima website|http://maxima.sourceforge.net]]

<<tiddler [[include_tiddlers/MOND.html#"MOND"]]>>

The ''\MacWilliams Identity'' establishes a relationship between the [[weight enumerator|Weight Enumerator]] of a code $C$ and its [[dual code|Dual Code]] $C^\bot$. It is  given by
\[
W(C^\perp;x,y) = \frac{1}{\operatorname{ord}(C)} W(C;y-x,y+x).
\]

According to ''Mach's Principle'' inertial effects are due to the distribution of matter in the universe.

In case of a rotating bucket filled with water, Mach's principle implies that one could equally well maintain the bucket fixed and rotate all the universe around the bucket axis, obtaining the same result: water with parabolic shape.


Papers:
* [[A Rotating Vacuum and the Quantum Mach’s Principle (2000) - R. D. M. De Paola, N. F. Svaiter|http://arxiv.org/pdf/gr-qc/0009058v3]] [[local|papers/0009058v3.pdf]] [[pct. 4|http://scholar.google.de/scholar?hl=de&lr=&cites=14810738744211657987&um=1&ie=UTF-8&ei=SzB2TbPhGY7Bswbg95z9BA&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB8QzgIwAA]]
* [[Dark Energy, Inertia and Mach’s Principle (2009) - C. Sivaram, K. Arun|http://arxiv.org/ftp/arxiv/papers/0912/0912.3049.pdf]] [[local|papers/0912.3049.pdf]] pct. 0

<<tiddler [[include_tiddlers/Magma.html#"Magma"]]>>

<<tiddler [[include_tiddlers/Magnus Expansion.html#"Magnus Expansion"]]>>

<<tiddler [[include_tiddlers/Majorana Mass Matrix.html#"Majorana Mass Matrix"]]>>

<<tiddler [[include_tiddlers/Malcev Algebra.html#"Malcev Algebra"]]>>

<<tiddler [[include_tiddlers/Malcev Identity.html#"Malcev Identity"]]>>

According to the ''Margolus\-Levitin Theorem'' (1998) the maximum rate $\nu_{\max}$ at which logical operations can be performed by a physical system with energy $E$ is
\[
\nu_{\max} = \frac{2E}{\pi\hbar}
\]
In other words, the rate at which a computer can compute is limited by its energy. This limit is independent of computer architecture. This implies that a (general) speed up by parallelization is impossible.

The theorem is based on the correct interpretation of Heisenberg's time-energy uncertainty principle $\Delta E \Delta t \ge \hbar$ not that it takes time $\Delta t$ to measure energy to an accuracy $\Delta E$ (a fallacy that was put to rest by Aharonov and Bohm), but rather that a quantum state with spread in energy $\Delta E$ takes time at least $\Delta t = \frac {\pi\hbar} {2 \Delta E}$ to evolve to an orthogonal (and hence distinguishable) state.
Margolus and Levitin extended this result to show that a quantum system with average energy $E$ takes time at least $t = \frac {\pi\hbar} {2E}$ to evolve to an orthogonal state.

The Margolus\-Levitin theorem, based on quantum mechanics, provides an answer to the question of how fast information can be processed using a given amount of energy.

Beyond of it, thermodynamics and statistical mechanics provide a fundamental limit to how many bits of information can be processed using a given amount of energy confined to a given volume. (See also [[Bekenstein bound|Bekenstein Bound]]).

In practice the limit given by the Margolus\-Levitin theorem will hardly be attainable due to the requirement for error correction.

Links:
* [[WIKIPEDIA - Margolus-Levitin-Theorem|http://de.wikipedia.org/wiki/Margolus-Levitin-Theorem]]

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<<tiddler [[include_tiddlers/Markus' Wisdom.html#"Markus' Wisdom"]]>>

Links:
* [[Mars Images|http://mars.lyleresearch.com/imagery/color/]]

<<tiddler [[include_tiddlers/Mass Gap.html#"Mass Gap"]]>>

<<tiddler [[include_tiddlers/Mass Inflation.html#"Mass Inflation"]]>>

<<tiddler [[include_tiddlers/Mass Shell Condition.html#"Mass Shell Condition"]]>>

<<tiddler [[include_tiddlers/Massive Graviton.html#"Massive Graviton"]]>>

<<tiddler [[include_tiddlers/Massive Photon.html#"Massive Photon"]]>>

/***
|Name|MatchTagsPlugin|
|Source|http://www.TiddlyTools.com/#MatchTagsPlugin|
|Documentation|http://www.TiddlyTools.com/#MatchTagsPluginInfo|
|Version|2.0.3|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|'tag matching' with full boolean expressions (AND, OR, NOT, and nested parentheses)|
!!!!!Documentation
> see [[MatchTagsPluginInfo]]
!!!!!Revisions
<<<
2010.03.02 2.0.3 added %6 format (tags)
| please see [[MatchTagsPluginInfo]] for additional revision details |
2008.02.28 1.0.0 initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.MatchTagsPlugin= {major: 2, minor: 0, revision: 3, date: new Date(2010,3,2)};

// store.getMatchingTiddlers() processes boolean expressions for tag matching
//    sortfield (optional) sets sort order for tiddlers - default=title
//    tiddlers (optional) use alternative set of tiddlers (instead of current store)
TiddlyWiki.prototype.getMatchingTiddlers = function(tagexpr,sortfield,tiddlers) {

	var debug=config.options.chkDebug; // abbreviation
	var cmm=config.macros.matchTags; // abbreviation
	var r=[]; // results are an array of tiddlers
	var tids=tiddlers||store.getTiddlers(sortfield||"title");
	if (tiddlers && sortfield) store.sortTiddlers(tids,sortfield);
	if (debug) displayMessage(cmm.msg1.format([tids.length]));

	// try simple lookup to quickly find single tags or tags that
	// contain boolean operators as literals, e.g. "foo and bar"
	for (var t=0; t<tids.length; t++)
		if (tids[t].isTagged(tagexpr)) r.pushUnique(tids[t]);
	if (r.length) {
		if (debug) displayMessage(cmm.msg4.format([r.length,tagexpr]));
		return r;
	}

	// convert expression into javascript code with regexp tests,
	// so that "tag1 AND ( tag2 OR NOT tag3 )" becomes
	// "/\~tag1\~/.test(...) && ( /\~tag2\~/.test(...) || ! /\~tag3\~/.test(...) )"

	// normalize whitespace, tokenize operators, delimit with "~"
	var c=tagexpr.trim(); // remove leading/trailing spaces
	c = c.replace(/\s+/ig," "); // reduce multiple spaces to single spaces
	c = c.replace(/\(\s?/ig,"~(~"); // open parens
	c = c.replace(/\s?\)/ig,"~)~"); // close parens
	c = c.replace(/(\s|~)?&&(\s|~)?/ig,"~&&~"); // &&
	c = c.replace(/(\s|~)AND(\s|~)/ig,"~&&~"); // AND
	c = c.replace(/(\s|~)?\|\|(\s|~)?/ig,"~||~"); // ||
	c = c.replace(/(\s|~)OR(\s|~)/ig,"~||~"); // OR
	c = c.replace(/(\s|~)?!(\s|~)?/ig,"~!~"); // !
	c = c.replace(/(^|~|\s)NOT(\s|~)/ig,"~!~"); // NOT
	c = c.replace(/(^|~|\s)NOT~\(/ig,"~!~("); // NOT(
	// change tag terms to regexp tests
	var terms=c.split("~"); for (var i=0; i<terms.length; i++) { var t=terms[i];
		if (/(&&)|(\|\|)|[!\(\)]/.test(t) || t=="") continue; // skip operators/parens/spaces
		if (t==config.macros.matchTags.untaggedKeyword)
			terms[i]="tiddlertags=='~~'"; // 'untagged' tiddlers
		else
			terms[i]="/\\~"+t+"\\~/.test(tiddlertags)";
	}
	c=terms.join(" ");
	if (debug) { displayMessage(cmm.msg2.format([tagexpr])); displayMessage(cmm.msg3.format([c])); }

	// scan tiddlers for matches
	for (var t=0; t<tids.length; t++) {
	 	// assemble tags from tiddler into string "~tag1~tag2~tag3~"
		var tiddlertags = "~"+tids[t].tags.join("~")+"~";
		try { if(eval(c)) r.push(tids[t]); } // test tags
		catch(e) { // error in test
			displayMessage(cmm.msg2.format([tagexpr]));
			displayMessage(cmm.msg3.format([c]));
			displayMessage(e.toString());
			break; // skip remaining tiddlers
		}
	}
	if (debug) displayMessage(cmm.msg4.format([r.length,tagexpr]));
	return r;
}
//}}}
//{{{
config.macros.matchTags = {
	msg1: "scanning %0 input tiddlers",
	msg2: "looking for '%0'",
	msg3: "using expression: '%0'",
	msg4: "found %0 tiddlers matching '%1'",
	noMatch: "no matching tiddlers",
	untaggedKeyword: "-",
	untaggedLabel: "no tags",
	untaggedPrompt: "show tiddlers with no tags",
	defTiddler: "MatchingTiddlers",
	defTags: "",
	defFormat: "[[%0]]",
	defSeparator: "\n",
	reportHeading: "Found %0 tiddlers tagged with: '{{{%1}}}'\n----\n",
	handler: function(place,macroName,params,wikifier,paramString,tiddler) {
		var mode=params[0]?params[0].toLowerCase():'';
		if (mode=="inline")
			params.shift();
		if (mode=="report" || mode=="panel") {
			params.shift();
			var target=params.shift()||this.defTiddler;
		}
		if (mode=="popup") {
			params.shift();
			if (params[0]&&params[0].substr(0,6)=="label:") var label=params.shift().substr(6);
			if (params[0]&&params[0].substr(0,7)=="prompt:") var prompt=params.shift().substr(7);
		} else {
			var fmt=(params.shift()||this.defFormat).unescapeLineBreaks();
			var sep=(params.shift()||this.defSeparator).unescapeLineBreaks();
		}
		var sortBy="+title";
		if (params[0]&&params[0].substr(0,5)=="sort:") sortBy=params.shift().substr(5);
		var expr = params.join(" ");
		if (mode!="panel" && (!expr||!expr.trim().length)) return;
		if (expr==this.untaggedKeyword)
			{ var label=this.untaggedLabel; var prompt=this.untaggedPrompt };
		switch (mode) {
			case "popup": this.createPopup(place,label,expr,prompt,sortBy); break;
			case "panel": this.createPanel(place,expr,fmt,sep,sortBy,target); break;
			case "report": this.createReport(target,this.defTags,expr,fmt,sep,sortBy); break;
			case "inline": default: this.createInline(place,expr,fmt,sep,sortBy); break;
		}
	},
	formatList: function(tids,fmt,sep) {
		var out=[];
		for (var i=0; i<tids.length; i++) { var t=tids[i];
			var title=t.title;
			var who=t.modifier;
			var when=t.modified.toLocaleString();
			var text=t.text;
			var first=t.text.split("\n")[0];
			var desc=store.getTiddlerSlice(t.title,"description");
			desc=desc||store.getTiddlerSlice(t.title,"Description");
			desc=desc||store.getTiddlerText(t.title+"##description");
			desc=desc||store.getTiddlerText(t.title+"##Description");
			var tags=t.tags.length?'[['+t.tags.join(']] [[')+']]':'';
			out.push(fmt.format([title,who,when,text,first,desc,tags]));
		}
		return out.join(sep);
	},
	createInline: function(place,expr,fmt,sep,sortBy) {
		wikify(this.formatList(store.sortTiddlers(store.getMatchingTiddlers(expr),sortBy),fmt,sep),place);
	},
	createPopup: function(place,label,expr,prompt,sortBy) {
		var btn=createTiddlyButton(place,
			(label||expr).format([expr]),
			(prompt||config.views.wikified.tag.tooltip).format([expr]),
			function(ev){ return config.macros.matchTags.showPopup(this,ev||window.event); });
		btn.setAttribute("sortBy",sortBy);
		btn.setAttribute("expr",expr);
	},
	showPopup: function(here,ev) {
		var p=Popup.create(here); if (!p) return false;
		var tids=store.getMatchingTiddlers(here.getAttribute("expr"));
		store.sortTiddlers(tids,here.getAttribute("sortBy"));
		var list=[]; for (var t=0; t<tids.length; t++) list.push(tids[t].title);
		if (!list.length) createTiddlyText(p,this.noMatch);
		else {
			var b=createTiddlyButton(createTiddlyElement(p,"li"),
				config.views.wikified.tag.openAllText,
				config.views.wikified.tag.openAllTooltip,
				function() {
					var list=this.getAttribute("list").readBracketedList();
					story.displayTiddlers(null,tids);
				});
			b.setAttribute("list","[["+list.join("]] [[")+"]]");
			createTiddlyElement(p,"hr");
		}
		var out=this.formatList(tids," &nbsp;[[%0]]&nbsp; ","\n"); wikify(out,p);
		Popup.show();
		ev.cancelBubble=true;
		if(ev.stopPropagation) ev.stopPropagation();
		return false;
	},
	createReport: function(target,tags,expr,fmt,sep,sortBy) {
		var tids=store.sortTiddlers(store.getMatchingTiddlers(expr),sortBy);
		if (!tids.length) { displayMessage('no matches for: '+expr); return false; }
		var msg=config.messages.overwriteWarning.format([target]);
		if (store.tiddlerExists(target) && !confirm(msg)) return false;
		var out=this.reportHeading.format([tids.length,expr])
		out+=this.formatList(tids,fmt,sep);
		store.saveTiddler(target,target,out,config.options.txtUserName,new Date(),tags,{});
		story.closeTiddler(target); story.displayTiddler(null,target);
	},
	createPanel: function(place,expr,fmt,sep,sortBy,tid) {
		var s=createTiddlyElement(place,"span"); s.innerHTML=store.getTiddlerText("MatchTagsPlugin##html");
		var f=s.getElementsByTagName("form")[0];
		f.expr.value=expr; f.fmt.value=fmt; f.sep.value=sep.escapeLineBreaks();
		f.tid.value=tid; f.tags.value=this.defTags;
	}
};
//}}}
/***
//{{{
!html
<form style='display:inline;white-space:nowrap'>
<input type='text'    name='expr' style='width:50%' title='tag expression'><!--
--><input type='text'    name='fmt'  style='width:10%' title='list item format'><!--
--><input type='text'    name='sep'  style='width:5%'  title='list item separator'><!--
--><input type='text'    name='tid'  style='width:12%' title='target tiddler title'><!--
--><input type='text'    name='tags' style='width:10%' title='target tiddler tags'><!--
--><input type='button'  name='go'   style='width:8%'  value='go' onclick="
	var expr=this.form.expr.value;
	if (!expr.length) { alert('Enter a boolean tag expression'); return false; }
	var fmt=this.form.fmt.value;
	if (!fmt.length) { alert('Enter the list item output format'); return false; }
	var sep=this.form.sep.value.unescapeLineBreaks();
	var tid=this.form.tid.value;
	if (!tid.length) { alert('Enter a target tiddler title'); return false; }
	var tags=this.form.tags.value;
	config.macros.matchTags.createReport(tid,tags,expr,fmt,sep,'title');
	return false;">
</form>
!end
//}}}
***/
//{{{
// SHADOW TIDDLER for displaying default panel input form
config.shadowTiddlers.MatchTags="<<matchTags panel>>";
//}}}
//{{{
// TWEAK core filterTiddlers() for enhanced boolean matching in [tag[...]] syntax:
// use getMatchingTiddlers instead getTaggedTiddlers
var fn=TiddlyWiki.prototype.filterTiddlers;
fn=fn.toString().replace(/getTaggedTiddlers/g,"getMatchingTiddlers");
eval("TiddlyWiki.prototype.filterTiddlers="+fn);
//}}}
//{{{
// REDEFINE core handler for enhanced boolean matching in tag:"..." paramifier
// use filterTiddlers() instead of getTaggedTiddlers() to get list of tiddlers.
config.paramifiers.tag = {
	onstart: function(v) {
		var tagged = store.filterTiddlers("[tag["+v+"]]");
		story.displayTiddlers(null,tagged,null,false,null);
	}
};
//}}}

<<tiddler [[include_tiddlers/MathJax.html#"MathJax"]]>>

$A_B$ now renders correctly, so does $A_{B}$.
Similarly, $ABc$, $AB c$, $A Bc$, $aBCd$, $ABC$ work now.
!The Lorenz Equations
\[\begin{eqnarray}
\dot{x} & = & \sigma(y-x) \\
\dot{y} & = & \rho x - y - xz \\
\dot{z} & = & -\beta z + xy
\end{eqnarray}\]
!The ~Cauchy-Schwarz Inequality
\[ \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2
\right) \left( \sum_{k=1}^n b_k^2 \right) \]
!A Cross Product Formula
\[\mathbf{V}_1 \times \mathbf{V}_2 =  \left|\begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial X}{\partial u} &  \frac{\partial Y}{\partial u} & 0 \\
            \frac{\partial X}{\partial v} &  \frac{\partial Y}{\partial v}
& 0
\end{array}\right|  \]
!The probability of getting \(k\) heads when flipping \(n\) coins is:
\[P(E)   = {n \choose k} p^k (1-p)^{ n-k} \]
!An Identity of Ramanujan
\[ \frac{1}{\left(\sqrt{\phi \sqrt{5}}-\phi\right) e^{\frac25 \pi}} =
1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}
{1+\frac{e^{-8\pi}} {1+\ldots} } } } \]
!A ~Rogers-Ramanujan Identity
\[  1 +  \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =
\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},
\quad\quad \textrm{for}\quad |q|<1. \]
!Maxwell's Equations
\[\begin{eqnarray}
\nabla \times \vec{\mathbf{B}} -\, \frac1c\,\frac{\partial\vec{\mathbf{E}}}{\partial t} & = & \frac{4\pi}{c}\vec{\mathbf{j}} \\
\nabla \cdot \vec{\mathbf{E}} & = & 4 \pi \rho \\
\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\,\frac{\partial\vec{\mathbf{B}}}{\partial t} & = &\vec{\mathbf{0}} \\
\nabla \cdot \vec{\mathbf{B}} & = & 0
\end{eqnarray}\]
Finally, while display equations look good for a page of samples, the ability to mix math and text
in a paragraph is also important. This expression \(\sqrt{3x-1}+(1+x)^2\) is an example of an
inline equation.  As you see, ~MathJax equations can be used this way as well, without unduly
disturbing the spacing between lines.

A ''([[Polyvector|Polyvector Space]]) Vielbein'' (or ''Geobein'') \(H_\alpha^A(\mathbf X)\) generalizes the concept of a [[tetrad|Tetrad]] and is defined by:
\[
\mathbf E_\alpha(\mathbf X)  \equiv H_\alpha^A(\mathbf X) \mathbf E_A (\mathbf X)
\]
or in Dirac notation
\[
|\mathbf E_\alpha(\mathbf X) \rangle  = H_\alpha^A(\mathbf X) | \mathbf E_A (\mathbf X) \rangle
\]
It depends on the polyvector event $\Sigma$ in the manifold ($\mathbf X = \mathbf X(\Sigma)$) and establishes the relationship between a local orthonormal frame $\{\mathbf E_A(\mathbf X) \}$ (fiducial frame) and a non-orthonormal frame \(\{ \mathbf E_\alpha(\mathbf X) \}\). For the orthonormal frame one has $ \langle \mathbf E_A(\mathbf X) | \mathbf E_B (\mathbf (X)\rangle \equiv N_{AB} $ with $N_{AB}$ a diagonal matrix (the pseudo orthonormal metric) with "+" and "-" entries depending on the signature of the underlying algebra (i.e. the polyvectorial generalization of the $\eta_{ab}$ of special relativity).

Resolving for the vielbeins one gets:
\[
H_\alpha^A (\mathbf X)  = \frac{ \langle \mathbf E_B (\mathbf X)| \mathbf E_\alpha (\mathbf X) \rangle}{\langle \mathbf E_A (\mathbf X)|\mathbf E_B(\mathbf X) \rangle } = \frac{ \langle \mathbf{E}_B(\mathbf X) |\mathbf E_\alpha (\mathbf X) \rangle}{N_{AB}}
\]
!!!Properties
To keep the notation compact, we'll sometimes suppress the argument $\mathbf X$ in the following.

''Inverse:''
The ''Inverse Vielbeins'' \( (H_\alpha^A)^{-1}\) are defined via
\[
\mathbf{E}_A = (H_\alpha^A)^{-1} \mathbf E_\alpha \equiv \bar H_A^\alpha \mathbf E_\alpha
\]
One has
\[
\mathbf E_A = \bar H_A^\alpha  H_\alpha^B \mathbf E_B
\]
and
\[
\langle \mathbf E_A | \mathbf E_A \rangle  =  \bar H_A^\alpha (\mathbf X) H_\alpha^B \langle \mathbf{E}_A | \mathbf{E}_B \rangle \delta^{AB} = (H_\alpha^A)^{-1} H_\alpha^A  \langle \mathbf E_A | \mathbf{E}_B \rangle \delta^{AB}
\]
thus
\[
(H_\alpha^A)^{-1} H_\alpha^B = \delta_{AB}
\]

The other way around we get
\[
\mathbf E_\alpha = H_\alpha^A \mathbf E_A = H_\alpha^A  H^\beta_A \mathbf E_\beta = \bar H_A^\alpha H^\beta_A \mathbf E_\beta
\]

''Metric tensor:''
\begin{eqnarray}
G_{\alpha\beta} & = &\langle \mathbf E_\alpha|\mathbf E_\beta \rangle  = H_\alpha^A H_\beta^B \langle \mathbf E_A | \mathbf E_B \rangle \\
&= &H_\alpha^A(\mathbf X) H_\beta^B (\mathbf X) N_{AB} = H_\alpha^A(\mathbf X) H_\beta^A N_A
\end{eqnarray}

''Coordinate change:''
Given a polyvectorial coordinate transformation $ \{ X^\alpha \} \rightarrow \{X^A\} \equiv \mathbf X' (\mathbf X)$,
\[
\partial_\alpha \mathbf E_A(\mathbf X) = \frac{\partial X^B} {\partial X^\alpha} \partial_B \mathbf E_A(\mathbf X')
\]
therefore
\[
H^B_\alpha =\frac{\partial X^B} {\partial X^\alpha}
\]
!!!!''[[Polycector Jacobi Matrix|Polyvector Jacobi Matrix]]:''
Given a general polyvectorial coordinate transformation $ \{ X^\alpha \} \rightarrow \{X^\beta\} \equiv \mathbf X' (\mathbf X) $, the associated polyvector Jacobi matrix $J_{\mathbf X} (\mathbf X')_{\alpha\beta}$ is given by
\[
J_{\mathbf X} (\mathbf X')_{\alpha\beta} = \frac{\partial X^\beta} {\partial X^\alpha}
\]
In case that the transformation is related to an orthonormal polyvector frame (see "Coordinate change:"), $J$ coincides with the vielbeins, thus
\[
J_{\mathbf X} (\mathbf X')_{\alpha B} = H^B_\alpha
\]
If one only considers the vector part of the polyvector, this amounts to
\[
J_{\mathbf x} (\mathbf x')_{\alpha b} = h^b_\alpha
\]
which reproduces the relationship between the conventional [[Jacobi matrix|Jacobi Matrix]] and the [[tetrads|Tetrad]].

''[[Structure constants|Structure Constants]]:''
\[
\mathbf{E}_\alpha \mathbf E_\beta = H_\alpha^A H_\beta^B \mathbf E_A \mathbf E_B = H_\alpha^A  H_\beta^B C_{AB}^D \mathbf E_D
\]
or in Dirac notation
\[
|\mathbf E_\alpha \rangle | \mathbf E_\beta \rangle = H_\alpha^A  H_\beta^B | \mathbf E_A  \rangle | \mathbf E_B  \rangle = H_\alpha^A  H_\beta^B  C_{AB}^D | \mathbf{E}_D \rangle
\]

''[[Total differential|Polyvector Total Differential]]:''
\begin{eqnarray}
D \mathbf E_\alpha = D (H_\alpha^A (\mathbf X) \mathbf E_A (\mathbf X)) & = &(d H_\alpha^A(\mathbf X)) \mathbf E_A (\mathbf X) + H_\alpha^A(\mathbf X) D\mathbf E_A (\mathbf X)  \\
&= & \frac{\partial H_\alpha^C(\mathbf X)}{\partial X_\beta} dX^\beta \mathbf E_C (\mathbf X) + H_\alpha^B(\mathbf X) dX^\beta \frac{\partial X^A} {\partial X^\beta}  \frac{\partial \mathbf E_B (\mathbf X) }{\partial X_A} \\
&= & \left ( \frac{\partial H_\alpha^C(\mathbf X)}{\partial X_\beta} + H_\alpha^B(\mathbf X)  H_\beta^A(\mathbf X) C_{AB}^C \right ) \mathbf E_C (\mathbf X) dX^\beta \\
\end{eqnarray}
On the other hand, using the definition of the [[polyvector total differential|Polyvector Total Differential]], we have
\[
D\mathbf E_\alpha(\mathbf X) = \frac{\partial \mathbf E_\alpha (\mathbf X)}{\partial X_\beta} dX^\beta = \Gamma^\gamma_{\alpha\beta}(\mathbf X) H_\gamma^C(\mathbf X) \mathbf E_C dX^\beta
\]
therefore
\[
 \Gamma^\gamma_{\alpha\beta}(\mathbf X) H_\gamma^C(\mathbf X)  = \frac{\partial H_\alpha^C(\mathbf X)}{\partial X_\beta} + H_\alpha^B(\mathbf X)  H_\beta^A(\mathbf X) C_{AB}^C
\]
and the [[polyvector connection coefficients|Polyvector Connection Coefficients]] can be expressed in terms of the vielbeins according to
\[
\Gamma^\gamma_{\alpha\beta}(\mathbf X) =\left ( \frac{\partial H_\alpha^C(\mathbf X)}{\partial X_\beta} + H_\alpha^B(\mathbf X)  H_\beta^A(\mathbf X) C_{AB}^C   \right )  H_\gamma^C(\mathbf X)^{-1}
\]
Thus the polyvector connection coefficients are completely determined by the vielbeins and the structure constants.

''Partial derivative:''
Using the results obtained for the polyvector total differential, the polyvector partial derivative can be written as
\[
\frac{\partial \mathbf E_\alpha(\mathbf X)} {\partial X_\beta} = \left ( \frac{\partial H_\alpha^C(\mathbf X)}{\partial X_\beta} + H_\alpha^B(\mathbf X)  H_\beta^A(\mathbf X) C_{AB}^C \right ) H_\gamma^C(\mathbf X)^{-1}  \mathbf E_\gamma (\mathbf X)
\]

!!!!Possible physical relevance
Lifting the tetrads $h^\mu_a$ to the vielbeins $H^A_\alpha$ in polyvector space makes the weakness of the symmetry of spacetime, namely [[diffeomorphism invariance|Diffeomorphism]], even worse. On the other hand the $H^A_\alpha$ now have the potential to code the gauge degrees of freedom of all the forces of nature. These correspond with the degrees of freedom to do transformations between equivalent [[multiplication tables|Multiplication Tables]]. For associative algebras these are [[automorphisms|Automorphism]] only.

Yet polyvector geometry allows for nonassociativity, which can restrict automorphisms considerably. (E.g. for the [[octonions|Octonion]] from [[SO(7)]] to [[G2]]). However, a further kind of symmetry comes into play, [[autotopism invariance|Autotopism]]. Contrary to automorphisms which mix the basis elements in the multiplication table, autotopisms permute the elements of an "orthonormal" multiplication table (i.e. they change the [["handedness"|Chirality]]). In case of the octonions one has [[480 such transformations|480 Octonion Multiplication Tables]]. One could regard them as "discrete gauge transformations" (see [[X-product|X-Product]]). The classical gauge fields should correspond with the automorphisms (what does this mean in case of gravity ?), i.e. transformations to a canonical frame (polyvector reference frame). Autotopisms on the other hand are "reflections" in polyvector space ([[CPT transformations|CPT Transformations]] being special cases thereof).

Algebraic products thus are constrained such that they can to justice to the $U(1)$, [[SU(2)]] and [[SU(3)]] gauge symmetries of the [[standard model|Standard Model]]. (It is tempting to assume that $G_2$ will also play an important role in this respect).

/***
|''Name''|MathJaxTiddlyPlugin|
|''Description''|This plugin enables Tiddlywiki to use [[MathJax|http://www.mathjax.org/]] to render mathematical expressions.|
|''Author''|[[Lizao Li|http://math.umn.edu/~lixx1445]]|
|''Version''|1.51|
|''Created''|Jan 23, 2011|
|''Last updated'' |Oct 12, 2011|
|''Status''|stable|
|''Source''|http://math.umn.edu/~lixx1445|
|''Copyright''|[[GPLv3|http://www.gnu.org/licenses/gpl-3.0.html]]|
|''CoreVersion''|2.6.1|
|''Feedback''|[[creatorlarryli@gmail.com|mailto:creatorlarryli@gmail.com]]|
|''Keywords''|MathJax, Tiddlywiki, MathJaxTiddlyPlugin, ~LaTeX|
!Installation
Backstage (top-right corner) -> import -> use the URL of this page -> choose "MathJaxTiddlyPlugin" -> import
!Configuration
* <<option chkMathJaxUseCustomizedSettings>> Enable the dollar sign for inline math.
* Configuration file to use <<option txtMathJaxConfigurationFile>>.
** There are 4 possible choices: @@TeX-AMS-MML_HTMLorMML@@ (default), @@TeX-AMS_HTML@@, @@MML_HTMLorMML@@, @@Accessible@@
** You can find details about these settings from [[MathJax website|http://www.mathjax.org/docs/1.1/config-files.html#common-configurations]].
* Use MathJax to render tiddlers with tag <<option txtMathJaxTag>>.
* MathJax url to use: <<option txtMathJaxURL>>.
''A note for authors:'' The above settings are actually for readers of the webpage, which are only saved in the local cookies. To save these settings so that all the readers will see the same, authors should set these settings persistently. You can add the following to the system tiddler [[SystemSettings]]:
{{{
chkMathJaxUseCustomizedSettings: false
txtMathJaxConfigurationFile: TeX-AMS-MML_HTMLorMML
txtMathJaxTag: math
txtMathJaxURL: http://cdn.mathjax.org/mathjax/latest/MathJax.js
}}}
More about persistent settings can be found on the [[official website|http://tiddlywiki.com/#PersistentOptions]].
!Usage
Tag the tiddlers need math rendering with "math" or the name set in the configuration.
!Examples
{{{
*Inline math test
**syntax 1: $\int_1^4 \sqrt{1+\sin(x)}\,dx$
**syntax 2: \(\alpha^{\beta^\gamma}\)
*Display-style math test:
**syntax 1:$$\frac{\partial v}{\partial t}+(v\cdot\nabla)v=-\nabla p+v\Delta v+f(x,t)$$
**syntax 2:\[\int_{\mathbb{R}}\frac{f(y)}{|x-y|^{3/2}}\,dy\]
}}}
The result should look like this:
*Inline math test
**syntax 1: $\int_1^4 \sqrt{1+\sin(x)}\,dx$
**syntax 2: \(\alpha^{\beta^\gamma}\)
*Display-style math test:
**syntax 1:$$\frac{\partial v}{\partial t}+(v\cdot\nabla)v=-\nabla p+v\Delta v+f(x,t)$$
**syntax 2:\[\int_{\mathbb{R}}\frac{f(y)}{|x-y|^{3/2}}\,dy\]
* There is also a more complicated [[demo|MathJaxTiddlyPluginDemo]]
!Revision History
* v1.51 (2011-10-12) - By Tim Bradshaw. An option to choose which MathJax to load was added. Bug fixed: Test whether the tiddler is null before refreshing it.
* v1.50 (2011-04-18) - Bugs fixed. Thanks to James Montaldi to point out that WikiTexts inside the math blocks are wrongly rendered. The fix was inspired by [[Bob McElrath's jsMath|http://bob.mcelrath.org/tiddlyjsmath-2.0.3.html]] and [[Martin Budden's FormatterPlugin|http://www.martinswiki.com/]].
* v1.15 (2011-03-31) - Added configurations. Now renders math according to the tag (Thanks to Patrick Ion for the suggestions).
* v1.1 (2011-03-17) - Updated after MathJax was updated to 1.1. Now MathJax is loading from the CDN, instead of the local server.
* v1.0 (2011-01-23) - Created.
!Code
***/
//{{{
if(!version.extensions.MathJaxPlugin) { // [ensure that the plugin is only installed once]
    version.extensions.MathJaxPlugin = { installed: true };

    if(config.options.chkMathJaxUseCustomizedSettings == undefined) config.options.chkMathJaxUseCustomizedSettings = false;
    if(config.options.txtMathJaxConfigurationFile == undefined) config.options.txtMathJaxConfigurationFile = 'TeX-AMS-MML_HTMLorMML';
    if(config.options.txtMathJaxTag == undefined) config.options.txtMathJaxTag = 'math';
    if(config.options.txtMathJaxURL == undefined) config.options.txtMathJaxURL = 'http://cdn.mathjax.org/mathjax/latest/MathJax.js';


    // [hijack refreshTiddler() to render mathjax]
    (function() {
	// [backup the original function]
	Story.prototype.oldrefreshfunction = Story.prototype.refreshTiddler;
	// [override the original function]
	Story.prototype.refreshTiddler = function(title,template,force,customFields,defaultText) {
	    Story.prototype.oldrefreshfunction.apply(this,arguments);
	    if(tiddler && window.MathJax) {
		for(var i=0;i<tiddler.tags.length;i++){
		    if(tiddler.tags[i] == config.options.txtMathJaxTag){
			window.MathJax.Hub.Queue(["Typeset",window.MathJax.Hub]);
		    }
		}
	    }
	}
    })();

    (function () {
	// [add the mathjax settings to the header]
	var mathjax_config = document.createElement("script");
	mathjax_config.type = "text/x-mathjax-config";
	var content_mathjax_config = "";

	// [enable the dollar sign for inline math rendering]
	if(config.options.chkMathJaxUseCustomizedSettings == true){
	    content_mathjax_config = content_mathjax_config +
		"MathJax.Hub.Config({'HTML-CSS': {webFont: 'TeX', scale: 74},TeX: {Macros: {bs: ['{\\\\overline #1}', 1], mb: ['{\\\\overline #1}', 1]}},tex2jax: { inlineMath: [ ['$','$'], ['\\\\(','\\\\)'] ], displayMath: [ ['$$','$$'], ['\\\\[','\\\\]'] ], processEscapes: true }});";
	}

	content_mathjax_config = content_mathjax_config + 'MathJax.Hub.Startup.onload();';

	if (window.opera) {mathjax_config.innerHTML = content_mathjax_config}
	else {mathjax_config.text = content_mathjax_config}

	document.getElementsByTagName("head")[0].appendChild(mathjax_config);

	//# Add the mathjax main script to the header
	var script = document.createElement("script");
	script.type = "text/javascript";
	script.src = config.options.txtMathJaxURL+"?config="+config.options.txtMathJaxConfigurationFile;
	document.getElementsByTagName("head")[0].appendChild(script);
    })();

    config.extensions.MathJaxPlugin = {
    };

    config.MathJax = {};

    // Render matched text as plain text without wikification (adapted from Bob McElrath's jsMath and Martin Budden's FormatterPlugin)
    config.MathJax.OutputPlainText = function(w) {
	createTiddlyElement(w.output);
	var e = document.createElement(this.element);
	var endRegExp = new RegExp(this.termRegExp, "mg");
	endRegExp.lastIndex = w.matchStart+w.matchLength;
	var matched = endRegExp.exec(w.source);
	if(matched) {
	    var txt = w.source.substr(w.matchStart, matched.index+matched[0].length-w.matchStart);
	    if(this.keepdelim) {
		txt = w.source.substr(w.matchStart, matched.index+matched[0].length-w.matchStart);
	    }
	    e.appendChild(document.createTextNode(txt));
	    w.output.appendChild(e);
	    w.nextMatch = endRegExp.lastIndex;
	}
    }

    config.MathJax.formatters = config.formatters;

    // Match the math blocks
    config.MathJax.formatters.push({
	    name: "mathjaxDisplay1",
		match: "\\\$\\\$",
		termRegExp: "\\\$\\\$",
		handler: config.MathJax.OutputPlainText
		});

    config.MathJax.formatters.push({
	    name: "mathjaxDisplay2",
		match: "\\\\\\\[",
		termRegExp: "\\\\\\\]",
		handler: config.MathJax.OutputPlainText
		});

    config.MathJax.formatters.push({
	    name: "mathjaxInline1",
		match: "\\\$",
		termRegExp: "\\\$",
		handler: config.MathJax.OutputPlainText
		});

    config.MathJax.formatters.push({
	    name: "mathjaxInline2",
		match: "\\\\\\\(",
		termRegExp: "\\\\\\\)",
		handler: config.MathJax.OutputPlainText
		});

    config.parsers.MathJaxFormatter = new Formatter(config.MathJax.formatters);
    config.parsers.MathJaxFormatter.format = 'MathJax';
    config.parsers.MathJaxFormatter.formatTag = config.options.txtMathJaxTag;

}//# end of "install only once"

//}}}

Links:
* [[INTERNATIONAL CONFERENCE OF MATHEMATICIANS MADRID 2006|http://www.dmmm.uniroma1.it/~prastaro/POSTER_ICM2006_APDEG.html]] - Tags of mathematicians.
* [[FamousWhy|http://www.famouswhy.com/Tags/mathematician]] - Dto.

<<tiddler [[include_tiddlers/Mathieu Group.html#"Mathieu Group"]]>>

<<tiddler [[include_tiddlers/Mathisson-Papapetrou Equations.html#"Mathisson-Papapetrou Equations"]]>>

Links:
* [[The Matrix Reference Manual|http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html#Intro]]

<<tiddler [[include_tiddlers/Maximal Torus.html#"Maximal Torus"]]>>

According to Julian Schwinger [1] the appropriate algebra for a quantum measurement is constrained as follows:

"We define the addition of such symbols to signify less specific selective measurements that produce a subensemble associated with any of the values in the summation, none of these being distinguished by the measurement. The multiplication of the measurement symbols represents the successive performance of measurements (read from right to left). It follows from the physical meaning of these operations that __addition is commutative and associative__, while __multiplication is associative__."

Furthermore he states:
"But a probability is a real, nonnegative number. Hence __we shall impose an admissible restriction__ on the numbers appearing in the measurement algebra, by requiring that $\langle a'| b' \rangle$ and $\langle b' |a' \rangle$ form a pair of __complex__ conjugate __numbers__".

This is not in general accordance with the opinion of others.
Pascual Jordan and John von Neumann for example consider weaker algebraic constraints [2], leading to [[Jordan algebras|Jordan Algebra]] as the appropriate algebras for quantum measurements.
J. v. Neumann states:
"Addition ($a+b$) is __commutative__ and __associative__. Jordan pointed out that a "quasi"-multiplication $a \circ b$ can be defined. ... $a \circ b$ is obviously __commutative__, but [[not necessarily associative|Nonassociative Algebra]]".
He proceeds:
"On the other hand, an algebraic discussion will be scarcely possible, if the [[distributive law|Nondistributive Algebra]] does not hold for $a \circ b$. ...
We require distributivity merely on the basis of its truth in the present system of quantum mechanics, and its algebraic rôle in connection with the distributive law. It seems to be one of the essential features of quantum theory, although __its true (phenomenological) meaning is obscure__".

Papers:
* [[[1] The Algebra of Microscopic Measurement - J. Schwinger|http://www.ncbi.nlm.nih.gov/pmc/articles/PMC222753/]] [[local|papers/pnas00197-0092.pdf]] [[pct. 18|http://scholar.google.de/scholar?cites=9329205194451544274&hl=de&as_sdt=2000]]
* [[[2] On an Algebraic Generalization of the Quantum Mechanical Formalism. I. (1936) - J. v. Neumann|http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN477674380_0043&DMDID=DMDLOG_0045]] [[local|papers/AlgebraicGeneralisationOfQMI.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=8458537790388091765&as_sdt=2005&sciodt=2000&hl=de]]
* [[Über eine Klasse Nichtassoziativer Hyperkomplexer Algebren - P. Jordan|http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=64263]] pct. 0

Abstracts:
* [[Über die Multiplikation Quantenmechanischer Größen - P. Jordan|http://www.springerlink.com/content/v0th72087k872828/]] [[pct. 32|http://scholar.google.de/scholar?cites=9336359917466676733&as_sdt=2005&sciodt=2000&hl=de]]

<<tiddler [[include_tiddlers/Measurement Induced Mass Generation.html#"Measurement Induced Mass Generation"]]>>

A ''Menon Design'' is a $2-(4u^2, 2u^2 \pm u, u^2 \pm u)$ [[design|Balanced Incomplete Block Design]].

In [[complex analysis|Complex Analysis]] a ''Meromorphic Function'' on an open subset $D$ of the complex plane is a function that is [[holomorphic|Holomorphic Function]] on all $D$ except a set of isolated points, which are poles for the function.

Every meromorphic function on $D$ can be expressed as the __ratio between two holomorphic functions__ (with the denominator not constant $0$) defined on $D$: any pole must coincide with a zero of the denominator.

Some examples:
* Elliptic functions,
* Gamma functions,
* Hurwitz's zeta\-functions,
* Modular forms,
* Riemann's $\zeta$-functions,
* Special functions.

Important theorems about meromorphic functions:
* Theorem of Mittag\-Leffler ([[the mathematician who allegedly is responsible for mathematicians not getting a nobel prize|http://www.fields.utoronto.ca/aboutus/jcfields/fieldsnobel.html]]),
* Residue theorem,
* Theorem of Riemann\-Roch.

Links:
* [[WIKIPEDIA - Meromorphic Function|http://en.wikipedia.org/wiki/Meromorphic_function]]

A ''Mersenne Prime'' $M_n$ is a prime number of the form
\[
 M_n = 2^n - 1
\]
with $n \in \mathbb N\,$. It can be shown that $M_n$ can only be prime if $n$ is prime.
Currently only $47$ Mersenne primes are known. The sequence goes like this: $7,31, 127, 8.191, 131.071, 524.287, 2.147.483.647, \ldots$.

Mersenne primes are related to [[perfect numbers|Perfect Number]]. In both cases it is still unknown if there exist infinitely many such numbers.

Links:
* [[WIKIPEDIA - Mersenne Primes|http://en.wikipedia.org/wiki/Mersenne_prime]]

<<tiddler [[include_tiddlers/Metamaterial.html#"Metamaterial"]]>>

<<tiddler [[include_tiddlers/Metric Affine Gravity.html#"Metric Affine Gravity"]]>>

<<tiddler [[include_tiddlers/Metric Affine Space.html#"Metric Affine Space"]]>>

<<tiddler [[include_tiddlers/Metric Compatibility.html#"Metric Compatibility"]]>>

<<tiddler [[include_tiddlers/Metric Tensor.html#"Metric Tensor"]]>>

<<tiddler [[include_tiddlers/Michael Atiyah.html#"Michael Atiyah"]]>>

<<tiddler [[include_tiddlers/Microtubule.html#"Microtubule"]]>>

<<tiddler [[include_tiddlers/Minimal Coupling.html#"Minimal Coupling"]]>>

<<tiddler [[include_tiddlers/Minimal Mass.html#"Minimal Mass"]]>>

<<tiddler [[include_tiddlers/Minisuperspace Model.html#"Minisuperspace Model"]]>>

<<tiddler [[include_tiddlers/Miracle Ocatad Generator (MOG).html#"Miracle Ocatad Generator (MOG)"]]>>

The concept of a ''Module'' is a generalization of the notion of a vector space. The difference is that a module is defined over a ring, whereas a vector space is defined over a field.

A module, like a vector space, is an additive [[abelian group|Group]]; a product is defined between elements of the ring and elements of the module, and this multiplication is associative and distributive.

A [[loop|Loop]] is called ''monoassociative'' (or ''3-''[[power-associative|Power Associative Algebra]], short ''3PA'')  if each of its elements generates an associative subloop.
This implies that the [[associators|Associator]] of its local algebras satisfy the condition
\[
[\mb A, \mb A, \mb A] = 0
\]

A [[quasigroup|Quasigroup]] is said to be local if all its local loops are monoassociative.
It follows that any tangent vector $\bs \zeta$ of such a local loop satisfies
\begin{eqnarray}
[\bs \zeta, \bs \zeta, \bs \zeta] = \zeta^\mu \zeta^\nu \zeta^\rho [\bs e_\mu, \bs e_\nu, \bs e_\rho] = \zeta^\mu \zeta^\nu \zeta^\rho A_{\mu\nu\rho}^\sigma \mb e_\sigma= 0
\end{eqnarray}
As
\[
\zeta^\mu \zeta^\nu \zeta^\rho A_{\mu\nu\rho}^\sigma \mb e_\sigma = 0 = \zeta^\mu \zeta^\nu \zeta^\rho A_{(\mu\nu\rho)}^\sigma \mb e_\sigma
\]
the monoassociativity condition in tensorial form reads
\[
A_{(\mu\nu\rho)}^\sigma = 0
\]

Monoassociativity is a weaker condition than is [[power-associativity|Power Associative Algebra]].

Monoassociativity is akin to [[alternativity|Alternative Algebra]] in that in the former case the associator vanishes if all three arguments are identical, whereas in the latter case the associator vanishes if two arguments are identical.
Thus monoassociativity is also weaker than is alternativity.

!!!!Example
Any [[flexible algebra|Flexible Algebra]] is monoassociative, as it satisfies $[\mb A,\mb B,\mb A] = 0$.

<<tiddler [[include_tiddlers/Monster Group.html#"Monster Group"]]>>

In 1978/79, J. \McKay, J. Thompson, [[J. Conway|John Conway]] and S. Norton had discovered astounding "numerology" culminating in the ''Monstrous Moonshine'' conjectures relating the not-yet-proved-to-exist [[Monster group|Monster Group]] to modular functions in number theory, namely:
There should exist a (natural) infinite-dimensional $\mathbb{Z}$-graded module for $M$ (i.e., representation of $M$)
\[
V=\bigoplus_{n=-1,0,1,2,3,\dots}V_n
\]
such that
\[
\sum_{n=-1,0,1,2,3,\dots}({\rm dim}V_n)q^n=J(q)
\]
where
\[
J(q)=q^{-1}+0+196884q+{\mbox {higher-order terms}}
\]
is the classical modular function with its constant term set to $0$.

Videos:
* [[What is Moonshine? - R. Borcherds|http://www.newton.ac.uk/webseminars/vault/borcherds/]]

<<tiddler [[include_tiddlers/Monty Hall Problem.html#"Monty Hall Problem"]]>>

The following table shows the four known ''Moore Graphs'' (and fifth that might exist):
<html>
<center>
<table width="600"  border="0" align="center">
  <tr>
    <th scope="col"><div align="center">Name</div></th>
    <th scope="col"><div align="center">Degree (edges per vertex) </div></th>
    <th scope="col"><div align="center">Diameter </div></th>
    <th scope="col"><div align="center">Vertices </div></th>

  </tr>
  <tr>
    <td><div align="center">Pentagon</div></td>
    <td><div align="center">2</div></td>
    <td><div align="center">2</div></td>
    <td><div align="center">5</div></td>
  </tr>

  <tr>
    <td><div align="center">Heptagon</div></td>
    <td><div align="center">2</div></td>
    <td><div align="center">3</div></td>
    <td><div align="center">7</div></td>
  </tr>
  <tr>

    <td><div align="center">Petersen Graph </div></td>
    <td><div align="center">3</div></td>
    <td><div align="center">2</div></td>
    <td><div align="center">10</div></td>
  </tr>
  <tr>
    <td><div align="center">Hoffman-Singleton Graph </div></td>

    <td><div align="center">7</div></td>
    <td><div align="center">2</div></td>
    <td><div align="center">50</div></td>
  </tr>
  <tr>
    <td><div align="center">???</div></td>
    <td><div align="center">57</div></td>

    <td><div align="center">2</div></td>
    <td><div align="center">3250</div></td>
  </tr>
</table> </center>
</html>

<<tiddler [[include_tiddlers/Moore's Law.html#"Moore's Law"]]>>

Papers:
* [[Morita Equivalence in Geometry and Algebra - R. Mayer|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.35.3449&rep=rep1&type=pdf]]

Reformulate physics and mathematics and recast it in a (coherent and consistent) form, that at least oneself can understand.

Why ?
* A lot of redundancies. Things are rediscovered over and over again and often it is not realised that they are the same. This corresponds to the problem of different representations for the same thing and different "brain wirings" of different authors, prefering different representations.
* One cannot learn mathematics and physics, one has to discover it.
* A lot is copied and pasted by authors, who do not really understand what the true meaning of the content is. This is even worse when things are copied over and over again. Errors accumulate and propagate. (Therefore it is no wonder, that original works are often way more joyful to read).
* A lot is too abstract, too far from an application. A priorisation of the relevance of mathematical structures in respect to their applicability in physics is needed. Classification of structures in mathematics is essential but one deals a lot with practically uninteresting structures (one can easily get lost).
* Reading and writing (or WIKIing these days) is better what concerns understanding than just reading.

<<tiddler [[include_tiddlers/Moufang Loop.html#"Moufang Loop"]]>>

{{center{[img(169px+, )[images/multifunction.jpg]]}}}
The theory of ''Multivalued Functions'' was fairly systematically developed for the first time in [1].

!!!!Applications
In physics, multivalued functions are used to describe
* [[Dirac's magnetic monopoles|Monopole]],
* defects in crystals,
* [[plasticity|Plasticity]] of materials,
* vortices and phase transitions in superfluids and superconductors,
* melting,
* quark confinement,
* [[gauge field structures|Gauge Theory]].

Links:
* [[WIKIPEDIA - Multivalued Function|http://en.wikipedia.org/wiki/Multivalued_function]]

Books:
* [1] Topological Spaces (1963) - C. Berge {{t1000Cite{[[bct. 1214|http://scholar.google.de/scholar?cites=1660586246030524267&as_sdt=2005&sciodt=2000&hl=de]]}}}

<<tiddler [[include_tiddlers/Multivector Derivative.html#"Multivector Derivative"]]>>

<<tiddler [[include_tiddlers/Multiverse.html#"Multiverse"]]>>

<html><center><img src="images/math_clock.jpg" style="width: 280px; "/></center></html>

Papers:
* [[My Favorite Integer Sequences N. J. A. Sloane|http://www.research.att.com/~njas/doc/sg.pdf]] [[pct. 9|http://scholar.google.com/scholar?hl=de&lr=&cites=15076201712377063024&um=1&ie=UTF-8&ei=2A2lSpX8B5m4sgaHtoXTBA&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]

Links:
* [[Cognitive Theoretic Model of the Universe (CMTU) - C. Langan|http://www.megafoundation.org/CTMU/]] - How a man with an alledged I.Q. of around 200 "sees" the universe.
* [[Ben Goertzel Essays|http://www.goertzel.org/essays.htm]]
* [[University of Toronto Mathematics Network - Question Corner and Discussion Area|http://www.math.toronto.edu/mathnet/questionCorner/qc.ps]]
* [[Articles by S. M. Phillips|http://www.smphillips.8m.com/html/articles.html]] - Interesting stuff, but to be taken with a grain of salt. ("Octonion Algebra is isomorphic to E8 Lie Algebra").
* [[Bitmaps for a Digital Theory of Everything - R. Aschheim|http://www.cs.indiana.edu/~dgerman/2008midwestNKSconference/rasch.pdf]]
* [[Strings and Loops in Event Symmetric Space-Time - P. Gibbs|http://arxiv.org/PS_cache/hep-th/pdf/9407/9407136v1.pdf]]
* [[Rafiki Inc.|http://www.codefun.com/]]
* [[The Cellular Universe website - C. Ranzan|http://www.cellularuniverse.org/]]
* [[Tony Smith's Homepage, 240 Thoughts|http://www.valdostamuseum.org/hamsmith/SWTxt.html]]
* [[Verman University Mathematical Quotations Server|http://math.furman.edu/~mwoodard/mqs/mquot.shtml]]
* [[God does not play Dice|http://www.god-does-not-play-dice.net/]]
* [[Gennady I. Shipov|http://www.shipov.com/]] - Torsion, warp drives and all that ...
* [[The Orientation Congruent Algebra and the Native Exterior Calculus of Twisted Differential Forms|http://felicity.freeshell.org/math/index.htm#vis-tw-objs]]
* [[viXra.org|http://www.vixra.org/]] - The alternative arXiv.

<<tiddler [[include_tiddlers/Mysterious Analogies.html#"Mysterious Analogies"]]>>

<<tiddler [[include_tiddlers/Möbius Transformation.html#"Möbius Transformation"]]>>

<<tiddler [[include_tiddlers/N-Cochain.html#"N-Cochain"]]>>

<<tiddler [[include_tiddlers/N-Cocycle.html#"N-Cocycle"]]>>

<<tiddler [[include_tiddlers/N-Curvature.html#"N-Curvature"]]>>

<<tiddler [[include_tiddlers/N-Quasigroup.html#"N-Quasigroup"]]>>

<<tiddler [[include_tiddlers/Naked Singularity.html#"Naked Singularity"]]>>

The (even, selfdual) ''Narain Lattice'' is given by $\Lambda_{6,22} = \Lambda_{6,6} \otimes \Lambda_{16}$ with $\Lambda_{16}$ the $Spin(32)/\mathbb{Z}_2$ lattice ([[Barnes-Wall Lattice|Barnes-Wall Lattice]]).

<<tiddler [[include_tiddlers/Negative Mass.html#"Negative Mass"]]>>

To get used to the very similar looking tensor expressions for nested [[commutators|Commutator]] and [[associators|Associator]] (i.e. [[torsion-|Torsion]] and [[nonassociativity tensors|Nonassociativity Tensor]]), in the following a list of some objects occuring:
!!!!Inner commutators
\begin{eqnarray}
[[\mb A, \mb B], \mb C, \mb D]  &= & A^i B^j C^k D^l [[\mb e_i, \mb e_j], \mb e_k, \mb e_l] \\
&= & A^i B^j C^k D^l  A_{mkl} T_{ij}^m
\end{eqnarray}
accordingly
\begin{eqnarray}
[\mb A, [\mb B, \mb C], \mb D]  &= & A^i B^j C^k D^l  A_{iml} T_{jk}^m
\end{eqnarray}
and
\begin{eqnarray}
[\mb A, \mb B, [\mb C, \mb D]]  &= & A^i B^j C^k D^l  A_{ijm} T_{kl}^m
\end{eqnarray}
Hence the position of the commutator equals the position of the summation index of the associator.

!!!!Inner associators
\begin{eqnarray}
[[\mb A, \mb B, \mb C], \mb D]  &= & A^i B^j C^k D^l [[\mb e_i, \mb e_j, \mb e_k], \mb e_l] \\
&= & A^i B^j C^k D^l  A_{ijk}^m T_{ml}
\end{eqnarray}
and accordingly
\begin{eqnarray}
[\mb A, [\mb B, \mb C, \mb D]] &= & A^i B^j C^k D^l [\mb e_i, [\mb e_j, \mb e_k, \mb e_l]] \\
&= & A^i B^j C^k D^l  A_{jkl}^m T_{im}
\end{eqnarray}
Hence the position of the summation index of the torsion tensor indicates, if the associator is to the left or to the right.

Furthermore as a mnemonic, the tensor with the additional index is the inner one.

Papers:
* [[Evolution of Networks - S.N. Dorogovtsev, J. F. F. Mendes|http://arxiv.org/PS_cache/cond-mat/pdf/0106/0106144v2.pdf]] {{t1000Cite{ [[pct. 2514|http://scholar.google.de/scholar?cites=7757634102586467395&hl=de]]

<<tiddler [[include_tiddlers/Neutrino.html#"Neutrino"]]>>

<<tiddler [[include_tiddlers/News.html#"News"]]>>

<<tiddler [[include_tiddlers/Nilpotency.html#"Nilpotency"]]>>

Links:
* [[WIKIPEDIA - Nilradical of a Ring|http://en.wikipedia.org/wiki/Nilradical_of_a_ring]]

<<tiddler [[include_tiddlers/No-Ghost Theorem.html#"No-Ghost Theorem"]]>>

<<tiddler [[include_tiddlers/No-boundary Proposal.html#"No-boundary Proposal"]]>>

<br>{{center{[img(419px+, )[images/noise.jpg]]}}}

Papers:
* [[A Computer Scientist's View of Life, the Universe, and Everything (1997) - J. Schmidhuber|ftp://ftp.idsia.ch/pub/juergen/everything.pdf]] [[local|papers/everything.pdf]] [[pct. 2|http://scholar.google.de/scholar?hl=de&lr=&cites=14344535664583841865&um=1&ie=UTF-8&ei=9XZfTY2WKIqYOu7lwMcN&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCgQzgIwAA]]

Links:
* [[WIKIPEDIA - Pink Noise|http://en.wikipedia.org/wiki/Pink_noise]]

<<tiddler [[include_tiddlers/Non-Metricity Tensor.html#"Non-Metricity Tensor"]]>>

There is a fundamental conflict between the theories of [[General Relativity]] and [[quantum mechanics|Quantum Mechanics]], which forbids a straightforward [[quantization|Quantization]] of the gravitational field:
Quantum mechanics is based on the assumption that the state of a system can be represented by a vector in [[Hilbert space|Hilbert Space]] which undergoes unitary time evolution. The linearity of Hilbert space allows for the existence of superpositions, while the unitarity of time evolution guarantees that such superpositions will not be lost in time (or, more precisely, that quantum information cannot be destroyed).
General Relativity on the other hand is founded on the principle of general covariance which states that the laws of physics should in no way depend on the choice of the coordinate system to describe them in. As a consequence, time and space not only look different for different observers (as in special relativity), but even the very concept of assigning globally valid space and time coordinates becomes impossible in general.
As it turns out, these two fundamental building blocks, unitarity and general covariance, are mutually incompatible. All theories trying to bridge the gap between General Relativity and quantum mechanics have to abandon either one or both of these principles.

!!!!My (current) take on this issue ...
The covariance principle is a deeply mathematical principle, whereas the unitarity principle is just a physical conjecture, stemming from the fact that quantum mechanics is assumed to be linear. One is therefore tempted to opt for giving up unitarity. Yet, the flip side of the coin is that the covariance principle is a principle of a presumably effective theory requiring a refinement. It is not clear if this principle can survive ensuing modifications. Furthermore there is still the well known incredible experimental robustness of quantum mechanics.
Yet I can well imagine that in the end the unitarity principle has to be given up in the most general case, as, for whatever the ultimate modification of gravity will be (resulting in a theory of [[quantum gravity|Quantum Gravity]]), it might turn out to be impossible to map the currently nonlinear theory of gravity to a theory being linear through and through.

See also:
* [[Collapse of the Wavefunction]]

Papers:
* [[Fundamental Decoherence from Quantum Gravity: a Pedagogical Review (2006) - R. Gambini, R. A. Porto, J. Pullin|http://arxiv.org/PS_cache/gr-qc/pdf/0603/0603090v1.pdf]] [[local|papers/0603090v1.pdf]] [[pct. 20|http://scholar.google.de/scholar?hl=de&lr=&cites=8790213088517583231&um=1&ie=UTF-8&ei=v3kZTfL4NYy28QPljLGBBw&sa=X&oi=science_links&ct=sl-citedby&resnum=4&ved=0CEsQzgIwAw]]
* [[A Model Independent Approach to Non Dissipative Decoherence (2000) - R. Bonifacio, S. Olivares, P. Tombesi, D. Vitali|http://arxiv.org/pdf/quant-ph/9911100v3]] [[local|papers/9911100v3.pdf]] [[pct. 9|http://scholar.google.de/scholar?hl=de&lr=&cites=7026624224849748975&um=1&ie=UTF-8&ei=dKQZTYrSBYSa8QPgk9mEBw&sa=X&oi=science_links&ct=sl-citedby&resnum=2&ved=0CCoQzgIwAQ]]
* [[Quantum Mechanics meets General Relativity in Nanoscale Experiments (2009) - J. van Wezel, T. H. Oosterkamp|http://arxiv.org/pdf/0912.3675v2]] [[local|papers/0912.pdf]] pct. 0
* [[Superposition Principle, Spontaneous Decoherence and C60 Molecule Interference (2002) - S. Olivares|http://qinf.fisica.unimi.it/~olivares/paperi/2002/olivares_JOptB_4_438.pdf]] [[local|papers/olivares_JOptB_4_438.pdf]] pct. 0

<<tiddler [[include_tiddlers/Nonassociative Algebra.html#"Nonassociative Algebra"]]>>

<<tiddler [[include_tiddlers/Nonassociative Geometry.html#"Nonassociative Geometry"]]>>

<<tiddler [[include_tiddlers/Nonassociative Physics.html#"Nonassociative Physics"]]>>

See also: [[quantum field theory|Quantum Field Theory]].

Papers:
* [[One-loop Unitarity of Scalar Field Theories on Poincaré Invariant Commutative Nonassociative Spacetimes - Y. Sasai, N. Sasakura|http://arxiv.org/PS_cache/hep-th/pdf/0604/0604194v2.pdf]] [[pct. 8|http://scholar.google.de/scholar?cites=15008719768523439994&hl=de]]
* [[Particle Scattering in Nonassociative Quantum Field Theory - V. D. Dzhunushaliev|http://arxiv.org/PS_cache/hep-th/pdf/9606/9606125v1.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=2693808319832489859&hl=de]]
* [[Nonperturbative Operator Quantization of Strongly Nonlinear Fields - V. Dzhunushaliev|http://arxiv.org/PS_cache/hep-th/pdf/0103/0103172v4.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=4059072563260696001&hl=de]]

<<tiddler [[include_tiddlers/Nonassociativity Tensor.html#"Nonassociativity Tensor"]]>>

<<tiddler [[include_tiddlers/Noncommutative Geometry.html#"Noncommutative Geometry"]]>>

<<tiddler [[include_tiddlers/Noncommutative Geometry and Quantization.html#"Noncommutative Geometry and Quantization"]]>>

<<tiddler [[include_tiddlers/Noncommutative Jordan Algebra.html#"Noncommutative Jordan Algebra"]]>>

<<tiddler [[include_tiddlers/Noncommutative Spacetime.html#"Noncommutative Spacetime"]]>>

<<tiddler [[include_tiddlers/Noncommutative Standard Model Dirac Operator.html#"Noncommutative Standard Model Dirac Operator"]]>>

Papers:
* [[On the Notion of Lower Central Series for Loops - J. Mostovoy|http://arxiv.org/PS_cache/math/pdf/0410/0410515v1.pdf]]

<<tiddler [[include_tiddlers/Nonholonomic Mapping Principle.html#"Nonholonomic Mapping Principle"]]>>

<<tiddler [[include_tiddlers/Nonlinear Quantum Mechanics.html#"Nonlinear Quantum Mechanics"]]>>

<<tiddler [[include_tiddlers/Nonlinear Sigma Model.html#"Nonlinear Sigma Model"]]>>

''Nonsymmetric Gravity Theory (NGT)'' allows for antisymmetric metrics. Nonsymmetric metrics were already studied by Einstein & Straus (1946) in their search for a unified theory for gravity and electromagnetism, yet this unification was not successful.

Links:
* [[WIKIPEDIA - Nonsymmetric Gravitational Theory|http://en.wikipedia.org/wiki/Nonsymmetric_gravitational_theory]]

<<tiddler [[include_tiddlers/Nordstrom-Robinson Code.html#"Nordstrom-Robinson Code"]]>>

<<tiddler [[include_tiddlers/Norm.html#"Norm"]]>>

A ''Normal Subgroup'' (''Normalteiler'') $\mathcal N$ of a [[group|Group]] $\mathcal G$ (written as $\mathcal N \triangleleft \mathcal G$) is a subgroup with the property, that for every element $n \in \mathcal N$ any [[conjugate|Conjugation]] element $g^{-1}\mathcal Ng$ is still in $\mathcal N$. A normal subgroup is the union of conjugacy classes.

''Example:''
$SU(N)$ is normal subgroup of $U(N)$.

See also: [[Normal subloop|Normal Subloop]].

A subloop $\mathcal N$ of a [[loop|Loop]] $\mathcal L$ is called ''Normal'' (written $\mathcal N \triangleleft \mathcal L$) if, $\forall \mb A, \mb B \in \mathcal L$, it satisfies
\begin{eqnarray}
\mb A \mathcal N &=& \mathcal N \mb A &\Leftrightarrow &[\mb A, \mathcal N] = 0  & \Leftrightarrow & \mb T(\mb A)  = \mb L^{-1}_{\mb A}  \mb R_{\mb A}  = \mb e  \\
\mb B (\mb A\mathcal N) &= &(\mb{BA}) \mathcal N & \Leftrightarrow &[\mb B, \mb A, \mathcal N] = 0 & \Leftrightarrow & \mb L(\mb A, \mb B) = \mb L^{?1}_{\mb{AB}} \mb L_{\mb B} \mb L_{\mb A} = \mb e\\
(\mathcal N \mb A) \mb B & = & \mathcal N (\mb{AB}) & \Leftrightarrow &[\mathcal N, \mb A, \mb B] = 0 &\Leftrightarrow & \mb R(\mb A, \mb B) = \mb R^{?1}_{\mb{AB}} \mb R_{\mb B} \mb R_{\mb A} = \mb e\\
&&
\end{eqnarray}
$\mb T(\mb A)$, $\mb L(\mb A, \mb B)$ and $\mb R(\mb A, \mb B)$ are called ''Middle-'', ''Left-'', and ''Right\-Inner Mappings'' of a loop. $\mb T(\mb A)$ is identical with [[conjugation|Conjugacy Class]].

$\mathcal N$ induces a [[homomorphism|Homomorphism]]  $L \rightarrow \mathcal L / \mathcal N$ exactly as in group theory. In particular, no nontrivial congruence can be defined on a loop having no proper normal subloop except for $\{\mb e\}$ and $\mathcal L$. Such a loop is said to be ''Simple''.

See also: [[Normal subgroup|Normal Subgroup]].

Papers:
* [[The Complexity of Computing over Quasigroups - H. Caussinus, F. Lemieux|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.66.5819&rep=rep1&type=pdf]] [[local|papers/10.1.1.66.5819-2.pdf]] [[pct. 17|http://scholar.google.de/scholar?cites=9187406233338423510&hl=de]] prl. 9

<<tiddler [[include_tiddlers/Nucleus.html#"Nucleus"]]>>

<<tiddler [[include_tiddlers/Null Geodesic.html#"Null Geodesic"]]>>

The ''Octacode'' is a [[linear|Linear Blockcode]] [[self-dual code|Dual Code]] over $\mathbb Z_4$ of length $8$ and minimal Lee distance $6$.
It is the $\mathbb Z_4$-analogue of a [[Hamming code|Hamming Code]].

<<tiddler [[include_tiddlers/Octonion.html#"Octonion"]]>>

<<tiddler [[include_tiddlers/Octonion Multiplication Tables.html#"Octonion Multiplication Tables"]]>>

The ''Octonion Projective Plane'' $\mathbb{OP}2$ is an example of a geometry not satisfying the full [[Desargue's theorem|Desargues' Theorem]]. This was shown by Ruth Moufang in 1933. However, as the octonions are alternative, the "small Desargue's theorem" is still valid.

Papers:
* [[On the Cell Structure of the Octonion Projective Plane - I. Yokota|http://dlisv03.media.osaka-cu.ac.jp/infolib/user_contents/sugaku/DB-M-006-01-03.pdf]]
* [[Submanifolds of the Cayley Projective Plane with bounded Second Fundamental Form - P. Coulton, J. Glazenbrook|http://www.springerlink.com/content/l370j13w60666g39/fulltext.pdf]]
* [[The Octonions - J. C. Baez|http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf]] [[local|papers/S0273-0979-01-00934-X.pdf]]

Papers:
* [[Fermions and Octonions (1987) - P. Goddard, W. Nahm, D. I. Olive, H. Ruegg, A. Schwimmer|http://www.projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1104159976]] [[local|papers/Fermions and Octonions.pdf]] [[pct. 29|http://scholar.google.com/scholar?hl=de&lr=&cites=10872601125204906990&um=1&ie=UTF-8&ei=J_lAT-6rN5Dwsga2mezRBA&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCwQzgIwAA]]
* [[Octonionic Quark Confinement (1978) - H. Ruegg|http://th-www.if.uj.edu.pl/acta/vol09/pdf/v09p1037.pdf]] [[local|papers/v09p1037.pdf]] [[pct. 5|http://scholar.google.de/scholar?hl=de&lr=&cites=7571302676604388954&um=1&ie=UTF-8&ei=ufZyTq23AsWxhQfBypWrDA&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCQQzgIwAA]]
- Gravitation
* [[Geometrical Properties of an Internal Local Octonionic Space in Curved Space Time (1986) - S. Marques, C. G. Oliveira|http://lss.fnal.gov/archive/1986/pub/Pub-86-060-A.pdf]] [[local|papers/Pub-86-060-A.pdf]] [[pct. 13|http://scholar.google.com/scholar?hl=de&lr=&cites=714899899457658264&um=1&ie=UTF-8&ei=yvlAT_b5OIHetAaX7-CBCw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCQQzgIwAA]]
* [[Natural Octonionic Generalization of General Relativity (2007) - J. Fredsted|http://arxiv.org/abs/0707.0554v1]] [[local|papers/0707.0554v1.pdf]] pct. 0

Videos:
* [[Fields and Strings Seminar: Octonions and Quantum Gravity (2010) - H. Nicolai|https://cast.itunes.uni-muenchen.de/vod/clips/n3TgkSb3QZ/flash.html]] - Note, that the definition of [[alternativity|Alternative Algebra]] he gives (14:20) is incorrect. What he writes down is actually the [[flexibility|Flexible Algebra]] condition.

The ''Octonionic Projective Plane'' $\mathbb{OP}^2$ is an example of a geometry not satisfying the full [[Desargue's theorem|Desargues' Theorem]]. This was shown by Ruth Moufang in 1933. However, as the [[octonions|Octonion]] are [[alternative|Alternative Algebra]], the "small Desargue's theorem" is still valid.

Papers:
* [[The Octonions - J. C. Baez|http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf]] [[local|papers/S0273-0979-01-00934-X.pdf]]  {{t100Cite{[[pct. 318|http://scholar.google.de/scholar?cites=12472142092836538554&hl=de]]}}}
* [[Submanifolds of the Cayley Projective Plane with bounded Second Fundamental Form - P. Coulton, J. Glazenbrook|http://www.springerlink.com/content/l370j13w60666g39/fulltext.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=17130347132186689525&hl=de]]
* [[OP2 Bundles in M-theory - H. Sati|http://www.hausdorff-research-institute.uni-bonn.de/files/preprints/OP2paper-Aug10.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=801582279879132875&hl=de]]
* [[Plurisubharmonic Functions on the Octonionic Plane and Spin(9)-Invariant Valuations on Convex Sets - S. Alesker|http://www.springerlink.com/content/50340775283p6712/fulltext.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=9030141090597490996&hl=de]]
* [[On the Geometry of the Supermultiplet in M-theory - H. Sati|http://arxiv.org/PS_cache/arxiv/pdf/0909/0909.4737v1.pdf]] pct. 0
* [[On the Cell Structure of the Octonion Projective Plane - I. Yokota|http://dlisv03.media.osaka-cu.ac.jp/infolib/user_contents/sugaku/DB-M-006-01-03.pdf]] pct. 0

<<tiddler [[include_tiddlers/Octooctonionic Projective Plane.html#"Octooctonionic Projective Plane"]]>>

Links:
* [[WIKIPEDIA - Olber's Paradox|http://en.wikipedia.org/wiki/Olbers'_paradox]]

Videos:
* [[Physics Lecture:- Olbers Paradox and the Distant Universe - R. Nemiroff|http://www.youtube.com/watch?v=sIDSf8-aHXI&feature=results_main&playnext=1&list=PLCAB7A268352F7836]]

<<tiddler [[include_tiddlers/Omega Point.html#"Omega Point"]]>>

An operad consists of operations shaped like
{{center{[img(80px+, )[images/operad.jpg]]}}}
with many inputs (five, here) and one output. Any tree of operations, such as

{{center{[img(125px+, )[images/operad_chain.jpg]]}}}
can be composed to give a single operation.

<<tiddler [[include_tiddlers/Orbifold.html#"Orbifold"]]>>

Given a set $X$ and a [[group|Group]] $\mathcal G$, then for a $x \in X$ the set
\[
\operatorname{Orb}(x) = \left\{ g x \mid g \in \mathcal G \right\}
\]
is called the ''Orbit'' of $x$. (It is the set of elements of $X$ to which $x$ can be "moved to" by the elements of $\mathcal G$).

The number of elements of an orbit is called the ''length of the orbit''.

The orbits of this action are exactly the [[left cosets|Coset]] of $X$ in $\mathcal G$.

Orbits form equivalence classes. The equivalence relation $x \sim y$ for $x,y \in X$ is given by the group action, i. e. $y = gx$ with a $g \in \mathcal G$.

The set of all orbits of $X$ under the action of $\mathcal G$ forms a partition of $X$. It is written as $X/\mathcal G$ (or, less frequently: $\mathcal G$\\$X$) and is called the quotient of the action; in geometric situations it may be called the ''Orbit Space''.

<<tiddler [[include_tiddlers/Orch-OR Model.html#"Orch-OR Model"]]>>

In [[group theory|Group]] the term ''Order'' is used in two closely related senses:
#  the order of a group is the number of its elements.
#  the order of an element $\mb A$ of a group is the smallest number $n \in \mathbb N$ such that $\mb A^n = \mb e$. All elements of finite groups have finite order.

<<tiddler [[include_tiddlers/Organic Universe.html#"Organic Universe"]]>>

A ''(Real) Orthogonal Design (ROD)'' is a $m\times n$-matrix $G$ in $k$ real variables $x_1, x_2, \ldots , x_k$ such that each non-zero entry of the matrix is a real linear combinations of the variables $G G^T = (x_1^2+x_2^2+ \ldots +x_k^2)I_n$
with $I_n$ the $n\times n$ identity matrix.

Papers:
* [[Algebraic Methods in the Design of Space-time Codes - J- Hiltunen|http://users.utu.fi/jakahi/lisuri.pdf]] - pct. 0
* [[Delay-optimal Rectangular Real Orthogonal Designs from Cayley-Dickson Algebra - S. Das, B. S. Rajan|http://www.ncc.org.in/download.php?f=NCC2009/file1.pdf]] pct. 0

The ''r+s-dimensional Orthogonal Group $O(r, s)$'' is the [[automorphism group|Automorphism]] of the [[norm|Norm]], i.e. the norm is preserved under its action.

It is defined by:
\[
O(r, s) \equiv \{\mb O \in GL(n,\mathbb R) :  \langle \mb O \mb A | \mb O \mb B \rangle = \langle \mb A | \mb B \rangle \;\, \forall \mb A, \mb B \in \mathbb R^{r+s}\}
\]

The ''Special Orthogonal $SO(r, s)$'' is defined by:
\[
SO(r, s) \equiv \{\mb A \in O(r, s) : \det(\mb A) = 1\}
\]
Orthogonal transforms are [[isometries|Isometry]], i.e. they preserve distances and angles, hence they are [[conformal maps|Conformal Transformation]]. However not all conformal linear transforms are orthogonal.

!!!!Orthogonal group over finite fields
Orthogonal groups can also be defined over [[finite fields|Galois Field]] $\mathbb F_q$, where $q$ is a power of a prime $p$. In this case they come in two types in even dimension: $O^+ (2n,q)$ and $O^? (2n,q)$ and one type in odd dimension: $O(2n + 1,q)$.
Their [[order|Order]] is given by:
\[
\operatorname{ord}(O(2n,q))=2(q^n+(-1)^{n+1})\prod_{i=1}^{n-1}(q^{2n}-q^{2i}).
\]
\[
\operatorname{ord}(O(2n+1,q))=2q^n\prod_{i=0}^{n-1}(q^{2n}-q^{2i})
\]

An ''Orthogonal Tranfsormation'' $\mb O$ is a linear transformation that preserves the scalar product
\begin{equation}
\langle \mb O(\mb A)|\mb O (\mb B)\rangle = \langle \mb A| \mb B \rangle
\end{equation}

<<tiddler [[include_tiddlers/P-Brane.html#"P-Brane"]]>>

Links:
* [[PARI/GP Website|http://pari.math.u-bordeaux.fr/]]

<<tiddler [[include_tiddlers/PG(3,2).html#"PG(3,2)"]]>>

<<tiddler [[include_tiddlers/PG(5,2).html#"PG(5,2)"]]>>

<<tiddler [[include_tiddlers/PSL(2,7).html#"PSL(2,7)"]]>>

According to the ''Palatini Principle (or first order formalism)'', [[metric|Metric Tensor]] and [[connection|Connection]] are regarded as independent. For a Lagrangian this means that the variation is carried out for both of them independently.
For the [[Einstein Hilbert action|Einstein-Hilbert Action]] this leads - besides the [[Einstein equations|Einstein Field Equations]] when varying in respect to the [[metric|Metric Tensor]] - to a second set of equations, namely
\[
\partial_\lambda g_{\mu\nu} - \Gamma^\eta_{\lambda\mu} g_{\eta\nu} - \Gamma^\eta_{\lambda\nu} g_{\mu\eta} = 0
\]
which is the condition of [[metric compatibility|Metric Compatibility]]. These equations besides the Einstein equations can be regarded as additional field equations describing the gravitational field.

In general the Palatini principle allows the geometry to have an affine structure, that is that the associated space-time is a [[metric affine space|Metric Affine Space]]. In four dimensions, i.e. for a $(\mathbb L_4, g)$-space, there are $10$ equations for the metric tensor and $64$ for the connection.

The advantage of deriving the field equations using the Palatini method is that the geometry of space-time is less restrictive. In general the Palatini variation allows for the existence of [[torsion|Torsion]] and [[non-metricity|Non-Metricity Tensor]].

Although the principle carries the name of Palatini its introduction is usually ascribed to Einstein.

Papers:
* [[The Universality of Einstein Equations (1993) - M. Ferraris, M. Francaviglia, I. Volovich|http://arxiv.org/PS_cache/gr-qc/pdf/9303/9303007v2.pdf]] [[pct. 72|http://scholar.google.de/scholar?cites=9818932703691021574&hl=de&as_sdt=2000]]
*[[On the so-called "Palatini Method" of Variation in Covariant Gravitational Theories (1973) - A. A. El-Kholy, R. U. Sexl, H. K. Urbantke|http://archive.numdam.org/ARCHIVE/AIHPA/AIHPA_1973__18_2/AIHPA_1973__18_2_121_0/AIHPA_1973__18_2_121_0.pdf]] pct. 0
*[[Palatini Variational Principle for an Extended Einstein-Hilbert Action (1997) - H. Burton, R. B. Mann|http://arxiv.org/PS_cache/gr-qc/pdf/9711/9711003v1.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=9052327792572994752&hl=de&as_sdt=2000]]

Theses:
* [[On the Palatini Variation and Connection Theories of Gravity - H. S. Burton (1998)|http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape9/PQDD_0009/NQ38225.pdf]] [[local|theses/NQ38225.pdf]] tct. 0

The conjecture is that the days of ''Paper and Pencil Physics and Mathematics'' are over.

Most of the reasonable work has been done, often long time ago. Everybody who is doing paper and pencil mathematics or physics these days is competing with those famous figures in history, that have already pushed the envelope as far as one can do by means of the (classical) method - at least in respect to reasonable mathematics and physics.
To get a paper published these days, in physics for example, often requires a very exotic topic to avoid duplicating old results, thus often having little to do with reality, i.e. nature in the first place, rather with the fact that as a scientist one is urged to publish on a regular basis.

On the other hand, due to [[Moore's law|Moore's Law]], computational mathematics and physics opens up new possibilities every day and therefore appears more promising.

Compare this to the game of chess. These days the best chess players are computers ...

<html><center><img src="images/paralleltransport.jpg" style="width: 320px; "/></center></html>
!!!!Historical
The concept of parallel transport of vectors was introduced independently by Levi\-Civita and Schouten in 1917 (two years after Einstein's general relativity theory was published).

!!!!Generalisations
Instead of vectors one can also transport oriented autoparallel segments on a manifold. For details see [[geodesic loop|Geodesic Loop]].

See also: [[autoparallelity|Autoparallelity]].

Papers:
* [[On the Parallel Transport of Tetrad in a Riemann-Cartan Spacetime - D.-C. Chern|http://psroc.phys.ntu.edu.tw/cjp/v19/45.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=13757775929773306320&as_sdt=2005&sciodt=2000&hl=de]]

A manifold $\mathcal M$ is said to be ''parallelizable'' if there exists a [[torsion|Torsion]] $T_{\nu\rho\sigma}$ which \"flattens\” it, i.e. makes the [[Riemann curvature tensor|Riemann Tensor]] $\tilde{R}_{\mu\nu\sigma\lambda}$ vanish. (This definition is due to Cartan and Schouten and is mostly used in physics literature. In mathematics it is the definition for absolute parallelism and is a stronger condition than parallelism. That is, that in general absolute parallelism implies parallelism, but not vice-versa.
One has the condition:
\begin{equation}
R_{\mu\nu\rho\sigma} =\tilde{R}_{\mu\nu\rho\sigma} + D_\rho T_{\mu\nu\sigma} - D_\sigma T_{\mu\nu\rho} + T_{\mu\sigma\lambda} {T^{\lambda}}_{\nu\rho} - T_{\mu\rho\lambda} {T^{\lambda}}_{\nu\sigma} = 0
\end{equation}
with $\tilde{R}_{\mu\nu\rho\sigma}$ stemming from the [[Christoffel part|Christoffel Symbols]] of the [[connection|Connection]] and the rest from the torsion part.

Analogously a manifold is said to be ''Ricci-parallizable'' if the [[Ricci tensor|Ricci Tensor]] $R_{\mu\nu}$ vanishes, i.e.
\begin{equation}
R_{\mu\nu} =\tilde{R}_{\mu\nu}  + D_\lambda {T^{\lambda}}_{\mu\nu} + T_{\mu\sigma\lambda} {T^{\sigma\lambda}}_{\nu} = 0
\end{equation}
This implies that the symmtric and the antisymmetric parts of the Ricci tensor vanish, i.e.
\begin{equation}
\tilde{R}_{\mu\nu}  + T_{\mu\sigma\lambda} {T^{\sigma\lambda}}_{\nu} = 0
\end{equation}
\begin{equation}
D_\lambda {T^{\lambda}}_{\mu\nu} = 0
\end{equation}

<<tiddler [[include_tiddlers/Paramecium.html#"Paramecium"]]>>

<<tiddler [[include_tiddlers/Particle Generation.html#"Particle Generation"]]>>

<<tiddler [[include_tiddlers/Partition Function.html#"Partition Function"]]>>

A ''Passive Transformation'' only changes the representation of any of its elements in respect to coordinates and a basis (i.e. the way it is decomposed into a basis and coordinates). It leaves the physical state of the system unchanged. It implies merely a change in the method of describing it.

In concrete: Given an element $\mb A$ of an algebra and a basis $\{\mb e_\mu \}$ one has the coordinate representation $\mb A = A_\mu \mb e_\mu$. If a transformation to new coordinates  $A'_\mu$ and a new basis $\{\mb e'_\mu \}$ is carried out such that $\mb A = A'_\mu \mb e'_\mu$ it satisfies the requirements for a passive transformation.
Obviously an infinitesimal line element isn't changed either as it is just the scalar product of two elements an hence the physical action stays the same.

Examples:
[[Passive Lorentz transformations|Lorentz Transformation]] that only change the reference frame for the description of the same phenomena. However Lorentz transformations can also be used to describe [[active transformations|Active Transformation]], so called particle boost transformations.

!!!!Polyvectors
Given a [[polyvector|Polyvector Space]] $\mb A = A_\alpha \mb E_\alpha$, a passive transformation is required to not change it. Therefore
\begin{eqnarray}
d\mb A & = & 0\\
& = & dA_\alpha \mb E_\alpha + A_\alpha d  \mb E_\alpha \\
& = & dA_\alpha \mb E_\alpha + A_\alpha \Gamma_{\beta\alpha}  \mb E_\alpha \\
& = & ( dA_\alpha + A_\beta \Gamma_{\beta\alpha}) \mb E_\alpha \\
\end{eqnarray}
Hence the [[covariant polyvector derivative|Polyvector Covariant Derivative]] must be zero.

<<tiddler [[include_tiddlers/Past Life Regression.html#"Past Life Regression"]]>>

<<tiddler [[include_tiddlers/Path Integral.html#"Path Integral"]]>>

<<tiddler [[include_tiddlers/Pati-Salam Model.html#"Pati-Salam Model"]]>>

<<tiddler [[include_tiddlers/Paul Adrien Maurice Dirac.html#"Paul Adrien Maurice Dirac"]]>>

The ''Pauli Matrices'' are defined by
\begin{equation}
\bs \sigma_1 = \left(\begin{matrix}0 & 1\\
1 & 0\end{matrix}\right), \quad \bs \sigma_2=\left(\begin{matrix}0 & - \mb i\\
\mb i & 0\end{matrix}\right), \quad \bs \sigma_3=\left(\begin{matrix}1 & 0\\
0 & -1\end{matrix}\right)
\end{equation}

* [[Find-pdF|http://www.find-pdf.com]]  <html>
 <div id="DIV_PREVIEW5">
      <form id="FRM_SEARCH" name="FRM_SEARCH" method="post" action="http://www.find-pdf.com/search.html">
      <input type="text" name="TXT_SEARCH" class="searchBox" id="TXT_SEARCH" width:331px; left: 17px; top: 29px; height:18px" value = ""/>          	<input type="submit" name="BTN_VALIDATE2" id="BTN_VALIDATE2" value="Search" style=" left: 370px; top: 26px; width: 95px; height: 26px;"/>
		</form></div></html>
* [[Pdf Search Engine|http://www.pdf-search-engine.com]] (Very comprehensive but slow) <html>
<form id="search" name="form-test" action="http://www.pdf-search-engine.com/pdf-search.php">
<table width="50%" height="20" align="center">
<tr> <td width="55%" height="34"><input name="search" class="recherche" id="search2" style="background: transparent url(images/bookIcon.gif) no-repeat scroll left; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; height: 30px; font-size: 20px; color: rgb(204, 51, 0); padding-left:0px;"  size="40" maxlength="40" type="text" /></td>
                      <td width="45%"><input value="Search" style="height: 34px; background-color:#F0F0F0; font-weight:bold" class="submit" type="submit" />
</td></tr></table>
</form>
</html>
* [[Pdfgeni|http://www.pdfgeni.com/]] <html>
<div id="searchk">
  <form action="http://www.pdfgeni.com/search.php" method="get" id="search" name="form-test" >
	<table width="50%" border="0" cellspacing="2" cellpadding="0">
	<tr>
		<td><input type="text" name="s" id="s" class="form_text" size="50" accesskey="f" maxlength="80"/></td>

		<td width="20"></td>
		<td><input type="submit" value="&nbsp;Search&nbsp;" class="submit" /></td>
	  </tr>
	</table>
  </form>
</div>
</html>

The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as Gentzen's proof. The small number of mathematicians who advocate [[ultrafinitism|Ultrafinitism]] reject Peano's axioms because the axioms require an infinite set of natural numbers.

<<tiddler [[include_tiddlers/Pentagon Identity.html#"Pentagon Identity"]]>>

<<tiddler [[include_tiddlers/People.html#"People"]]>>

A ''Perfect Number'' is a number which is the sum of all its positive divisors except for itself. (I.e. it cannot be a prime number).
The first four perfect numbers were already described by Euklid and obey the formula $2^{n?1}(2^n ? 1) = 2^{n?1} M_n$ with $M_n$ a [[Mersenne prime number|Mersenne Prime]].
Hence
\begin{eqnarray}
6 & = &2^1(2^2-1) = 1+2+3  \\
28 &=& 2^2(2^3-1) = 1+2+3+4+5+6+7 = 1^3+3^3 \\
496 & =& 2^4(2^5-1) = 1+2+3+\cdots+29+30+31 = 1^3+3^3+5^3+7^3  \\
8128 &= & 2^6(2^7-1) = 1+2+3+\cdots+125+126+127 = 1^3+3^3+5^3+7^3+9^3+11^3+13^3+15^3
\end{eqnarray}

Currently $47$ perfect numbers are known which equals the number of Mersenne primes.

Links:
* [[WIKIPEDIA - Perfect Number|http://en.wikipedia.org/wiki/Perfect_number]]
* [[The MacTutor History of Mathematics Archive - Perfect Numbers|http://www-history.mcs.st-andrews.ac.uk/HistTopics/Perfect_numbers.html]]

<<tiddler [[include_tiddlers/Permittivity.html#"Permittivity"]]>>

<<tiddler [[include_tiddlers/Perron-Frobenius Theorem.html#"Perron-Frobenius Theorem"]]>>

A ''Pfaffian Equation'' is defined by relationship between [[forms|Form]]. The theory of Pfaffian equations is a rather difficult subject.
!!!!Example
$xdy ? ydx = 0$

''Pfister forms'' are a special class of [[quadratic forms|Quadratic Form]] and play a key role in the theory of quadratic forms.

A ''$n$-Pfister Form'' (a.k.a ''$n$-fold Pfister Form'' or '' Pfister $n$-Form'') is defined by
\[
\langle \langle a_1, a_2, ... , a_n \rangle \rangle  \equiv \otimes_{i = 1}^n  \langle 1, -a_i \rangle
\]
i.e. it is a the quadratic form with coefficients obtained by an $n$-fold tensor product of the coefficients $\{1, a_i\}$ of a quadratic forms given by $ 1\cdot x_1^2 - a_ix_2^2$.

$n$-Pfister forms for $n \le 3$ are [[norm forms|Norm]] of [[composition algebras|Composition Algebra]].

!!!!Examples
$n = 1$: $\langle\langle a_1\rangle\rangle = \langle 1, -a_1 \rangle = x_1^2 - a_1x_2^2$.
Depending on the sign of $a_1$ one has either [[signature|Signature]] $(2,0)$ or $(1,1)$.

$n = 2$: $\langle\langle a_1, a_2 \rangle\rangle = \langle 1, -a_1 \rangle \otimes \langle 1, -a_2 \rangle =  x_1^2 - a_1x_2^2 - a_2x_3^2 + a_1a_2 x_4^2$.
Depending on the signs of $a_1$ and $a_2$  one has either signature $(4,0)$ or $(2,2)$.

$n = 3$: $\langle\langle a_1, a_2, a_3 \rangle\rangle = \langle 1, -a_1 \rangle \otimes \langle 1, -a_2 \rangle  \otimes  \langle 1, -a_3 \rangle =  x_1^2 - a_1x_2^2 - a_2x_3^2 -  a_3x_4^2 + a_1 a_2 x_5^2 + a_1 a_3 x_6^2 + a_2 a_3 x_7^2 - a_1a_2a_3 x_8^2$.

The quadratic form of an [[octonion algebra|Octonion]] is always a Pfister $3$-form. There are two Pfister $3$-forms over $\mathbb R$ in dimension $8$, the one is [[hyperbolic|Isotropy]] (signature $(4,4)$) and the other one is anisotropic (signature $(8,0)$).

Lectures:
* [[Topics on Quadratic Forms - A. Vishik|http://users.ictp.it/~pub_off/lectures/lns023/Vishik/Vishik.pdf]]

Papers:
* [[Sums of Squares - O. Taussky|http://www.dm.unito.it/~cerruti/ntlab2007/taussky-squares.pdf]] [[pct. 16|http://scholar.google.de/scholar?cites=5052403810380338764&hl=de]]
* [[The Degen-Graves-Cayley Eight-Square Identity - T. Piezas III|http://www.geocities.com/titus_piezas/DegenGraves1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=16420083384206827699&hl=de]]
* [[Pfister's Theorem on Sums of Squares - K. Conrad|http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/pfister.pdf]] pct. 0

<<tiddler [[include_tiddlers/Phase Space.html#"Phase Space"]]>>

<<tiddler [[include_tiddlers/Phase Transition.html#"Phase Transition"]]>>

Papers:
* [[The Fundamental Constants and their Variation: Observational Status and Theoretical Motivations (2002) - J.-P. Uzan|http://arxiv.org/PS_cache/hep-ph/pdf/0205/0205340v1.pdf]] [[local|papers/0205340v1.pdf]] {{t500Cite{[[pct. 501|http://scholar.google.de/scholar?cites=14660009318965666691&as_sdt=2005&sciodt=2000&hl=de]]}}}
* [[Trialogue on the Number of Fundamental Constants (2002) - M. J. Duff, L. B. Okun, G. Veneziano|http://arxiv.org/pdf/physics/0110060v3]] [[local|papers/0110060v3.pdf]] [[pct. 68|http://scholar.google.de/scholar?hl=de&lr=&cites=195580220291175074&um=1&ie=UTF-8&ei=scUhTfTKBMXIswaOwpH8DQ&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCgQzgIwAA]]

Links:
* [[WIKIPEDIA - Physical Constant|http://en.wikipedia.org/wiki/Physical_constant]]
* [[Particle Data Group|http://pdg.lbl.gov/]]

<<tiddler [[include_tiddlers/Physics and Mathematics Blogs.html#"Physics and Mathematics Blogs"]]>>

<<tiddler [[include_tiddlers/Physics and the Soul.html#"Physics and the Soul"]]>>

Given an algebra $\mathcal A$, its ''Pierce Decomposition'' relative to an [[idempotent|Idempotency]] $\mb E$ is given by
\[
\mathcal A = \mathcal A_0 \oplus \mathcal A_{1/2} \oplus  \mathcal A_1
\]
with
\begin{eqnarray}
\mathcal A_k &= &\{\mb A \in \mathcal A: \mb{AE} = \mb{EA} = k \mb A\}, \quad k = 0,1 \\
\mathcal A_{1/2} &= &\{\mb A \in \mathcal A:  \mb{AE} + \mb{EA}  = \mb A\}
\end{eqnarray}

<<tiddler [[include_tiddlers/Pin Group.html#"Pin Group"]]>>

<<tiddler [[include_tiddlers/Pion.html#"Pion"]]>>

<<tiddler [[include_tiddlers/Pioneer Anomaly.html#"Pioneer Anomaly"]]>>

<<tiddler [[include_tiddlers/Planck Length.html#"Planck Length"]]>>

<<tiddler [[include_tiddlers/Planck Mass.html#"Planck Mass"]]>>

Links:
* [[WIKIPEDIA - Planck Temperature|http://en.wikipedia.org/wiki/Planck_temperature]]
* [[WIKIPEDIA - Absolute Hot|http://en.wikipedia.org/wiki/Absolute_hot]]

<<tiddler [[include_tiddlers/Planck Units.html#"Planck Units"]]>>

<<tiddler [[include_tiddlers/Planck's Constant.html#"Planck's Constant"]]>>

The five ''Platonic Solids'' are the
* tetrahedron (tetra = $4$),
* cube (or hexahedron) (hexa = $6$),
* octahedron (octa = $8$),
* dodecahedron (dodeca = $12$),
* icosahedron (icosa = $20$).
The names are derived from the number of surfaces they have (see also: [[Greek number names|Greek Number Names]]).

Links:
* [[WIKIPEDIA - Platonic Solid|http://en.wikipedia.org/wiki/Platonic_solid]]

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|Name|PlayerPlugin|
|Source|http://www.TiddlyTools.com/#PlayerPlugin|
|Version|1.1.4|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|Embed a media player in a tiddler|
!!!!!Usage
<<<
{{{<<player [id=xxx] [type] [URL] [width] [height] [autoplay|true|false] [showcontrols|true|false] [extras]>>}}}

''id=xxx'' is optional, and specifies a unique identifier for each embedded player.  note: this is required if you intend to display more than one player at the same time.

''type'' is optional, and is one of the following: ''windows'', ''realone'', ''quicktime'', ''flash'', ''image'' or ''iframe''.  If the media type is not specified, the plugin automatically detects Windows, Real, QuickTime, Flash video or JPG/GIF images by matching known file extensions and/or specialized streaming-media transfer protocols (such as RTSP:).  For unrecognized media types, the plugin displays an error message.

''URL'' is the location of the media content

''width'' and ''height'' are the dimensions of the video display area (in pixels)

''autoplay'' or ''true'' or ''false'' is optional, and specifies whether the media content should begin playing as soon as it is loaded, or wait for the user to press the "play" button.  Default is //not// to autoplay.

''showcontrols'' or ''true'' or ''false'' is optional, and specifies whether the embedded media player should display its built-in control panel (e.g., play, pause, stop, rewind, etc), if any.  Default is to display the player controls.

''extras'' are optional //pairs// of parameters that can be passed to the embedded player, using the {{{<param name=xxx value=yyy>}}} HTML syntax.

''If you use [[AttachFilePlugin]] to encode and store a media file within your document, you can play embedded media content by using the title of the //attachment tiddler//'' as a parameter in place of the usual reference to an external URL.  When playing an attached media content, you should always explicitly specify the media type parameter, because the name used for the attachment tiddler may not contain a known file extension from which a default media type can be readily determined.
<<<
!!!!!Configuration
<<<
Default player size:
width: <<option txtPlayerDefaultWidth>> height: <<option txtPlayerDefaultHeight>>
<<<
!!!!!Revisions
<<<
2008.05.10 [1.1.4] in handlers(), immediately return if no params (prevents error in macro).  Also, refactored auto-detect code to make type mapping configurable.
2007.10.15 [1.1.3] in loadURL(), add recognition for .PNG (still image), fallback to iframe for unrecognized media types
2007.08.31 [1.1.2] added 'click-through' link for JPG/GIF images
2007.06.21 [1.1.1] changed "hidecontrols" param to "showcontrols" and recognize true/false values in addition to 'showcontrols', added "autoplay" param (also recognize true/false values), allow "auto" as value for type param
2007.05.22 [1.1.0] added support for type=="iframe" (displays src URL in an IFRAME)
2006.12.06 [1.0.1] in handler(), corrected check for config.macros.attach (instead of config.macros.attach.getAttachment) so that player plugin will work when AttachFilePlugin is NOT installed.  (Thanks to Phillip Ehses for bug report)
2006.11.30 [1.0.0] support embedded media content using getAttachment() API defined by AttachFilePlugin or AttachFilePluginFormatters.  Also added support for 'image' type to render JPG/GIF still images
2006.02.26 [0.7.0] major re-write.  handles default params better.  create/recreate player objects via loadURL() API for use with interactive forms and scripts.
2006.01.27 [0.6.0] added support for 'extra' macro params to pass through to object parameters
2006.01.19 [0.5.0] Initial ALPHA release
2005.12.23 [0.0.0] Started
<<<
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	if (p=="auto" || p=="windows" || p=="realone" || p=="quicktime" || p=="flash" || p=="image" || p=="iframe")
		type=params.shift().toLowerCase();
	var url=params.shift(); if (!url || !url.trim().length) url="";
	if (url.length && config.macros.attach!=undefined) // if AttachFilePlugin is installed
		if ((tid=store.getTiddler(url))!=null && tid.isTagged("attachment")) // if URL is attachment
			url=config.macros.attach.getAttachment(url); // replace TiddlerTitle with URL
	var width=params.shift();
	var height=params.shift();
	var autoplay=false;
	if (params[0]=='autoplay'||params[0]=='true'||params[0]=='false')
		autoplay=(params.shift()!='false');
	var show=true;
	if (params[0]=='showcontrols'||params[0]=='true'||params[0]=='false')
		show=(params.shift()!='false');
	var extras="";
	while (params[0]!=undefined)
		extras+="<param name='"+params.shift()+"' value='"+params.shift()+"'> ";
	this.loadURL(place,id,type,url,width,height,autoplay,show,extras);
}

if (config.options.txtPlayerDefaultWidth==undefined) config.options.txtPlayerDefaultWidth="100%";
if (config.options.txtPlayerDefaultHeight==undefined) config.options.txtPlayerDefaultHeight="480"; // can't use "100%"... player height doesn't stretch right :-(

config.macros.player.typeMap={
	windows: ['mms', '.asx', '.wvx', '.wmv', '.mp3'],
	realone: ['rtsp', '.ram', '.rpm', '.rm', '.ra'],
	quicktime: ['.mov', '.qt'],
	flash: ['.swf', '.flv'],
	image: ['.jpg', '.gif', '.png'],
	iframe: ['.htm', '.html', '.shtml', '.php']
};

config.macros.player.loadURL=function(place,id,type,url,width,height,autoplay,show,extras) {

	if (id==undefined) id="tiddlyPlayer";
	if (!width) var width=config.options.txtPlayerDefaultWidth;
	if (!height) var height=config.options.txtPlayerDefaultHeight;
	if (url && (!type || !type.length || type=="auto")) { // determine type from URL
		u=url.toLowerCase();
		var map=config.macros.player.typeMap;
		for (var t in map) for (var i=0; i<map[t].length; i++)
			if (u.indexOf(map[t][i])!=-1) var type=t;
	}
	if (!type || !config.macros.player.html[type]) var type="none";
	if (!url) var url="";
	if (show===undefined) var show=true;
	if (!extras) var extras="";
	if (type=="none" && url.trim().length) type="iframe"; // fallback to iframe for unrecognized media types

	// adjust parameter values for player-specific embedded HTML
	switch (type) {
		case "windows":
			autoplay=autoplay?"1":"0"; // player-specific param value
			show=show?"1":"0"; // player-specific param value
			break;
		case "realone":
			autoplay=autoplay?"true":"false";
			show=show?"block":"none";
			height-=show?60:0; // leave room for controls
			break;
		case "quicktime":
			autoplay=autoplay?"true":"false";
			show=show?"true":"false";
			break;
		case "image":
			show=show?"block":"none";
			break;
		case "iframe":
			show=show?"block":"none";
			break;
	}

	// create containing div for player HTML
	// and add or replace player in TW DOM structure
	var newplayer = document.createElement("div");
	newplayer.playerType=type;
	newplayer.setAttribute("id",id+"_div");
	var existing = document.getElementById(id+"_div");
	if (existing && !place) place=existing.parentNode;
	if (!existing)
		place.appendChild(newplayer);
	else {
		if (place==existing.parentNode) place.replaceChild(newplayer,existing)
		else { existing.parentNode.removeChild(existing); place.appendChild(newplayer); }
	}

	var html=config.macros.player.html[type];
	html=html.replace(/%i%/mg,id);
	html=html.replace(/%w%/mg,width);
	html=html.replace(/%h%/mg,height);
	html=html.replace(/%u%/mg,url);
	html=html.replace(/%a%/mg,autoplay);
	html=html.replace(/%s%/mg,show);
	html=html.replace(/%x%/mg,extras);
	newplayer.innerHTML=html;
}
//}}}

// // Player-specific API functions: isReady(id), isPlaying(id), toggleControls(id), showControls(id,flag)

//{{{
// status values:
// Windows: 0=Undefined, 1=Stopped, 2=Paused, 3=Playing, 4=ScanForward, 5=ScanReverse
//          6=Buffering, 7=Waiting, 8=MediaEnded, 9=Transitioning, 10=Ready, 11=Reconnecting
// RealOne: 0=Stopped, 1=Contacting, 2=Buffering, 3=Playing, 4=Paused, 5=Seeking
// QuickTime: 'Waiting', 'Loading', 'Playable', 'Complete', 'Error:###'
// Flash: 0=Loading, 1=Uninitialized, 2=Loaded, 3=Interactive, 4=Complete
config.macros.player.isReady=function(id)
{
	var d=document.getElementById(id+"_div"); if (!d) return false;
	var p=document.getElementById(id); if (!p) return false;
	if (d.playerType=='windows') return !((p.playState==0)||(p.playState==7)||(p.playState==9)||(p.playState==11));
	if (d.playerType=='realone') return (p.GetPlayState()>1);
	if (d.playerType=='quicktime') return !((p.getPluginStatus()=='Waiting')||(p.getPluginStatus()=='Loading'));
	if (d.playerType=='flash') return (p.ReadyState>2);
	return true;
}
config.macros.player.isPlaying=function(id)
{
	var d=document.getElementById(id+"_div"); if (!d) return false;
	var p=document.getElementById(id); if (!p) return false;
	if (d.playerType=='windows') return (p.playState==3);
	if (d.playerType=='realone') return (p.GetPlayState()==3);
	if (d.playerType=='quicktime') return (p.getPluginStatus()=='Complete');
	if (d.playerType=='flash') return (p.ReadyState<4);
	return false;
}
config.macros.player.showControls=function(id,flag) {
	var d=document.getElementById(id+"_div"); if (!d) return false;
	var p=document.getElementById(id); if (!p) return false;
	if (d.playerType=='windows') { p.ShowControls=flag; p.ShowStatusBar=flag; }
	if (d.playerType=='realone') { alert('show/hide controls not available'); }
	if (d.playerType=='quicktime')      // if player not ready, retry in one second
		{ if (this.isReady(id)) p.setControllerVisible(flag); else setTimeout('config.macros.player.showControls("'+id+'",'+flag+')',1000); }
	if (d.playerType=='flash') { alert('show/hide controls not available'); }
}
config.macros.player.toggleControls=function(id) {
	var d=document.getElementById(id+"_div"); if (!d) return false;
	var p=document.getElementById(id); if (!p) return false;
	if (d.playerType=='windows') var flag=!p.ShowControls;
	if (d.playerType=='realone') var flag=true; // TBD
	if (d.playerType=='quicktime') var flag=!p.getControllerVisible();
	if (d.playerType=='flash') var flag=true; // TBD
	this.showControls(id,flag);
}
config.macros.player.fullScreen=function(id) {
	var d=document.getElementById(id+"_div"); if (!d) return false;
	var p=document.getElementById(id); if (!p) return false;
	if (d.playerType=='windows') p.DisplaySize=3;
	if (d.playerType=='realone') p.SetFullScreen();
	if (d.playerType=='quicktime') { alert('full screen not available'); }
	if (d.playerType=='flash') { alert('full screen not available'); }
}
//}}}

// // Player HTML

//{{{
// placeholder (no player)
config.macros.player.html.none=' \
	<table id="%i%" width="%w%" height="%h%" style="background-color:#111;border:0;margin:0;padding:0;"> \
	<tr style="background-color:#111;border:0;margin:0;padding:0;"> \
	<td width="%w%" height="%h%" style="background-color:#111;color:#ccc;border:0;margin:0;padding:0;text-align:center;"> \
	&nbsp; \
	%u% \
	&nbsp; \
	</td></tr></table>';
//}}}

//{{{
// JPG/GIF/PNG still images
config.macros.player.html.image='\
	<a href="%u%" target="_blank"><img width="%w%" height="%h%" style="display:%s%;" src="%u%"></a>';
//}}}

//{{{
// IFRAME web page viewer
config.macros.player.html.iframe='\
	<iframe id="%i%" width="%w%" height="%h%" style="display:%s%;background:#fff;" src="%u%"></iframe>';
//}}}

//{{{
// Windows Media Player
// v7.1 ID: classid=CLSID:6BF52A52-394A-11d3-B153-00C04F79FAA6
// v9	ID: classid=CLSID:22d6f312-b0f6-11d0-94ab-0080c74c7e95
config.macros.player.html.windows=' \
	<object id="%i%" width="%w%" height="%h%" style="margin:0;padding:0;width:%w%;height:%h%px;" \
		classid="CLSID:22d6f312-b0f6-11d0-94ab-0080c74c7e95" \
		codebase="http://activex.microsoft.com/activex/controls/mplayer/en/nsmp2inf.cab#Version=6,4,5,715" \
		align="baseline" border="0" \
		standby="Loading Microsoft Windows Media Player components..." \
		type="application/x-oleobject"> \
		<param name="FileName" value="%u%"> <param name="ShowControls" value="%s%"> \
		<param name="ShowPositionControls" value="1"> <param name="ShowAudioControls" value="1"> \
		<param name="ShowTracker" value="1"> <param name="ShowDisplay" value="0"> \
		<param name="ShowStatusBar" value="1"> <param name="AutoSize" value="1"> \
		<param name="ShowGotoBar" value="0"> <param name="ShowCaptioning" value="0"> \
		<param name="AutoStart" value="%a%"> <param name="AnimationAtStart" value="1"> \
		<param name="TransparentAtStart" value="0"> <param name="AllowScan" value="1"> \
		<param name="EnableContextMenu" value="1"> <param name="ClickToPlay" value="1"> \
		<param name="InvokeURLs" value="1"> <param name="DefaultFrame" value="datawindow"> \
		%x% \
		<embed src="%u%" style="margin:0;padding:0;width:%w%;height:%h%px;" \
			align="baseline" border="0" width="%w%" height="%h%" \
			type="application/x-mplayer2" \
			pluginspage="http://www.microsoft.com/windows/windowsmedia/download/default.asp" \
			name="%i%" showcontrols="%s%" showpositioncontrols="1" \
			showaudiocontrols="1" showtracker="1" showdisplay="0" \
			showstatusbar="%s%" autosize="1" showgotobar="0" showcaptioning="0" \
			autostart="%a%" autorewind="0" animationatstart="1" transparentatstart="0" \
			allowscan="1" enablecontextmenu="1" clicktoplay="0" invokeurls="1" \
			defaultframe="datawindow"> \
		</embed> \
	</object>';
//}}}

//{{{
// RealNetworks' RealOne Player
config.macros.player.html.realone=' \
	<table width="%w%" style="border:0;margin:0;padding:0;"><tr style="border:0;margin:0;padding:0;"><td style="border:0;margin:0;padding:0;"> \
	<object id="%i%" width="%w%" height="%h%" style="margin:0;padding:0;" \
		CLASSID="clsid:CFCDAA03-8BE4-11cf-B84B-0020AFBBCCFA"> \
		<PARAM NAME="CONSOLE" VALUE="player"> \
		<PARAM NAME="CONTROLS" VALUE="ImageWindow"> \
		<PARAM NAME="AUTOSTART" Value="%a%"> \
		<PARAM NAME="MAINTAINASPECT" Value="true"> \
		<PARAM NAME="NOLOGO" Value="true"> \
		<PARAM name="BACKGROUNDCOLOR" VALUE="#333333"> \
		<PARAM NAME="SRC" VALUE="%u%"> \
		%x% \
		<EMBED width="%w%" height="%h%" controls="ImageWindow" type="audio/x-pn-realaudio-plugin" style="margin:0;padding:0;" \
			name="%i%" \
			src="%u%" \
			console=player \
			maintainaspect=true \
			nologo=true \
			backgroundcolor=#333333 \
			autostart=%a%> \
		</OBJECT> \
	</td></tr><tr style="border:0;margin:0;padding:0;"><td style="border:0;margin:0;padding:0;"> \
	<object id="%i%_controls" width="%w%" height="60" style="margin:0;padding:0;display:%s%" \
		CLASSID="clsid:CFCDAA03-8BE4-11cf-B84B-0020AFBBCCFA"> \
		<PARAM NAME="CONSOLE" VALUE="player"> \
		<PARAM NAME="CONTROLS" VALUE="All"> \
		<PARAM NAME="NOJAVA" Value="true"> \
		<PARAM NAME="MAINTAINASPECT" Value="true"> \
		<PARAM NAME="NOLOGO" Value="true"> \
		<PARAM name="BACKGROUNDCOLOR" VALUE="#333333"> \
		<PARAM NAME="SRC" VALUE="%u%"> \
		%x% \
		<EMBED WIDTH="%w%" HEIGHT="60" NOJAVA="true" type="audio/x-pn-realaudio-plugin" style="margin:0;padding:0;display:%s%" \
			controls="All" \
			name="%i%_controls" \
			src="%u%" \
			console=player \
			maintainaspect=true \
			nologo=true \
			backgroundcolor=#333333> \
		</OBJECT> \
	</td></tr></table>';
//}}}

//{{{
// QuickTime Player
config.macros.player.html.quicktime=' \
	<OBJECT ID="%i%" WIDTH="%w%" HEIGHT="%h%" style="margin:0;padding:0;" \
		CLASSID="clsid:02BF25D5-8C17-4B23-BC80-D3488ABDDC6B" \
		CODEBASE="http://www.apple.com/qtactivex/qtplugin.cab"> \
		<PARAM name="SRC" VALUE="%u%"> \
		<PARAM name="AUTOPLAY" VALUE="%a%"> \
		<PARAM name="CONTROLLER" VALUE="%s%"> \
		<PARAM name="BGCOLOR" VALUE="#333333"> \
		<PARAM name="SCALE" VALUE="aspect"> \
		<PARAM name="SAVEEMBEDTAGS" VALUE="true"> \
		%x% \
		<EMBED name="%i%" WIDTH="%w%" HEIGHT="%h%" style="margin:0;padding:0;" \
			SRC="%u%" \
			AUTOPLAY="%a%" \
			SCALE="aspect" \
			CONTROLLER="%s%" \
			BGCOLOR="#333333" \
			EnableJavaSript="true" \
			PLUGINSPAGE="http://www.apple.com/quicktime/download/"> \
		</EMBED> \
	</OBJECT>';
//}}}

//{{{
// Flash Player
config.macros.player.html.flash='\
	<object id="%i%" width="%w%" height="%h%" style="margin:0;padding:0;" \
		classid="clsid:D27CDB6E-AE6D-11cf-96B8-444553540000" \
		codebase="http://download.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=6,0,29,0"> \
		<param name="movie" value="%u%"> \
		<param name="quality" value="high"> \
		<param name="SCALE" value="exactfit"> \
		<param name="bgcolor" value="333333"> \
		%x% \
		<embed name="%i%" src="%u%" style="margin:0;padding:0;" \
			height="%h%" width="%w%" quality="high" \
			pluginspage="http://www.macromedia.com/go/getflashplayer" \
			type="application/x-shockwave-flash" scale="exactfit"> \
		</embed> \
	</object>';
//}}}

In [[coding theory|Coding Theory]] the ''Plotkin Construction'' is a method for constructing a binary code $C(2n,k,d)$ of length $2n$  from two binary codes $C_1(n,k_1,d_1)$ and $C_2(n,k_2,d_2)$, both having length $n$. It applies to linear as well as non-linear codes.

The explicit recursion formula is given by:
\[
C \equiv \{(u, u+v), u \in C_1, v \in C_2\}
\]
The generator matrix $G$ of the resulting code is given in terms of the generator matrices $G_1$ and $G_2$ of the original codes by
\[
G = \begin{pmatrix} G_1 & G_1 \\ 0 & G_2 \end{pmatrix}
\]
!!!!Examples:
A classical application are [[Reed-Muller codes|Reed-Muller Code]].
Here the construction reads:
\[
\operatorname{RM}(r,m) = \{(x, x + y): x \in \operatorname{RM}(r,m ? 1), y \in \operatorname{RM}(r ? 1,m ? 1)\} \; \text {with} \; r = 1,2,\ldots,m ? 1
\]

Notice the similarity with the recursion formula for the [[binomial coefficient|Binomial Coefficient]].


<html><center><img src="images/plotkin.jpg" style="width: 715px; "/></center></html>

Papers:
* [[Plotkin Construction: Rank and Kernel - J. Borges, C. Fernandez|http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.3878v1.pdf]] pct. 0

Theses:
* [[Rekursive Codes mit der Plotkin-Konstruktion und ihre Decodierung - N. Stolte|http://tuprints.ulb.tu-darmstadt.de/183/1/stolte.pdf]]

Links:
* [[Lab Course: Channel Coding - G. Klotz|http://tait.e-technik.uni-ulm.de/~klotz/Lehre/KC08/RMProject_Description.pdf]]

/***

''Inspired by [[TiddlyPom|http://www.warwick.ac.uk/~tuspam/tiddlypom.html]]''

|Name|SplashScreenPlugin|
|Created by|SaqImtiaz|
|Location|http://lewcid.googlepages.com/lewcid.html#SplashScreenPlugin|
|Version|0.21 |
|Requires|~TW2.08+|
!Description:
Provides a simple splash screen that is visible while the TW is loading.

!Installation
Copy the source text of this tiddler to your TW in a new tiddler, tag it with systemConfig and save and reload. The SplashScreen will now be installed and will be visible the next time you reload your TW.

!Customizing
Once the SplashScreen has been installed and you have reloaded your TW, the splash screen html will be present in the MarkupPreHead tiddler. You can edit it and customize to your needs.

!History
* 20-07-06 : version 0.21, modified to hide contentWrapper while SplashScreen is displayed.
* 26-06-06 : version 0.2, first release

!Code
***/
//{{{
var old_lewcid_splash_restart=restart;

restart = function()
{   if (document.getElementById("SplashScreen"))
        document.getElementById("SplashScreen").style.display = "none";
      if (document.getElementById("contentWrapper"))
        document.getElementById("contentWrapper").style.display = "block";

    old_lewcid_splash_restart();

    if (splashScreenInstall)
       {if(config.options.chkAutoSave)
			{saveChanges();}
        displayMessage("TW SplashScreen has been installed, please save and refresh your TW.");
        }
}


var oldText = store.getTiddlerText("MarkupPreHead");
if (oldText.indexOf("SplashScreen")==-1)
   {var siteTitle = store.getTiddlerText("SiteTitle");
   var splasher='\n\n<style type="text/css">#contentWrapper {display:none;}</style><div id="SplashScreen" style="border: 3px solid #ccc; display: block; text-align: center; width: 320px; margin: 100px auto; padding: 50px; color:#000; font-size: 28px; font-family:Tahoma; background-color:#eee;"><b>'+siteTitle +'</b> is loading<blink> ...</blink><br><br><span style="font-size: 14px; color:red;">Requires Javascript.</span></div>';
   if (! store.tiddlerExists("MarkupPreHead"))
       {var myTiddler = store.createTiddler("MarkupPreHead");}
   else
      {var myTiddler = store.getTiddler("MarkupPreHead");}
      myTiddler.set(myTiddler.title,oldText+splasher,config.options.txtUserName,null,null);
      store.setDirty(true);
      var splashScreenInstall = true;
}
//}}}

/***
|Name|Plugin: jsMath|
|Created by|BobMcElrath|
|Email|my first name at my last name dot org|
|Location|http://bob.mcelrath.org/tiddlyjsmath.html|
|Version|1.5.1|
|Requires|[[TiddlyWiki|http://www.tiddlywiki.com]] &ge; 2.0.3, [[jsMath|http://www.math.union.edu/~dpvc/jsMath/]] &ge; 3.0|
!Description
LaTeX is the world standard for specifying, typesetting, and communicating mathematics among scientists, engineers, and mathematicians.  For more information about LaTeX itself, visit the [[LaTeX Project|http://www.latex-project.org/]].  This plugin typesets math using [[jsMath|http://www.math.union.edu/~dpvc/jsMath/]], which is an implementation of the TeX math rules and typesetting in javascript, for your browser.  Notice the small button in the lower right corner which opens its control panel.
!Installation
In addition to this plugin, you must also [[install jsMath|http://www.math.union.edu/~dpvc/jsMath/download/jsMath.html]] on the same server as your TiddlyWiki html file.  If you're using TiddlyWiki without a web server, then the jsMath directory must be placed in the same location as the TiddlyWiki html file.

I also recommend modifying your StyleSheet use serif fonts that are slightly larger than normal, so that the math matches surrounding text, and \\small fonts are not unreadable (as in exponents and subscripts).
{{{
.viewer {
  line-height: 125%;
  font-family: serif;
  font-size: 12pt;
}
}}}

If you had used a previous version of [[Plugin: jsMath]], it is no longer necessary to edit the main tiddlywiki.html file to add the jsMath <script> tag.  [[Plugin: jsMath]] now uses ajax to load jsMath.
!History
* 11-Nov-05, version 1.0, Initial release
* 22-Jan-06, version 1.1, updated for ~TW2.0, tested with jsMath 3.1, editing tiddlywiki.html by hand is no longer necessary.
* 24-Jan-06, version 1.2, fixes for Safari, Konqueror
* 27-Jan-06, version 1.3, improved error handling, detect if ajax was already defined (used by ZiddlyWiki)
* 12-Jul-06, version 1.4, fixed problem with not finding image fonts
* 26-Feb-07, version 1.5, fixed problem with Mozilla "unterminated character class".
* 27-Feb-07, version 1.5.1, Runs compatibly with TW 2.1.0+, by Bram Chen
!Examples
|!Source|!Output|h
|{{{The variable $x$ is real.}}}|The variable $x$ is real.|
|{{{The variable \(y\) is complex.}}}|The variable \(y\) is complex.|
|{{{This \[\int_a^b x = \frac{1}{2}(b^2-a^2)\] is an easy integral.}}}|This \[\int_a^b x = \frac{1}{2}(b^2-a^2)\] is an easy integral.|
|{{{This $$\int_a^b \sin x = -(\cos b - \cos a)$$ is another easy integral.}}}|This $$\int_a^b \sin x = -(\cos b - \cos a)$$ is another easy integral.|
|{{{Block formatted equations may also use the 'equation' environment \begin{equation}  \int \tan x = -\ln \cos x \end{equation} }}}|Block formatted equations may also use the 'equation' environment \begin{equation}  \int \tan x = -\ln \cos x \end{equation}|
|{{{Equation arrays are also supported \begin{eqnarray} a &=& b \\ c &=& d \end{eqnarray} }}}|Equation arrays are also supported \begin{eqnarray} a &=& b \\ c &=& d \end{eqnarray} |
|{{{I spent \$7.38 on lunch.}}}|I spent \$7.38 on lunch.|
|{{{I had to insert a backslash (\\) into my document}}}|I had to insert a backslash (\\) into my document|
!Code
***/
//{{{

// AJAX code adapted from http://timmorgan.org/mini
// This is already loaded by ziddlywiki...
if(typeof(window["ajax"]) == "undefined") {
  ajax = {
      x: function(){try{return new ActiveXObject('Msxml2.XMLHTTP')}catch(e){try{return new ActiveXObject('Microsoft.XMLHTTP')}catch(e){return new XMLHttpRequest()}}},
      gets: function(url){var x=ajax.x();x.open('GET',url,false);x.send(null);return x.responseText}
  }
}

// Load jsMath
jsMath = {
  Setup: {inited: 1},          // don't run jsMath.Setup.Body() yet
  Autoload: {root: new String(document.location).replace(/[^\/]*$/,'jsMath/')}  // URL to jsMath directory, change if necessary
};
var jsMathstr;
try {
  jsMathstr = ajax.gets(jsMath.Autoload.root+"jsMath.js");
} catch(e) {
  alert("jsMath was not found: you must place the 'jsMath' directory in the same place as this file.  "
       +"The error was:\n"+e.name+": "+e.message);
  throw(e);  // abort eval
}

try {
  window.eval(jsMathstr);
} catch(e) {
  alert("jsMath failed to load.  The error was:\n"+e.name + ": " + e.message + " on line " + e.lineNumber);
}

jsMath.Setup.inited=0;  //  allow jsMath.Setup.Body() to run again

// Define wikifers for latex
config.formatterHelpers.mathFormatHelper = function(w) {
    var e = document.createElement(this.element);
    e.className = this.className;
    var endRegExp = new RegExp(this.terminator, "mg");
    endRegExp.lastIndex = w.matchStart+w.matchLength;
    var matched = endRegExp.exec(w.source);
    if(matched) {
        var txt = w.source.substr(w.matchStart+w.matchLength,
            matched.index-w.matchStart-w.matchLength);
        if(this.keepdelim) {
          txt = w.source.substr(w.matchStart, matched.index+matched[0].length-w.matchStart);
        }
        e.appendChild(document.createTextNode(txt));
        w.output.appendChild(e);
        w.nextMatch = endRegExp.lastIndex;
    }
}

config.formatters.push({
  name: "displayMath1",
  match: "\\\$\\\$",
  terminator: "\\\$\\\$\\n?", // 2.0 compatability
  termRegExp: "\\\$\\\$\\n?",
  element: "div",
  className: "math",
  handler: config.formatterHelpers.mathFormatHelper
});

config.formatters.push({
  name: "inlineMath1",
  match: "\\\$",
  terminator: "\\\$", // 2.0 compatability
  termRegExp: "\\\$",
  element: "span",
  className: "math",
  handler: config.formatterHelpers.mathFormatHelper
});

var backslashformatters = new Array(0);

backslashformatters.push({
  name: "inlineMath2",
  match: "\\\\\\\(",
  terminator: "\\\\\\\)", // 2.0 compatability
  termRegExp: "\\\\\\\)",
  element: "span",
  className: "math",
  handler: config.formatterHelpers.mathFormatHelper
});

backslashformatters.push({
  name: "displayMath2",
  match: "\\\\\\\[",
  terminator: "\\\\\\\]\\n?", // 2.0 compatability
  termRegExp: "\\\\\\\]\\n?",
  element: "div",
  className: "math",
  handler: config.formatterHelpers.mathFormatHelper
});

backslashformatters.push({
  name: "displayMath3",
  match: "\\\\begin\\{equation\\}",
  terminator: "\\\\end\\{equation\\}\\n?", // 2.0 compatability
  termRegExp: "\\\\end\\{equation\\}\\n?",
  element: "div",
  className: "math",
  handler: config.formatterHelpers.mathFormatHelper
});

// These can be nested.  e.g. \begin{equation} \begin{array}{ccc} \begin{array}{ccc} ...
backslashformatters.push({
  name: "displayMath4",
  match: "\\\\begin\\{eqnarray\\}",
  terminator: "\\\\end\\{eqnarray\\}\\n?", // 2.0 compatability
  termRegExp: "\\\\end\\{eqnarray\\}\\n?",
  element: "div",
  className: "math",
  keepdelim: true,
  handler: config.formatterHelpers.mathFormatHelper
});

// The escape must come between backslash formatters and regular ones.
// So any latex-like \commands must be added to the beginning of
// backslashformatters here.
backslashformatters.push({
    name: "escape",
    match: "\\\\.",
    handler: function(w) {
        w.output.appendChild(document.createTextNode(w.source.substr(w.matchStart+1,1)));
        w.nextMatch = w.matchStart+2;
    }
});

jsMath.Extension.Require("AMSmath");
jsMath.Extension.Require('underset-overset');
jsMath.Extension.Require("boldsymbol");
jsMath.Extension.Require("autobold");
jsMath.Extension.Require("moreArrows");
jsMath.Extension.Require("AMSsymbols");

/*
jsMath.Font.Load("eufm10");
jsMath.Font.Load("cmmib10");  */

/* insert jsMath LaTeX macros here */

/* jsMath.Macro('Chr','{\\small\\genfrac{\\{}{\\}}{0pt}{}{#1}{#2}}',2); */
jsMath.Macro('Chr','   \\left\\{ {\\begin{array}{*{20}c}  #1  \\\\   {#2}  \\\\ \\end{array} } \\right\\}      ',2);
jsMath.Macro('mathfrak','\\eufm #1 ',1);
jsMath.Macro('mb','\\mathbf{#1}',1);
jsMath.Macro('bs','\\boldsymbol{#1}',1);
jsMath.Macro('slash', '{#1}\\!\\!\\!\/',1);
jsMath.Macro('cunit','\\mathbf i_{\\small \\mathbb C}');


config.formatters=backslashformatters.concat(config.formatters);

window.wikify = function(source,output,highlightRegExp,tiddler)
{
    if(source && source != "") {
        if(version.major == 2 && version.minor > 0) {
            var wikifier = new Wikifier(source,getParser(tiddler),highlightRegExp,tiddler);
            wikifier.subWikifyUnterm(output);
        } else {
            var wikifier = new Wikifier(source,formatter,highlightRegExp,tiddler);
            wikifier.subWikify(output,null);
        }
        jsMath.ProcessBeforeShowing();
    }
}
//}}}

/***
|''Name''|PluginMathJax|
|''Description''|Displays TeX math using MathJax|
|''Author''|[[Canada East|http://tiddlywiki.canada-east.ca/]]|
|''Version''|1.3|
|''Date''|2010-10-07|
|''CodeRepository''|[[PluginMathJax|http://tiddlywiki.canada-east.ca/#PluginMathJax]]|
|''CoreVersion''|[[2.6.1|http://www.tiddlywiki.com]]|
|''Requires''|[[MathJax v1.01|http://www.mathjax.org/]]|
|''Feedback''|[[Contact|https://spreadsheets.google.com/viewform?formkey=dGg2RkpxZW5zWTh6QjZxOXgzZUlfakE6MQ]]|
|''Tweaks''|~MathJax location and default HTML-CSS scale changed by Gareth Davies.  I've also added a hook to a ~MathJax extension for managing local definitions written by Richard Lupton and I've set the newcommand extension to load on start-up.|
!Description
This plugin uses [[MathJax|http://www.mathjax.org/]] to typeset ([[AMS|http://www.ams.org/publications/authors/tex/amslatex]]) [[LaTeX|http://www.latex-project.org/]]  math. It can also be configured to use additional MathJax functionality.
>"MathJax is an open source JavaScript display engine for mathematics that works in all modern browsers."
!Notes
Right click any math display for a MathJax menu. The user can select the renderer and zoom settings. It performs best in [[Webkit|http://en.wikipedia.org/wiki/List_of_web_browsers#WebKit-based_browsers]] based browsers. Larger math displays such as the additional examples tiddler below can put quite a load on IE. PluginMathJax is based on: [[Plugin: jsMath|http://bob.mcelrath.org/tiddlyjsmath.html]]
!Installation
#''Backup'' your TiddlyWiki!
#It is required that the MathJax directory is installed in '''js/MathJax/''' in the same location as the TiddlyWiki html file.<br>(Or edit the script source where commented in the plugin code below after installation to match the location of your MathJax install.)
#Install this plugin (and examples tiddler linked below if desired).
!Usage
|!Source|!Output|h
|{{{The variable $x$ is real.}}}|The variable $x$ is real.|
|{{{The variable \(y\) is complex.}}}|The variable \(y\) is complex.|
|{{{This \[\int_a^b x = \frac{1}{2}(b^2-a^2)\] is an easy integral.}}}|This \[\int_a^b x = \frac{1}{2}(b^2-a^2)\] is an easy integral.|
|{{{This $$\int_a^b \sin x = -(\cos b - \cos a)$$ is another easy integral.}}}|This $$\int_a^b \sin x = -(\cos b - \cos a)$$ is another easy integral.|
|{{{Block formatted equations may also use the 'equation' environment \begin{equation}  \int \tan x = -\ln \cos x \end{equation} }}}|Block formatted equations may also use the 'equation' environment \begin{equation}  \int \tan x = -\ln \cos x \end{equation}|
|{{{Equation arrays are also supported \begin{eqnarray} a &=& b \\ c &=& d \end{eqnarray} }}}|Equation arrays are also supported \begin{eqnarray} a &=& b \\ c &=& d \end{eqnarray} |
|{{{I spent \$7.38 on lunch.}}}|I spent \$7.38 on lunch.|
|{{{I had to insert a backslash (\\) into my document}}}|I had to insert a backslash (\\) into my document|
| <br>[[Complete list of supported LaTeX commands|http://www.mathjax.org/resources/docs/?tex.html#supported-latex-commands]] |>|
!Examples
[[Additional MathJax Examples|MathJax Examples]]
!Configuration
MathJax can be manually configured if desired by editing the code below (advanced). See the [[MathJax documentation|http://www.mathjax.org/resources/docs/?configuration.html#configuration-options-by-component]] for details.
!Revision History
*v1.3, 2010-10-07, returned to original formatters design, kept modified wikify and recommended way of loading MathJax dynamically, removed the tex2jax extension and corrected several browser compatibility issues (InnerHTML for Opera and IE9).
*v1.2, 2010-10-05, removed some redundant MathJax config entries, moved modified wikify and MathJax.Hub.Queue call.
*v1.1, 2010-10-03, autoLinkWikiWords disabled in absence of DisableWikiLinksPlugin, modifed wikify.
*v1.0, 2010-09-26, Initial Release
!Code
***/
//{{{

if(!version.extensions.PluginMathJax) {
    version.extensions.PluginMathJax = { installed: true };

    config.extensions.PluginMathJax = {

        install: function() {

            var script = document.createElement("script");
            script.type = "text/javascript";

            // *** Use the location of your MathJax! *** :
            /*
             * Gareth: The following line assumes you have
             * MathJax installed on your server in a sensible
             * location.  I've commented this out.
             */
            //script.src = "js/MathJax/MathJax.js";

            /*
             * Because this tiddlywiki is currently hosted on
             * tiddlyspace.com I've had to point to the 'MathJax
             * Content Delivery Network' instead.
             */
            script.src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"
            // EndGareth

            /*
             * Gareth: Richard's local definition ~MathJax
             * extension (implementation of TeXs \let command)
             * has been added to the list of extensions along
             * with the newcommand extension upon which it
             * depends.  Also, the scale option for HTML-CSS was
             * changed from 115 to 100.
             */
            var mjconfig = 'MathJax.Hub.Config({' +
            'jax: ["input/TeX","output/HTML-CSS"],' +
            'extensions: ["TeX/AMSmath.js", "TeX/AMSsymbols.js", "TeX/newcommand.js", "http://oxkunengroup.tiddlyspace.com/localTeX.js"],' +
            '"HTML-CSS": {' +
                'scale: 100' +
                '}' +
            '});' +

            'MathJax.Hub.Startup.onload();';

            var ie9RegExp = /^9\./;
            var UseInnerHTML = (config.browser.isOpera || config.browser.isIE && ie9RegExp.test(config.browser.ieVersion[1]));

            if (UseInnerHTML) {script.innerHTML = mjconfig;}
                else {script.text = mjconfig;}

            script.text = mjconfig;

            document.getElementsByTagName("head")[0].appendChild(script);

            // Define wikifers for latex
            config.formatterHelpers.mathFormatHelper = function(w) {
                var e = document.createElement(this.element);
                e.type = this.type;
                var endRegExp = new RegExp(this.terminator, "mg");
                endRegExp.lastIndex = w.matchStart+w.matchLength;
                var matched = endRegExp.exec(w.source);
                if(matched) {
                    var txt = w.source.substr(w.matchStart+w.matchLength,
                        matched.index-w.matchStart-w.matchLength);
                    if(this.keepdelim) {
                      txt = w.source.substr(w.matchStart, matched.index+matched[0].length-w.matchStart);
                    }
                    if (UseInnerHTML) {
                        e.innerHTML = txt;
                    } else {
                        e.text = txt;
                    }
                    w.output.appendChild(e);
                    w.nextMatch = endRegExp.lastIndex;
                }
            }

            config.formatters.push({
              name: "displayMath1",
              match: "\\\$\\\$",
              terminator: "\\\$\\\$\\n?",
              termRegExp: "\\\$\\\$\\n?",
              element: "script",
              type: "math/tex; mode=display",
              handler: config.formatterHelpers.mathFormatHelper
            });

            config.formatters.push({
              name: "inlineMath1",
              match: "\\\$",
              terminator: "\\\$",
              termRegExp: "\\\$",
              element: "script",
              type: "math/tex",
              handler: config.formatterHelpers.mathFormatHelper
            });

            var backslashformatters = new Array(0);

            backslashformatters.push({
              name: "inlineMath2",
              match: "\\\\\\\(",
              terminator: "\\\\\\\)",
              termRegExp: "\\\\\\\)",
              element: "script",
              type: "math/tex",
              handler: config.formatterHelpers.mathFormatHelper
            });

            backslashformatters.push({
              name: "displayMath2",
              match: "\\\\\\\[",
              terminator: "\\\\\\\]\\n?",
              termRegExp: "\\\\\\\]\\n?",
              element: "script",
              type: "math/tex; mode=display",
              handler: config.formatterHelpers.mathFormatHelper
            });

            backslashformatters.push({
              name: "displayMath3",
              match: "\\\\begin\\{equation\\}",
              terminator: "\\\\end\\{equation\\}\\n?",
              termRegExp: "\\\\end\\{equation\\}\\n?",
              element: "script",
              type: "math/tex; mode=display",
              handler: config.formatterHelpers.mathFormatHelper
            });

            // These can be nested.  e.g. \begin{equation} \begin{array}{ccc} \begin{array}{ccc} ...
            backslashformatters.push({
              name: "displayMath4",
              match: "\\\\begin\\{eqnarray\\}",
              terminator: "\\\\end\\{eqnarray\\}\\n?",
              termRegExp: "\\\\end\\{eqnarray\\}\\n?",
              element: "script",
              type: "math/tex; mode=display",
              keepdelim: true,
              handler: config.formatterHelpers.mathFormatHelper
            });

            // The escape must come between backslash formatters and regular ones.
            // So any latex-like \commands must be added to the beginning of
            // backslashformatters here.
            backslashformatters.push({
                name: "escape",
                match: "\\\\.",
                handler: function(w) {
                    w.output.appendChild(document.createTextNode(w.source.substr(w.matchStart+1,1)));
                    w.nextMatch = w.matchStart+2;
                }
            });

          config.formatters=backslashformatters.concat(config.formatters);

          old_wikify = wikify;
          wikify = function(source,output,highlightRegExp,tiddler)
          {
              old_wikify.apply(this,arguments);
              if (window.MathJax) {MathJax.Hub.Queue(["Typeset",MathJax.Hub,output])}
          };

        }
    };

  config.extensions.PluginMathJax.install();

}

//}}}

Papers:
* [[Theoretical Analysis of a Reported Weak Gravitational Shielding Effect.(1996) - G. Modanese|http://arxiv.org/pdf/hep-th/9505094v2]] [[local|papers/9505094v2.pdf]] [[pct. 26|http://scholar.google.de/scholar?hl=de&lr=&cites=800284828379922753&um=1&ie=UTF-8&ei=LNU2Tb3pJcjrOYamqJ8E&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCMQzgIwAA]] - "These values support our hypothesis that the total cosmological term is positive in the superconductor."
* [[Impulse Gravity Generator Based on Charged Y Ba2Cu3O7-y Superconductor with Composite Crystal Structure (2001) - E. Podkletnov, G. Modanese|http://arxiv.org/PS_cache/physics/pdf/0108/0108005v2.pdf]] [[local|papers/0108005v2.pdf]] [[pct. 16|http://scholar.google.de/scholar?cites=1784556933495283544&as_sdt=2005&sciodt=2000&hl=de]]
* [[A Theory of the Podkletnov Effect based on General Relativity: Anti-Gravity Force due to the Perturbed Non-Holonomic Background of Space (2007) - D. Rabounski, L. Borissova|http://www.ptep-online.com/index_files/2007/PP-10-13.PDF]] [[local|papers/PP-10-13.PDF]] [[pct. 1|http://scholar.google.de/scholar?hl=de&lr=&cites=12392183758488447834&um=1&ie=UTF-8&ei=9WEsTautLoqVswbPmIzaBw&sa=X&oi=science_links&ct=sl-citedby&resnum=5&ved=0CDIQzgIwBA]]

See also:
* [[Gravitomagnetism]]
* [[Bore hole anomaly|Bore Hole Anomaly]]

Videos:
* [[Rare Podkletnov Interview 2004|http://www.youtube.com/watch?v=jQBb76Snx50&feature=mfu_in_order&list=UL]]
* [[Podkletnov - Rotating Superconductors|http://video.google.com/videoplay?docid=8902110659306746348#docid=-6561466449871868116]]

The ''Poincaré Gauge Theory of Gravity (PGT)'' is a [[gauge theory of gravity|Gauge Theory of Gravity]] based on the [[Poincaré group|Poincaré Transformation]].

The description naturally includes [[torsion|Torsion]]. It is now generally accepted that torsion is an inevitable feature of a gauge theory based on the Poincaré group.

PGT has the geometric structure of a [[Riemann-Cartan space|Riemann-Cartan Space]] and therefore encompasses gauge theories of gravity based on [[teleparallel spaces|Teleparallel Gravity]] and [[Riemann spaces|Riemann Space]].

Given the generators of Lorentz-rotations $\mb M_{ab}$ and the generators of spacetime translations $\mb P_a$ of the Poincaré group, the gauge fields $\mb A_\mu$ are given by
\[
\mb A_\mu = h_\mu^a \mb P_a  - \frac 12 \omega_\mu^{ab} \mb M_{ab}
\]
with $\omega_\mu^{ab}$ the coefficients of the [[spin connection|Spin Connection]].
The field strength $\mb F_{\mu\nu}$ (or curvature $\mb R_{\mu\nu}$) reads
\[
\mb F_{\mu\nu} \equiv \mb R_{\mu\nu} = R_{\mu\nu}^a \mb P_a - \frac 12 R_{\mu\nu}^{ab} \mb M_{ab}
\]
A Lagrangian invariant under local translations and local Lorentz transformations is not invariant under general coordinate transformations unless a curvature constraint is imposed. This can be done via
\[
R_{\mu\nu}^a = 0
\]
This way the translations and rotations cease to be independent, i.e. $\omega_\mu^{ab} = \omega_\mu^{ab}(h)$.

The most general Poincaré gauge theory Lagrangian is given by
\[
\mathcal{L} =  \text{const. + (curvature scalar) + (torsion)}^2 + \text{(curvature)}^2
\]
Choosing only the first two possible terms for the Lagrangians, namely the constant and the curvature scalar, one obtains the [[Einstein-Cartan-Sciama-Kibble (ECSK) theory|Einstein-Cartan Theory]] which therefore is it is a specific realisation of a PGT.

Papers:
* [[Three Lectures on Poincaré Gauge Theory (2003) - M. M. Blagojevi?|http://arxiv.org/PS_cache/gr-qc/pdf/0302/0302040v1.pdf]] [[pct. 11|http://scholar.google.de/scholar?cites=3112120635154274210&as_sdt=2005&sciodt=2000&hl=de]]

Presentations:
* [[Supergravity - A.B. Lahanas|http://www.physics.ntua.gr/corfu2009/Talks/alahanas@phys_uoa_gr_03.pdf]] [[local|presentations/alahanas_phys_uoa_gr_03.pdf]]

>All of the recent theories of elementary particles have been shaped by the paper by Wigner, containing the classification of the irreducible representations of $ SL(2,\mathbb R) \rtimes \mathbb R^{1,3}$.... It is difficult to overestimate the "importance of this paper, which will certainly stand as one of the great intellectual achievements of our century.
> - Shlomo Sternberg - Group Theory and Physics

A ''Poincaré Transformation'' is an inhomogeneous [[Lorentz transformation|Lorentz Transformation]] given by
\[
x'^a={\Lambda^a}_b x^b+t^a
\]
The ''Poincaré Group''  $ISO(1,3)$ (or  ''Inhomogeneous Lorentz Group'') is the [[semi-direct product|Semi Direct Product]] of the [[Lorentz group|Lorentz Group]] and the translation group ($\mathbb R^{1,3} \rtimes O(1,3)$), such that the translations form an invariant subgroup but the the Lorentz-transformations do not. It is a $10$-parameter [[Lie group|Lie Group]] and the associated [[Lie algebra|Lie Algebra]] is spanned by the $4$ generators of translations $\mb P_a$ and the $6$ Lorentz-generators $\mb M_{ab}$ which satisfy
\begin{eqnarray}
[\mb P_a, \mb P_b] &=& \mb 0 \\
[\mb M_{ab}, \mb P_c] &= &\eta_{ac} \mb P_b-\eta_{bc} \mb P_a \\
&= & 2 \eta_{[a|c|} \mb P_{b]} \\
[\mb M_{ab}, \mb M_{cd}] &= &\eta_{ac} \mb M_{bd}- \eta_{ad} \mb M_{bc} + \eta_{bd} \mb M_{ac} - \eta_{bc}  \mb M_{ad} \\
& = & 2 \eta_{a[c} \mb M_{|b|d]} + 2 \eta_{b[d} \mb M_{|a|c]}
\end{eqnarray}
The full Poincaré group which includes time reversal and space reflections is the [[isometry group|Isometry]] of Minkowski space. The conservation laws of [[energy-momentum|Stress Energy Tensor]] and angular-momentum in special relativity are connected with the Poincaré group. In fact, according to [[Noether’s theorem the invariance of a physical system under a spacetime translation leads to the conservation of the canonical energy-momentum tensor, whereas the invariance under a Lorentz transformation leads to the conservation of the canonical angular-momentum tensor. When passing to general relativity, these two tensors are modified by the presence of gravitation.

The Lie algebra of the Poincaré group has two basic invariants, interpreted physically as mass and spin.
The Poincaré group provides a correct description of masses and spin of elementary particles which is not the case for the mere Lorentz group.

!!!!Generalisations
* See: [[Twisted Poincaré symmetry|Twisted Poincaré Symmetry]]

<html><center><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_60.html" width=51% height=86></iframe></center></html>Lectures:

Lectures:
* [[The Lorentz and Poincaré Groups and their Representations - R. M. D. Delgado|http://www.hep.manchester.ac.uk/u/rosa/Symmetries.pdf]] [[local|lectures/Symmetries.pdf]]
* [[Lie Groups and Lie Algebras - I. Runkel|http://www.mth.kcl.ac.uk/~bdoyon/Lie2010/Runkel.pdf]] [[local|lectures/Runkel.pdf]]

Documents:
* [[Lorentz Group and the Dirac Algebra|http://www.physics.mcgill.ca/~guymoore/appendixC.pdf]] - Appendix C of the book "The Standard Model: A Primer" by C. P. Burgess and Guy D. Moore.@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/Poincare.jpg][Comments]]}}}&nbsp; @@

The ''Poincaré\-Birkhoff\-Witt (PBW) Theorem'' implies that any [[Lie algebra|Lie Algebra]] $L$ is isomorphic to a subalgebra of the commutator algebra of some associative algebra. This result is established by constructing an associative universal enveloping algebra $U(L)$ for $L$, together with an injective Lie algebra [[homomorphism|Homomorphism]] $L \to U(L)$.

<<tiddler [[include_tiddlers/Point Defect.html#"Point Defect"]]>>

Die ''Poisson Distribution'' (or ''Poisson Law of Small Numbers'') is a discrete probability distribution one gets approximately when doing many Bernoulli ("yes"-/"no"-) experiments in a row where the probability for one of the two outcomes is small. It is therefore a special case of the exact binomial distribution.

The probability of having $n$ "positive" outcomes, doing $N$ Bernoulli experiments is given by
\[
p_N (n) = \frac{\lambda_N^n}{n!}\, \mathrm{e}^{-\lambda_N}
\]
where
\[
\lambda_N = \sqrt N = \mu = \sigma^2
\]
with $\mu$ the mean value and $\sigma$ the standard deviation.

Thus, the number $n$ fluctuates about its mean $λ_N$ with a standard deviation $\sigma =\sqrt{\lambda}$. These fluctuations are denoted as ''Poisson [[Noise]]'' or (particularly in electronics) as shot noise.

Links:
* [[WIKIPEDIA - Poisson Distribution|http://en.wikipedia.org/wiki/Poisson_distribution]]
* [[WIKIPEDIA - Shot Noise|http://en.wikipedia.org/wiki/Shot_noise]]

<<tiddler [[include_tiddlers/Poisson Manifold.html#"Poisson Manifold"]]>>

Theses:
* [[Vierdimensionale Plytope - M. Holzbauer|http://www.ub.tuwien.ac.at/dipl/2007/AC05034559.pdf]]

The ''Polyvector Action Principle'' or ''Principle of least Polyvector Action'' is the conventional action principle applied to a [[polyvector action |Polyvector Action]] $S[\mb{X}]$:
\[
\delta S[\mb{X}] = 0
\]
The (minimum-) solutions of this equation are the [[Polyvector Euler-Lagrange equations|Polyvector Euler-Lagrange Equations]].

The minimum of the polyvector action represents the "shortest path" in polyvector space (polyvector geodesic). Contrary to a classical vector space this path is represented by a linear combination of points, curves, surfaces, volumes, etc.
This generalizes the concept of a [[geodesic|Geodesic Equation]] of a point particle in a curved spacetime background.
If one also considers second grades in the polyvector action (as is done in the Clifford algebra approach) one gets the description of a spinning particle in a curved background. Such particles obey an extended equivalence principle, the [[polyvector equivalence principle|Polyvector Equivalence Principle]]. Their trajectory can be interpreted as one of an extended object and might be seen as an alternative to string theory where the Polyakov and [[Nambu-Goto actions|Dirac-Nambu-Goto Action]] for example describe the trajectory of a string that sweeps out a worldsheet.
If one furthermore takes into account all relevant grades in $4$ spacetime dimensions, i.e. grades up to order $4$, one gets the geodesic of the whole universe including the "subgeodesics" of matter within it: ''The Trajectory of the Universe''. This opens up the possibility of a description of the dynamics of the whole universe which is somewhat reminiscent of the Hartle Hawking [[wavefunction of the universe|Wavefunction of the Universe]] description.

<<tiddler [[include_tiddlers/Polyvector Analysis.html#"Polyvector Analysis"]]>>

<<tiddler [[include_tiddlers/Polyvector Autoparallelity.html#"Polyvector Autoparallelity"]]>>

<<tiddler [[include_tiddlers/Polyvector Derivative.html#"Polyvector Derivative"]]>>

<<tiddler [[include_tiddlers/Polyvector Dirac Equation.html#"Polyvector Dirac Equation"]]>>

<html><center><img src="images/spin.gif" style="width: 340px; "/></center></html>
The ''Weak Polyvector Equivalence Principle'' or ''Generalized Weak Equivalence Principle'' states that all particles follow the same path in a gravitational field independent of their [[polyvector mass|Physical Polyvectors]].
This generalizes the conventional [[equivalence principle|Equivalence Principles]] of general relativity which only holds for point particles. If e.g. a spinning particle is included, which at least classically can be interpreted as an extended object, the particle is supposed to carry out a Zitterbewegung instead of following the more straight path of the point particle. The ''Weak Polyvector Equivalence Principle'' restores the equivalence in such situations in that it introduces additional [[polyvector mass terms|Physical Polyvectors]].

The ''Strong Polyvector Equivalence Principle'' states that, given fields, an [[autoparallel|Autoparallelity]] system in [[P-space|Polyvector Space]] can be found that can be regarded as a generalized inertial system, i.e. one in which the laws of a "flat" P-space are valid. In this system it is not only compensated for linear accelerations (as is the case for the strong equivalence principle of general relativity, but also for angular and higher order ones). In general relativity one gets to a local inertial system by carrying out a boost which transforms away the gravitational field. It is conjectured that a generalization of boosts to P-space allows for transforming away all other fields as well. A boost in polyvector space can be understood as a rotation in this space which generalizes a Lorentz boost, which is a rotation in the Minkowski subspace (grade 1 subspace) only. (One can therefore regard it as an "enhanced" or "extended" Lorentz transformation). The algebraic consequences are, that the grades of physical entities depend on the observer's polyvector frame.
The principle was suggested in the context of [[Clifford algebras|Clifford Algebra]] by [[W. M. Pezzaglia Jr.|http://www.clifford.org/~wpezzag/index.html]] under the name [[local automorphism invariance|Principle of Local Polyvector Invariance]]. As a Clifford space is a special P-space, a generalization to P-spaces is therefore suggestive.

Note, that the polyvector equivalence principle cannot apply to a spacetime described in terms of [[Riemannian geometry|Riemann Space]] (only), as "the best one can do" in this case is choose local [[Riemann normal coordinates|Riemann Normal Coordinates]], leaving one with third- and higher order deviations (corresponding with "tidal forces").

An idea is to realise the polyvector equivalence principle by introducing [[canonical coordinates|Canonical Coordinates]] instead and trying to describe spacetime by means of algebras (symmetries) dictated by them.

<<tiddler [[include_tiddlers/Polyvector Gauge Fields.html#"Polyvector Gauge Fields"]]>>

<<tiddler [[include_tiddlers/Polyvector Gauge Gravity.html#"Polyvector Gauge Gravity"]]>>

<<tiddler [[include_tiddlers/Polyvector Hilbert Space.html#"Polyvector Hilbert Space"]]>>

<<tiddler [[include_tiddlers/Polyvector Invariant Mass.html#"Polyvector Invariant Mass"]]>>

The ''Polyvector Jacobi Matrix'' $(J_\mb{F})_{ij}(\mb{X})$ generalizes the the conventional [[Jacobi Matrix]] in that a real valued function $f_i(x_1, x_2, \dots, x_n), i = 1,...,m$  is replaced by a polyvector valued function $F_i(X_1, X_2, \dots, X_{2^n}), i = 1,...,2^m$.

<<tiddler [[include_tiddlers/Polyvector Klein-Gordon Equation.html#"Polyvector Klein-Gordon Equation"]]>>

<<tiddler [[include_tiddlers/Polyvector Lagrange Function.html#"Polyvector Lagrange Function"]]>>

<<tiddler [[include_tiddlers/Polyvector Partial Derivative.html#"Polyvector Partial Derivative"]]>>

<<tiddler [[include_tiddlers/Polyvector Path Length.html#"Polyvector Path Length"]]>>

<<tiddler [[include_tiddlers/Polyvector Propagator.html#"Polyvector Propagator"]]>>

<<tiddler [[include_tiddlers/Polyvector Proper Time.html#"Polyvector Proper Time"]]>>

<<tiddler [[include_tiddlers/Polyvector Space.html#"Polyvector Space"]]>>

<<tiddler [[include_tiddlers/Polyvector Total Differential.html#"Polyvector Total Differential"]]>>

<<tiddler [[include_tiddlers/Polyvector Vielbein.html#"Polyvector Vielbein"]]>>

<<tiddler [[include_tiddlers/Polyvector Worldline.html#"Polyvector Worldline"]]>>

The ''Pomeransky\-Khriplovich Equations'' describe the motion of a spinning particle in an electromagnetic and gravitational field. Spin is considered as being linear and quadratic (pole-dipole approximation).

Papers:
* [[Spinning Relativistic Particles in External Fields - I.B. Khriplovich|http://arxiv.org/PS_cache/arxiv/pdf/0801/0801.1881v1.pdf]]
* [[Classical and Quantum Spins in Curved Spacetimes - A. J. Silenko|http://th-www.if.uj.edu.pl/acta/sup1/pdf/s1p0087.pdf]]   [[Transparencies|http://www.fuw.edu.pl/~krp/mathisson/wyklady/Silenko.ppt]]

>... power-associativity is the very least assumption we have to require to be even able to speak about the exponential function.
>- K. H. Hofmann, K. Strambach -

An algebra $\mathcal A$ is called ''Power Associative'' if every $\mb A \in \mathcal A$ satisfies
\[
\mb A^m \mb A^n = \mb A^{m + n}
\]

If an algebra satisfies the [[flexibility|Flexible Algebra]] law (a.k.a [[monoassociative identity|Monoassociativity]])
\[
\mb A \mb A^2 = \mb A^2 \mb A \,\Leftrightarrow \,[\mb A, \mb A, \mb A] = 0
\]
and the [[Jordan Identity]]
\[
\mb A^2 \mb A^2 = \mb A \mb A^3 = \mb A^3 \mb A
\]
it is power associative.

Powerassociativity is required to define the series expansion of the exponential function in the usual way
\[
\exp(\mb A) = \sum_{n = 0}^\infty \frac{\mb A^n}{n!}
\]
as the powers of $\mb A$ are unambiguous in this case.

Any power associative algebra satisfies:
\[
[\mb A, \mb A, \mb B] + [\mb A, \mb B, \mb A] + [\mb B, \mb A, \mb A] = 0
\]

All semisimple commutative power-associative algebras of characteristic $0$ are [[Jordan algebras|Jordan Algebra]].

Papers:
* [[Power-Associative Rings (1947) - A. A. Albert|http://www.ams.org/journals/tran/1948-064-03/S0002-9947-1948-0027750-7/S0002-9947-1948-0027750-7.pdf]] [[local|papers/S0002-9947-1948-0027750-7.pdf]] {{t100Cite{[[pct. 218|http://scholar.google.de/scholar?cites=3800344251440980725&hl=de&as_sdt=2000]]}}}

<<tiddler [[include_tiddlers/Power Law.html#"Power Law"]]>>

The ''Poynting Theorem'' in electrodynamics states:
\begin{equation}
\vec E \cdot \vec  j + \vec \nabla \cdot \vec  S  -\frac{\partial \rho}{\partial t} = 0
\end{equation}
With $\rho$ the energy density of the electromagnetic field:
\begin{equation}
\rho = \frac{1}{2} ( \vec E  \cdot \vec D + \vec B \cdot \vec H)
\end{equation}
and $\vec S$ the Poynting vector
\begin{equation}
\vec S = \vec E \times \vec H
\end{equation}

<<tiddler [[include_tiddlers/Prime Number.html#"Prime Number"]]>>

<<tiddler [[include_tiddlers/Primitive Element.html#"Primitive Element"]]>>

Papers:
* [[This Time – What a Strange Turn of Events! (2010)- P. E. Gibbs|http://prespacetime.com/index.php/pst/article/viewFile/12/9]] pct. 0

Links:
* [[WIKIPEDIA - Event Symmetry|http://en.wikipedia.org/wiki/Event_symmetry]]

[[Cellular automata|Cellular Automaton]] are usually assumed to be entirely deterministic. One may however introduce random noise directly into the cellular automaton rules, making cellular automata analogous to lattice spin systems at nonzero temperature.
Such ''Probabilistic Cellular Automata'' (''PCA'') are found to exhibit phase transitions as a function of noise level. It is usually assumed that at each time step, the values of all the sites in a PCA are updated together.

Papers:
* [[Growth of Surfaces Generated by a Probabilistic Cellular Automaton - P. Bhattacharyya|http://arxiv.org/PS_cache/cond-mat/pdf/9811/9811160v2.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=7935521626988383074&hl=de]]

<<tiddler [[include_tiddlers/Problems in Physics.html#"Problems in Physics"]]>>

> All is flux, nothing stays still.
> - Heraclit -

''Process Physics'' models reality as self-organising relational information and takes account of the limitations of logic, discovered by [[Gödel|Gödel's Theorems]] and extended by [[Chaitin|Gregory Chaitin]] by using the concept of self-referential noise (SNR). It provides a dynamical model where space and matter are seen to emerge from a fundamentally random but self-organising system. The system operates by forming a dissipative structure, driven by the SRN, and which is characterised by an emergent and expanding three-dimensional [[fractal space|Fractal Spacetime]] in which are embedded self-replicating fractal topological defects. This emergence is a non-algorithmic increase in complexity in the system. The emergent space is continually undergoing replacement of its components. The key behavioural mode for defects which are sufficiently large is that their existence, as identified by their topological properties, will survive the ongoing process of mutation, decay and regeneration; they are topologically self-replicating.
In process physics the collapse of the wavefunction finds its explanation in Gödel's incompleteness theorem and its associated SRN within a process-system. Process physics predicts both fermionic and bosonic quantum modes, but identified as topologically encoded information and not with objects or 'particles'. Unlike conventional quantum field theory the fermionic/bosonic modes are fractal in nature. At all levels the model exhibits evolved processes for self-replicating information.

Links:
* [[Website|http://www.flinders.edu.au/science_engineering/caps/our-school/staff-postgrads/academic-staff/cahill-reg/process-physics/home.cfm]]

Papers:
* [[Process Physics (2003) - R. T. Cahill|http://www.mountainman.com.au/process_physics/HPS13.pdf]] [[local|papers/HPS13.pdf]] [[pct. 53|http://scholar.google.de/scholar?cites=11399556518260808129&hl=de]]
* [[Self-Referential Noise as a Fundamental Aspect of Reality (1999) - R. T. Cahill, C. M. Klinger|http://arxiv.org/PS_cache/gr-qc/pdf/9905/9905082v1.pdf]] [[local|papers/9905082v1.pdf]] [[pct. 30|http://scholar.google.de/scholar?cites=13250125711378012782&hl=de]]
* [[Process Physics: From Quantum Foam to General Relativity (2002) - R. T. Cahill|http://arxiv.org/PS_cache/gr-qc/pdf/0203/0203015v1.pdf]] [[local|papers/0203015v1.pdf]] [[pct. 23|http://scholar.google.de/scholar?cites=9301220453371540843&hl=de]]
* [[Process Physics: Modelling Reality as Self-Organising Information (2000) - R. T. Cahill, C. M. Klinger, K. Kitto|http://arxiv.org/PS_cache/gr-qc/pdf/0009/0009023v1.pdf]] [[local|papers/0009023v1.pdf]] [[pct. 19|http://scholar.google.de/scholar?cites=14094756012964523909&hl=de]]

Articles:
* [[Random Reality - M. Chown|http://www.newscientist.com/article/mg16522274.300-random-reality.html]]

<<tiddler [[include_tiddlers/Projective General Linear Group.html#"Projective General Linear Group"]]>>

Papers:
* [[Incidence Projective Space (a Reduction Theorem in a Plane) - E. Kusak, W. Leonczuk|http://www.cs.ualberta.ca/~piotr/Mizar/mirror/httpd/JFM/pdf/projred1.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=6322663170675441136&hl=de]]

Google books:
* [[Compact Projective Planes - H. Salzmann|http://books.google.de/books?hl=de&lr=&id=5yqGmM7ajDoC&oi=fnd&pg=PR5&dq=Kleinfeld+proof+of+the+Bruck-Kleinfeld-Skornyakov+theorem&ots=pBwEzZPmzJ&sig=hJLTTUfoEF_emXCBOXGIqP2lUrg#v=onepage&q=&f=false]] [[local|google_books/CompactProjectivePlanes.pdf]] {{t100Cite{[[bct. 140|http://scholar.google.de/scholar?cites=3092334794607053342&hl=de]]}}}

Papers:
* [[Projective Relativity: Present Status and Outlook - B. Fauser|http://arxiv.org/PS_cache/gr-qc/pdf/0011/0011015v1.pdf]]

<<tiddler [[include_tiddlers/Propagator.html#"Propagator"]]>>

<<tiddler [[include_tiddlers/Proper Time.html#"Proper Time"]]>>

A ''Pseudo-'' or ''Semi- Riemannian Space'' is a generalization of a [[Riemannian space|Riemann Space]] in that the [[metric tensor|Metric Tensor]] need not be positive-definite. Instead a weaker condition of nondegeneracy is imposed.

Given a [[linear code|Linear Blockcode]] $C = [n,k,d]_q$ with $d>1$, a ''Punctured Code'' is defined as a $[n-1,k,d-1]_q \equiv C^*$-code and is obtained from $C$ by what is called a ''Projection'' of $C$. This means that the codewords of $C$ are projected onto the first $n-1$ coordinates, i.e.
\[
C^* = \{(x_1,\ldots,x_{n-1}) :  (x_1,\ldots,x_{n-1}, x) \in C\}
\]
The converse is called ''Extension'' of a code. This is not generally possible, yet for odd binary codes (i.e. $q=2$ and $d$ odd) it always is.
The entry in the extra coordinate is known as ''Parity Check Bit''.

Examples of punctured codes are [[punctured Reed-Muller codes|Reed-Muller Code]].
Examples for extensions are the [[extended Hamming codes|Hamming Code]] and the extended binary [[Golay code|Golay Code]].

For [[Cayley-Dickson algebras|Cayley-Dickson Algebra]] $\mathbb A_n$ one defines:

''Pure Elements''
An element $\mb A \in \mathbb A_n$ is called ''Pure'' if $\mb A + \mb A^* \equiv \operatorname{tr} (\mathbb A) = \text{Re}(\mb A) = 0$.
Saying an element is a pure element is equivalent to saying that it is an imaginary element (e.g. an imaginary [[octonion|Octonion]] = pure octonion). Sometimes one also speaks of a purely imaginary element.

''Doubly Pure Elements''
An element $\mb C = (\mb A, \mb B) \in \mathbb A_n$  is called ''Doubly Pure'' in $\mathbb A_n$ if $\mb A$ and $\mb B$ are both pure in $\mathbb A_{n-1}$.

A ''Pyrochlore Lattice'' is built up of tetrahedra.
<html><center><img src="images/pyrochlore.jpg" style="width: 265px; "/></center></html>

<<tiddler [[include_tiddlers/QCD.html#"QCD"]]>>

A [[nonassociative algebra|Nonassociative Algebra]] $\mathcal A$ (with unity $\mb e$ over a commutative ring) is called a ''Quadratic Algebra'' if $\mathcal A$ admits a [[quadratic form|Quadratic Form]] $\mathcal Q$ such that
#$\mathcal Q (\mb e) = 1$,
#the ''Quadratic Equation'' $\mb A^2 + b(\mb A, \mb e) \mb A + \mathcal Q(\mb A) \mb e = \mb 0$  is satisfied by all $\mb A \in \mathcal A$ where $b$ is the associated [[symmetric bilinear form|Scalar Product]] given by $b(\mb A, \mb B) \equiv \mathcal Q(\mb A + \mb B) - \mathcal Q(\mb A) - \mathcal Q( \mb B)$.

Videos:
* [[Arizona Winter School 2009: Quadratic Forms|http://swc.math.arizona.edu/aws/09/index.html]]

<<tiddler [[include_tiddlers/Quantization.html#"Quantization"]]>>

<<tiddler [[include_tiddlers/Quantum Biology.html#"Quantum Biology"]]>>

<<tiddler [[include_tiddlers/Quantum Black Hole.html#"Quantum Black Hole"]]>>

<<tiddler [[include_tiddlers/Quantum Brain Dynamics.html#"Quantum Brain Dynamics"]]>>

<<tiddler [[include_tiddlers/Quantum Chaos.html#"Quantum Chaos"]]>>

<<tiddler [[include_tiddlers/Quantum Computer.html#"Quantum Computer"]]>>

See also:
* [[Quantum brain dynamics|Quantum Brain Dynamics]]
* [[Orch-OR model|Orch-OR Model]]
* [[David Bohm]]


Papers:
* [[Overextension of Conjunctive Concepts: Evidence for a Unitary Model of Concept Typicality and Class Inclusion (1988) - J. A. Hampton|http://www.staff.city.ac.uk/~sc318/PDF%20files/hamptonjep88.pdf]] [[local|papers/hamptonjep88.pdf]] {{t100Cite{[[pct. 136|http://scholar.google.de/scholar?hl=de&lr=&cites=8034426220452485829&um=1&ie=UTF-8&ei=PxJvToHlCuPV4QTO8vyCCQ&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCAQzgIwAA]]}}}
* [[Disjunction of Natural Concepts (1988) - J. A. Hampton|http://www.staff.city.ac.uk/hampton/PDF%20files/Hampton%20Disjunction1988b.pdf]] [[local|papers/Hampton Disjunction1988b.pdf]] [[pct. 41|http://scholar.google.de/scholar?hl=de&lr=&cites=5568130222976748828&um=1&ie=UTF-8&ei=-qNtTpmOIIy2hAfcvtHaDg&sa=X&oi=science_links&ct=sl-citedby&resnum=2&ved=0CDQQzgIwAQ]]
* [[Quantum Structure in Cognition (2009) - D. Aerts|http://arxiv.org/PS_cache/arxiv/pdf/0805/0805.3850v2.pdf]] [[local|papers/0805.3850v2.pdf]] [[pct. 35|http://scholar.google.de/scholar?hl=de&lr=&cites=4571391476686151738&um=1&ie=UTF-8&ei=oaRtTsSTDsyLhQfXt5WDDA&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCkQzgIwAA]]
* [[Theory of Brain Function, Quantum Mechanics and Superstrings (1995) - D. Nanopoulos|http://arxiv.org/PS_cache/hep-ph/pdf/9505/9505374v1.pdf]] [[local|papers/9505374v1.pdf]] [[pct. 32|http://scholar.google.de/scholar?hl=de&lr=&cites=8462979060079347020&um=1&ie=UTF-8&ei=QcJtTq3MIuqF4gSnkemiAw&sa=X&oi=science_links&ct=sl-citedby&resnum=2&ved=0CCYQzgIwAQ]]
* [[Quantum Dissipation and Information: A Route to Consciousness Modeling (2007 - G. Vitiello |http://www.library.utoronto.ca/see/SEED/Vol1-2/Vitiello.pdf]] [[local|papers/Vitiello-2.pdf]] [[pct. 20|http://scholar.google.de/scholar?hl=de&lr=&cites=14350893280662123570&um=1&ie=UTF-8&ei=6jwITuiGPMWLswaQ6oDGDA&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCMQzgIwAA]]
* [[Mental States Follow Quantum Mechanics During Perception and Cognition of Ambiguous Figures. (2009) - E. Conte, A. Y. Khrennikov, O. Todarello, A. Federici, L. Mendolicchio, J. P. Zbilut|http://philpapers.org/archive/CONMSF.1.pdf]] [[local|papers/CONMSF.1.pdf]] [[pct. 17|http://scholar.google.de/scholar?hl=de&lr=&cites=11260127474830020408&um=1&ie=UTF-8&ei=hKZtTryGGsi2hAea4LSDDA&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCIQzgIwAA]]
* [[Experimental Evidence for Quantum Structure in Cognition (2008) - D. Aerts, S. Aerts, L. Gabora|http://arxiv.org/PS_cache/arxiv/pdf/0810/0810.5290v2.pdf]] [[local|papers/0810.5290v2.pdf]] [[pct. 16|http://scholar.google.de/scholar?hl=de&lr=&cites=6631926220255369188&um=1&ie=UTF-8&ei=UBBvTpjCB6r_4QSewbCgCg&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCMQzgIwAA]]
* [[Quantum Models of Consciousness (2008) - A. Vannini|http://www.quantumbiosystems.org/admin/files/QBS2%20165-184.pdf]] [[local|papers/QBS2 165-184.pdf]]  [[pct. 15|http://scholar.google.de/scholar?cites=12386766698508987334&as_sdt=2005&sciodt=0,5&hl=de]]
* [[Empirical Comparison of Markov and Quantum Models of Decision Making (2009) - J. R. Busemeyer, Z. Wang, A. Lambert-Mogilian|http://wongzheng.web.officelive.com/Documents/Busemeyer%20Wang%20LambertMogiliansky%20%282009%29%20empirical%20comparison%20of%20Markov%20and%20quantum%20models%20of%20desion%20making.pdf]] [[local|papers/MarkovAndQuantum.pdf]] [[pct. 13|http://scholar.google.de/scholar?hl=de&lr=&cites=6199756610782636057&um=1&ie=UTF-8&ei=ixBvTpORCab64QTwr6iBCQ&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCMQzgIwAA]]
* [[A Non-Critical String (Liouville) Approach to Brain Microtubules: State Vector Reduction, Memory Coding and Capacity (1995) - N.E. Mavromatos, D.V. Nanopoulos|http://arxiv.org/pdf/quant-ph/9512021v2]] [[local|papers/9512021v2.pdf]] [[pct. 9|http://scholar.google.de/scholar?hl=de&lr=&cites=2917089343887323932&um=1&ie=UTF-8&ei=n6ZtTvf5FcmohAel04iDDA&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB8QzgIwAA]]
* [[A Quantum Theory of Consciousness (2008) - S. Gao|http://www.sciencenet.cn/upload/blog/file/2010/9/2010910142946571464.pdf]] [[local|papers/2010910142946571464.pdf]] [[pct. 4|http://scholar.google.de/scholar?hl=de&lr=&cites=8518598590086051582&um=1&ie=UTF-8&ei=wxFvTun7KemP4gSJzMDOCQ&sa=X&oi=science_links&ct=sl-citedby&resnum=2&ved=0CCsQzgIwAQ]]
* [[Quantum Processes, Space-time Representation and Brain Dynamics (2003) - S. Roy, M. Kafatos|http://arxiv.org/PS_cache/quant-ph/pdf/0304/0304137v2.pdf]] [[local|papers/0304137v2.pdf]] [[pct. 1|http://scholar.google.de/scholar?hl=de&lr=&cites=12818098890424813348&um=1&ie=UTF-8&ei=hMRtTpbEF6Wg4gSorJjMBA&sa=X&oi=science_links&ct=sl-citedby&resnum=2&ved=0CCcQzgIwAQ]]
* [[Quantum Structure in Cognition: Fundamentals and Applications (2011) - D. Aerts, L. Gabora, S. Sozzo, T. Veloz|http://arxiv.org/PS_cache/arxiv/pdf/1104/1104.3344v1.pdf]] [[local|papers/1104.3344v1.pdf]] pct. 0

Links:
* [[WIKIPEDIA - Quantum Mind|http://en.wikipedia.org/wiki/Quantum_mind]]
* [[PHYSORG: Study Rules Out Fröhlich Condensates in Quantum Consciousness Model|http://www.physorg.com/news155904395.html]]

<<tiddler [[include_tiddlers/Quantum Cosmology.html#"Quantum Cosmology"]]>>

<<tiddler [[include_tiddlers/Quantum Entanglement.html#"Quantum Entanglement"]]>>

<<tiddler [[include_tiddlers/Quantum Field Theory.html#"Quantum Field Theory"]]>>

<<tiddler [[include_tiddlers/Quantum Gravity.html#"Quantum Gravity"]]>>

See [[Hopf Algebra]].

<<tiddler [[include_tiddlers/Quantum Harmonic Oscillator.html#"Quantum Harmonic Oscillator"]]>>

<<tiddler [[include_tiddlers/Quantum Information.html#"Quantum Information"]]>>

<<tiddler [[include_tiddlers/Quantum Mechanics.html#"Quantum Mechanics"]]>>

<<tiddler [[include_tiddlers/Quantum Neuronal Network.html#"Quantum Neuronal Network"]]>>

<<tiddler [[include_tiddlers/Quantum Polyvector Geometrodynamics.html#"Quantum Polyvector Geometrodynamics"]]>>

<br><<tiddler [[include_tiddlers/Quantum State.html#"Quantum State"]]>>

<<tiddler [[include_tiddlers/Quantum Teleportation.html#"Quantum Teleportation"]]>>

<<tiddler [[include_tiddlers/Quantum Tunneling.html#"Quantum Tunneling"]]>>

Papers:
* [[A Conceptual Analysis of Quantum Zeno; Paradox, Measurement, and Experiment - D. Home, M. A. B. Whitaker|http://www.quniverse.sk/buzek/zaujimave/home.pdf]] [[local|papers/home.pdf]] [[pct. 85|http://scholar.google.de/scholar?hl=de&lr=&cites=7482614870574404983&um=1&ie=UTF-8&ei=0I8TTdGtDpKTjAeM76T4BQ&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCUQzgIwAA]]

<<tiddler [[include_tiddlers/Quark.html#"Quark"]]>>

A ''Quasi\-Hopf Algebra'' is a generalization of a [[Hopf algebra|Hopf Algebra]], which was introduced by Vladimir Drinfeld in 1989.
Quasi\-Hopf algebras have coproducts which are coassociative only up to a [[3-cocycle|3-Cocycle]].

Papers:
* [[Quasialgebra Structure of the Octonions - H. Albuquerque, S. Majid|http://arxiv.org/PS_cache/math/pdf/9802/9802116v1.pdf]] [[pct. 49|http://scholar.google.de/scholar?cites=15697425696231458277&hl=de&as_sdt=2000]]

<<tiddler [[include_tiddlers/Quasicrystal.html#"Quasicrystal"]]>>

<<tiddler [[include_tiddlers/Quasigroup.html#"Quasigroup"]]>>

<<tiddler [[include_tiddlers/Quasigroup Manifold.html#"Quasigroup Manifold"]]>>

<<tiddler [[include_tiddlers/Quaternator.html#"Quaternator"]]>>

<<tiddler [[include_tiddlers/Quaternator Identities.html#"Quaternator Identities"]]>>

<<tiddler [[include_tiddlers/Quaternion.html#"Quaternion"]]>>

The multiplication table of the ''Quaternion Group'' $\mathcal Q$ is given by:
|* |! e |! -e |! i |! -i |! j |! -j | !k |! -k|
|! e | e | -e | i | -i | j | -j | k |-k|
|!-e| -e | e | -i | i | -j | j | -k | k|
|! i | i | -i | -e | e | k | -k | -j | j|
|!-i | -i | i | e | -e | -k | k | j | -j|
|! j| j | -j | -k | k | -e | e | i | -i|
|! -j | -j | j | k | -k | e | -e | -i | i|
| !k | k | -k | j | -j | -i | i | -e | e|
|! -k | -k | k | -j | j | i | -i | e | -e|

If one uses the [[XOR table|XOR Tables]] of the quaternions and extends it by assigning "0" to a positive sign and "1" to a negative sign one can express the table as an XOR table which looks as follows:

|!* |! 000 |! 100 |! 001 |! 101 |! 010 |! 110 | !011 |! 111|
|! 000 | 000 | 100 | 001 | 101 | 010 | 110 | 011 |111|
|!100| 100 | 000 | 101 | 001 | 110 | 010 | 111 | 011|
|! 001 | 001 | 101| 100 | 000 | 011| 111 | 110 | 010|
|!101 | 101 | 001 | 000 | 100 |111 | 011| 010 | 110|
|! 010| 010 | 110 | 111 | 011| 100 | 000 | 001 | 101|
|! 110 |110 | 010 | 011 | 111 | 000 | 100 | 101 | 001|
| !011| 011 | 111 | 010 | 110 | 101 | 001 | 100 | 000|
|! 111 | 111 | 011 | 110 | 010 | 001 | 101 | 000 | 100|

This table differs considerably from an XOR\-table associated with the [[Fano plane|Fano Plane]] which is commutative.
// Question: Is there a way to represent the table in binary form such that the multiplication can be carried out by means of a known boolean operation ? //

The [[factor group|Quotient Group]] $Q/\{\pm 1\}$ is isomorphic to the [[Klein four-group|Klein Four-group]].

<<tiddler [[include_tiddlers/Quaternion Multiplication Tables.html#"Quaternion Multiplication Tables"]]>>

<<tiddler [[include_tiddlers/Quaternionic Analysis.html#"Quaternionic Analysis"]]>>

<br><<tiddler [[include_tiddlers/Quaternionic Quantum Mechanics.html#"Quaternionic Quantum Mechanics"]]>>

<<tiddler [[include_tiddlers/Quateroctonion.html#"Quateroctonion"]]>>

<<tiddler [[include_tiddlers/Prime Number.html#"Prime Number"]]>>

<<tiddler [[include_tiddlers/Qubit.html#"Qubit"]]>>

<<tiddler [[include_tiddlers/Quintessence.html#"Quintessence"]]>>

Verba volant, scripta manent.
----
A good part of science is distinguishing between useful crazy ideas and those that are just plain nutty. - Princeton University book advertisement -
----
An expert is a person who has made all the mistakes that can be made in a very narrow field. - Niels Bohr -
----
The ordinary man wonders at marvellous things; the wise man wonders at the commonplace. - Confucius -
----
I hear and I forget. I see and I remember. I do and I understand - Confucius -
----
One who learns but does not think, is lost. One who thinks but does not learn is in great danger. - Confucius -
----
Nothing in life is to be feared, it is only to be understood. Now is the time to understand more, so that we may fear less. - Marie Curie -
----
Any fool can know. The point is to understand. - Albert Einstein -
----
Not everything that can be counted counts, and not everything that counts can be counted. - Albert Einstein -
----
Everything should be made as simple as possible, but not simpler. - Albert Einstein -
----
Solange man jung ist, gehören alle Gedanken der Liebe - später gehört alle Liebe den Gedanken. - Albert Einstein -
----
Geniale Menschen sind selten ordentlich, ordentliche selten genial. - Albert Einstein -
----
Reading, after a certain age, diverts the mind too much from its creative pursuits. Any man who reads too much and uses his own brain too little falls into lazy habits of thinking. - Albert Einstein -
----
Only two things are infinite, the universe and human stupidity, and I'm not sure about the former. - Albert Einstein -
----
Strive not to be a success, but rather to be of value. - Albert Einstein -
----
Only those who will risk going too far can possibly find out how far one can go. - Thomas Stearns Eliot -
----
I wonder why I wonder why. I wonder why I wonder. I wonder why I wonder why I wonder why I wonder! - Richard Feynman -
----
I think I can safely say that nobody understands quantum mechanics. - Richard Feynman -
----
Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? - Stephen Hawking -
----
When I hear of Schrödinger's cat, I reach for my gun. - Stephen Hawking -
----
Der Horizont vieler Menschen ist ein Kreis mit Radius Null - und das nennen sie ihren Standpunkt. - David Hilbert -
----
Physics is much too hard for physicists. - David Hilbert -
----
The whole purpose of physics is to find a number, with decimal points, etc! Otherwise you haven't done anything. - Richard Feynman -
----
Every scientific statement must remain tentative for ever. - Karl Raimund Popper -
----
When all else fails, you can always tell the truth. - Abdus Salam -
----
Die Grenzen meiner Sprache bedeuten die Grenzen meiner Welt. - Ludwig Wittgenstein -
----
Nicht wie die Welt ist, ist das Mystische, sondern dass sie ist. - Ludwig Wittgenstein -
----
Wovon man nicht sprechen kann, darüber muss man schweigen. - Ludwig Wittgenstein -
----
The effort to understand the universe is one of the very few things that lifts human life a little above the level of farce, and gives it some of the grace of tragedy. - Steven Weinberg -
----
We're all lying in the gutter, but some of us are gazing at the stars. - Oscar Wilde -
----
The whole purpose of science is to find meaningful simplicity in the midst of disorderly complexity - Herbert Simon -
----
The unreasonable effectiveness of mathematics in the natural sciences - Eugene Wigner -
----
Science is a wonderful thing if one does not have to earn one's living at it - Albert Einstein -
----
As far as I see, all a priori statements in physics have their origin in symmetry. - Hermann Weyl -
----
Number rules the universe. - Pythagoras -
----
Use examples; that such as thou teachest may understand thee the better! - Pythagoras -
----
A fool is known by his Speech; and a wise man by Silence. - Pythagoras -
----
Time is the soul of this world. - Pythagoras -
----
The oldest, shortest words - "yes" and "no" - are those which require the most thought. - Pythagoras -
----
You cannot teach a man anything; you can only help him discover it in himself. - Galileo Galilei -
----
The great book of nature can be read only by those who know the language in which it was written. And this language is mathematics. - Galileo Galilei -
----
All is flux, nothing stays still. - Heraclit -
----
Nothing endures but change. - Heraklit -
----
Enthusiasm is followed by disappointment and even depression, and then by renewed enthusiasm. - Murray Gell-Mann -
----
Available energy is the main object at stake in the struggle for existence and the evolution of the world. - Ludwig Boltzmann -
----
What then is time? If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know. - Saint Augustine -
----
Omnibus ex nihil decendis sufficit unum. (One suffices to derive all out of nothing.) - Gottfried Leibniz
----
Never make a calculation until you know the answer. - John Archibald Wheeler
----
The most powerful method of advance is to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics. - Paul Adrien Maurice Dirac -
----
The notion of existence is one of the primitive concepts with which we must begin as given. It is the clearest concept we have. - Kurt Gödel
----
Numquam ponenda est pluralitas sine necessitate. - William of Ockham
----
If A equals success, then the formula is A = X + Y + Z. X is work. Y is play. Z is keep your mouth shut. - Albert Einstein -
----
Ich behaupte aber, dass in jeder besonderen Naturlehre nur so viel eigentliche Wissenschaft angetroffen werden könne, als darin Mathematik anzutreffen ist. - Immanuel Kant -
----
Mathemata mathematicis scribuntur (mathematics is written for mathematicians) - Nicolaus Copernicus -
----
The art of doing mathematics consists in finding that special case which contains all the germs of generality. - David Hilbert -
----
Fundamental concepts are rare. - Shiing Shen Chern -
----
If I could explain it to the average person, I wouldn't have been worth the Nobel Prize. - Richard Feynman -
----
If you haven't found something strange during the day, it hasn't been much of a day. - Archibald Wheeler -
----
Publish or perish. - J. C. Polkinghorne -
----
Shut up and calculate! - David Mermin -
----
The most incomprehensible thing about the world is that it is comprehensible. - Albert Einstein -
----
Ich brauche mehr Informationen - Dieter Hallervorden -
----
Basic research is what I am doing when I don't know what I am doing. - Wernher von Braun -
----
Life is good for only two things, discovering mathematics and teaching mathematics. - Siméon Poisson -
----
In the middle of difficulty lies opportunity. - Albert Einstein -
----
Infinite space is the sensorium of the Deity - Sir Isaac Newton -
----
Never contract friendship with a man that is not better than thyself. - Confucius -
----
Life's a piece of shit when you look at it. - Monthy Python -
----
All science is either physics or stamp collecting. - Ernest Rutherford -

/***
|Name|QuoteOfTheDayPlugin|
|Source|http://www.TiddlyTools.com/#QuoteOfTheDayPlugin|
|Documentation|http://www.TiddlyTools.com/#QuoteOfTheDayPluginInfo|
|Version|1.4.1|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements <br>and [[Creative Commons Attribution-ShareAlike 2.5 License|http://creativecommons.org/licenses/by-sa/2.5/]]|
|~CoreVersion|2.1|
|Type|plugin|
|Requires||
|Overrides||
|Description|Display a randomly selected "quote of the day" from a list defined in a separate tiddler|

!!!!!Documentation
>see [[QuoteOfTheDayPluginInfo]]
!!!!!Revisions
<<<
2008.03.21 [1.4.1] in showNextItem(), corrected handling for random selection so that //initial// index value will randomized correctly instead of always showing first item, even when randomizing.  Thanks to Riccardo Gherardi for finding this.
| Please see [[QuoteOfTheDayPluginInfo]] for previous revision details |
2005.10.21 [1.0.0] Initial Release.  Based on a suggestion by M.Russula
<<<
!!!!!Code
***/
//{{{
version.extensions.QuoteOfTheDayPlugin= {major: 1, minor: 4, revision: 1, date: new Date(2008,3,21)};
config.macros.QOTD = {
	clickTooltip: "click to view another item",
	timerTooltip: "auto-timer stopped...  'mouseout' to restart timer",
	timerClickTooltip: "auto-timer stopped...  click to view another item, or 'mouseout' to restart timer",
	handler:
	function(place,macroName,params) {
		var tid=params.shift(); // source tiddler containing HR-separated quotes
		var p=params.shift();
		var click=true; // allow click for next item
		var inline=false; // wrap in slider for animation effect
		var random=true; // pick an item at random (default for "quote of the day" usage)
		var folder=false; // use local filesystem folder list
		var cookie=""; // default to no cookie
		var next=0; // default to first item (or random item)
		while (p) {
			if (p.toLowerCase()=="noclick") var click=false;
			if (p.toLowerCase()=="inline") var inline=true;
			if (p.toLowerCase()=="norandom") var random=false;
			if (p.toLowerCase().substr(0,7)=="cookie:") var cookie=p.substr(8);
			if (!isNaN(p)) var delay=p;
			p=params.shift();
		}
		if ((click||delay) && !inline) {
			var panel = createTiddlyElement(null,"div",null,"sliderPanel");
			panel.style.display="none";
			place.appendChild(panel);
			var here=createTiddlyElement(panel,click?"a":"span",null,"QOTD");
		}
		else
			var here=createTiddlyElement(place,click?"a":"span",null,"QOTD");
		here.id=(new Date()).convertToYYYYMMDDHHMMSSMMM()+Math.random().toString(); // unique ID
		// get items from tiddler or file list
		var list=store.getTiddlerText(tid,"");
		if (!list||!list.length) { // not a tiddler... maybe an image directory?
			var list=this.getImageFileList(tid);
			if (!list.length) { // maybe relative path... fixup and try again
				var h=document.location.href;
				var p=getLocalPath(decodeURIComponent(h.substr(0,h.lastIndexOf("/")+1)));
				var list=this.getImageFileList(p+tid);
			}
		}
		if (!list||!list.length) return false; // no contents... nothing to display!
		here.setAttribute("list",list);
		if (delay) here.setAttribute("delay",delay);
		here.setAttribute("random",random);
		here.setAttribute("cookie",cookie);
		if (click) {
			here.title=this.clickTooltip
			if (!inline) here.style.display="block";
			here.setAttribute("href","javascript:;");
			here.onclick=function(event)
				{ config.macros.QOTD.showNextItem(this); }
		}
		if (config.options["txtQOTD_"+cookie]!=undefined) next=parseInt(config.options["txtQOTD_"+cookie]);
		here.setAttribute("nextItem",next);
		config.macros.QOTD.showNextItem(here);
		if (delay) {
			here.title=click?this.timerClickTooltip:this.timerTooltip
			here.onmouseover=function(event)
				{ clearTimeout(this.ticker); };
			here.onmouseout=function(event)
				{ this.ticker=setTimeout("config.macros.QOTD.tick('"+this.id+"')",this.getAttribute("delay")); };
			here.ticker=setTimeout("config.macros.QOTD.tick('"+here.id+"')",delay);
		}
	},
	tick: function(id) {
		var here=document.getElementById(id); if (!here) return;
		config.macros.QOTD.showNextItem(here);
		here.ticker=setTimeout("config.macros.QOTD.tick('"+id+"')",here.getAttribute("delay"));
	},
	showNextItem:
	function (here) {
		// hide containing slider panel (if any)
		var p=here.parentNode;
		if (p.className=="sliderPanel") p.style.display = "none"
		// get a new quote
		var index=here.getAttribute("nextItem");
		var items=here.getAttribute("list").split("\n----\n");
		if (index<0||index>=items.length) index=0;
		if (here.getAttribute("random")=="true") index=Math.floor(Math.random()*items.length);
		var txt=items[index];
		// re-render quote display element, and advance index counter
		removeChildren(here); wikify(txt,here);
		index++; here.setAttribute("nextItem",index);
		var cookie=here.getAttribute("cookie");
		if (cookie.length) {
			config.options["txtQOTD_"+cookie]=index.toString();
			saveOptionCookie("txtQOTD_"+cookie);
		}
		// redisplay slider panel (if any)
		if (p.className=="sliderPanel") {
			if(anim && config.options.chkAnimate)
				anim.startAnimating(new Slider(p,true,false,"none"));
			else p.style.display="block";
		}
	},
	getImageFileList: function(cwd) { // returns HR-separated list of image files
		function isImage(fn) {
			var ext=fn.substr(fn.length-3,3).toLowerCase();
			return ext=="jpg"||ext=="gif"||ext=="png";
		}
		var files=[];
		if (config.browser.isIE) {
			cwd=cwd.replace(/\//g,"\\");
			// IE uses ActiveX to read filesystem info
			var fso = new ActiveXObject("Scripting.FileSystemObject");
			if(!fso.FolderExists(cwd)) return [];
			var dir=fso.GetFolder(cwd);
			for(var f=new Enumerator(dir.Files); !f.atEnd(); f.moveNext())
				if (isImage(f.item().path)) files.push("[img[%0]]".format(["file:///"+f.item().path.replace(/\\/g,"/")]));
		} else {
			// FireFox (mozilla) uses "components" to read filesystem info
			// get security access
			if(!window.Components) return;
			try { netscape.security.PrivilegeManager.enablePrivilege("UniversalXPConnect"); }
			catch(e) { alert(e.description?e.description:e.toString()); return []; }
			// open/validate directory
			var file=Components.classes["@mozilla.org/file/local;1"].createInstance(Components.interfaces.nsILocalFile);
			try { file.initWithPath(cwd); } catch(e) { return []; }
			if (!file.exists() || !file.isDirectory()) { return []; }
			var folder=file.directoryEntries;
			while (folder.hasMoreElements()) {
				var f=folder.getNext().QueryInterface(Components.interfaces.nsILocalFile);
				if (f instanceof Components.interfaces.nsILocalFile)
					if (isImage(f.path)) files.push("[img[%0]]".format(["file:///"+f.path.replace(/\\/g,"/")]));
			}
		}
		return files.join("\n----\n");
	}
}
//}}}

A ''Quotient Group'' (or ''Factor Group'') $G/N$ of a group $G$ is a partition of $G$ defined by a [[normal subgroup|Normal Subgroup]] $N$ where $G/N$ inherits the group operation of $G$, i.e. for  $g, h \in G , N \trianglelefteq G$:
\[
(gN)*(hN) = (g*h)N
\]
!!!!Examples
* $\mathbb{Z}_n \equiv \mathbb{Z}/n\mathbb{Z}\;$ under addition. $n\mathbb{Z}\,$ is a [[normal subgroup|Normal Subgroup]] of $\mathbb{Z}$ as addition is abelian. The [[order|Order]] of this factor group is $n$. It is isomorphic to the set $\{0, 1, ..., n-1\}$ with addition modulo $n$. The algebras $\mathbb Z_n$ are called ''n-ary Algebras''. For the ''Binary Algebra'' with elements $\{0,1\}$ one therefore has $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}\;$.
* $\mathbb R/\mathbb Z\,$ is isomorphic to the circle group $S^1$, the group of complex numbers of absolute value 1.
* $SO(3) = SU(2)/\mathbb Z_2$ which means that $SU(2)$ is a "double cover" of $SO(3)$.

<<tiddler [[include_tiddlers/Randall-Sundrum Model.html#"Randall-Sundrum Model"]]>>

<<tiddler [[include_tiddlers/Random Walk.html#"Random Walk"]]>>

The ''Rank'' of a [[group|Group]] is the minimal number of elements needed to generate it.

It is equal to the dimension of its [[maximal torus|Maximal Torus]]. The rank is well-defined since all maximal tori are [[conjugate|Conjugation]] to one another.

For semi-simple groups the rank is equal to the number of nodes in the associated [[Dynkin diagram|Dynkin Diagram]].

Links:
* [[WIKIPEDIA - Raychaudhuri Equation|http://en.wikipedia.org/wiki/Raychaudhuri_equation]]

View of Saturn's north pole:
<html><center><img src="images/saturn.jpg" style="width: 360px; "/></center></html>

<<tiddler [[include_tiddlers/Real Structure.html#"Real Structure"]]>>

/***
|Name|RecentChangesPlugin|
|Source|http://www.TiddlyTools.com/#RecentChangesPlugin|
|Version|2.1.0|
|Author|Eric Shulman - ELS Design Studios|
|License|http://www.TiddlyTools.com/#LegalStatements <br>and [[Creative Commons Attribution-ShareAlike 2.5 License|http://creativecommons.org/licenses/by-sa/2.5/]]|
|~CoreVersion|2.1|
|Type|plugin|
|Requires||
|Overrides||
|Description|display droplist of recently changed tiddlers with goto, edit, and preview buttons|
!!!!!Usage
<<<
The {{{<<recentChanges>>}}} macro displays a droplist of all tiddlers that have been changed within the last N days (default=10 days).
<<<
!!!!!Examples
<<<
{{smallform{
{{{<<recentChanges>>}}}
><<recentChanges>>
or
{{{<<recentChanges #ofdays summary>>}}}
>where:
>* #ofdays specifies the time limit for list changed tiddlers.  Use 0 (zero) to list all tiddlers in the document
>* "summary" is a keyword that outputs only the summary text (without the droplist or buttons)
>{{{<<recentChanges 14 summary>>}}}
><<recentChanges 14 summary>>
or
{{{<<recentChanges #ofdays previewheight previewclass>>}}}
>where:
>* #ofdays specifies the time limit for list changed tiddlers.  Use 0 (zero) to list all tiddlers in the document
>* previewheight is a CSS height measurement and sets the FIXED height of the tiddler preview area (default is 15em)
>* previewclass is any CSS classname, and can be used to apply custom styles to the preview area (default is to use the standard 'viewer' class)
>{{{<<recentChanges 14 10em groupbox>>}}}
><<recentChanges 14 10em groupbox>>
}}}
<<<
!!!!!Revisions
<<<
2008.07.01 [2.1.0] added optional "summary" keyword for simply text output
2008.05.01 [2.0.1] fixup for titles with double-quotes
2007.07.26 [2.0.0] re-written as plugin
2006.10.02 [1.0.0] initial release (as inline script ShowRecentChanges)
<<<
!!!!!Code
***/
//{{{
version.extensions.RecentChangesPlugin= {major: 2, minor: 1, revision: 0, date: new Date(2008,7,1)};

config.shadowTiddlers.RecentChanges="<<recentChanges>>";

config.macros.recentChanges = {
	layout: '<form><!--\
		--><select size=1 name="list" style="width:69.5%" \
			onchange=" \
				this.form.goto.disabled=this.form.edit.disabled=this.form.preview.disabled=!this.value.length; \
				var target=this.parentNode.parentNode.nextSibling; removeChildren(target); \
				if (!this.value.length) \
					{ target.style.display=\'none\'; this.form.preview.value=\'preview\'; } \
				else if (target.style.display==\'block\') { \
					wikify(\'<\'+\'<tiddler [[\'+this.value+\']]>\'+\'>\',target); \
					target.style.display=\'block\'; \
					this.form.preview.value=\'done\'; \
				} \
			"><!--\
		-->%options%<!--\
		--></select><!--\
		--><input type="button" name="goto" value="goto" disabled title="view selected tiddler" style="width:10%" \
			onclick="var target=this.parentNode.parentNode.nextSibling; removeChildren(target); \
				target.style.display=\'none\'; this.form.preview.value=\'preview\'; \
				story.displayTiddler(story.findContainingTiddler(this),this.form.list.value); \
			"><!--\
		--><input type="button" name="edit" value="edit" disabled title="edit selected tiddler" style="width:10%" \
			onclick="var target=this.parentNode.parentNode.nextSibling; removeChildren(target); \
				target.style.display=\'none\'; this.form.preview.value=\'preview\'; \
				story.displayTiddler(story.findContainingTiddler(this),this.form.list.value,DEFAULT_EDIT_TEMPLATE); \
			"><!--\
		--><input type="button" name="preview" value="preview" disabled title="show/hide tiddler preview" style="width:10%" \
			onclick="var target=this.parentNode.parentNode.nextSibling; \
				if (this.value==\'preview\') { \
					removeChildren(target); \
					wikify(\'<\'+\'<tiddler [[\'+this.form.list.value+\']]>\'+\'>\',target); \
					target.style.display=this.form.list.value.length?\'block\':\'none\'; this.value=\'done\'; \
				} else { \
					removeChildren(target); \
					target.style.display=\'none\'; this.value=\'preview\'; \
				} \
			"><!--\
		--></form>',
	handler: function(place,macroName,params,wikifier,paramString,tiddler) {
		var days=10; if (!isNaN(params[0])) days=parseInt(params[0]); // time limit in days (use 0 for all tiddlers)
		var summary=params[1]&&params[1].toLowerCase()=="summary"; if (summary) params.shift();
		var height='15em'; if (params[1]) height=params[1]; // preview area fixed height
		var previewclass='viewer'; if (params[2]) previewclass=params[2]; // preview area CSS class
		var tiddlers=store.getTiddlers('modified','excludeLists').reverse();
		var count=tiddlers.length;
		if (days) {
			var timelimit=(new Date()).getTime()-86400000*days;
			for (var count=0; count<tiddlers.length && tiddlers[count].modified>timelimit; count++);
		}
		var s=count+' tiddlers have changed since ';
		s+=new Date(timelimit).formatString("DDD, MMM DDth YYYY 0hh:0mm");
		s+=' ('+days+' days ago)';
		if (summary)
			{ wikify(s,place); return; }
		var opts='<option value="">'+s+'</option>';
		for (var i=0; i<count; i++) { var t=tiddlers[i];
			opts+='<option value="'+t.title.replace(/"/g,"&#x22;")+'">';
			opts+=t.modified.formatString('YYYY.0MM.0DD 0hh:0mm')+' - '+t.title;
			opts+='</option>';
		}
		createTiddlyElement(place,"div").innerHTML=this.layout.replace(/%options%/,opts);
		var preview=createTiddlyElement(place,"div",null,previewclass);
		preview.style.display='none';
		preview.style.whiteSpace='normal';
		preview.style.overflow='auto';
		preview.style.height=height;
	}
}
//}}}

<<tiddler [[include_tiddlers/Reductio Ad Absurbum.html#"Reductio Ad Absurbum"]]>>

A (locally) ''Reductive Space'' is a generalisation of a [[symmetric space|Symmetric Space]] and a [[Lie group manifold|Lie Group Manifold]]. Given an affine connection and a covariant derivative $\nabla$, one distinguishes

$\quad\quad$ Lie Group: $\;\quad \quad \quad \;\,\nabla T = 0, \quad \;R = 0$
$\quad\quad$ Symmetric space: $\;\; \,T = 0, \;\, \nabla R = 0$
$\quad\quad$ Reductive space: $\;\nabla T=0, \;\, \nabla R = 0$

with $T$ denoting the [[torsion tensor|Torsion]] and $R$ the [[Riemann tensor|Riemann Tensor]].

Reductive spaces were introduced by K. Nomizu in 1954.

[[Tangent algebras|Tangent Algebra]] of locally reductive spaces are [[Lie triple algebras|Bol Algebra]]. A construction was given by K. Yamaguti.

Papers:
* [[A Generalization of Lie Groups and Symmetric Spaces - N. Hitotsuyanag|http://ir.kagoshima-u.ac.jp/bitstream/10232/7165/1/AN00408518_1986_001.pdf]] [[local|papers/AN00408518_1986_001.pdf]] pct. 0

<<tiddler [[include_tiddlers/Reed-Muller Code.html#"Reed-Muller Code"]]>>

/%
!info
|Name|RefreshPageDisplay|
|Source|http://www.TiddlyTools.com/#RefreshPageDisplay|
|Version|2.0.0|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|transclusion|
|Description|create a link to redraw all page elements without restarting|
Usage
<<<
{{{
<<tiddler RefreshPageDisplay>>
<<tiddler RefreshPageDisplay with: label>>
}}}
<<<
Example
<<<
{{{<<tiddler RefreshPageDisplay with: "click me">>}}}
<<tiddler RefreshPageDisplay##show with: "click me">>
<<<
!end
!show
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	with: {{'$1'=='$'+'1'?'refresh page display':'$1'}}>>

<<tiddler [[include_tiddlers/Regge Trajectory.html#"Regge Trajectory"]]>>

{{center{[img(302px+, )[images/Reissner-Nordstrom_black_hole.gif]]}}}
The ''Reissner\-Nordström Metric'' is an exact, asymptotically flat, stationary and spherically symmetric solution of the [[Einstein-Maxwell equations|Einstein-Maxwell Equations]] for a non-rotating object having mass $M$ and electric charge $Q_E$.
The associated line element in spherical coordinates is given by:
\[
ds^2 = - \Delta_r ~ d (ct)^2 + \Delta_r^{-1} dr^2 + r^2(\sin^2\theta d \phi^2+ d\phi^2)
\]
with
\begin{eqnarray}
\Delta_r & = &\left(1-\frac{R_S} r+\frac{R_Q^2}{r^2}\right) = \frac 1 {r^2} \left(r^2- R_S r + R_Q^2 \right)  \\
R_Q & = & \frac G{4\pi\epsilon_0 c^4}  Q_E
\end{eqnarray}
$R_S$ denotes the [[Schwarzschild radius|Schwarzschild Radius]] and $R_Q$ defines a length-scale corresponding to $Q_E$.
Setting $Q_e = 0$, one obtains the [[Schwarzschild metric|Schwarzschild Metric]].
!!!![[Event horizons|Event Horizons]]
...

!!!!Astronomical relevance
Real black holes probably have spin but almost no electric charge because our universe appears to be electrically neutral, and a charged black hole would quickly neutralize by attracting charge of the opposite sign.

See also:
* [[Penrose-Carter diagram|Penrose-Carter Diagram]]

Links:
* [[WIKIPEDIA - Reissner-Nordström Metric|http://en.wikipedia.org/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric]]
* [[WIKIPEDIA - Black Hole Electron|http://en.wikipedia.org/wiki/Black_hole_electron]]
* [[The Encyclopedia of Science - Reissner-Nordstrom Black Hole|http://www.daviddarling.info/encyclopedia/R/Reissner-Nordstrom_black_hole.html]]

Videos:
* [[Lecture 11B - Advanced General Relativity (2008) - E. Poisson|http://streamer.perimeterinstitute.ca/Flash/f15f09f2-b71e-42fc-b081-dc09b8d76731/viewer.html]] - Excellent lecture, learned a lot.
* [[Journey into and through a Reissner-Nordström Black Hole - A. Hamilton|http://jila.colorado.edu/~ajsh/insidebh/rn.html]]

<<tiddler [[include_tiddlers/Relational Quantum Mechanics.html#"Relational Quantum Mechanics"]]>>

<<tiddler [[include_tiddlers/Renormalization.html#"Renormalization"]]>>

<<tiddler [[include_tiddlers/Renormalization Group.html#"Renormalization Group"]]>>

A $t-(v, k, \lambda)$ [[design|T-Design]] is called ''Resolvable'' (or a ''Design with Parallelism'') if it can be partitioned into so called ''Parallel Classes''. (Such a partition is also called a ''Resolution''). A parallel class is a subset of the blocks of the design, consisting of $v/k$ __disjoint blocks that contain each point of the design exactly once__.
A necessary condition for the resolvability of a design thus is that $v \operatorname{mod} k = 0$.
If a resolvable t-design is a BIBD it is also refered to as ''RBIBD''.

Examples of resolvable t-designs are:
* [[Affine planes|Affine Plane]] (any one is resolvable).
* [[Kirkman-Steiner Triple Sytems|Kirkman Triple System]], i.e. $2-(v, 3, 1)$ designs.
* $3$-designs:
** $3–(8,4,1)$
** $3–(16,4,1)$
** $3–(20,4,1)$
** $3–(48,6,30)$
** $3–(65,13,396)$
** $3–(96,6,30)$
*$5$-designs:
** $5–(12,6,1)$
** $5–(24,6,1)$
** $5–(24,8,1)$
** $5–(48,6,1)$
!!!!Example
The $(9,3,1)$-BIBD consisting of the points $\{1,2,3,4, 5,6,7,8, 9\}$ and the blocks $\{123,456,789,147,258,369,159,267,346, 168,249,357\}$ is resolvable.
The $4$ parallel classes are given by
\begin{eqnarray}
\{123,456,789\} \\
\{147,258,369\} \\
\{159,267,348\} \\
\{168, 249, 357\}
\end{eqnarray}

A subspace $S$ of $X$ is called a ''Retract'' of $X$ if there is a continuous map $ f: X \rightarrow X$ (called a retraction) such that $\forall x \in X$ and $\forall s \in S$:
# $f(x) \in S$ and
# $f(s)=s$
(i.e. the subspace is preserved under the map and the rest is mapped into the subspace).

Equivalently, a subspace $S$ of $X$ is called a retract of $X$ if there is a continuous map $ f: X \rightarrow S$ (called a retraction) such that $\forall s \in S$: $f(s)=s$.

<<tiddler [[include_tiddlers/Ricci Tensor.html#"Ricci Tensor"]]>>

<<tiddler [[include_tiddlers/Richard Feynman.html#"Richard Feynman"]]>>

<<tiddler [[include_tiddlers/Riemann Hypothesis.html#"Riemann Hypothesis"]]>>

<<tiddler [[include_tiddlers/Riemann Normal Coordinates.html#"Riemann Normal Coordinates"]]>>

> Um den Tatsachen der Metrik und der Gravitation gerecht zu werden , nehmen wir die Existenz einer Riemann\-Metrik an. In der Natur existieren aber auch die elektromagnetischen Felder, welche durch die Riemann\-Metrik nicht dargestellt werden können.
> - Albert Einstein -

A ''Riemannian Space'' (which is a [[symmetric space|Symmetric Space]]) is a [[Riemann-Cartan space|Riemann-Cartan Space]] with vanishing [[torsion|Torsion]]. The latter implies that the [[connection|Connection]] is symmetrical. In contrast to a general [[metric affine space|Metric Affine Space]] where the [[connection-|Connection]] and the [[metric-|Metric Tensor]]fields are completely independent, in a Riemann space one of them completely determines the other one. The additional conditions that have to be imposed to get from a metric affine space to a Riemann space are [[metricity|Metric Compatibility]] and vanishing of [[torsion|Torsion]].
If both torsion and [[curvature|Curvature]] vanish, the space has to be a Minkowskian since the Minkowski metric $g_{\mu\nu} = \eta_{\mu\nu}$  is the only one that makes the full [[Riemann tensor|Riemann Tensor]] vanish in the absence of torsion.

A $4$-dimensional Riemannian space-time is commonly denoted $\mathbb V_4$.

<<tiddler [[include_tiddlers/Riemann Tensor.html#"Riemann Tensor"]]>>

<<tiddler [[include_tiddlers/Riemann Zeta Function.html#"Riemann Zeta Function"]]>>

A ''Riemann\-Cartan Space (= Einstein\-Cartan Space)'' is the most general [[metric compatible|Metric Compatibility]], [[metric affine space|Metric Affine Space]], commonly denoted $\mathbb U_4$. It is a special case of a [[Weyl-Cartan space|Weyl Space]].

For such a space a general [[connection|Connection]] simplifies to:
\[
\Gamma_{\mu\nu}^\sigma = \Chr{\sigma}{\mu\nu} + K_{\mu\nu}^\sigma
\]
called ''Riemann\-Cartan Connection'', with $\Chr{\sigma}{\mu\nu} $ the [[Chrisoffel symbols|Christoffel Symbols]] and $K_{\mu\nu}^\sigma$ the [[contorsion|Contorsion]] tensor.

Thus the differential geometry of $\mathbb U_4$ is completely determined by two tensor fields:
# the [[metric tensor|Metric Tensor]] $g_{\mu\nu}$ and,
# the contorsion tensor $K_{\mu\nu}^\sigma$, or equivalently the torsion tensor $T_{\mu\nu}^\sigma$.

For vanishing torsion and hence contorsion, one arrives at a [[Riemann Space]].

!!!! Applications
* [[Einstein-Cartan theory of gravity|Einstein-Cartan Theory]].
* Plastic flow and material fatigue can be described by Riemann\-Cartan spaces as they require [[curvature|Curvature]] and [[torsion|Torsion]] due to [[disclinations|Disclination]] and [[dislocations|Dislocation]].

<<tiddler [[include_tiddlers/Ring.html#"Ring"]]>>

<<tiddler [[include_tiddlers/Robotic Telescopes.html#"Robotic Telescopes"]]>>

<<tiddler [[include_tiddlers/Roger Penrose.html#"Roger Penrose"]]>>

A [[root|Root Vector]] of an even integral [[lattice|Lattice]] $L$ is a vector $r$ of norm $\sqrt 2$ (or squared norm $2$).

The set $R(L)$ of all such vectors is called a ''(Simply Laced) Root Lattice'' or ''(Simply Laced) Root System'' and is given by
\[
R(L) = \{r \in L : \langle r|r\rangle = 2\}
\]
An integral lattice is called ''Rootless'' if it has no roots.

More generally, every lattice with vectors of equal length defines a root lattice, as it can be rescaled appropriately.

''Theorem''
A simply laced root lattice is the orthogonal sum of one of the following indecomposable root lattices:

|!L                | !ord (R(L))| !h         | !det(L) |
|$\mathbb A_n \;(n \ge 1)$|$n(n+1)$|$n+1$|$n+1$  |
|$\mathbb D_n \; (n \ge 4, \text{even})$ |$2n(n-1)$|$2(n-1)$ |$ 4$|
|$\mathbb D_n \; (n \ge 5, \text{odd})$ |$2n(n-1)$|$2(n-1)$  |$4$|
|$\mathbb E_6$|$72$|$12$|$3$|
|$\mathbb E_7$ |$126$|$18$|$2$|
|$\mathbb E_8$|$240$|$30$|$1$|

$h = \operatorname{ord}(R(L)/n$ is called ''Coxeter Number''.

Some important cases are:
* $ \operatorname{ord}(R(\mathbb D_3)) = 12$
* $ \operatorname{ord}(R(\mathbb D_4)) = 24$
* $ \operatorname{ord}(R(\mathbb D_8)) = 112$
* $ \operatorname{ord}(R(\mathbb D_{16})) = 480$
The lattices above are called ''$\mathbb{ADE}$-Series''. $\mathbb B$, $\mathbb C$, $\mathbb G$ and $\mathbb F$ do not appear in the list due to the fact that they are not simply laced lattices, i.e. they consist of vectors of two different lengths, called ''Short Roots'' and ''Long Roots''.

In general an irreducible root system can only consist of roots all having the same length or of roots having two values for their lengths.
Any two roots of the same length lie in the same [[orbit|Orbit]] of the [[Weyl group|Weyl Group]].

See also: [[Dn+ Lattices|Checkerboard Lattice]].

Google books:
* [[Perfect lattices in Euclidean Spaces - J. Martinet|http://books.google.com/books?id=gd9CcFclBRIC&pg=PA489&lpg=PA489&dq=Perfect+lattices+in+Euclidean+Spaces&source=bl&ots=Kf1_uYhQc3&sig=5CYfXVDs_GKbc9wK_KI1gDnsao8&hl=de&ei=LhUFS5GCG8eHsAad_6m6Cg&sa=X&oi=book_result&ct=result&resnum=3&ved=0CBcQ6AEwAg]] {{t100Cite{[[bct. 112|http://scholar.google.de/scholar?cites=1604944218385646456&hl=de&as_sdt=2000]]}}}

> A personal advice: Let's go back to the roots of physics and have a look at the root cause of our universe ...

''Root Vectors'' are [[weight vectors|Weight Vectors]] of the [[adjoint representation|Adjoint]]. Hence they are ''eigenvalues'' of a special representation of the algebra.

!!!! Construction:
Every simple [[Lie algebra|Lie Algebra]] $\mathfrak H$ contains a [[Cartan subalgebra|Cartan Subalgebra]] $\mathfrak h \subset \mathfrak H$ (of rank $l$) for which one can choose an orthonormal basis $\{\mb h_1, \ldots, \mb h_l\}$. This basis can be extended to a basis of the whole algebra  $\{\mb h_1, \ldots, \mb h_l, \mb g_1,  \mb g_{-1},  \mb g_2,  \mb g_{-2}, \ldots,  \mb g_{\frac{n-l}{2}}, \mb g_{-\frac{n-l}{2}} \}$ with the basis elements satisfying
\begin{eqnarray}
&&(1) \quad [\mb h_i, \mb g_j ] = \lambda^j_i  \mb g_j \; \text{(no sum)}, \; \lambda^j_i \in \mathbb R \\
&&(2) \quad [\mb h_i, \mb h_j] = 0 \\
&&(3) \quad [\mb g_j , \mb g_{?j}] \in \mathfrak H\\
\end{eqnarray}
Using (1) one defines $l$-tuples $r_j = (\lambda^j_1, \lambda^j_2, \ldots, \lambda^j_l )$ with $j = \frac{n-l}{2}, \ldots,\frac{n+l}{2}$ which are called roots of the algebra. They satisfy $ r_{?j} = ?r_j$.
Analogously due to (2) one defines $l$-tuples $r^0_i = (0,\ldots,0) \equiv r^0$ with $i = 1, \ldots, l$,  which are called ''Zero Roots''.
(3) leads to the relationship $r_j + r_{?j} = r^0$.
This kind of representation of a Lie algebra is also referred to as its ''Cartan\-Weyl Representation'' or ''Cartan\-Weyl Form''.

When plotted in $\mathbb R^l$, the set of roots provides a geometric description of the algebra.
The algebra’s ''Root System'', and hence its ''Root Diagram'' (or more generally weight diagram), determines the algebra uniquely up to [[isomorphisms|Homomorphism]].
For this it is sufficient to know the length and angle relations among [[simple roots|Simple Root]]. From them the whole set of roots can be generated by means of [[Weyl reflections|Weyl Reflection]]. Therefore the whole root system of $\mathfrak H$ is invariant with respect to the Weyl reflections.

!!!!Physics
While Dynkin diagrams are useful for classifying Lie algebras, root (and weight) diagrams are most often used in applications, such as when describing the properties of fundamental particles.

Links:
* [[Visualizing Lie Subalgebras Using Root and Weight Diagrams - A. Wangenberg, T. Dray|http://mathdl.maa.org/mathDL/23/?pa=content&sa=viewDocument&nodeId=3287]] [[local|papers/joma_paper.pdf]]

Lectures:
* [[Notes by W. A. de Graaf|http://www.science.unitn.it/~degraaf/notes.pdf]]

<<tiddler [[include_tiddlers/Rényi Entropy.html#"Rényi Entropy"]]>>

A [[Steiner system|Steiner System]] ''$S(5, 6, 12)$'' is a collection of special six point subsets (hexads) of a twelve point set $\Omega$, such that every five point subset of $\Omega$ is contained in exactly one special hexad. There are $132$ such hexads.

<<tiddler [[include_tiddlers/S(5,8,24).html#"S(5,8,24)"]]>>

The ''Scattering Matrix'' (or ''S-matrix'') relates the initial state and the final state of a physical system undergoing a scattering process. These are also refered to as "in"- and "out"-state $|\alpha \rangle$ and $|\beta \rangle$, where $\alpha$ and $\beta$ denote sets of quantum numbers, characterizing momenta, spin z-components and types of particles (e.g., photons, leptons, etc).

More formally, the S-matrix $S$ is defined as the [[unitary|Unitarity]] matrix (i.e. $S^ \dagger S = 1$) connecting asymptotic particle states $\Psi$ in the [[Hilbert space|Hilbert Space]] of physical states (scattering channels), i.e. $\Psi(\infty) = S \Psi(-\infty)$.
It is given by
\[
S = \lim_{t \rightarrow + \infty\atop t' \rightarrow -\infty} e^{it H_0 } e^{-i (t-t') H} e^{-it' H_0}
\]
where $H_0$ is the free Hamiltonian and $H$ the Hamiltonian containing the interaction term $V (\Psi)$.

While the S-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no horizons, it has a simple form in the case of the Minkowski space where the Hilbert space is a space of irreducible unitary representations of the [[inhomogeneous Lorentz group|Poincaré Transformation]]; the S-matrix is the evolution operator between time equal to minus infinity, and time equal to plus infinity. It is defined only in the limit of zero energy density (or infinite particle separation distance). It can be shown that if a quantum field theory in Minkowski space has a [[mass gap|Mass Gap]], the state in the asymptotic past and in the asymptotic future are both described by Fock spaces.

!!!!Limitations
The S-matrix formulation is possible only if the assumption of the existence of non-interacting asymptotic states or fields before and after the scattering process is justified.
This is not the case in quantum electrodynamics and in [[quantum field theories|Quantum Field Theory]] in a curved spacetime background, where one encounters the co called "infrared problem". Here one is seeking for alternatives to the S-matrix description.
In all these cases no formulation of a Fock space for asymptotic states for very early and very late times is possible.

In [[conformal|Conformal Transformation]] quantum field theories the sheer definition of a S-matrix is not possible. Asymptotic states and fields are ill defined as far removed points can be mapped to close points through a dilation transformation.

!!!!Historical
The S-matrix was introduced independently by John Archibald Wheeler in 1937 and by Werner Heisenberg in the 1940s.

@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/Heisenberg.jpg][Comments]]}}}&nbsp;@@

<<tiddler [[include_tiddlers/SL(2,C).html#"SL(2,C)"]]>>

Papers:
* [[A New Family Symmetry for SO(10) GUTs (2009) - S. F. King, C. Luhn|http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.1686v1.pdf]] [[local|papers/0905.1686v1.pdf]] [[pct. 15|http://scholar.google.de/scholar?cites=17881056879842397469&as_sdt=2005&sciodt=2000&hl=de]]
* [[A Supersymmetric Grand Unified Theory of Flavour with PSL2(7) × SO(10) (2010) - S. F. King, C. Luhn|http://arxiv.org/PS_cache/arxiv/pdf/0912/0912.1344v3.pdf]] [[local|papers/0912.1344v3.pdf]] [[pct. 6|http://scholar.google.de/scholar?cites=8133791727243503799&as_sdt=2005&sciodt=2000&hl=de]]
* [[Renormalisable SO (10) Models and Neutrino Masses and Mixing (2007) - W. Grimus, H. Kühböck|http://th-www.if.uj.edu.pl/acta/vol38/pdf/v38p3373.pdf]] [[local|papers/v38p3373.pdf]] pct. 0

Papers:
* [[Going down from a 3-form in 16 Dimensions - L. Baulieu |http://arxiv.org/PS_cache/hep-th/pdf/0207/0207184v1.pdf]]

<<tiddler [[include_tiddlers/SO(32).html#"SO(32)"]]>>

See [[conformal group|Conformal Group]].

Papers:
* [[Representation Theory and Projective Geometry (2004) - J. M. Landsberg, L. Manivel|http://www.math.tamu.edu/~jml/LMpopreprint.pdf]] [[local|papers/LMpopreprint.pdf]] [[pct. 25|http://scholar.google.de/scholar?cites=12280133377652367576&hl=de&as_sdt=2000]]

<<tiddler [[include_tiddlers/SO(8).html#"SO(8)"]]>>

''STEP'' (the ''S''atellite ''T''est of the ''E''quivalence ''P''rinciple) will advance experimental limits on violations of [[Einstein’s equivalence principle|Equivalence Principles]] from their present sensitivity of $2$ parts in $10^{13}$ to $1$ part in $10^{18}$ through multiple comparison of the motions of four pairs of test masses of different compositions in an earth-orbiting drag-free satellite.

Papers:
* [[The Science Case for STEP - J. Overduin, F. Everitt, J. Mester, P. Worden|http://arxiv.org/PS_cache/arxiv/pdf/0902/0902.2247v2.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=10364899760222302819&hl=de]]

<<tiddler [[include_tiddlers/STS(15).html#"STS(15)"]]>>

''$SU(1,1)$'' is a non-compact [[Lie group|Lie Group]] which is isomorphic to $SL(2,\mathbb R)$ and $Sp(2,\mathbb R)$.

The $SU(1,1)$ group manifold corresponds to a two-dimensional conformally invariant sigma model (Wess\-Zumino\-Witten model), with torsion.

Papers:
* [[Quantization of the Optical Phase Space S^2 = {phi mod 2pi, I > 0} in Terms of the Group SO(1,2) - H. A. Kastrup|http://arxiv.org/PS_cache/quant-ph/pdf/0307/0307069v4.pdf]] [[local|papers/0307069v4.pdf]] [[pct. 22|http://scholar.google.de/scholar?cites=14215644449933176298&hl=de]]
* [[Some Basics of su(1,1) - M. Novaes|http://www.sbfisica.org.br/rbef/pdf/040704.pdf]] [[pct. 3|http://scholar.google.com/scholar?hl=de&lr=&cites=3668612177166917623&um=1&ie=UTF-8&ei=-T2uSqjXHZmF_AbQrJG0Bg&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]

The group ''$SU(2,2)$'', which is non-compact, is locally isomorphic to [[SO(4,2)]]; as such it is the group of conformal relativity.

Papers:
* [[ On the Unitary Representations of SU (1, 1) and SU (2, 1) - L. C. Biedenharn, J. Nuyts, N. Straumann|http://archive.numdam.org/article/AIHPA_1965__3_1_13_0.pdf]] pct. 0

$SU(3) \in$ [[SO(8)]], as generally one has $SU(n) \in SO(n^2 -1)$.

Papers:
* [[Semi-Simple Lie Algebras and their Representations - R. N. Cahn|http://www.scribd.com/document_downloads/87692?extension=pdf&secret_password=]]

<<tiddler [[include_tiddlers/Sabinin Algebra.html#"Sabinin Algebra"]]>>

''[[Sage|http://www.sagemath.org]]'' is a coherent consolidation platform of over 64 of the best open source scientific packages in pure and applied mathematics, working under a command line and a browser interface. Its internal scripting language is the Python.

*<<LaunchApplicationButton "Launch SUN Virtual Box" "System" "file:///C:\Program Files\Sun\VirtualBox\VirtualBox.exe">>
* [[Launch Browser Worksheet|http://192.168.56.101]]
* [[SAGE Online|http://www.sagenb.org/]]

SAGE manuals:
* [[Sage Reference Manual|http://www.sagemath.org/doc/reference/index.html]]

Systems included:
* [[GAP]]
* [[PARI GP]]
* Singular
* [[MAXIMA|MAXIMA]]

(Some) optional systems:
* [[Mathematica]]
* [[MAGMA]]
* [[Maple|Maple]]
* [[MATLAB|MATLAB]]

Examples:
* [[Short Sage-Combinatorics Demo|http://alpha.sagenb.org/home/pub/8/]]

Google books:
* [[Computational Group Theory and the Theory of Groups: AMS Special Session on Computational Group Theory, March 3-4, 2007, Davidson College, Davidson, North Carolina - L.-C. Kappe, A. Magidin, R. F. Morse|http://books.google.com/books?id=3i-XzHwgHGIC&pg=PA133&lpg=PA133&dq=gap+orthogonal+group+finite+fields&source=bl&ots=8cL1FgTdDR&sig=PdaUTJ7s7uTasW4_fsqI9JkUq0U&hl=de&ei=xxEtS_yrDYej_Ab--ISNCQ&sa=X&oi=book_result&ct=result&resnum=7&ved=0CDQQ6AEwBg#v=onepage&q=gap%20orthogonal%20group%20finite%20fields&f=false]] pct. 0

Videos:
* [[Sage Days: Algebraic Geometry - MSRI (Mathematical Sciences Institute)|http://www.msri.org/calendar/workshops/WorkshopInfo/502/show_workshop]]

<<tiddler [[include_tiddlers/Sagle Identity.html#"Sagle Identity"]]>>

A ''Scalar Product'' $\langle\, .\,|\,.\, \rangle_\mathbb{K}\,$ (a.k.a. ''Inner Product'' or ''Dot Product'') is a symmetric bilinear form $\langle\, .\,|\,.\,\rangle_\mathbb{K}: ( \mb A, \mb B)  \mapsto \mathbb{K}\;$ defined by:
\begin{eqnarray}
\langle \mb A| \mb B \rangle_\mathbb{K} = \langle \mb B| \mb A \rangle_\mathbb{K} &\equiv & \frac{1}{2} ( \mb{AB}^{*} +  \mb{BA}^{*} ) \\

& =& \operatorname{Tr} (\mb{AB}^*) = \operatorname{Tr} (\mb{BA}^*) \\

& =& \frac{1}{2} (( \mb A+ \mb B)( \mb A+ \mb B)^{*} -  \mb{AA}^{*} -  \mb{BB}^{*})\\
& =& \langle\mb A+\mb B|\mb A+\mb B\rangle_\mathbb{K} - \langle\mb A|\mb A\rangle_\mathbb{K} - \langle\mb B|\mb B\rangle_\mathbb{K}\\
& \equiv& \mathcal Q(\mb A +  \mb B) - \mathcal Q(\mb A) - \mathcal Q(\mb B) \\
&= &\sum_{i}   \mb A_i \mb B_i \\
\end{eqnarray}
where $\mathcal{Q} (\mb A) \equiv ||\mb A||^2$ is a [[quadratic form|Quadratic Form]] (or ''Quadratic [[Norm]]'') which is the special case of a scalar product with $\mb A = \mb B$. $\operatorname{Tr}$ is the [[trace-|Trace]]function.

For reasons of brevity we'll usually omit the index $\mathbb{K}$.

<<tiddler [[include_tiddlers/Scale Invariance.html#"Scale Invariance"]]>>

<<tiddler [[include_tiddlers/Scale Relativity.html#"Scale Relativity"]]>>

The (time-dependent) ''Schrödinger Equation'' is given by
\[
i\hbar\frac{\partial}{\partial t} \Psi(\mb r,\,t) = \left(-\frac{\hbar^2}{2m}\bs \nabla^2 + V(\mb r)\right)\Psi(\mb r,\,t) = \hat H \Psi
\]

See also:
* [[Discrete Schrödinger operator|Discrete Schrödinger Operator]]

Links:
* [[WIKIPEDIA - Schrödinger Equation|http://en.wikipedia.org/wiki/Schrödinger_equation]]
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<<tiddler [[include_tiddlers/Schrödinger's Cat.html#"Schrödinger's Cat"]]>>

The ''Schrödinger\-Helmholtz Equation'' is given by
\[
\bs \nabla^2 \Phi (\mb r) + \lambda \Phi (\mb r) = 0
\]

It is the radial part of the time-dependent [[Schrödinger equation|Schrödinger Equation]], applying the ansatz  $\Psi(\mb r, t) \equiv \Phi(\mb r) e^{-iEt /\hbar}$, in case that $\lambda = \frac {2mE}{\hbar^2}$.
@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/Schroedinger.jpg][Comments]]}}}&nbsp;@@

<<tiddler [[include_tiddlers/Schwarzschild Metric.html#"Schwarzschild Metric"]]>>

<<tiddler [[include_tiddlers/Schwarzschild Radius.html#"Schwarzschild Radius"]]>>

<<tiddler [[include_tiddlers/Schwarzschild-De Sitter Metric.html#"Schwarzschild-De Sitter Metric"]]>>

The following search keywords are used:

"pct." - paper citations (Google scholar)
"jct." - journal citations (Google scholar)
"bct." - book citations (Google scholar)

"prl." - paper relevance, scale 1,...,10, with 10 the best rating (a personal estimate)
"jrl." -  journal relevance, scale 1,...,10, with 10 the best rating (a personal estimate)
"trl." - presentation/talk relevance, scale 1,...,10, with 10 the best rating (a personal estimate)
"brl." - book relevance, scale 1,...,10, with 10 the best rating (a personal estimate)

/***
|Name|SearchOptionsPlugin|
|Source|http://www.TiddlyTools.com/#SearchOptionsPlugin|
|Documentation|http://www.TiddlyTools.com/#SearchOptionsPluginInfo|
|Version|3.0.6|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|extend core search function with additional user-configurable options|
Adds extra options to core search function including selecting which data items to search, enabling/disabling incremental key-by-key searches, and generating a ''list of matching tiddlers'' instead of immediately displaying all matches.  This plugin also adds syntax for rendering 'search links' within tiddler content to embed one-click searches using pre-defined 'hard-coded' search terms.
!!!!!Documentation
>see [[SearchOptionsPluginInfo]]
!!!!!Configuration
<<<
Search in:
<<option chkSearchTitles>> titles <<option chkSearchText>> text <<option chkSearchTags>> tags <<option chkSearchFields>> fields <<option chkSearchShadows>> shadows
<<option chkSearchHighlight>> Highlight matching text in displayed tiddlers
<<option chkSearchList>> Show list of matches
<<option chkSearchListTiddler>> Write list to [[SearchResults]] tiddler
<<option chkSearchTitlesFirst>> Show title matches first
<<option chkSearchByDate>> Sort matching tiddlers by modification date (most recent first)
<<option chkIncrementalSearch>> Incremental key-by-key search: {{twochar{<<option txtIncrementalSearchMin>>}}} or more characters,  {{threechar{<<option txtIncrementalSearchDelay>>}}} msec delay
<<option chkSearchOpenTiddlers>> Search only in tiddlers that are currently displayed
<<option chkSearchExcludeTags>> Exclude tiddlers tagged with: <<option txtSearchExcludeTags>>
<<<
!!!!!Revisions
<<<
2009.09.22 [3.0.6] in TiddlyWiki.prototype.search, added 'match' param for core compatibility
2009.01.16 [3.0.5] added chkSearchOpenTiddlers option to limit searches to displayed tiddlers only
|please see [[SearchOptionsPluginInfo]] for additional revision details|
2005.10.18 [1.0.0] Initial Release
<<<
!!!!!Code
***/
//{{{
version.extensions.SearchOptionsPlugin= {major: 3, minor: 0, revision: 6, date: new Date(2009,9,22)};

var defaults={
	chkSearchTitles:	true,
	chkSearchText:		true,
	chkSearchTags:		true,
	chkSearchFields:	true,
	chkSearchTitlesFirst:	true,
	chkSearchList:		true,
	chkSearchHighlight:	true,
	chkSearchListTiddler:	false,
	chkSearchByDate:	false,
	chkIncrementalSearch:	true,
	chkSearchShadows:	true,
	chkSearchOpenTiddlers:	false,
	chkSearchExcludeTags:	true,
	txtSearchExcludeTags:	'excludeSearch',
	txtIncrementalSearchDelay:	500,
	txtIncrementalSearchMin:	3
}; for (var id in defaults) if (config.options[id]===undefined)
	config.options[id]=defaults[id];

if (config.macros.search.reportTitle==undefined)
	config.macros.search.reportTitle="SearchResults"; // note: not a cookie!
config.macros.search.label+="\xa0"; // a little bit of space just because it looks better
//}}}
// // searchLink: {{{[search[text to find]] OR [search[text to display|text to find]]}}}
//{{{
config.formatters.push( {
	name: "searchLink",
	match: "\\[search\\[",
	lookaheadRegExp: /\[search\[(.*?)(?:\|(.*?))?\]\]/mg,
	prompt: "search for: '%0'",
	handler: function(w)
	{
		this.lookaheadRegExp.lastIndex = w.matchStart;
		var lookaheadMatch = this.lookaheadRegExp.exec(w.source);
		if(lookaheadMatch && lookaheadMatch.index == w.matchStart) {
			var label=lookaheadMatch[1];
			var text=lookaheadMatch[2]||label;
			var prompt=this.prompt.format([text]);
			var btn=createTiddlyButton(w.output,label,prompt,
				function(){story.search(this.getAttribute("searchText"))},"searchLink");
			btn.setAttribute("searchText",text);
			w.nextMatch = this.lookaheadRegExp.lastIndex;
		}
	}
});
//}}}
// // incremental search uses option settings instead of hard-coded delay and minimum input values
//{{{
var fn=config.macros.search.onKeyPress;
fn=fn.toString().replace(/500/g, "config.options.txtIncrementalSearchDelay||500");
fn=fn.toString().replace(/> 2/g, ">=(config.options.txtIncrementalSearchMin||3)");
eval("config.macros.search.onKeyPress="+fn);
//}}}
// // REPLACE story.search() for option to "show search results in a list"
//{{{
Story.prototype.search = function(text,useCaseSensitive,useRegExp)
{
	var co=config.options; // abbrev
	var re=new RegExp(useRegExp ? text : text.escapeRegExp(),useCaseSensitive ? "mg" : "img");
	if (config.options.chkSearchHighlight) highlightHack=re;
	var matches = store.search(re,co.chkSearchByDate?"modified":"title","");
	if (co.chkSearchByDate) matches=matches.reverse(); // most recent first
	var q = useRegExp ? "/" : "'";
	clearMessage();
	if (!matches.length) {
		if (co.chkSearchListTiddler) discardSearchResults();
		displayMessage(config.macros.search.failureMsg.format([q+text+q]));
	} else {
		if (co.chkSearchList||co.chkSearchListTiddler)
			reportSearchResults(text,matches);
		else {
			var titles = []; for(var t=0; t<matches.length; t++) titles.push(matches[t].title);
			this.closeAllTiddlers(); story.displayTiddlers(null,titles);
			displayMessage(config.macros.search.successMsg.format([matches.length, q+text+q]));
		}
	}
	highlightHack = null;
}
//}}}
// // REPLACE store.search() for enhanced searching/sorting options
//{{{
TiddlyWiki.prototype.search = function(searchRegExp,sortField,excludeTag,match)
{
	var co=config.options; // abbrev
	var tids = this.reverseLookup("tags",excludeTag,!!match,sortField);
	var opened=[]; story.forEachTiddler(function(tid,elem){opened.push(tid);});

	// eliminate tiddlers tagged with excluded tags
	if (co.chkSearchExcludeTags&&co.txtSearchExcludeTags.length) {
		var ex=co.txtSearchExcludeTags.readBracketedList();
		var temp=[]; for(var t=tids.length-1; t>=0; t--)
			if (!tids[t].tags.containsAny(ex)) temp.push(tids[t]);
		tids=temp;
	}

	// scan for matching titles first...
	var results = [];
	if (co.chkSearchTitles) {
		for(var t=0; t<tids.length; t++) {
			if (co.chkSearchOpenTiddlers && !opened.contains(tids[t].title)) continue;
			if(tids[t].title.search(searchRegExp)!=-1) results.push(tids[t]);
		}
		if (co.chkSearchShadows)
			for (var t in config.shadowTiddlers) {
				if (co.chkSearchOpenTiddlers && !opened.contains(t)) continue;
				if ((t.search(searchRegExp)!=-1) && !store.tiddlerExists(t))
					results.push((new Tiddler()).assign(t,config.shadowTiddlers[t]));
			}
	}
	// then scan for matching text, tags, or field data
	for(var t=0; t<tids.length; t++) {
		if (co.chkSearchOpenTiddlers && !opened.contains(tids[t].title)) continue;
		if (co.chkSearchText && tids[t].text.search(searchRegExp)!=-1)
			results.pushUnique(tids[t]);
		if (co.chkSearchTags && tids[t].tags.join(" ").search(searchRegExp)!=-1)
			results.pushUnique(tids[t]);
		if (co.chkSearchFields && store.forEachField!=undefined)
			store.forEachField(tids[t],
				function(tid,field,val) {
					if (val.search(searchRegExp)!=-1) results.pushUnique(tids[t]);
				},
				true); // extended fields only
	}
	// then check for matching text in shadows
	if (co.chkSearchShadows)
		for (var t in config.shadowTiddlers) {
			if (co.chkSearchOpenTiddlers && !opened.contains(t)) continue;
			if ((config.shadowTiddlers[t].search(searchRegExp)!=-1) && !store.tiddlerExists(t))
				results.pushUnique((new Tiddler()).assign(t,config.shadowTiddlers[t]));
		}

	// if not 'titles first', or sorting by modification date,
	// re-sort results to so titles, text, tag and field matches are mixed together
	if(!sortField) sortField = "title";
	var bySortField=function(a,b){
		if(a[sortField]==b[sortField])return(0);else return(a[sortField]<b[sortField])?-1:+1;
	}
	if (!co.chkSearchTitlesFirst || co.chkSearchByDate) results.sort(bySortField);

	return results;
}
//}}}
// // HIJACK core {{{<<search>>}}} macro to add "report" and "simple inline" output
//{{{
config.macros.search.SOP_handler=config.macros.search.handler;
config.macros.search.handler = function(place,macroName,params)
{
	// if "report", use SearchOptionsPlugin report generator for inline output
	if (params[1]&&params[1].substr(0,6)=="report") {
		var keyword=params[0];
		var options=params[1].split("=")[1]; // split "report=option+option+..."
		var heading=params[2]?params[2].unescapeLineBreaks():"";
		var matches=store.search(new RegExp(keyword.escapeRegExp(),"img"),"title","excludeSearch");
		if (matches.length) wikify(heading+window.formatSearchResults(keyword,matches,options),place);
	} else if (params[1]) {
		var keyword=params[0];
		var heading=params[1]?params[1].unescapeLineBreaks():"";
		var seperator=params[2]?params[2].unescapeLineBreaks():", ";
		var matches=store.search(new RegExp(keyword.escapeRegExp(),"img"),"title","excludeSearch");
		if (matches.length) {
			var out=[];
			for (var m=0; m<matches.length; m++) out.push("[["+matches[m].title+"]]");
			wikify(heading+out.join(seperator),place);
		}
	} else
		config.macros.search.SOP_handler.apply(this,arguments);
};
//}}}
// // SearchResults panel handling
//{{{
setStylesheet(".searchResults { padding:1em 1em 0 1em; }","searchResults"); // matches std tiddler padding

config.macros.search.createPanel=function(text,matches,body) {

	function getByClass(e,c) { var d=e.getElementsByTagName("div");
		for (var i=0;i<d.length;i++) if (hasClass(d[i],c)) return d[i]; }
	var panel=createTiddlyElement(null,"div","searchPanel","searchPanel");
	this.renderPanel(panel,text,matches,body);
	var oldpanel=document.getElementById("searchPanel");
	if (!oldpanel) { // insert new panel just above tiddlers
		var da=document.getElementById("displayArea");
		da.insertBefore(panel,da.firstChild);
	} else { // if panel exists
		var oldwrap=getByClass(oldpanel,"searchResults");
		var newwrap=getByClass(panel,"searchResults");
		// if no prior content, just insert new content
		if (!oldwrap) oldpanel.insertBefore(newwrap,null);
		else {	// swap search results content but leave containing panel intact
			oldwrap.style.display='block'; // unfold wrapper if needed
			var i=oldwrap.getElementsByTagName("input")[0]; // get input field
			if (i) { var pos=this.getCursorPos(i); i.onblur=null; } // get cursor pos, ignore blur
			oldpanel.replaceChild(newwrap,oldwrap);
			panel=oldpanel; // use existing panel
		}
	}
	this.showPanel(true,pos);
	return panel;
}

config.macros.search.renderPanel=function(panel,text,matches,body) {

	var wrap=createTiddlyElement(panel,"div",null,"searchResults");
	wrap.onmouseover = function(e){ addClass(this,"selected"); }
	wrap.onmouseout = function(e){ removeClass(this,"selected"); }
	// create toolbar: "open all", "fold/unfold", "close"
	var tb=createTiddlyElement(wrap,"div",null,"toolbar");
	var b=createTiddlyButton(tb, "open all", "open all matching tiddlers", function() {
		story.displayTiddlers(null,this.getAttribute("list").readBracketedList()); return false; },"button");
	var list=""; for(var t=0;t<matches.length;t++) list+='[['+matches[t].title+']] ';
	b.setAttribute("list",list);
	var b=createTiddlyButton(tb, "fold", "toggle display of search results", function() {
		config.macros.search.foldPanel(this); return false; },"button");
	var b=createTiddlyButton(tb, "close", "dismiss search results",	function() {
		config.macros.search.showPanel(false); return false; },"button");
	createTiddlyText(createTiddlyElement(wrap,"div",null,"title"),"Search for: "+text); // title
	wikify(body,createTiddlyElement(wrap,"div",null,"viewer")); // report
	return panel;
}

config.macros.search.showPanel=function(show,pos) {
	var panel=document.getElementById("searchPanel");
	var i=panel.getElementsByTagName("input")[0];
	i.onfocus=show?function(){config.macros.search.stayFocused(true);}:null;
	i.onblur=show?function(){config.macros.search.stayFocused(false);}:null;
	if (show && panel.style.display=="block") { // if shown, grab focus, restore cursor
		if (i&&this.stayFocused()) { i.focus(); this.setCursorPos(i,pos); }
		return;
	}
	if(!config.options.chkAnimate) {
		panel.style.display=show?"block":"none";
		if (!show) { removeChildren(panel); config.macros.search.stayFocused(false); }
	} else {
		var s=new Slider(panel,show,false,show?"none":"children");
		s.callback=function(e,p){e.style.overflow="visible";}
		anim.startAnimating(s);
	}
	return panel;
}

config.macros.search.foldPanel=function(button) {
	var d=document.getElementById("searchPanel").getElementsByTagName("div");
	for (var i=0;i<d.length;i++) if (hasClass(d[i],"viewer")) var v=d[i]; if (!v) return;
	var show=v.style.display=="none";
	if(!config.options.chkAnimate)
		v.style.display=show?"block":"none";
	else {
		var s=new Slider(v,show,false,"none");
		s.callback=function(e,p){e.style.overflow="visible";}
		anim.startAnimating(s);
	}
	button.innerHTML=show?"fold":"unfold";
	return false;
}

config.macros.search.stayFocused=function(keep) { // TRUE/FALSE=set value, no args=get value
	if (keep===undefined) return this.keepReportInFocus;
	this.keepReportInFocus=keep;
	return keep
}

config.macros.search.getCursorPos=function(i) {
	var s=0; var e=0; if (!i) return { start:s, end:e };
	try {
		if (i.setSelectionRange) // FF
			{ s=i.selectionStart; e=i.selectionEnd; }
		if (document.selection && document.selection.createRange) { // IE
			var r=document.selection.createRange().duplicate();
			var len=r.text.length; s=0-r.moveStart('character',-100000); e=s+len;
		}
	}catch(e){};
	return { start:s, end:e };
}
config.macros.search.setCursorPos=function(i,pos) {
	if (!i||!pos) return; var s=pos.start; var e=pos.end;
	if (i.setSelectionRange) //FF
		i.setSelectionRange(s,e);
	if (i.createTextRange) // IE
		{ var r=i.createTextRange(); r.collapse(true); r.moveStart("character",s); r.select(); }
}
//}}}
// // SearchResults report generation
// note: these functions are defined globally, so they can be more easily redefined to customize report formats//
//{{{
if (!window.reportSearchResults) window.reportSearchResults=function(text,matches)
{
	var cms=config.macros.search; // abbrev
	var body=window.formatSearchResults(text,matches);
	if (!config.options.chkSearchListTiddler) // show #searchResults panel
		window.scrollTo(0,ensureVisible(cms.createPanel(text,matches,body)));
	else { // write [[SearchResults]] tiddler
		var title=cms.reportTitle;
		var who=config.options.txtUserName;
		var when=new Date();
		var tags="excludeLists excludeSearch temporary";
		var tid=store.getTiddler(title); if (!tid) tid=new Tiddler();
		tid.set(title,body,who,when,tags);
		store.addTiddler(tid);
		story.closeTiddler(title);
		story.displayTiddler(null,title);
	}
}

if (!window.formatSearchResults) window.formatSearchResults=function(text,matches,opt)
{
	var body='';
	var title=config.macros.search.reportTitle
	var q = config.options.chkRegExpSearch ? "/" : "'";
	if (!opt) var opt="all";
	var parts=opt.split("+");
	for (var i=0; i<parts.length; i++) { var p=parts[i].toLowerCase();
		if (p=="again"||p=="all")   body+=window.formatSearchResults_again(text,matches);
		if (p=="summary"||p=="all") body+=window.formatSearchResults_summary(text,matches);
		if (p=="list"||p=="all")    body+=window.formatSearchResults_list(text,matches);
		if (p=="buttons"||p=="all") body+=window.formatSearchResults_buttons(text,matches);
	}
	return body;
}

if (!window.formatSearchResults_again) window.formatSearchResults_again=function(text,matches)
{
	var title=config.macros.search.reportTitle
	var body='';
	// search again
	body+='{{span{<<search "'+text.replace(/"/g,'&#x22;')+'">> /%\n';
	body+='%/<html><input type="button" value="search again"';
	body+=' onclick="var t=this.parentNode.parentNode.getElementsByTagName(\'input\')[0];';
	body+=' config.macros.search.doSearch(t); return false;">';
	body+=' <a href="javascript:;" onclick="';
	body+=' var e=this.parentNode.nextSibling;';
	body+=' var show=e.style.display!=\'block\';';
	body+=' if(!config.options.chkAnimate) e.style.display=show?\'block\':\'none\';';
	body+=' else anim.startAnimating(new Slider(e,show,false,\'none\'));';
	body+=' return false;">options...</a>';
	body+='</html>@@display:none;border-left:1px dotted;margin-left:1em;padding:0;padding-left:.5em;font-size:90%;/%\n';
	body+='	%/<<option chkSearchTitles>>titles /%\n';
	body+='	%/<<option chkSearchText>>text /%\n';
	body+='	%/<<option chkSearchTags>>tags /%\n';
	body+='	%/<<option chkSearchFields>>fields /%\n';
	body+='	%/<<option chkSearchShadows>>shadows\n';
	body+='	<<option chkCaseSensitiveSearch>>case-sensitive /%\n';
	body+='	%/<<option chkRegExpSearch>>text patterns /%\n';
	body+='	%/<<option chkSearchByDate>>sorted by date\n';
	body+='	<<option chkSearchHighlight>> highlight matching text in displayed tiddlers\n';
	body+='	<<option chkIncrementalSearch>>incremental key-by-key search: /%\n';
	body+='	%/{{twochar{<<option txtIncrementalSearchMin>>}}} or more characters, /%\n';
	body+='	%/{{threechar{<<option txtIncrementalSearchDelay>>}}} msec delay\n';
	body+='	<<option chkSearchOpenTiddlers>> search only in tiddlers that are currently displayed\n';
	body+='	<<option chkSearchExcludeTags>>exclude tiddlers tagged with:\n';
	body+='	{{editor{<<option txtSearchExcludeTags>>}}}/%\n';
	body+='%/@@}}}\n\n';
	return body;
}

if (!window.formatSearchResults_summary) window.formatSearchResults_summary=function(text,matches)
{
	// summary: nn tiddlers found matching '...', options used
	var body='';
	var co=config.options; // abbrev
	var title=config.macros.search.reportTitle
	var q = co.chkRegExpSearch ? "/" : "'";
	body+="''"+config.macros.search.successMsg.format([matches.length,q+"{{{"+text+"}}}"+q])+"''\n";
	var opts=[];
	if (co.chkSearchTitles) opts.push("titles");
	if (co.chkSearchText) opts.push("text");
	if (co.chkSearchTags) opts.push("tags");
	if (co.chkSearchFields) opts.push("fields");
	if (co.chkSearchShadows) opts.push("shadows");
	if (co.chkSearchOpenTiddlers) body+="^^//search limited to displayed tiddlers only//^^\n";
	body+="~~&nbsp; searched in "+opts.join(" + ")+"~~\n";
	body+=(co.chkCaseSensitiveSearch||co.chkRegExpSearch?"^^&nbsp; using ":"")
		+(co.chkCaseSensitiveSearch?"case-sensitive ":"")
		+(co.chkRegExpSearch?"pattern ":"")
		+(co.chkCaseSensitiveSearch||co.chkRegExpSearch?"matching^^\n":"");
	return body;
}

if (!window.formatSearchResults_list) window.formatSearchResults_list=function(text,matches)
{
	// bullet list of links to matching tiddlers
	var body='';
	var pattern=co.chkRegExpSearch?text:text.escapeRegExp();
	var sensitive=co.chkCaseSensitiveSearch?"mg":"img";
	var link='{{tiddlyLinkExisting{<html><nowiki><a href="javascript:;" onclick="'
		+'if(config.options.chkSearchHighlight)'
		+'	highlightHack=new RegExp(\x27'+pattern+'\x27.escapeRegExp(),\x27'+sensitive+'\x27);'
		+'story.displayTiddler(null,\x27%0\x27);'
		+'highlightHack = null; return false;'
		+'" title="%2">%1</a></html>}}}';
	for(var t=0;t<matches.length;t++) {
		body+="* ";
		if (config.options.chkSearchByDate)
			body+=matches[t].modified.formatString('YYYY.0MM.0DD 0hh:0mm')+" ";
		var title=matches[t].title;
		var fixup=title.replace(/'/g,"\\x27").replace(/"/g,"\\x22");
		var tid=store.getTiddler(title);
		var tip=tid?tid.getSubtitle():''; tip=tip.replace(/"/g,"&quot;");
		body+=link.format([fixup,title,tip])+'\n';
	}
	return body;
}

if (!window.formatSearchResults_buttons) window.formatSearchResults_buttons=function(text,matches)
{
	// embed buttons only if writing SearchResults to tiddler
	if (!config.options.chkSearchListTiddler) return "";
	// "open all" button
	var title=config.macros.search.reportTitle;
	var body="";
	body+="@@diplay:block;<html><input type=\"button\" href=\"javascript:;\" "
		+"onclick=\"story.displayTiddlers(null,[";
	for(var t=0;t<matches.length;t++)
		body+="'"+matches[t].title.replace(/\'/mg,"\\'")+"'"+((t<matches.length-1)?", ":"");
	body+="],1);\" accesskey=\"O\" value=\"open all matching tiddlers\"></html> ";
	// "discard SearchResults" button
	body+="<html><input type=\"button\" href=\"javascript:;\" "
		+"onclick=\"discardSearchResults()\" value=\"discard "+title+"\"></html>";
	body+="@@\n";
	return body;
}

if (!window.discardSearchResults) window.discardSearchResults=function()
{
	// remove the tiddler
	story.closeTiddler(config.macros.search.reportTitle);
	store.deleteTiddler(config.macros.search.reportTitle);
	store.notify(config.macros.search.reportTitle,true);
}
//}}}

''Second\-Order Arithmetic'' is a collection of axiomatic systems that formalize the natural numbers and sets thereof.
Pairs of natural numbers can be coded in the usual way as natural numbers. In this way arbitrary integers or rational numbers can be represented.
Real numbers can be defined as Cauchy sequences of rational numbers. Real functions are legitimate objects of study, since they are defined by their values on the rationals.

<<tiddler [[include_tiddlers/Sedenion.html#"Sedenion"]]>>

<<tiddler [[include_tiddlers/Sedenion Multiplication Tables.html#"Sedenion Multiplication Tables"]]>>

<<tiddler [[include_tiddlers/Sedenion Projective Geometry.html#"Sedenion Projective Geometry"]]>>

<<tiddler [[include_tiddlers/Sedenion Subalgebras.html#"Sedenion Subalgebras"]]>>

<<tiddler [[include_tiddlers/Sedenion Zero Divisor.html#"Sedenion Zero Divisor"]]>>

Seiberg and Witten have argued that certain noncommutative [[gauge theories|Gauge Theory]] are equivalent to commutative ones and in particular that there exists a map from a commutative gauge field to a noncommutative one, which is compatible with the gauge structure of each. This map has become known as the ''Seiberg\-Witten (SW) Map''.

<<tiddler [[include_tiddlers/Selberg Trace Formula.html#"Selberg Trace Formula"]]>>

''Self Organized Criticality'' (= ''SOC'').

Links:
* [[WIKIPEDIA - Self-organized Criticality|http://en.wikipedia.org/wiki/Self-organized_criticality]]

Papers:
* [[Phase Transitions and Complex Systems - R. V. Solé, S. C. Manrubia, B. Luque, J. Delgado|http://complex.upf.es/~ricard/COMPLEXITY-96.pdf]] [[pct. 66|http://scholar.google.de/scholar?cites=15790333541130288044&hl=de]]
* [[Cellular Automata and Self-organized Criticality - M. Creutz|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.50.5288&rep=rep1&type=pdf]] [[pct. 8|http://scholar.google.com/scholar?cites=9131077436801048589&as_sdt=2005&sciodt=2000&hl=de]] TRD
* [[Symmetropy and Self-Organized Criticality - K. Nanjo, H. Nagahama, E. Yodogawa|http://www.scipress.org/journals/forma/pdf/1603/16030213.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=2013446146386239124&hl=de]]
* [[Y-Bias and Angularity: The Dynamics of Self-Organizing Criticality From the Zero Point to Infinity - D. G. Yurth, D. Ayres|http://www.pureenergysystems.com/academy/papers/Y-Bias_and_Angularity/Y-Bias_Monograph_Preview_Edition_25Apr06c.doc]] pct. 0

A ''Semigroup'' is an algebraic structure consisting of a nonempty set together with an associative binary operation. A semigroup differs from a [[group|Group]] in that for each of its elements there may not exist an inverse and an identity element.

<<tiddler [[include_tiddlers/Separable State.html#"Separable State"]]>>

<<tiddler [[include_tiddlers/Sextenion.html#"Sextenion"]]>>

<<tiddler [[include_tiddlers/Shannon Entropy.html#"Shannon Entropy"]]>>

<<tiddler [[include_tiddlers/Shear.html#"Shear"]]>>

<<tiddler [[include_tiddlers/Ship of Theseus.html#"Ship of Theseus"]]>>

The ''Shuffle Product'' $\mb A \sqcup \! \sqcup \mb B$ of two words $\mb A = a_1 \ldots a_m$ and $\mb B = b_{m+1} \ldots b_{m+n}$ is defined as the sum of all words it is possible to construct from $\mb A$ and $\mb B$ by preserving the order of all the letters in each of the words.
I.e.
\[
\mb A \sqcup \! \sqcup \mb B \equiv \sum_\sigma  a_{\sigma(1)} a_{\sigma(2)}\ldots  a_{\sigma(m)}  b_{\sigma(m+1)} b_{\sigma(m+2)}\ldots b_{\sigma(m+n)}
\] with $\sigma(1) < \sigma(2) < \ldots < \sigma(m)$ and $\sigma(m + 1) < \sigma(m + 2) < \ldots < \sigma(m + n)$, where $\sigma$ is the operation of permutation.

The product is also called a ''(m, n)-Shuffle''.

!!!!Examples
# $12 \sqcup\!\sqcup ab= 12ab + 1a2b + 1ab2 + a12b + a1b2 + ab12 $.  <br><br>
# $123 \sqcup\!\sqcup ab = 123ab + 12a3b + 12ab3 + 1a23b + 1a2b3 + 1ab23 + a123b + a12b3 + a1b23 + ab123$.

!!!!Properties
* Commutative, i.e. $\mb A \sqcup \! \sqcup \mb B = \mb B \sqcup \! \sqcup \mb A$. <br><br>
* The number of terms of the product is given by ${m + n \choose m} = {m + n \choose n}$.

If one extracts the signs of a multiplication table one gets what will be called a ''Sign Table'' (or ''Sign Matrix'').

In the following the sign tables of the [[Cayley-Dickson Algebras|Cayley-Dickson Algebra]] obtained by the conventional [[doubling process|Cayley-Dickson Doubling]] are listed.
In the non-split cases the signs are "balanced" and one gets [[Hadamard matrices|Hadamard Matrix]]. Therefore these sign matrices have maximally possible absolute determinant.
In case of the split-algebras (hyperbolic algebras) one always has sign matrices with a determinant equal to zero.

!!!Complex Numbers
!!!!CD(1) ([[Complex Numbers|Complex Number]])
~~
|+|+|
|+|-|
~~
!!!!CD(-1) ([[Split complex numbers|Split Complex Number]])
~~
|+|+|
|+|+|
~~
!!!Quaternions
!!!!CD(1,1)  ([[Classical Quaternions|Quaternion]])
~~
|+|+|+|+|
|+|-|+|-|
|+|-|-|+|
|+|+|-|-|
~~
!!!!CD(1,-1) ([[Split quaternions|Split Quaternion]])
~~
|+|+|+|+|
|+|-|+|-|
|+|-|+|-|
|+|+|+|+|
~~
!!!!CD(-1,1)
~~
|+|+|+|+|
|+|+|+|+|
|+|-|-|+|
|+|-|-|+|
~~
!!!!CD(-1,-1)
~~
|+|+|+|+|
|+|+|+|+|
|+|-|+|-|
|+|-|+|-|
~~
!!!Octonions
!!!!CD(1,1,1)
~~
|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|-|+|
|+|-|-|+|+|+|-|-|
|+|+|-|-|+|-|+|-|
|+|-|-|-|-|+|+|+|
|+|+|-|+|-|-|-|+|
|+|+|+|-|-|+|-|-|
|+|-|+|+|-|-|+|-|
~~
!!!!CD(1,1,-1)
~~
|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|-|+|
|+|-|-|+|+|+|-|-|
|+|+|-|-|+|-|+|-|
|+|-|-|-|+|-|-|-|
|+|+|-|+|+|+|+|-|
|+|+|+|-|+|-|+|+|
|+|-|+|+|+|+|-|+|
~~
!!!!CD(1,-1,1)
~~
|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|-|+|
|+|-|+|-|+|+|+|+|
|+|+|+|+|+|-|-|+|
|+|-|-|-|-|+|+|+|
|+|+|-|+|-|-|-|+|
|+|+|-|+|-|+|+|+|
|+|-|-|-|-|-|-|+|
~~
!!!!CD(-1,1,1)
~~
|+|+|+|+|+|+|+|+|
|+|+|+|+|+|+|-|-|
|+|-|-|+|+|+|-|-|
|+|-|-|+|+|+|+|+|
|+|-|-|-|-|+|+|+|
|+|-|-|-|-|+|-|-|
|+|+|+|-|-|+|-|-|
|+|+|+|-|-|+|+|+|
~~
!!!!CD(1,-1,-1)
~~
|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|-|+|
|+|-|+|-|+|+|+|+|
|+|+|+|+|+|-|-|+|
|+|-|-|-|+|-|-|-|
|+|+|-|+|+|+|+|-|
|+|+|-|+|+|-|-|-|
|+|-|-|-|+|+|+|-|
~~
!!!!CD(-1,1,-1)
~~
|+|+|+|+|+|+|+|+|
|+|+|+|+|+|+|-|-|
|+|-|-|+|+|+|-|-|
|+|-|-|+|+|+|+|+|
|+|-|-|-|+|-|-|-|
|+|-|-|-|+|-|+|+|
|+|+|+|-|+|-|+|+|
|+|+|+|-|+|-|-|-|
~~
!!!!CD(-1,-1,1)
~~
|+|+|+|+|+|+|+|+|
|+|+|+|+|+|+|-|-|
|+|-|+|-|+|+|+|+|
|+|-|+|-|+|+|-|-|
|+|-|-|-|-|+|+|+|
|+|-|-|-|-|+|-|-|
|+|+|-|+|-|+|+|+|
|+|+|-|+|-|+|-|-|
~~
!!!!CD(-1,-1,-1)
~~
|+|+|+|+|+|+|+|+|
|+|+|+|+|+|+|-|-|
|+|-|+|-|+|+|+|+|
|+|-|+|-|+|+|-|-|
|+|-|-|-|+|-|-|-|
|+|-|-|-|+|-|+|+|
|+|+|-|+|+|-|-|-|
|+|+|-|+|+|-|+|+|
~~

!!!Sedenions
!!!!CD(1,1,1,1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|-|+|+|-|-|+|-|+|+|-|
|+|-|-|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|+|-|-|+|-|+|-|+|-|+|-|-|+|-|+|
|+|-|-|-|-|+|+|+|+|+|+|+|-|-|-|-|
|+|+|-|+|-|-|-|+|+|-|+|-|+|-|+|-|
|+|+|+|-|-|+|-|-|+|-|-|+|+|-|-|+|
|+|-|+|+|-|-|+|-|+|+|-|-|+|+|-|-|
|+|-|-|-|-|-|-|-|-|+|+|+|+|+|+|+|
|+|+|-|+|-|+|+|-|-|-|-|+|-|+|+|-|
|+|+|+|-|-|-|+|+|-|+|-|-|-|-|+|+|
|+|-|+|+|-|+|-|+|-|-|+|-|-|+|-|+|
|+|+|+|+|+|-|-|-|-|+|+|+|-|-|-|-|
|+|-|+|-|+|+|+|-|-|-|+|-|+|-|+|-|
|+|-|-|+|+|-|+|+|-|-|-|+|+|-|-|+|
|+|+|-|-|+|+|-|+|-|+|-|-|+|+|-|-|
~~
!!!!CD(1,1,1,-1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|-|+|+|-|-|+|-|+|+|-|
|+|-|-|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|+|-|-|+|-|+|-|+|-|+|-|-|+|-|+|
|+|-|-|-|-|+|+|+|+|+|+|+|-|-|-|-|
|+|+|-|+|-|-|-|+|+|-|+|-|+|-|+|-|
|+|+|+|-|-|+|-|-|+|-|-|+|+|-|-|+|
|+|-|+|+|-|-|+|-|+|+|-|-|+|+|-|-|
|+|-|-|-|-|-|-|-|+|-|-|-|-|-|-|-|
|+|+|-|+|-|+|+|-|+|+|+|-|+|-|-|+|
|+|+|+|-|-|-|+|+|+|-|+|+|+|+|-|-|
|+|-|+|+|-|+|-|+|+|+|-|+|+|-|+|-|
|+|+|+|+|+|-|-|-|+|-|-|-|+|+|+|+|
|+|-|+|-|+|+|+|-|+|+|-|+|-|+|-|+|
|+|-|-|+|+|-|+|+|+|+|+|-|-|+|+|-|
|+|+|-|-|+|+|-|+|+|-|+|+|-|-|+|+|
~~
!!!!CD(1,1,-1,1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|-|+|+|-|-|+|-|+|+|-|
|+|-|-|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|+|-|-|+|-|+|-|+|-|+|-|-|+|-|+|
|+|-|-|-|+|-|-|-|+|+|+|+|+|+|+|+|
|+|+|-|+|+|+|+|-|+|-|+|-|-|+|-|+|
|+|+|+|-|+|-|+|+|+|-|-|+|-|+|+|-|
|+|-|+|+|+|+|-|+|+|+|-|-|-|-|+|+|
|+|-|-|-|-|-|-|-|-|+|+|+|+|+|+|+|
|+|+|-|+|-|+|+|-|-|-|-|+|-|+|+|-|
|+|+|+|-|-|-|+|+|-|+|-|-|-|-|+|+|
|+|-|+|+|-|+|-|+|-|-|+|-|-|+|-|+|
|+|+|+|+|-|+|+|+|-|+|+|+|+|+|+|+|
|+|-|+|-|-|-|-|+|-|-|+|-|-|+|-|+|
|+|-|-|+|-|+|-|-|-|-|-|+|-|+|+|-|
|+|+|-|-|-|-|+|-|-|+|-|-|-|-|+|+|
~~
!!!!CD(1,-1,1,1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|-|+|+|-|-|+|-|+|+|-|
|+|-|+|-|+|+|+|+|+|+|+|+|-|-|-|-|
|+|+|+|+|+|-|-|+|+|-|-|+|-|+|+|-|
|+|-|-|-|-|+|+|+|+|+|+|+|-|-|-|-|
|+|+|-|+|-|-|-|+|+|-|+|-|+|-|+|-|
|+|+|-|+|-|+|+|+|+|-|+|-|+|-|+|-|
|+|-|-|-|-|-|-|+|+|+|+|+|+|+|+|+|
|+|-|-|-|-|-|-|-|-|+|+|+|+|+|+|+|
|+|+|-|+|-|+|+|-|-|-|-|+|-|+|+|-|
|+|+|-|+|-|-|-|-|-|+|+|+|-|-|-|-|
|+|-|-|-|-|+|+|-|-|-|-|+|-|+|+|-|
|+|+|+|+|+|-|-|-|-|+|+|+|-|-|-|-|
|+|-|+|-|+|+|+|-|-|-|+|-|+|-|+|-|
|+|-|+|-|+|-|-|-|-|-|+|-|+|-|+|-|
|+|+|+|+|+|+|+|-|-|+|+|+|+|+|+|+|
~~
!!!!CD(-1,1,1,1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|+|+|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|-|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|-|+|+|+|+|+|+|+|+|+|-|-|-|-|
|+|-|-|-|-|+|+|+|+|+|+|+|-|-|-|-|
|+|-|-|-|-|+|-|-|+|+|+|+|+|+|+|+|
|+|+|+|-|-|+|-|-|+|-|-|+|+|-|-|+|
|+|+|+|-|-|+|+|+|+|-|-|+|+|-|-|+|
|+|-|-|-|-|-|-|-|-|+|+|+|+|+|+|+|
|+|-|-|-|-|-|+|+|-|+|-|-|-|-|+|+|
|+|+|+|-|-|-|+|+|-|+|-|-|-|-|+|+|
|+|+|+|-|-|-|-|-|-|+|+|+|-|-|-|-|
|+|+|+|+|+|-|-|-|-|+|+|+|-|-|-|-|
|+|+|+|+|+|-|+|+|-|+|+|+|+|+|+|+|
|+|-|-|+|+|-|+|+|-|-|-|+|+|-|-|+|
|+|-|-|+|+|-|-|-|-|-|-|+|+|-|-|+|
~~
!!!!CD(1,1,-1,-1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|-|+|+|-|-|+|-|+|+|-|
|+|-|-|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|+|-|-|+|-|+|-|+|-|+|-|-|+|-|+|
|+|-|-|-|+|-|-|-|+|+|+|+|+|+|+|+|
|+|+|-|+|+|+|+|-|+|-|+|-|-|+|-|+|
|+|+|+|-|+|-|+|+|+|-|-|+|-|+|+|-|
|+|-|+|+|+|+|-|+|+|+|-|-|-|-|+|+|
|+|-|-|-|-|-|-|-|+|-|-|-|-|-|-|-|
|+|+|-|+|-|+|+|-|+|+|+|-|+|-|-|+|
|+|+|+|-|-|-|+|+|+|-|+|+|+|+|-|-|
|+|-|+|+|-|+|-|+|+|+|-|+|+|-|+|-|
|+|+|+|+|-|+|+|+|+|-|-|-|-|-|-|-|
|+|-|+|-|-|-|-|+|+|+|-|+|+|-|+|-|
|+|-|-|+|-|+|-|-|+|+|+|-|+|-|-|+|
|+|+|-|-|-|-|+|-|+|-|+|+|+|+|-|-|
~~
!!!!CD(1,-1,1,-1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|-|+|+|-|-|+|-|+|+|-|
|+|-|+|-|+|+|+|+|+|+|+|+|-|-|-|-|
|+|+|+|+|+|-|-|+|+|-|-|+|-|+|+|-|
|+|-|-|-|-|+|+|+|+|+|+|+|-|-|-|-|
|+|+|-|+|-|-|-|+|+|-|+|-|+|-|+|-|
|+|+|-|+|-|+|+|+|+|-|+|-|+|-|+|-|
|+|-|-|-|-|-|-|+|+|+|+|+|+|+|+|+|
|+|-|-|-|-|-|-|-|+|-|-|-|-|-|-|-|
|+|+|-|+|-|+|+|-|+|+|+|-|+|-|-|+|
|+|+|-|+|-|-|-|-|+|-|-|-|+|+|+|+|
|+|-|-|-|-|+|+|-|+|+|+|-|+|-|-|+|
|+|+|+|+|+|-|-|-|+|-|-|-|+|+|+|+|
|+|-|+|-|+|+|+|-|+|+|-|+|-|+|-|+|
|+|-|+|-|+|-|-|-|+|+|-|+|-|+|-|+|
|+|+|+|+|+|+|+|-|+|-|-|-|-|-|-|-|
~~
!!!!CD(-1,1,1,-1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|+|+|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|-|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|-|+|+|+|+|+|+|+|+|+|-|-|-|-|
|+|-|-|-|-|+|+|+|+|+|+|+|-|-|-|-|
|+|-|-|-|-|+|-|-|+|+|+|+|+|+|+|+|
|+|+|+|-|-|+|-|-|+|-|-|+|+|-|-|+|
|+|+|+|-|-|+|+|+|+|-|-|+|+|-|-|+|
|+|-|-|-|-|-|-|-|+|-|-|-|-|-|-|-|
|+|-|-|-|-|-|+|+|+|-|+|+|+|+|-|-|
|+|+|+|-|-|-|+|+|+|-|+|+|+|+|-|-|
|+|+|+|-|-|-|-|-|+|-|-|-|+|+|+|+|
|+|+|+|+|+|-|-|-|+|-|-|-|+|+|+|+|
|+|+|+|+|+|-|+|+|+|-|-|-|-|-|-|-|
|+|-|-|+|+|-|+|+|+|+|+|-|-|+|+|-|
|+|-|-|+|+|-|-|-|+|+|+|-|-|+|+|-|
~~
!!!!CD(1,-1,-1,1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|-|+|+|-|-|+|-|+|+|-|
|+|-|+|-|+|+|+|+|+|+|+|+|-|-|-|-|
|+|+|+|+|+|-|-|+|+|-|-|+|-|+|+|-|
|+|-|-|-|+|-|-|-|+|+|+|+|+|+|+|+|
|+|+|-|+|+|+|+|-|+|-|+|-|-|+|-|+|
|+|+|-|+|+|-|-|-|+|-|+|-|-|+|-|+|
|+|-|-|-|+|+|+|-|+|+|+|+|-|-|-|-|
|+|-|-|-|-|-|-|-|-|+|+|+|+|+|+|+|
|+|+|-|+|-|+|+|-|-|-|-|+|-|+|+|-|
|+|+|-|+|-|-|-|-|-|+|+|+|-|-|-|-|
|+|-|-|-|-|+|+|-|-|-|-|+|-|+|+|-|
|+|+|+|+|-|+|+|+|-|+|+|+|+|+|+|+|
|+|-|+|-|-|-|-|+|-|-|+|-|-|+|-|+|
|+|-|+|-|-|+|+|+|-|-|+|-|-|+|-|+|
|+|+|+|+|-|-|-|+|-|+|+|+|-|-|-|-|
~~
!!!!CD(-1,1,-1,1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|+|+|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|-|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|-|+|+|+|+|+|+|+|+|+|-|-|-|-|
|+|-|-|-|+|-|-|-|+|+|+|+|+|+|+|+|
|+|-|-|-|+|-|+|+|+|+|+|+|-|-|-|-|
|+|+|+|-|+|-|+|+|+|-|-|+|-|+|+|-|
|+|+|+|-|+|-|-|-|+|-|-|+|-|+|+|-|
|+|-|-|-|-|-|-|-|-|+|+|+|+|+|+|+|
|+|-|-|-|-|-|+|+|-|+|-|-|-|-|+|+|
|+|+|+|-|-|-|+|+|-|+|-|-|-|-|+|+|
|+|+|+|-|-|-|-|-|-|+|+|+|-|-|-|-|
|+|+|+|+|-|+|+|+|-|+|+|+|+|+|+|+|
|+|+|+|+|-|+|-|-|-|+|+|+|-|-|-|-|
|+|-|-|+|-|+|-|-|-|-|-|+|-|+|+|-|
|+|-|-|+|-|+|+|+|-|-|-|+|-|+|+|-|
~~
!!!!CD(-1,-1,1,1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|+|+|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|+|-|+|+|+|+|+|+|+|+|-|-|-|-|
|+|-|+|-|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|-|-|-|+|+|+|+|+|+|+|-|-|-|-|
|+|-|-|-|-|+|-|-|+|+|+|+|+|+|+|+|
|+|+|-|+|-|+|+|+|+|-|+|-|+|-|+|-|
|+|+|-|+|-|+|-|-|+|-|+|-|+|-|+|-|
|+|-|-|-|-|-|-|-|-|+|+|+|+|+|+|+|
|+|-|-|-|-|-|+|+|-|+|-|-|-|-|+|+|
|+|+|-|+|-|-|-|-|-|+|+|+|-|-|-|-|
|+|+|-|+|-|-|+|+|-|+|-|-|-|-|+|+|
|+|+|+|+|+|-|-|-|-|+|+|+|-|-|-|-|
|+|+|+|+|+|-|+|+|-|+|+|+|+|+|+|+|
|+|-|+|-|+|-|-|-|-|-|+|-|+|-|+|-|
|+|-|+|-|+|-|+|+|-|-|+|-|+|-|+|-|
~~
!!!!CD(1,-1,-1,-1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|-|+|+|-|-|+|-|+|+|-|
|+|-|+|-|+|+|+|+|+|+|+|+|-|-|-|-|
|+|+|+|+|+|-|-|+|+|-|-|+|-|+|+|-|
|+|-|-|-|+|-|-|-|+|+|+|+|+|+|+|+|
|+|+|-|+|+|+|+|-|+|-|+|-|-|+|-|+|
|+|+|-|+|+|-|-|-|+|-|+|-|-|+|-|+|
|+|-|-|-|+|+|+|-|+|+|+|+|-|-|-|-|
|+|-|-|-|-|-|-|-|+|-|-|-|-|-|-|-|
|+|+|-|+|-|+|+|-|+|+|+|-|+|-|-|+|
|+|+|-|+|-|-|-|-|+|-|-|-|+|+|+|+|
|+|-|-|-|-|+|+|-|+|+|+|-|+|-|-|+|
|+|+|+|+|-|+|+|+|+|-|-|-|-|-|-|-|
|+|-|+|-|-|-|-|+|+|+|-|+|+|-|+|-|
|+|-|+|-|-|+|+|+|+|+|-|+|+|-|+|-|
|+|+|+|+|-|-|-|+|+|-|-|-|+|+|+|+|
~~
!!!!CD(-1,1,-1,-1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|+|+|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|-|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|-|+|+|+|+|+|+|+|+|+|-|-|-|-|
|+|-|-|-|+|-|-|-|+|+|+|+|+|+|+|+|
|+|-|-|-|+|-|+|+|+|+|+|+|-|-|-|-|
|+|+|+|-|+|-|+|+|+|-|-|+|-|+|+|-|
|+|+|+|-|+|-|-|-|+|-|-|+|-|+|+|-|
|+|-|-|-|-|-|-|-|+|-|-|-|-|-|-|-|
|+|-|-|-|-|-|+|+|+|-|+|+|+|+|-|-|
|+|+|+|-|-|-|+|+|+|-|+|+|+|+|-|-|
|+|+|+|-|-|-|-|-|+|-|-|-|+|+|+|+|
|+|+|+|+|-|+|+|+|+|-|-|-|-|-|-|-|
|+|+|+|+|-|+|-|-|+|-|-|-|+|+|+|+|
|+|-|-|+|-|+|-|-|+|+|+|-|+|-|-|+|
|+|-|-|+|-|+|+|+|+|+|+|-|+|-|-|+|
~~
!!!!CD(-1,-1,1,-1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|+|+|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|+|-|+|+|+|+|+|+|+|+|-|-|-|-|
|+|-|+|-|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|-|-|-|+|+|+|+|+|+|+|-|-|-|-|
|+|-|-|-|-|+|-|-|+|+|+|+|+|+|+|+|
|+|+|-|+|-|+|+|+|+|-|+|-|+|-|+|-|
|+|+|-|+|-|+|-|-|+|-|+|-|+|-|+|-|
|+|-|-|-|-|-|-|-|+|-|-|-|-|-|-|-|
|+|-|-|-|-|-|+|+|+|-|+|+|+|+|-|-|
|+|+|-|+|-|-|-|-|+|-|-|-|+|+|+|+|
|+|+|-|+|-|-|+|+|+|-|+|+|+|+|-|-|
|+|+|+|+|+|-|-|-|+|-|-|-|+|+|+|+|
|+|+|+|+|+|-|+|+|+|-|-|-|-|-|-|-|
|+|-|+|-|+|-|-|-|+|+|-|+|-|+|-|+|
|+|-|+|-|+|-|+|+|+|+|-|+|-|+|-|+|
~~
!!!!CD(-1,-1,-1,1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|+|+|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|+|-|+|+|+|+|+|+|+|+|-|-|-|-|
|+|-|+|-|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|-|-|+|-|-|-|+|+|+|+|+|+|+|+|
|+|-|-|-|+|-|+|+|+|+|+|+|-|-|-|-|
|+|+|-|+|+|-|-|-|+|-|+|-|-|+|-|+|
|+|+|-|+|+|-|+|+|+|-|+|-|-|+|-|+|
|+|-|-|-|-|-|-|-|-|+|+|+|+|+|+|+|
|+|-|-|-|-|-|+|+|-|+|-|-|-|-|+|+|
|+|+|-|+|-|-|-|-|-|+|+|+|-|-|-|-|
|+|+|-|+|-|-|+|+|-|+|-|-|-|-|+|+|
|+|+|+|+|-|+|+|+|-|+|+|+|+|+|+|+|
|+|+|+|+|-|+|-|-|-|+|+|+|-|-|-|-|
|+|-|+|-|-|+|+|+|-|-|+|-|-|+|-|+|
|+|-|+|-|-|+|-|-|-|-|+|-|-|+|-|+|
~~
!!!!CD(-1,-1,-1,-1)
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|+|+|+|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|+|-|+|+|+|+|+|+|+|+|-|-|-|-|
|+|-|+|-|+|+|-|-|+|+|-|-|-|-|+|+|
|+|-|-|-|+|-|-|-|+|+|+|+|+|+|+|+|
|+|-|-|-|+|-|+|+|+|+|+|+|-|-|-|-|
|+|+|-|+|+|-|-|-|+|-|+|-|-|+|-|+|
|+|+|-|+|+|-|+|+|+|-|+|-|-|+|-|+|
|+|-|-|-|-|-|-|-|+|-|-|-|-|-|-|-|
|+|-|-|-|-|-|+|+|+|-|+|+|+|+|-|-|
|+|+|-|+|-|-|-|-|+|-|-|-|+|+|+|+|
|+|+|-|+|-|-|+|+|+|-|+|+|+|+|-|-|
|+|+|+|+|-|+|+|+|+|-|-|-|-|-|-|-|
|+|+|+|+|-|+|-|-|+|-|-|-|+|+|+|+|
|+|-|+|-|-|+|+|+|+|+|-|+|+|-|+|-|
|+|-|+|-|-|+|-|-|+|+|-|+|+|-|+|-|
~~

!!!![[MAGMA|http://magma.maths.usyd.edu.au/calc/]]^^[[Help|MAGMA]]^^ examples
* [[Code File|code/MAGMASignMatrices.txt]]

<<tiddler [[include_tiddlers/Signature.html#"Signature"]]>>

<<tiddler [[include_tiddlers/Signature Change.html#"Signature Change"]]>>

Given a set of [[root vectors|Root Vector]], a ''Simple Root'' is a positive root which is not the sum of any two positive roots. The simple roots form a basis of a vector space. Any positive root is a positive integer linear combination of simple roots.

/***
|Name|SinglePageModePlugin|
|Source|http://www.TiddlyTools.com/#SinglePageModePlugin|
|Documentation|http://www.TiddlyTools.com/#SinglePageModePluginInfo|
|Version|2.9.6|
|Author|Eric Shulman - ELS Design Studios|
|License|http://www.TiddlyTools.com/#LegalStatements <br>and [[Creative Commons Attribution-ShareAlike 2.5 License|http://creativecommons.org/licenses/by-sa/2.5/]]|
|~CoreVersion|2.1|
|Type|plugin|
|Requires||
|Overrides|Story.prototype.displayTiddler(), Story.prototype.displayTiddlers()|
|Options|##Configuration|
|Description|Show tiddlers one at a time with automatic permalink, or always open tiddlers at top/bottom of page.|
This plugin allows you to configure TiddlyWiki to navigate more like a traditional multipage web site with only one tiddler displayed at a time.
!!!!!Documentation
>see [[SinglePageModePluginInfo]]
!!!!!Configuration
<<<
<<option chkSinglePageMode>> Display one tiddler at a time
><<option chkSinglePagePermalink>> Automatically permalink current tiddler
><<option chkSinglePageKeepFoldedTiddlers>> Don't close tiddlers that are folded
><<option chkSinglePageKeepEditedTiddlers>> Don't close tiddlers that are being edited
<<option chkTopOfPageMode>> Open tiddlers at the top of the page
<<option chkBottomOfPageMode>> Open tiddlers at the bottom of the page
<<option chkSinglePageAutoScroll>> Automatically scroll tiddler into view (if needed)

Notes:
* The "display one tiddler at a time" option can also be //temporarily// set/reset by including a 'paramifier' in the document URL: {{{#SPM:true}}} or {{{#SPM:false}}}.
* If more than one display mode is selected, 'one at a time' display takes precedence over both 'top' and 'bottom' settings, and if 'one at a time' setting is not used, 'top of page' takes precedence over 'bottom of page'.
* When using Apple's Safari browser, automatically setting the permalink causes an error and is disabled.
<<<
!!!!!Revisions
<<<
2008.10.17 [2.9.6] changed chkSinglePageAutoScroll default to false
| Please see [[SinglePageModePluginInfo]] for previous revision details |
2005.08.15 [1.0.0] Initial Release.  Support for BACK/FORWARD buttons adapted from code developed by Clint Checketts.
<<<
!!!!!Code
***/
//{{{
version.extensions.SinglePageModePlugin= {major: 2, minor: 9, revision: 6, date: new Date(2008,10,17)};
//}}}
//{{{
config.paramifiers.SPM = { onstart: function(v) {
	config.options.chkSinglePageMode=eval(v);
	if (config.options.chkSinglePageMode && config.options.chkSinglePagePermalink && !config.browser.isSafari) {
		config.lastURL = window.location.hash;
		if (!config.SPMTimer) config.SPMTimer=window.setInterval(function() {checkLastURL();},1000);
	}
} };
//}}}
//{{{
if (config.options.chkSinglePageMode==undefined)
	config.options.chkSinglePageMode=true;
if (config.options.chkSinglePagePermalink==undefined)
	config.options.chkSinglePagePermalink=true;
if (config.options.chkSinglePageKeepFoldedTiddlers==undefined)
	config.options.chkSinglePageKeepFoldedTiddlers=false;
if (config.options.chkSinglePageKeepEditedTiddlers==undefined)
	config.options.chkSinglePageKeepEditedTiddlers=false;
if (config.options.chkTopOfPageMode==undefined)
	config.options.chkTopOfPageMode=false;
if (config.options.chkBottomOfPageMode==undefined)
	config.options.chkBottomOfPageMode=false;
if (config.options.chkSinglePageAutoScroll==undefined)
	config.options.chkSinglePageAutoScroll=false;
//}}}
//{{{
config.SPMTimer = 0;
config.lastURL = window.location.hash;
function checkLastURL()
{
	if (!config.options.chkSinglePageMode)
		{ window.clearInterval(config.SPMTimer); config.SPMTimer=0; return; }
	if (config.lastURL == window.location.hash) return; // no change in hash
	var tids=decodeURIComponent(window.location.hash.substr(1)).readBracketedList();
	if (tids.length==1) // permalink (single tiddler in URL)
		story.displayTiddler(null,tids[0]);
	else { // restore permaview or default view
		config.lastURL = window.location.hash;
		if (!tids.length) tids=store.getTiddlerText("DefaultTiddlers").readBracketedList();
		story.closeAllTiddlers();
		story.displayTiddlers(null,tids);
	}
}


if (Story.prototype.SPM_coreDisplayTiddler==undefined)
	Story.prototype.SPM_coreDisplayTiddler=Story.prototype.displayTiddler;
Story.prototype.displayTiddler = function(srcElement,tiddler,template,animate,slowly)
{
	var title=(tiddler instanceof Tiddler)?tiddler.title:tiddler;
	var tiddlerElem=document.getElementById(story.idPrefix+title); // ==null unless tiddler is already displayed
	var opt=config.options;
	var single=opt.chkSinglePageMode && !startingUp;
	var top=opt.chkTopOfPageMode && !startingUp;
	var bottom=opt.chkBottomOfPageMode && !startingUp;
	if (single) {
		story.forEachTiddler(function(tid,elem) {
			// skip current tiddler and, optionally, tiddlers that are folded.
			if (	tid==title
				|| (opt.chkSinglePageKeepFoldedTiddlers && elem.getAttribute("folded")=="true"))
				return;
			// if a tiddler is being edited, ask before closing
			if (elem.getAttribute("dirty")=="true") {
				if (opt.chkSinglePageKeepEditedTiddlers) return;
				// if tiddler to be displayed is already shown, then leave active tiddler editor as is
				// (occurs when switching between view and edit modes)
				if (tiddlerElem) return;
				// otherwise, ask for permission
				var msg="'"+tid+"' is currently being edited.\n\n";
				msg+="Press OK to save and close this tiddler\nor press Cancel to leave it opened";
				if (!confirm(msg)) return; else story.saveTiddler(tid);
			}
			story.closeTiddler(tid);
		});
	}
	else if (top)
		arguments[0]=null;
	else if (bottom)
		arguments[0]="bottom";
	if (single && opt.chkSinglePagePermalink && !config.browser.isSafari) {
		window.location.hash = encodeURIComponent(String.encodeTiddlyLink(title));
		config.lastURL = window.location.hash;
		document.title = wikifyPlain("SiteTitle") + " - " + title;
		if (!config.SPMTimer) config.SPMTimer=window.setInterval(function() {checkLastURL();},1000);
	}
	if (tiddlerElem && tiddlerElem.getAttribute("dirty")=="true") { // editing... move tiddler without re-rendering
		var isTopTiddler=(tiddlerElem.previousSibling==null);
		if (!isTopTiddler && (single || top))
			tiddlerElem.parentNode.insertBefore(tiddlerElem,tiddlerElem.parentNode.firstChild);
		else if (bottom)
			tiddlerElem.parentNode.insertBefore(tiddlerElem,null);
		else this.SPM_coreDisplayTiddler.apply(this,arguments); // let CORE render tiddler
	} else
		this.SPM_coreDisplayTiddler.apply(this,arguments); // let CORE render tiddler
	var tiddlerElem=document.getElementById(story.idPrefix+title);
	if (tiddlerElem&&opt.chkSinglePageAutoScroll) {
		// scroll to top of page or top of tiddler
		var isTopTiddler=(tiddlerElem.previousSibling==null);
		var yPos=isTopTiddler?0:ensureVisible(tiddlerElem);
		// if animating, defer scroll until after animation completes
		var delay=opt.chkAnimate?config.animDuration+10:0;
		setTimeout("window.scrollTo(0,"+yPos+")",delay);
	}
}

if (Story.prototype.SPM_coreDisplayTiddlers==undefined)
	Story.prototype.SPM_coreDisplayTiddlers=Story.prototype.displayTiddlers;
Story.prototype.displayTiddlers = function() {
	// suspend single/top/bottom modes when showing multiple tiddlers
	var opt=config.options;
	var saveSPM=opt.chkSinglePageMode; opt.chkSinglePageMode=false;
	var saveTPM=opt.chkTopOfPageMode; opt.chkTopOfPageMode=false;
	var saveBPM=opt.chkBottomOfPageMode; opt.chkBottomOfPageMode=false;
	this.SPM_coreDisplayTiddlers.apply(this,arguments);
	opt.chkBottomOfPageMode=saveBPM;
	opt.chkTopOfPageMode=saveTPM;
	opt.chkSinglePageMode=saveSPM;
}
//}}}

''Singular Value Decomposition (SVD)'' is the decomposition of a given matrix $\mb M$ (not necessarily a square matrix) into a product of three special matrices:
\begin{equation}
\mb M = \mb U \mb D \mb V
\end{equation}
with $\mb D$ a diagonal matrix and $\mb U$ and $\mb V$ unitary matrices.
The diagonal entries of $\mb D$ are called ''singular values''. For certain matrices $\mb M$ the singular values coincide with the eigenvalues of the matrix.

 - A Mathematics, Physics and Philosophy Notebook
<<gradient vert #dddddd #ffffff >>color:white;text-align:center;<<QOTD Quotations 10000>>>>

Trajectory of the Universe

<<tiddler [[include_tiddlers/Skyrmion.html#"Skyrmion"]]>>

> Sleep is one of the great unresolved issues in biology.
> - [1] -

!!!!A personal remark
As sleep is related with an altered state of [[consciousness|Consciousness]] it is quite suggestive that the big enigma of sleep is just another facet of the big enigma consciousness.
Therefore, one approach to gain a better understanding of consciousness might be to have a closer look at sleep.


Videos:
* [[[1] Thesciencenetwork - Why Do We Sleep? - G. Tononi|http://thesciencenetwork.org/programs/waking-up-to-sleep/giulio-tononi]]
* [[Thesciencenetwork - Sleep in Invertebrates - R. Greenspan|http://thesciencenetwork.org/programs/waking-up-to-sleep/ralph-greenspan]]

>I have a great suspicion that for example Euler today would spend much more of his time on writing software because he spent so much of his time e.g., in efforts of calculating tables of moon positions. And I believe that Gauß as well would spend much more time sitting in front of the screen.
> - Yuri I. Manin -

!!!!Programs
* [[Program folder |programs/]]
!!!!!Computer algebra
* [[Sage]] -e.g. [[codes|Nordstrom-Robinson Code]]
* [[GAP|GAP]] - e.g. groups, codes (GUAVA package), loops (LOOPS package)
* [[MAGMA|MAGMA]] - e.g. [[codes|Reed-Muller Code]], [[Hadamard matrices|Hadamard Matrix]]
* [[Mathematica]]
* [[Albert|Albert]]
* [[MAXIMA|MAXIMA]]
* [[Maple]] - e.g. codes
* [[GRTensorII|http://grtensor.phy.queensu.ca/]] (Requires Maple. A limited version is also available for Mathematica).
!!!!!Visualisations
* [[vZome|http://www.vorthmann.org/zome/]]
* [[Jenn3D|http://www.math.cmu.edu/~fho/jenn/]]
!!!!!Miscellaneous
* [[HotBits: Genuine Random Numbers, Generated by Radioactive Decay|http://www.fourmilab.ch/hotbits/]]
* [[C Routines for Systematic Investigation of Complexes, Quaternions, Octonions, Seizeons/Sedenions/Hexons, etc. W. D. Smith |programs/szon3.txt]]

Papers:
* [[A Field-theory Motivated Approach to Symbolic Computer Algebra - K. Peeters|http://arxiv.org/PS_cache/cs/pdf/0608/0608005v2.pdf]]
* [[Introducing Cadabra: A Symbolic Computer Algebra System for Field Theory Problems - K. Peeters|http://arxiv.org/PS_cache/hep-th/pdf/0701/0701238v1.pdf]]

Links:
* [[SAL (Scientific Applications on Linux): Computer Algebra Systems|http://www.sai.msu.su/sal/A/1/index.shtml]]
* [[Comparison of Computer Algebra Systems|http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems]]
* [[WIKIPEDIA - Tensor|http://en.wikipedia.org/wiki/Tensor]] - Contains a list of links of tensor software.

Links:
* [[WIKIPEDIA - Solder Form|http://en.wikipedia.org/wiki/Solder_form]]

<<tiddler [[include_tiddlers/Spacetime Algebra.html#"Spacetime Algebra"]]>>

<<tiddler [[include_tiddlers/Spacetime Condensate.html#"Spacetime Condensate"]]>>

<<tiddler [[include_tiddlers/Spacetime Entropy.html#"Spacetime Entropy"]]>>

The ''Special Orthogonal Group $SO(r, s)$'' is defined by:
\[
SO(r, s) \equiv \{\mb A \in O(r, s) : \det(\mb A) = 1\}
\]
with $O(r, s)$ denoting the [[orthogonal group|Orthogonal Group]].

''Special Relativity'' (short ''SR'') is synonymous with [[Poincaré invariance|Poincaré Transformation]] of spacetime, which includes the Noether conservation of angular momentum.

Papers:
* [[Alternative-Algebra Methods in the Special Theory of Relativity - E. K. Loginov|http://www.springerlink.com/content/uqq32882562804h2/fulltext.pdf]] pct. 0

<<tiddler [[include_tiddlers/Speciality of the World.html#"Speciality of the World"]]>>

<<tiddler [[include_tiddlers/Spectral Action.html#"Spectral Action"]]>>

<<tiddler [[include_tiddlers/Spectral Geometry.html#"Spectral Geometry"]]>>

<<tiddler [[include_tiddlers/Spectral Theorem.html#"Spectral Theorem"]]>>

A ''Spectral Triple $(\mathcal A, H, D)$'' (a.k.a. ''Unbounded K\-Cycle'') consists of an algebra $\mathcal A$ (not necessarily commutative), a [[Hilbert space|Hilbert Space]] $H$ and a [[Dirac operator|Dirac Operator]] $D$ and defines a [[noncommutative space|Noncommutative Manifold]] or geometry.

Spectral triples allow for a (noncommutative) generalization of [[Riemannian geometry|Riemann Space]].

There are not many examples of spectral triples: so far all known are classical (commutative geometry), finite dimensional (over finite algebras) or based on noncommutative tori (including the recently introduced "isospectral deformations").

Papers:
* [[Equivariant Lorentzian Spectral Triples - M. Paschke, A. Sitarz|http://arxiv.org/PS_cache/math-ph/pdf/0611/0611029v1.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=6447584271726142251&hl=de]]

<<tiddler [[include_tiddlers/Spin Connection.html#"Spin Connection"]]>>

<<tiddler [[include_tiddlers/Spin Foam.html#"Spin Foam"]]>>

<<tiddler [[include_tiddlers/Spin Network.html#"Spin Network"]]>>

>... I wish to recall what I still consider to be a fundamental open question of path integration. This is the providing of a clean, intellectually satisfying relativistic path integral for particles with spin.
> - L. S. Schulman [1] -

See also:
* [[Feynman checkerboard|Feynman Checkerboard]]


Papers:
* [[A Path Integral for Spin (1968) - L. Schulman|http://people.clarkson.edu/~lschulma/1968PISpinPR.pdf]] [[local|papers/1968PISpinPR.pdf]] {{t100Cite{[[pct. 278|http://scholar.google.de/scholar?hl=de&lr=&cites=8352899487690416685&um=1&ie=UTF-8&ei=yiPdTZLJEIj5sgalxajHDg&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCQQzgIwAA]]}}}
* [[Path Integrals and Pseudoclassical Description for Spinning Particles in Arbitrary Dimensions (1997) - D. M. Gitman|http://www.dfn.if.usp.br/pesq/tqr/gitman/10.pdf]] [[local|papers/10.pdf]] [[pct. 29|http://scholar.google.de/scholar?hl=de&lr=&cites=15081806253202322783&um=1&ie=UTF-8&ei=JG5PTpDaIeqQ4gT147DrBw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCIQzgIwAA]]
* [[A Rigorous Path Integral for Quantum Spin using Flat-space Wiener Regularization - B. Bodmann, H. Leschke, S. Warzel|http://www.theorie1.physik.uni-erlangen.de/leschke/papers/JMP02549.pdf]] [[local|papers/JMP02549.pdf]] [[pct. 10|http://scholar.google.de/scholar?hl=de&lr=&cites=8992519366958274455&um=1&ie=UTF-8&ei=LLFPTpDTE-T04QS0uqHPBw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB8QzgIwAA]]
* [[Spin in Quantum Field Theory (2005) - S. Forte|http://arxiv.org/pdf/hep-th/0507291]] [[local|papers/0507291v4.pdf]] [[pct. 4|http://scholar.google.at/scholar?cites=1181921085743274482&as_sdt=2005&sciodt=0,5&hl=de]]
* [[[1] Frontiers of Path Integration (1994) - L. S. Schulman|http://people.clarkson.edu/~lschulma/1994FrontiersPathIntegration.pdf]] [[local|papers/1994FrontiersPathIntegration.pdf]] pct. 0

Links:
* [[L. S. Schulman|http://people.clarkson.edu/~lschulma/]]

The ''Spin\-Statistics Theorem'' states that integer spin particles are bosons, while half-integer spin particles are fermions.

In quantum field theory the spin-statistics theorem is a consequence of the microscopic causality implied by the causal structure of space-time.

<html><center><img src="images/spirograph.jpg" style="width: 225px; "/></center></html>
A tool that fascinated me so much in my youth ...

<html><center><img src="images/spirograph_designs.jpg" style="width: 265px; "/></center></html>
Links:
* [[WIKIPEDIA - Spirograph|http://en.wikipedia.org/wiki/Spirograph]]

<<tiddler [[include_tiddlers/Split Complex Number.html#"Split Complex Number"]]>>

The ''Exceptional Lie Group $\tilde G_2$'' is the split real form of the complex [[Lie group|Lie Group]] [[G2]]. Contrary to $G_2$, $\tilde G_2$ is non-compact.

$\tilde G_2$ is the [[automorphism group|Automorphism]] of the [[split octonions|Split Octonion]], i.e. $\tilde G_2 = Aut (\mathbb{\tilde O})$.

Furthermore: $\tilde G_2 \subset O(3,4) \subset O(4,4)$ and as $\tilde G_2$ is connected, even $\tilde G_2 \subset SO(3,4)$.

The associated [[Lie algebra|Lie Algebra]] $\mathfrak{\tilde g}_2$ is the algebra of [[derivations|Derivation]] of the split octonions: $\mathfrak{\tilde g}_2$$ = \operatorname {Der} (\mathbb{\tilde O}$).

$\tilde G_2$ has [[SU(3)]] as a subgroup (see [1]) which makes it interesting in respect to applications in physics.


Papers:
* [[Split Octonions and Generic Rank Two Distributions in Dimension Five (2006) - K. Sagerschnig|http://emis.bibl.cwi.nl/journals/AM/06-S/sager.pdf]] [[local|papers/sager.pdf]] [[pct. 7|http://scholar.google.de/scholar?cites=9233518920655789497&hl=de]] prl. 8

Lectures:
* [[What is ... G2 ? - M. H. Weissman|http://people.ucsc.edu/~weissman/WhatIsG2.pdf]]

Theses:
* [[Weyl Structures for Generic Rank Two Distributions in Dimension Five (2008) - K. Sagerschnig|http://othes.univie.ac.at/2186/1/2008-10-09_9702296.pdf]] [[local|theses/2008-10-09_9702296.pdf]] [[tct. 4|http://scholar.google.de/scholar?cites=9760992082226168904&as_sdt=2005&sciodt=2000&hl=de]]

Journals:
* [1] Quark Structure and Octonions (1973) - M. Günaydin, F. Gürsey {{t100Cite{[[jct. 127|http://scholar.google.de/scholar?cites=54600949756025659&hl=de]]}}}

<<tiddler [[include_tiddlers/Split Quaternion.html#"Split Quaternion"]]>>

<<tiddler [[include_tiddlers/Split-Biquaternion.html#"Split-Biquaternion"]]>>

<<tiddler [[include_tiddlers/Spontaneous Symmetry Breaking.html#"Spontaneous Symmetry Breaking"]]>>

<<tiddler [[include_tiddlers/Sporadic Group.html#"Sporadic Group"]]>>

A ''Spread'' in a three-dimensional [[projective space|Projective Space]] $PG(3, q)$ is a set of lines such that each point of the space is incident with exactly one line. (In other words: the lines partition the set of points). A spread in $PG(3, q)$ exists for any prime power $q$.

!!!!Example
[[PG(3,2)]] contains $56$ spreads with $5$ elements each.

<html><center><img src="images/spread.jpg" style="width: 280px; "/></center></html>

<<tiddler [[include_tiddlers/Squeezed State.html#"Squeezed State"]]>>

Papers:
* [[Incompatibility and Instability Based Size Eects in Crystals and Composites at Finite Elastoplastic Strains - M. Becker|http://elib.uni-stuttgart.de/opus/volltexte/2006/2610/pdf/Becker06.pdf]]  [[local|papers/Becker06.pdf]]

Given a group $G$ and a manifold $\mathcal M$, the ''Stabilizer'' (''Stationary Subgroup'' or ''Little Group'') of a point $x \in \mathcal M$ is the subgroup $G_x \subset G$ which contains only those elements $g \subset G$ which leave the point $x$ fixed
\begin{equation}
G_x \equiv \{g \in G | gx = x \}
\end{equation}

<<tiddler [[include_tiddlers/Standard Model.html#"Standard Model"]]>>

<<tiddler [[include_tiddlers/Standard Model Lagrangian.html#"Standard Model Lagrangian"]]>>

The following is a (supposedly incomplete) list of theories that manage to derive parameters of the [[Standard model|Standard Model]] to a greater or lesser degree of accuracy:
* [[Chaotic Quantisation|Chaotic Quantization]]
* [[Lattice QCD]]
* [[Bit strings|Bit String Physics]]
* [[Heim theory|Heim Theory]]
* [[Sidarth's pion model|Pion]]

Papers:
* [[A Logical Basis for the Standard Model and Quark Mass Ratios (2008) - J. A. de Wet|http://m-hikari.com/imf-password2008/13-16-2008/dewetIMF13-16-2008.pdf]] [[local|papers/dewetIMF13-16-2008.pdf]] pct. 0

<<tiddler [[include_tiddlers/Star Product.html#"Star Product"]]>>

The ''Stasheff Polytope'' is the [[associahedron|Associahedron]] $\mathcal K(5)$.
{{center{[img(280px+, )[images/stasheff.gif]]}}}
{{center{[img(441px+, )[images/stasheff2.jpg]]}}}

The Stasheff polytope constists of $9$ faces, $6$ of which are pentagons and $4$ of which are squares.

Papers:
* [[Realization of the Stasheff Polytope - J.-L. Loday|http://arxiv.org/PS_cache/math/pdf/0212/0212126v1.pdf]] [[pct. 41|http://scholar.google.de/scholar?cites=14315522305952591595&hl=de]]
* [[Realizations of the Associahedron and Cyclohedron - C. Hohlweg, C. Lange|http://arxiv.org/PS_cache/math/pdf/0510/0510614v3.pdf]] [[pct. 12|http://scholar.google.de/scholar?cites=11322852241104519155&hl=de]]

<<tiddler [[include_tiddlers/Statistical Operator.html#"Statistical Operator"]]>>

<<tiddler [[include_tiddlers/Steiner Quadruple System.html#"Steiner Quadruple System"]]>>

A ''Steiner System S(t,k,n)'' is a [[t-design|Design]] $t-(n,k,\lambda)$ with $\lambda=1$. It is a set $\Omega$ of $n$ elements together with subsets of $\Omega$ consisting of $k$ elements (called ''Blocks'') with the property that each $t$-element subset of $\Omega$ is contained in exactly one block.
!!!!Special cases
A Steiner system ...
* $S(2, \tilde k + 1, \tilde k^2 + \tilde k + 1)$ is called a [[Projective Plane|Projective Plane]] $PG(2, \tilde k)$.
* $S(2, k, k^2) $ is called an ''Affine Plane'' $AG(2,k)$.
* $S(2, 3, n)$ is called a [[Steiner triple system|Steiner Triple System]] $ST(n)$.
* $S(3, 4, n)$ is called a [[Steiner quadruple system|Steiner Quadruple System]].
* $S(4, 5, n)$ is called a ''Steiner Quintuple System''.

!!!!Existency
For given $t$-element subsets the number of known Steiner systems is
* $t = 3$: infinitely many,
* $t = 4$: only a few,
* $t = 5$: exactly $9$ which are $S(5, 6, n)$ for $n = 12, 24, 48, 72, 84, 108, 132$, $S(5, 7, 28)$ and [[S(5,8,24)]].
* $t > 5$: none.

!!!!Example
$S(2,3,7)$ which is equivalent to the [[Fano plane|Fano Plane]].

<html><center><img src="images/FanoAndSteiner.jpg" style="width: 380px; "/></center></html>
The $7$ elements of $\Omega$ are represented by the $7$ points of the Fano plane, the blocks consisting of $3$ elements by the $7$ lines. The fact that any $2$-element subset is contained in exactly one block corresponds to the fact that each pair of points of the Fano plane lies on exactly one line.

<<tiddler [[include_tiddlers/Steiner Triple System.html#"Steiner Triple System"]]>>

<<tiddler [[include_tiddlers/Stephen Hawking.html#"Stephen Hawking"]]>>

Links:
* [[WIKIPEDIA - Sterile Neutrino|http://en.wikipedia.org/wiki/Sterile_neutrino]]
* [[Naturenews - Hunt for the Sterile Neutrino Heats up|http://www.nature.com/news/2010/100317/full/464334a.html?s=news_rss]]
* [[PhysOrg - Physics Experiment Suggests Existence of new Particle|http://www.physorg.com/news/2010-11-physics-particle.html]]

<<tiddler [[include_tiddlers/Steven Weinberg.html#"Steven Weinberg"]]>>

Links:
* [[WIKIPEDIA - Stiefel Manifold|http://en.wikipedia.org/wiki/Stiefel_manifold]]

''Stochastic [[Quantization]]'' was originally proposed by G. Parisi and Y.S. Wu in 1981 and has since been developed further by many people. The method reproduces [[quantum mechanics|Quantum Mechanics]] as the thermal-equilibrium limit of a hypothetical stochastic process with respect to some fictitious evolution parameter (also called fictitious time, which is different from ordinary time).
It can be considered to be a third method of [[quantization|Quantization]], remarkably different from the conventional theories, i.e. the canonical and [[path-integral|Path Integral]] ones. For instance it does not require a gauge-fixing and Faddeev\-Popov ghosts.
Stochastic quantization has been used for quantizing gauge fields and in performing large numerical simulations, which have never been obtained by other methods. It seems to have the potential to extend the territory of quantum mechanics and of [[quantum field theory|Quantum Field Theory]].

Papers:
* [[Stochastic Quantization of Einstein Gravity (1985) - H. Rumpf|http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?198510230]] [[local|papers/StochasticGravity.pdf]] [[pct. 49|http://scholar.google.com/scholar?cites=64449343545504033&as_sdt=5&sciodt=0&hl=de]]
* [[Functional Integral Approach to Parisi-Wu Stochastic Quantization: Abelian Gauge Theory (1985) - E. Gozzi|http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?198501171]] [[local|papers/FunctionalIntegralApproach.pdf]] [[pct. 8|http://scholar.google.com/scholar?hl=de&lr=&cites=8417203777927721970&um=1&ie=UTF-8&ei=t5RXTY3lKYHvsgaRgemlCw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB0QzgIwAA]] - "We feel that our proof, even if very formal, clarifies once and for all the mechanism that is responsible for the generation of the gauge fixing and of the Faddev\-Popov determinant in stochastic quantization".

Google books:
* [[Stochastic Quantization (1988) - P. H. Damgaard, H. Hüffel|http://books.google.com/books?id=nxZB4BfDhzcC&pg=PA337&lpg=PA337&dq=%22stochastic+quantization%22&source=bl&ots=XaGMEKRyqu&sig=evveQ9_qk_HH4u1ydSLHVDCfA9s&hl=de&ei=YDe7SuO1BYyW_QaKrqiVDQ&sa=X&oi=book_result&ct=result&resnum=6]] [[local|google_books/Stochastic Quantization.pdf]] {{t100Cite{[[bct. 231|http://scholar.google.de/scholar?cites=2435139135058331786&hl=de]]}}} - contains the original paper by Parisi & Wu.

Books:
* Stochastic Quantization (1992) - M. Namiki [[bct. 97|http://scholar.google.de/scholar?cites=8866850923628482282&hl=de]]

<<tiddler [[include_tiddlers/Strain.html#"Strain"]]>>

<br><<tiddler [[include_tiddlers/Strangeness.html#"Strangeness"]]>>

<br>@@display:block;text-align:center;[img[images/stress.gif]]@@

See also:
* [[Strain|Strain]]

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<<tiddler [[include_tiddlers/Subalgebra.html#"Subalgebra"]]>>

An algebra $\mathcal A$ is called a ''Superalgebra'' if it is $\mathbb Z_2$-graded, i.e. there exist linear subspaces $\mathcal A_i$ with $i \in \{0,1\}$ such that
\begin{eqnarray}
\mathcal A = \mathcal A_0 \oplus \mathcal A_1 \\
\end{eqnarray}
and
\begin{eqnarray}
\mathcal A_ 0 \mathcal A_0 \subseteq \mathcal A_0, \quad \mathcal A_ 0 \mathcal A_1 \subseteq \mathcal A_1, \quad \mathcal A_1 \mathcal A_0 \subseteq \mathcal A_1, \quad \mathcal A_ 1 \mathcal A_1 \subseteq \mathcal A_0 \\
\end{eqnarray}
$\mathcal A_0$ is called ''even'' and $\mathcal A_1$ ''odd'' part of $\mathcal A$.
Notice that a superalgebra is neither supposed to be commutative nor associative.

!!!!!Examples
* [[Lie superalgebras|Supersymmetry]]
* [[Clifford algebras|Clifford Algebra]]
* Malcev superalgebras

Papers:
* [[Maximal Subalgebras of Simple Alternative Superalgebras - J. Laliena, S. Sacristan|https://belenus.unirioja.es/~laliena/Articulos/Maxaltern.pdf]] pct. 0
* [[Akivis Superalgebras and Speciality - H. Albuquerque, A. P. Santana|http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.4535v1.pdf]] pct. 0

In low-temperature superconductors (Pb, Nb, Nb${}_3$Sn, etc.) the phonon-mediated electron-electron interaction produces spin-singlet pairing with s-wave symmetry according to the theory of Bardeen, Cooper, and Schrieffer (BCS theory).

In high-temperature cuprate superconductors the structure of Cooper pairs is predominantly, but not exclusively, a d-wave pairing.

Near the critical temperature the Ginzburg\-Landau equations can be derived from the BCS theory and the order parameter can be identified as superposition of the wave-functions of all the pairs. In case of the high temperature superconductors the order parameter seems to agree with the presence of a superposition of wave-functions of two populations of Cooper pairs, one associated to the s-wave component of the order parameter and the other associated to the d-wave component.

The strong correlation among pairs is the main difference between superconductivity and other forms of Bose-condensation such as superfluidity and Bose-condensates of trapped atoms. Because of the strong correlation, fluctuations are averaged over the pairs. In fact, the Josephson effect and the related large literature on superconducting quantum interference devices (\SQUIDs) indicate that the coherence of the wave-function is maintained over very large distances, with no apparent local splitting of the ground state into fine structures or the appearance of drifts, which is also prohibited by BCS theory itself. Being the d-wave component the one with the highest binding energy, it represents the state with the highest population.


See also:
* [[Podkletnov experiment|Podkletnov Experiment]]

Papers:
* [[High Temperature Superconductors as Quantum Sources of Gravitational Waves: the HTSC GASER (2010) - G. Fontana|http://www.scribd.com/doc/24171073/High-Temperature-Superconductors-as-Quantum-Sources-of-Gravitational-Waves-HTSC-GASER]] pct. 0
* [[On the Critical Temperatures of Superconductors: A Quantum Gravity Approach (2010) - A. Gregori|http://arxiv.org/pdf/1007.3731]] [[local|papers/1007.3731v1.pdf]] pct. 0

Documents:
* [[Superconductors and Quantum Gravity - Ü. Onbaşlı, Z. G. Özdemir|http://www.intechopen.com/articles/show/title/superconductors-and-quantum-gravity]] [[local|documents/11875.pdf]]

<<tiddler [[include_tiddlers/Superconformal Algebra.html#"Superconformal Algebra"]]>>

Papers:
* [[Introduction to Supergravity - H. Samtleben|http://www.itp.uni-hannover.de/~lechtenf/Events/Lectures/samtleben.pdf]]
* [[Supergravities without an Action: Gauging the Trombone - A. L. Dion, H. Samtleben|http://prunel.ccsd.cnrs.fr/docs/00/32/54/44/PDF/trombone_Sep29.pdf]]

<<tiddler [[include_tiddlers/Superluminal Neutrino.html#"Superluminal Neutrino"]]>>

Many complex systems in various areas of science exhibit a spatio-temporal dynamics that is inhomogeneous and can be effectively described by a superposition of  several statistics on different scales, called ''Superstatistics''. The notion was introduced in 2003 by Christian Beck.

Papers:
* [[Recent Developments in Superstatistics (2009) - C. Beck|http://arxiv.org/PS_cache/arxiv/pdf/0811/0811.4363v2.pdf]] [[local|papers/0811.4363v2.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=11128355148008017349&as_sdt=2005&sciodt=2000&hl=de]]

<<tiddler [[include_tiddlers/Superstring Theory.html#"Superstring Theory"]]>>

<<tiddler [[include_tiddlers/Supersymmetric Quantum Mechanics.html#"Supersymmetric Quantum Mechanics"]]>>

<<tiddler [[include_tiddlers/Supersymmetry.html#"Supersymmetry"]]>>

<<tiddler [[include_tiddlers/Surreal Number.html#"Surreal Number"]]>>

A ''Sylvester Matrix'' is a special representation of a [[Hadamard Matrix]], based on the following construction (a.k.a. ''Sylvester Construction''):

If $H_n$ is a Hadamard matrix of order $n$ then
\[
H_{2n}  = \begin{pmatrix} H_n & H_n \\ H_n & -H_n \end{pmatrix}
\]
is a Hadamard matrix of order $2n$ (starting from $H_1 = (1))$.

!!!![[Block Designs|Balanced Incomplete Block Design]]
* The [[Hadamard 3-design|Hadamard Design]] obtained from a Sylvester matrix is the point-hyperplane design of an [[affine geometry|Affine Geometry]] over $GF(2)$.
* Its derived [[Hadamard 2-design|Hadamard Design]] is the point-hyperplane design of a [[projective geometry|Projective Geometry]] over $GF(2)$.

A ''Symmetric (Balanced Incomplete Block) Design'' (or short ''SBIBD'') is a $(v, k, \lambda, r, b)$- [[balanced incomplete block design|Balanced Incomplete Block Design]] with the property that __the number of blocks equals the number of points__, $b=v$. Equivalently: $r=k$.

The [[incidence matrix|Incidence Matrix]] $M$ of a symmetric design is a square matrix satisfying
\[
MM^T = M^TM
\]
This however does not imply that $M$ need to be symmetric.

In an incidence matrix of a symmetric $(v, k, \lambda)$-design each row and each column has exactly $k$ $1$'s. Furthermore every two columns and every two rows have $\lambda$ $1$'s in common.

''Theorem''
If $M$ is an incidence matrix of a symmetric $(v, k, \lambda)$-BIBD, then $M^T$ is also an incidence matrix of a symmetric $(v, k, \lambda)$-BIBD.

Notice, that due to Fisher's inequality, this property can only hold for a symmteric design (where $v = b$).

The SBIBD having an incidence matrix $M^T$ is called the ''Dual Design'' of the SBIBD having an incidence matrix $M$. A \SBIBD and its dual need not to be identical, not even need they to be isomorphic.

A dual design of a symmetric design is obtained by exchanging blocks and points. (This construction fails for nonsymmetric designs.) A special and well known case of such a duality is related to the exchange of points and lines of a [[projective plane|Projective Plane]].

All \SBIBDs satisfy $4m - 1\le v \le m^2 + m + 1$. [[Projective planes|Projective Plane]] attain the upper bound. [[Hadamard designs|Hadamard Design]] satisfy the lower one.

''Theorem:''
There exists a [[Hadamard matrix|Hadamard Matrix]] of order $4m$ if and only if there exists a (symmetric) $(4m-1,2m-1,m-1)$-BIBD. Such a design is a [[Hadamard 2-Design|Hadamard Design]].

Links:
* [[WIKIPEDIA - Symmetric Group|http://en.wikipedia.org/wiki/Symmetric_group]]

A ''Symmetric Space'' is a smooth manifold whose [[group|Group]] of symmetries contains an "inversion symmetry" (i.e. geodesic reversing [[isometry|Isometry]]) about every point.
There are two ways to make this precise, via [[Riemannian geometry|Riemann Space]] or via [[Lie theory|Lie Group]]; the Lie theoretic definition being the more general and algebraic one.

Riemannian symmetric spaces were completely classifiied by Élie Cartan in 1926.

chkAnimate: false
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<<tiddler [[include_tiddlers/T-Design.html#"T-Design"]]>>

''T-duality'' is a symmetry of [[string theory|Superstring Theory]] relating small and large distances.

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|Author|Eric Shulman|
|Original Author|Clint Checketts|
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* ''tag tag tag...'' (or ''title title title'' if ''links''/''references'' is used)<br>shows all tags/links in the document //except// for those listed as macro parameters
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* ''-TiddlerName''<br>show all tags/links //except// those read from a space-separated, bracketed list stored in a separate tiddler.
* ''=tagvalue'' (//only if type=''tags''//)<br>shows only tags that are themselves tagged with the indicated tag value (i.e., ~TagglyTagging usage)
//note: for backward-compatibility, you can also use the macro {{{<<tagCloud ...>>}}} in place of {{{<<cloud ...>>}}}//
<<<
!Examples
<<<
//all tags excluding<<tag systemConfig>>, <<tag excludeMissing>> and <<tag script>>//
{{{<<cloud systemConfig excludeMissing script>>}}}
{{groupbox{<<cloud systemConfig excludeMissing script>>}}}
//top 10 tags excluding<<tag systemConfig>>, <<tag excludeMissing>> and <<tag script>>//
{{{<<cloud limit:10 systemConfig excludeMissing script>>}}}
{{groupbox{<<cloud limit:10 systemConfig excludeMissing script>>}}}
//tags listed in// [[FavoriteTags]]
{{{<<cloud +FavoriteTags>>}}}
{{groupbox{<<cloud +FavoriteTags>>}}}
//tags NOT listed in// [[FavoriteTags]]
{{{<<cloud -FavoriteTags>>}}}
{{groupbox{<<cloud -FavoriteTags>>}}}
//links to tiddlers tagged with 'package'//
{{{<<cloud action:goto =package>>}}}
{{groupbox{<<cloud action:goto =package>>}}}
//top 20 most referenced tiddlers//
{{{<<cloud references limit:20>>}}}
{{groupbox{<<cloud references limit:20>>}}}
//top 20 tiddlers that contain the most links//
{{{<<cloud links limit:20>>}}}
{{groupbox{<<cloud links limit:20>>}}}
<<<
!Revisions
<<<
2009.07.17 [1.7.0] added {{{-TiddlerName}}} parameter to exclude tags that are listed in the indicated tiddler
2009.02.26 [1.6.0] added {{{action:...}}} parameter to apply popup vs. goto action when clicking cloud items
2009.02.05 [1.5.0] added ability to show links or back-links (references) instead of tags and renamed macro to {{{<<cloud>>}}} to reflect more generalized usage.
2008.12.16 [1.4.2] corrected group calculation to prevent 'group=0' error
2008.12.16 [1.4.1] revised tag filtering so excluded tags don't affect calculations
2008.12.15 [1.4.0] added {{{limit:...}}} parameter to restrict the number of tags displayed to the top N most popular
2008.11.15 [1.3.0] added {{{+TiddlerName}}} parameter to include only tags that are listed in the indicated tiddler
2008.09.05 [1.2.0] added '=tagname' parameter to include only tags that are themselves tagged with the specified value (i.e., ~TagglyTagging usage)
2008.07.03 [1.1.0] added 'segments' property to macro object.  Extensive code cleanup
<<<
!Code
***/
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version.extensions.TagCloudPlugin= {major: 1, minor: 7 , revision: 0, date: new Date(2009,7,17)};
//Originally created by Clint Checketts, contributions by Jonny Leroy and Eric Shulman
//Currently maintained and enhanced by Eric Shulman
//}}}
//{{{
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	groups: 9,
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			+'.tagCloud span {line-height: 3.5em; margin:3px;}\n'
			+'.tagCloud1{font-size: 80%;}\n'
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			if (params[0].substr(0,1)=='+') { // read taglist from tiddler
				inc=inc.concat(store.getTiddlerText(params[0].substr(1),'').readBracketedList());
			} else if (params[0].substr(0,1)=='-') { // exclude taglist from tiddler
				ex=ex.concat(store.getTiddlerText(params[0].substr(1),'').readBracketedList());
			} else if (params[0].substr(0,1)=='=') { // get tag list using tagged tags
				var tagged=store.getTaggedTiddlers(params[0].substr(1));
				for (var t=0; t<tagged.length; t++) inc.push(tagged[t].title);
			} else ex.push(params[0]); // exclude params
			params.shift();
		}
		// get all items, include/exclude specific items
		var items=[];
		var list=(links||refs)?store.getTiddlers('title','excludeLists'):store.getTags();
		for (var t=0; t<list.length; t++) {
			var title=(links||refs)?list[t].title:list[t][0];
			if (links)	var count=this.getLinks(list[t]).length;
			else if (refs)	var count=store.getReferringTiddlers(title).length;
			else 		var count=list[t][1];
			if ((!inc.length||inc.contains(title))&&(!ex.length||!ex.contains(title)))
				items.push({ title:title, count:count });
		}
		if(!items.length) return;
		// sort by decending count, limit results (optional)
		items=items.sort(function(a,b){return(a.count==b.count)?0:(a.count>b.count?-1:1);});
		while (limit && items.length>limit) items.pop();
		// find min/max and group size
		var most=items[0].count;
		var least=items[items.length-1].count;
		var groupSize=(most-least+1)/this.groups;
		// sort by title and draw the cloud of items
		items=items.sort(function(a,b){return(a.title==b.title)?0:(a.title>b.title?1:-1);});
		var cloudWrapper = createTiddlyElement(place,'div',null,'tagCloud',null);
		for (var t=0; t<items.length; t++) {
			cloudWrapper.appendChild(document.createTextNode(' '));
			var group=Math.ceil((items[t].count-least)/groupSize)||1;
			var className='tagCloudtag tagCloud'+group;
			var tip=refs?this.refstip:links?this.linkstip:this.tagstip;
			tip=tip.format([items[t].title,items[t].count]);
			if (action=='goto') { // TAG/LINK/REFERENCES GOTO
				var btn=createTiddlyLink(cloudWrapper,items[t].title,true,className);
				btn.title=tip;
				btn.style.fontWeight='normal';
			} else if (!links&&!refs) { // TAG POPUP
				var btn=createTiddlyButton(cloudWrapper,items[t].title,tip,onClickTag,className);
				btn.setAttribute('tag',items[t].title);
			} else { // LINK/REFERENCES POPUP
				var btn=createTiddlyButton(cloudWrapper,items[t].title,tip,
					function(ev) { var e=ev||window.event; var cmt=config.macros.cloud;
						var popup = Popup.create(this);
						var title = this.getAttribute('tiddler');
						var count = this.getAttribute('count');
						var refs  = this.getAttribute('refs')=='T';
						var links = this.getAttribute('links')=='T';
						var label = (refs?cmt.refslabel:cmt.linkslabel).format([count]);
						createTiddlyLink(popup,title,true);
						createTiddlyText(popup,label);
						createTiddlyElement(popup,'hr');
						if (refs) {
							popup.setAttribute('tiddler',title);
							config.commands.references.handlePopup(popup,title);
						}
						if (links) {
							var tiddler = store.fetchTiddler(title);
							var links=config.macros.cloud.getLinks(tiddler);
							for(var i=0;i<links.length;i++)
								createTiddlyLink(createTiddlyElement(popup,'li'),
									links[i],true);
						}
						Popup.show();
						e.cancelBubble=true; if(e.stopPropagation) e.stopPropagation();
						return false;
					}, className);
				btn.setAttribute('tiddler',items[t].title);
				btn.setAttribute('count',items[t].count);
				btn.setAttribute('refs',refs?'T':'F');
				btn.setAttribute('links',links?'T':'F');
				btn.title=tip;
			}
		}
	}
};
//}}}

<<tiddler [[include_tiddlers/Tangent Algebra.html#"Tangent Algebra"]]>>

Let $x^\mu = x^\mu(t)$ be a smooth curve passing through a point $\mb x$ of an $n$-dimensional differentiable manifold $\mathcal M$. One can parametrize this curve such that $x^\mu(0) = 0$. The quantities $ \frac{dx^\mu(\lambda)}{d\lambda} |_{\lambda =0} = \xi^\mu$ are called the coordinates of the tangent vector $\bs \xi$ to the  curve in the point $\mb x$. The parametric equations of the curve can be written as $x^\mu(\lambda) = \xi^\mu \lambda + o(\lambda)$, where $o(\lambda)$ are infinitesimals of orders higher than $\lambda$.

The set of tangent vectors to all curves passing through a point $\mb x \in \mathcal M$ forms a $n$-dimensional vector space. This space is called the ''Tangent Space'' of the manifold $\mathcal M$ in the point $\mb x$ and is denoted by $T_{\mb x}(\mathcal M)$.

The set of all tangent spaces of the manifold $\mathcal M$ is called its ''Tangent Bundle'' and is denoted by $T(\mathcal M)$. An element of the tangent bundle is a pair $(\mb x,\bs \xi)$, where $\mb x \in \mathcal M$ and $\bs \xi \in T_{\mb x}(\mathcal M)$. This explains why the tangent bundle is also a differentiable manifold, having dimension $2n$.

<<tiddler [[include_tiddlers/Taoism.html#"Taoism"]]>>

<<tiddler [[include_tiddlers/Tardigrade.html#"Tardigrade"]]>>

Given a scalar function $\Phi$, its Taylor series is given by
\[
\Phi (\mb x + \mb{dx}) = \sum_{n=0}^\infty \frac {1}{n!} \left (\mb{dx} \frac {\partial}{\partial \mb x} \right )^n \Phi (\mb x)
\]
where $\left (\mb{dx} \frac {\partial}{\partial \mb x} \right )$ is to be seen as an operator (i.e. it "lives" in a specific algebra). I.e. it may for example be noncommutative.

Papers:
* [[Lois de Groupes et Analyseurs (1955) - M. Lazard|http://www.numdam.org/item?id=ASENS_1955_3_72_4_299_0]] [[local|papers/ASENS_1955_3_72_4_299_0.pdf]] [[pct. 56|http://scholar.google.de/scholar?cites=7898688134542541111&hl=de&as_sdt=2000]]

<<tiddler [[include_tiddlers/Teleparallel Gravity.html#"Teleparallel Gravity"]]>>

<<tiddler [[include_tiddlers/Tensor.html#"Tensor"]]>>

<<tiddler [[include_tiddlers/Tensor Algebra.html#"Tensor Algebra"]]>>

See also:
* [[N-Webs|Web]]
* [[N-quasigroups|N-Quasigroup]]
* [[Comtrans algebra|Comtrans Algebra]]

Papers:
* [[On a Class of Ternary Composition Algebras (1996) - A. Elduque|http://www.mathnet.or.kr/mathnet/kms_tex/320.pdf]] [[pct. 7|http://scholar.google.de/scholar?cites=17826483022027317139&hl=de&as_sdt=2000]]
* [[Ternary Numbers and Algebras Reflexive Numbers and Berger Graphs (2007) - A. Dubrovski, G. Volkov|http://arxiv.org/PS_cache/hep-th/pdf/0608/0608073v3.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=3278358118796010998&hl=de&as_sdt=2000]]

A ''Tessellation'' or ''Tiling'' of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. Generalizations to higher dimensions are possible.

Links:
* [[Tilings - J. Baez|http://math.ucr.edu/home/baez/tilings/]]
* [[The Edges of Plausibility - Exploring the Boundaries of Filters, Animation, Gradients, Patterns and Script in SVG - D. Dailey|http://srufaculty.sru.edu/david.dailey/svg/SVGOpen2008/edges_of_plausibility.htm]]

A ''Tetracategory'' is related to the [[associahedron|Associahedron]] $\mathcal K(6)$.

Links:
* [[WIKIPEDIA - Tetracategory|http://en.wikipedia.org/wiki/Tetracategory]]

The ''Tetracode'' is a [[self-dual|Dual Code]] $(4,2,3)$-[[code|Blockcode]] over $GF(3)$.

Papers:
* [[Kittens, S(5, 6, 12), and Mathematical Blackjack in SAGE - D. Joyner, A. Casey|http://sage.math.washington.edu/home/wdj/expository/hexads_sage.pdf]] pct. 0

Links:
* [[Applied Abstract Algebra, The Tetracode Construction - D. Joyner, R. Kreminski, J. Turisco|http://www.usna.edu/Users/math/wdj/book/node224.html]]

> ...tetrads are more fundamental than metric tensors, because they are essential for introducing spinors and quantum mechanics into GR.
>  - David Hestenes -

> These components can be connected if the coordinate components of the orthonormal basis vectors and the orthonormal components of the coordinate basis vectors are both known. If you are lecturing, practice saying this quickly.
> - James B. Hartle -

A ''Tetrad'' (a.k.a. ''Vierbein'', ''4-leg'', ''Repère Mobile'') is a local linear transformation that mediates between an orthogonal ([[anholonomic|Anholonomy]]) frame and a non-othogonal (holonomic) frame.

A (holonomic) ''Tetrad'' ${h_\mu}^a$ can be defined as:
\[
{h_\mu} ^{a} (\mb x)  = \langle \mb{e}_\mu (\mb x) | \mb{e}^a (\mb x) \rangle
\]
where $\mb e_a (\mb x)$ is an orthonormal (reference) frame (a.k.a. coordinate frame, fiducial frame or anholonomic frame) and $\mb e_\mu (\mb x)$ is a local non-orthonormal frame (a.k.a. non-coordinate frame), depending on the location in a curved space.
Note that the tetrad components are not the components of a $(1,1)$-tensor because of the mixture of two different bases.
Its $16$ independent components carry two kinds of information: Firstly they contain all the information needed to construct the $10$ independent components of the metric tensor field. Secondly, each local inertial frame can be redefined by Lorentz transformations, involving three independent rotations and three boosts, and the remaining six degrees of freedom in the vierbein specify the choices we have actually made.

In general relativity the set of $4$ base vectors $\mb e_a$ depend on the wordline parameter $\tau$ and are given by
\[
\langle \mb{e}_a(\tau)| \mb e_a(\tau) \rangle = \eta_{ab}
\]
They define a "local laboratory" (Lorentz frame). For an observer that moves along the worldline the tetrad does not change, i.e. he experiences no gravity (to a first approximation). Quantities measured in this laboratory correspond to projections of the relevant physical $4$-vectors and $4$-tensors onto this orthonormal frame.
As the frame and the tetrad depend on the point in spacetime one can define a so called ''Frame Field'' $\mb e_\mu(\tau)$ and ''Tetrad Field'' $h_\mu^a(\tau)$. Gravity determines how the fields are transformed from one spacetime point to another and is therefore coded in the differential change of the fields. This change is given by the [[covariant derivative|Covariant Derivative]].

Generalisations: See [[polyvector vielbein|Polyvector Vielbein]].

Papers:
* [[Part III: Applications of Differential Geometry to Physics - G. W. Gibbons|http://www.damtp.cam.ac.uk/user/gr/about/members/dgnotes3.pdf]]
* [[Type D Gravitational Fields - W. M. Kinnersley|http://etd.caltech.edu/etd/available/etd-04272006-094112/unrestricted/Kinnersley_wm_1969.pdf]]
* [[An Ambiguous Statement Called ‘Tetrad Postulate’ and the Correct Field Equations Satisfied by the Tetrad Fields - W. A. Rodrigues Jr., Q. A. G. de Souza|http://arxiv.org/PS_cache/math-ph/pdf/0411/0411085v12.pdf]]
* [[The Mixmaster Model as a Cosmological Framework and Aspects of its Quantum Dynamics - G. Imponentey, G. Montaniz|http://icpr.snu.ac.kr/resource/wop.pdf/J01/2003/042/S01/J012003042S010054.pdf]]

<<tiddler [[include_tiddlers/Tetrad Postulate.html#"Tetrad Postulate"]]>>

<<tiddler [[include_tiddlers/The Landscape.html#"The Landscape"]]>>

!!!! A personal remark
If a simulation of our universe is run on the very fundamental (informational) level of nature (supposedly the spacetime structure at the Planck scale), there is no more need to distinguish between a simulation and reality. I.e. a simulation is not an approximation to the real world and the laws of nature any more. Instead __the very physical universe is calculating__ and it is the final computer.

Yet, for this argument to be viable, it has to be assumed that the fundamental level of reality of our universe is the same as that of the universe of our "simulators". (A quite reasonable assumption, in my opinion, because it is only consequent to think of the universe "running" us as being a simulation as well and so forth).

The vision that our universe is simulated on a computer comparable to the ones we have today (in particular a silicon based one) I regard as too primitive and a bit of a manifestation of a lack of imagination, making the whole argument - which in fact is a good one, I believe - way less convincing to me and appear rather artificial.

My proposal instead: [[Moore's law|Moore's Law]] will continue to hold until the computing power reaches its final limit, where information processing and reality converge.


Papers:
* [[The Physical World as a Virtual Reality (2007) - B. Whitworth|http://arxiv.org/pdf/0801.0337]] [[local|papers/0801.pdf]] [[pct. 5|http://scholar.google.de/scholar?hl=de&lr=&cites=1309691784552700873&um=1&ie=UTF-8&ei=en4ITeWLKYis8gP886X1Dw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCoQzgIwAA]]

Links:
* [[Are You Living In a Computer Simulation?|http://www.simulation-argument.com/]]
* [[How To Live In A Simulation - R. Hanson|http://www.transhumanist.com/volume7/simulation.html]]

<<tiddler [[include_tiddlers/The Smallest Particle in the Universe.html#"The Smallest Particle in the Universe"]]>>

Gauß's ''Theorema Egregium'' (Latin: "Remarkable Theorem") is a foundational result in differential geometry proved by Carl Friedrich Gauß that concerns the curvature of $2$-dimensional surfaces. Informally it says, that the [[Gaussian curvature|Gaussian Curvature]] of such a surface can be determined entirely by measuring angles and distances on the surface itself, without further reference to the particular way in which the surface is embedded in an ambient $3$-dimensional Euclidean space. Thus the [[Gaussian curvature|Gaussian Curvature]] is an intrinsic invariant of a surface.

Gauß presented the theorem in this way (translated from Latin):
>Thus the formula of the preceding article leads itself to the remarkable theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged.
In modern mathematical language, the theorem may be stated as follows: The Gaussian curvature of a surface is invariant under local [[isometry|Isometry]].

Distances in a curved Space:
In flat space (Euclidean space) an infinitesimal distance ds is given by
\[
ds^2 = dx_{int}^2 + dy_{int}^2
\]
In curved  (non-euclidean) space this formula no longer applies. Instead one has:
\[
ds^2 = Fdx_{int}^2 + 2 Gdx_{int}dy_{int} + Hdy_{int}^2
\]
$F$, $G$ and $H$ are called ''metric coefficients'' that generally depend on $x$ and $y$. This is expression is also referred to as [[first fundamental form|First Fundamental Form]].

Riemann has extended Gauß's work to higher dimensions introducing [[Riemannian curvature|Riemann Tensor]].
The number of metric coefficients in $3$ dimensions is $6$, in $4$ dimensions $10$ (which follows from the symmetry of the respective [[metric tensor|Metric Tensor]]).

<<tiddler [[include_tiddlers/Theory of Everything.html#"Theory of Everything"]]>>

The ''Theta Series $\theta_L (z)$'' of a [[lattice|Lattice]] $L$ is a generating function for the [[norms|Norm]] $||\vec x||$ of lattice vectors $\vec x$.
It is defined as
\[
\theta_L (z) \equiv \sum_{\vec x \in L}N(||\vec x||^2) \exp(\pi iz)^{\langle \vec x | \vec x \rangle}
\]
where $N(||\vec x||^2)$ is the number of lattice points of $L$ at squared distance $||\vec x||^2 = \langle \vec x | \vec x \rangle$ from the origin.

The series does not contain the information about the distribution of points in each concentric shell. Indeed, there are several examples of pairs of distinct lattices with the same theta series. But these lattices have different [[Voronoi polytopes|Voronoi Polytope]].

The theta series of an integral lattice is a [[modular form|Modular Form]].

<<tiddler [[include_tiddlers/Thoughts.html#"Thoughts"]]>>

<<tiddler [[include_tiddlers/Thoughts About Problems in Physics.html#"Thoughts About Problems in Physics"]]>>

<<tiddler [[include_tiddlers/Thoughts about a Discrete Structure of Spacetime.html#"Thoughts about a Discrete Structure of Spacetime"]]>>

/***
|Name|TiddlerTweakerPlugin|
|Source|http://www.TiddlyTools.com/#TiddlerTweakerPlugin|
|Version|2.4.4|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|select multiple tiddlers and modify author, created, modified and/or tag values|
~TiddlerTweaker is a 'power tool' for TiddlyWiki authors.  Select multiple tiddlers from a listbox and 'bulk modify' the creator, author, created, modified and/or tag values of those tiddlers using a compact set of form fields.  The values you enter into the fields simultaneously overwrite the existing values in all tiddlers you have selected.
!!!!!Usage
<<<
{{{<<tiddlerTweaker>>}}}
{{smallform{<<tiddlerTweaker>>}}}
By default, any tags you enter into the TiddlerTweaker will //replace// the existing tags in all the tiddlers you have selected.  However, you can also use TiddlerTweaker to quickly filter specified tags from the selected tiddlers, while leaving any other tags assigned to those tiddlers unchanged:
>Any tag preceded by a '+' (plus) or '-' (minus), will be added or removed from the existing tags //instead of replacing the entire tag definition// of each tiddler (e.g., enter '-excludeLists' to remove that tag from all selected tiddlers.  When using this syntax, care should be taken to ensure that //every// tag is preceded by '+' or '-', to avoid inadvertently overwriting any other existing tags on the selected tiddlers.  (note: the '+' or '-' prefix on each tag value is NOT part of the tag value, and is only used by TiddlerTweaker to control how that tag value is processed)
Important Notes:
* TiddlerTweaker is a 'power user' tool that can make changes to many tiddlers at once.  ''You should always have a recent backup of your document (or 'save changes' just *before* tweaking the tiddlers), just in case you accidentally 'shoot yourself in the foot'.''
* The date and author information on any tiddlers you tweak will ONLY be updated if the corresponding checkboxes have been selected.  As a general rule, after using TiddlerTweaker, always ''//remember to save your document//'' when you are done, even though the tiddler timeline tab may not show any recently modified tiddlers.
* Selecting and updating all tiddlers in a document can take a while.  Your browser may warn about an 'unresponsive script'.  Usually, if you allow it to continue, it should complete the processing... eventually.  Nonetheless, be sure to save your work before you begin tweaking lots of tiddlers, just in case something does get stuck.
<<<
!!!!!Revisions
<<<
2009.09.15 2.4.4 added 'edit' button. moved html definition to separate section
2009.09.13 2.4.3 in settiddlers(), convert backslashed chars (\n\b\s\t) in replacement text
2009.06.26 2.4.2 only add brackets around tags containing spaces
2009.06.22 2.4.1 in setFields(), add brackets around all tags shown tweaker edit field
2009.03.30 2.4.0 added 'sort by modifier'
2009.01.22 2.3.0 added support for text pattern find/replace
2008.10.27 2.2.3 in setTiddlers(), fixed Safari bug by replacing static Array.concat(...) with new Array().concat(...)
2008.09.07 2.2.2 added removeCookie() function for compatibility with [[CookieManagerPlugin]]
2008.05.12 2.2.1 replace built-in backstage tweak task with tiddler tweaker control panel (moved from BackstageTweaks)
2008.01.13 2.2.0 added 'auto-selection' links: all, changed, tags, title, text
2007.12.26 2.1.0 added support for managing 'creator' custom field (see [[CoreTweaks]])
2007.11.01 2.0.3 added config.options.txtTweakerSortBy for cookie-based persistence of list display order preference setting.
2007.09.28 2.0.2 in settiddlers() and deltiddlers(), added suspend/resume notification handling (improves performance when operating on multiple tiddlers)
2007.08.03 2.0.1 added shadow definition for [[TiddlerTweaker]] tiddler for use as parameter references with {{{<<tiddler>>, <<slider>> or <<tabs>>}}} macros.
2007.08.03 2.0.0 converted from inline script
2006.01.01 1.0.0 initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.TiddlerTweakerPlugin= {major: 2, minor: 4, revision: 4, date: new Date(2009,9,15)};

// shadow tiddler
config.shadowTiddlers.TiddlerTweaker='<<tiddlerTweaker>>';

// defaults
if (config.options.txtTweakerSortBy==undefined) config.options.txtTweakerSortBy='modified';

// backstage task
if (config.tasks) { // for TW2.2b3 or above
	config.tasks.tweak.tooltip='review/modify tiddler internals: dates, authors, tags, etc.';
	config.tasks.tweak.content='{{smallform small groupbox{<<tiddlerTweaker>>}}}';
}

// if removeCookie() function is not defined by TW core, define it here.
if (window.removeCookie===undefined) {
	window.removeCookie=function(name) {
		document.cookie = name+'=; expires=Thu, 01-Jan-1970 00:00:01 UTC; path=/;';
	}
}

config.macros.tiddlerTweaker = {
	handler: function(place,macroName,params,wikifier,paramString,tiddler) {
		var span=createTiddlyElement(place,'span');
		span.innerHTML=store.getTiddlerText('TiddlerTweakerPlugin##html');
		this.init(span.getElementsByTagName('form')[0],config.options.txtTweakerSortBy);
	},
	init: function(f,sortby) { // initialize form controls
		if (!f) return; // form might not be rendered yet...
		while (f.list.options[0]) f.list.options[0]=null; // empty current list content
		var tids=store.getTiddlers(sortby);
		if (sortby=='size') // descending order
			tids.sort(function(a,b) {return a.text.length > b.text.length ? -1 : (a.text.length == b.text.length ? 0 : +1);});
		var who='';
		for (i=0; i<tids.length; i++) { var t=tids[i];
			var label=t.title; var value=t.title;
			switch (sortby) {
				case 'modified':
				case 'created':
					var t=tids[tids.length-i-1]; // reverse order
					var when=t[sortby].formatString('YY.0MM.0DD 0hh:0mm ');
					label=when+t.title;
					value=t.title;
					break;
				case 'size':
					label='['+t.text.length+'] '+label;
					break;
				case 'modifier':
				case 'creator':
					if (who!=t[sortby]) {
						who=t[sortby];
						f.list.options[f.list.length]=new Option('by '+who+':','',false,false);
					}
					label='\xa0\xa0\xa0'+label; // indent
					break;
			}
			f.list.options[f.list.length]=new Option(label,value,false,false);
		}
		f.title.value=f.who.value=f.creator.value=f.tags.value='';
		f.cm.value=f.cd.value=f.cy.value=f.ch.value=f.cn.value='';
		f.mm.value=f.md.value=f.my.value=f.mh.value=f.mn.value='';
		f.stats.disabled=f.set.disabled=f.del.disabled=f.edit.disabled=f.display.disabled=true;
		f.settitle.disabled=false;
		config.options.txtTweakerSortBy=sortby;
		f.sortby.value=sortby; // sync droplist
		if (sortby!='modified') saveOptionCookie('txtTweakerSortBy');
		else removeCookie('txtTweakerSortBy');
	},
	selecttiddlers: function(here) { // enables/disables inputs based on #items selected
		var f=here.form; var list=f.list;
		var c=0; for (i=0;i<list.length;i++) if (list.options[i].selected) c++;
		if (c>1) f.title.disabled=true;
		if (c>1) f.settitle.checked=false;
		f.set.disabled=(c==0);
		f.del.disabled=(c==0);
		f.edit.disabled=(c==0);
		f.display.disabled=(c==0);
		f.settitle.disabled=(c>1);
		f.stats.disabled=(c==0);
		var msg=(c==0)?'select tiddlers':(c+' tiddler'+(c!=1?'s':'')+' selected');
		here.previousSibling.firstChild.firstChild.nextSibling.innerHTML=msg;
		if (c) clearMessage(); else displayMessage('no tiddlers selected');
	},
	setfields: function(here) { // set fields from first selected tiddler
		var f=here.form;
		if (!here.value.length) {
			f.title.value=f.who.value=f.creator.value=f.tags.value='';
			f.cm.value=f.cd.value=f.cy.value=f.ch.value=f.cn.value='';
			f.mm.value=f.md.value=f.my.value=f.mh.value=f.mn.value='';
			return;
		}
		var tid=store.getTiddler(here.value); if (!tid) return;
		f.title.value=tid.title;
		f.who.value=tid.modifier;
		f.creator.value=tid.fields['creator']||''; // custom field - might not exist
		f.tags.value=tid.tags.map(function(t){return String.encodeTiddlyLink(t)}).join(' ');
		var c=tid.created; var m=tid.modified;
		f.cm.value=c.getMonth()+1;
		f.cd.value=c.getDate();
		f.cy.value=c.getFullYear();
		f.ch.value=c.getHours();
		f.cn.value=c.getMinutes();
		f.mm.value=m.getMonth()+1;
		f.md.value=m.getDate();
		f.my.value=m.getFullYear();
		f.mh.value=m.getHours();
		f.mn.value=m.getMinutes();
	},
	settiddlers: function(here) {
		var f=here.form; var list=f.list;
		var tids=[];
		for (i=0;i<list.length;i++) if (list.options[i].selected) tids.push(list.options[i].value);
		if (!tids.length) { alert('please select at least one tiddler'); return; }
		var cdate=new Date(f.cy.value,f.cm.value-1,f.cd.value,f.ch.value,f.cn.value);
		var mdate=new Date(f.my.value,f.mm.value-1,f.md.value,f.mh.value,f.mn.value);
		if (tids.length>1 && !confirm('Are you sure you want to update these tiddlers:\n\n'+tids.join(', '))) return;
		store.suspendNotifications();
		for (t=0;t<tids.length;t++) {
			var tid=store.getTiddler(tids[t]); if (!tid) continue;
			var title=!f.settitle.checked?tid.title:f.title.value;
			var who=!f.setwho.checked?tid.modifier:f.who.value;
			var text=tid.text;
			if (f.replacetext.checked) {
				var r=f.replacement.value.replace(/\\t/mg,'\t').unescapeLineBreaks();
				text=text.replace(new RegExp(f.pattern.value,'mg'),r);
			}
			var tags=tid.tags;
			if (f.settags.checked) {
				var intags=f.tags.value.readBracketedList();
				var addtags=[]; var deltags=[]; var reptags=[];
				for (i=0;i<intags.length;i++) {
					if (intags[i].substr(0,1)=='+')
						addtags.push(intags[i].substr(1));
					else if (intags[i].substr(0,1)=='-')
						deltags.push(intags[i].substr(1));
					else
						reptags.push(intags[i]);
				}
				if (reptags.length)
					tags=reptags;
				if (addtags.length)
					tags=new Array().concat(tags,addtags);
				if (deltags.length)
					for (i=0;i<deltags.length;i++)
						{ var pos=tags.indexOf(deltags[i]); if (pos!=-1) tags.splice(pos,1); }
			}
			if (!f.setcdate.checked) cdate=tid.created;
			if (!f.setmdate.checked) mdate=tid.modified;
			store.saveTiddler(tid.title,title,text,who,mdate,tags,tid.fields);
			if (f.setcreator.checked) store.setValue(tid.title,'creator',f.creator.value); // set creator
			if (f.setcdate.checked) tid.assign(null,null,null,null,null,cdate); // set create date
		}
		store.resumeNotifications();
		this.init(f,f.sortby.value);
	},
	displaytiddlers: function(here,edit) {
		var f=here.form; var list=f.list;
		var tids=[];
		for (i=0; i<list.length;i++) if (list.options[i].selected) tids.push(list.options[i].value);
		if (!tids.length) { alert('please select at least one tiddler'); return; }
		story.displayTiddlers(story.findContainingTiddler(f),tids,edit?DEFAULT_EDIT_TEMPLATE:null);
	},
	deltiddlers: function(here) {
		var f=here.form; var list=f.list;
		var tids=[];
		for (i=0;i<list.length;i++) if (list.options[i].selected) tids.push(list.options[i].value);
		if (!tids.length) { alert('please select at least one tiddler'); return; }
		if (!confirm('Are you sure you want to delete these tiddlers:\n\n'+tids.join(', '))) return;
		store.suspendNotifications();
		for (t=0;t<tids.length;t++) {
			var tid=store.getTiddler(tids[t]); if (!tid) continue;
			if (tid.tags.contains('systemConfig')) {
				var msg=tid.title+' is tagged with systemConfig.'
					+'\n\nRemoving this tiddler may cause unexpected results.  Are you sure?';
				if (!confirm(msg)) continue;
			}
			store.removeTiddler(tid.title);
			story.closeTiddler(tid.title);
		}
		store.resumeNotifications();
		this.init(f,f.sortby.value);
	},
	stats: function(here) {
		var f=here.form; var list=f.list; var tids=[]; var out=''; var tot=0;
		var target=f.nextSibling;
		for (i=0;i<list.length;i++) if (list.options[i].selected) tids.push(list.options[i].value);
		if (!tids.length) { alert('please select at least one tiddler'); return; }
		for (t=0;t<tids.length;t++) {
			var tid=store.getTiddler(tids[t]); if (!tid) continue;
			out+='[['+tid.title+']] '+tid.text.length+'\n'; tot+=tid.text.length;
		}
		var avg=tot/tids.length;
		out=tot+' bytes in '+tids.length+' selected tiddlers ('+avg+' bytes/tiddler)\n<<<\n'+out+'<<<\n';
		removeChildren(target);
		target.innerHTML="<hr><font size=-2><a href='javascript:;' style='float:right' "
			+"onclick='this.parentNode.parentNode.style.display=\"none\"'>close</a></font>";
		wikify(out,target);
		target.style.display='block';
	}
};
//}}}
/***
//{{{
!html
<style>
.tiddlerTweaker table,
.tiddlerTweaker table tr,
.tiddlerTweaker table td
	{ padding:0;margin:0;border:0;white-space:nowrap; }
</style><form class='tiddlerTweaker'><!--
--><table style="width:100%"><tr valign="top"><!--
--><td style="text-align:center;width:99%;"><!--
	--><font size=-2><div style="text-align:left;"><span style="float:right"><!--
	-->&nbsp; <a href="javascript:;"
		title="select all tiddlers"
		onclick="
		var f=this; while (f&&f.nodeName.toLowerCase()!='form')f=f.parentNode;
		for (var t=0; t<f.list.options.length; t++)
			if (f.list.options[t].value.length) f.list.options[t].selected=true;
		config.macros.tiddlerTweaker.selecttiddlers(f.list);
		return false">all</a><!--
	-->&nbsp; <a href="javascript:;"
		title="select tiddlers that are new/changed since the last file save"
		onclick="
		var lastmod=new Date(document.lastModified);
		var f=this; while (f&&f.nodeName.toLowerCase()!='form')f=f.parentNode;
		for (var t=0; t<f.list.options.length; t++) {
			var tid=store.getTiddler(f.list.options[t].value);
			f.list.options[t].selected=tid&&tid.modified>lastmod;
		}
		config.macros.tiddlerTweaker.selecttiddlers(f.list);
		return false">changed</a><!--
	-->&nbsp; <a href="javascript:;"
		title="select tiddlers with at least one matching tag"
		onclick="
		var t=prompt('Enter space-separated tags (match ONE)');
		if (!t||!t.length) return false;
		var tags=t.readBracketedList();
		var f=this; while (f&&f.nodeName.toLowerCase()!='form')f=f.parentNode;
		for (var t=0; t<f.list.options.length; t++) {
			f.list.options[t].selected=false;
			var tid=store.getTiddler(f.list.options[t].value);
			if (tid&&tid.tags.containsAny(tags)) f.list.options[t].selected=true;
		}
		config.macros.tiddlerTweaker.selecttiddlers(f.list);
		return false">tags</a><!--
	-->&nbsp; <a href="javascript:;"
		title="select tiddlers whose titles include matching text"
		onclick="
		var txt=prompt('Enter a title (or portion of a title) to match');
		if (!txt||!txt.length) return false;
		var f=this; while (f&&f.nodeName.toLowerCase()!='form')f=f.parentNode;
		for (var t=0; t<f.list.options.length; t++) {
			f.list.options[t].selected=f.list.options[t].value.indexOf(txt)!=-1;
		}
		config.macros.tiddlerTweaker.selecttiddlers(f.list);
		return false">titles</a><!--
	-->&nbsp; <a href="javascript:;"
		title="select tiddlers containing matching text"
		onclick="
		var txt=prompt('Enter tiddler text (content) to match');
		if (!txt||!txt.length) return false;
		var f=this; while (f&&f.nodeName.toLowerCase()!='form')f=f.parentNode;
		for (var t=0; t<f.list.options.length; t++) {
			var tt=store.getTiddlerText(f.list.options[t].value,'');
			f.list.options[t].selected=(tt.indexOf(txt)!=-1);
		}
		config.macros.tiddlerTweaker.selecttiddlers(f.list);
		return false">text</a> &nbsp;<!--
	--></span><span>select tiddlers</span><!--
	--></div><!--
	--></font><select multiple name=list size="11" style="width:99.99%"
		title="use click, shift-click and/or ctrl-click to select multiple tiddler titles"
		onclick="config.macros.tiddlerTweaker.selecttiddlers(this)"
		onchange="config.macros.tiddlerTweaker.setfields(this)"><!--
	--></select><br><!--
	-->show<input type=text size=1 value="11"
		onchange="this.form.list.size=this.value; this.form.list.multiple=(this.value>1);"><!--
	-->by<!--
	--><select name=sortby size=1
		onchange="config.macros.tiddlerTweaker.init(this.form,this.value)"><!--
	--><option value="title">title</option><!--
	--><option value="size">size</option><!--
	--><option value="modified">modified</option><!--
	--><option value="created">created</option><!--
	--><option value="modifier">modifier</option><!--
	--></select><!--
	--><input type="button" value="refresh"
		onclick="config.macros.tiddlerTweaker.init(this.form,this.form.sortby.value)"<!--
	--> <input type="button" name="stats" disabled value="totals..."
		onclick="config.macros.tiddlerTweaker.stats(this)"><!--
--></td><td style="width:1%"><!--
	--><div style="text-align:left"><font size=-2>&nbsp;modify values</font></div><!--
	--><table style="width:100%;"><tr><!--
	--><td style="padding:1px"><!--
		--><input type=checkbox name=settitle unchecked
			title="allow changes to tiddler title (rename tiddler)"
			onclick="this.form.title.disabled=!this.checked">title<!--
	--></td><td style="padding:1px"><!--
		--><input type=text name=title size=35 style="width:98%" disabled><!--
	--></td></tr><tr><td style="padding:1px"><!--
		--><input type=checkbox name=setcreator unchecked
			title="allow changes to tiddler creator"
			onclick="this.form.creator.disabled=!this.checked">created by<!--
	--></td><td style="padding:1px;"><!--
		--><input type=text name=creator size=35 style="width:98%" disabled><!--
	--></td></tr><tr><td style="padding:1px"><!--
		--><input type=checkbox name=setwho unchecked
			title="allow changes to tiddler author"
			onclick="this.form.who.disabled=!this.checked">modified by<!--
	--></td><td style="padding:1px"><!--
		--><input type=text name=who size=35 style="width:98%" disabled><!--
	--></td></tr><tr><td style="padding:1px"><!--
		--><input type=checkbox name=setcdate unchecked
			title="allow changes to created date"
			onclick="var f=this.form;
				f.cm.disabled=f.cd.disabled=f.cy.disabled=f.ch.disabled=f.cn.disabled=!this.checked"><!--
		-->created on<!--
	--></td><td style="padding:1px"><!--
		--><input type=text name=cm size=2 style="width:2em;padding:0;text-align:center" disabled><!--
		--> / <input type=text name=cd size=2 style="width:2em;padding:0;text-align:center" disabled><!--
		--> / <input type=text name=cy size=4 style="width:3em;padding:0;text-align:center" disabled><!--
		--> at <input type=text name=ch size=2 style="width:2em;padding:0;text-align:center" disabled><!--
		--> : <input type=text name=cn size=2 style="width:2em;padding:0;text-align:center" disabled><!--
	--></td></tr><tr><td style="padding:1px"><!--
		--><input type=checkbox name=setmdate unchecked
			title="allow changes to modified date"
			onclick="var f=this.form;
				f.mm.disabled=f.md.disabled=f.my.disabled=f.mh.disabled=f.mn.disabled=!this.checked"><!--
		-->modified on<!--
	--></td><td style="padding:1px"><!--
		--><input type=text name=mm size=2 style="width:2em;padding:0;text-align:center" disabled><!--
		--> / <input type=text name=md size=2 style="width:2em;padding:0;text-align:center" disabled><!--
		--> / <input type=text name=my size=4 style="width:3em;padding:0;text-align:center" disabled><!--
		--> at <input type=text name=mh size=2 style="width:2em;padding:0;text-align:center" disabled><!--
		--> : <input type=text name=mn size=2 style="width:2em;padding:0;text-align:center" disabled><!--
	--></td></tr><tr><td style="padding:1px"><!--
		--><input type=checkbox name=replacetext unchecked
			title="find/replace matching text"
			onclick="this.form.pattern.disabled=this.form.replacement.disabled=!this.checked">replace text<!--
	--></td><td style="padding:1px"><!--
		--><input type=text name=pattern size=15 value="" style="width:40%" disabled
			title="enter TEXT PATTERN (regular expression)"> with<!--
		--><input type=text name=replacement size=15 value="" style="width:40%" disabled
			title="enter REPLACEMENT TEXT"><!--
	--></td></tr><tr><td style="padding:1px"><!--
		--><input type=checkbox name=settags checked
			title="allow changes to tiddler tags"
			onclick="this.form.tags.disabled=!this.checked">tags<!--
	--></td><td style="padding:1px"><!--
		--><input type=text name=tags size=35 value="" style="width:98%"
			title="enter new tags or use '+tag' and '-tag' to add/remove tags from existing tags"><!--
	--></td></tr></table><!--
	--><div style="text-align:center"><!--
	--><nobr><input type=button name=display disabled style="width:24%" value="display"
		title="show selected tiddlers"
		onclick="config.macros.tiddlerTweaker.displaytiddlers(this,false)"><!--
	--> <input type=button name=edit disabled style="width:23%" value="edit"
		title="edit selected tiddlers"
		onclick="config.macros.tiddlerTweaker.displaytiddlers(this,true)"><!--
	--> <input type=button name=del disabled style="width:24%" value="delete"
		title="remove selected tiddlers"
		onclick="config.macros.tiddlerTweaker.deltiddlers(this)"><!--
	--> <input type=button name=set disabled style="width:24%" value="update"
		title="update selected tiddlers"
		onclick="config.macros.tiddlerTweaker.settiddlers(this)"></nobr><!--
	--></div><!--
--></td></tr></table><!--
--></form><span style="display:none"><!--content replaced by tiddler "stats"--></span>
!end
//}}}
***/

<<tiddler [[include_tiddlers/TiddlyWiki ....html#"TiddlyWiki ..."]]>>

<<tiddler [[include_tiddlers/Tiling.html#"Tiling"]]>>

<<tiddler [[include_tiddlers/Time.html#"Time"]]>>

Papers:
* [[A Mathematical Theory of Communication (1949) - C. E. Shannon|http://faculty.kfupm.edu.sa/RI/abdallah/041/ee406/shannon.pdf]] [[local|papers/shannon.pdf]] {{t1000Cite{[[pct. 27097|http://scholar.google.at/scholar?cites=14966908346949486806&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[The Sequence of the Human Genome (2001) - J. C. Venter et al.|http://www.upch.edu.pe/facien/dbmbqf/gorjeda/cursos/geneticaavanzada%202007/articulos/Science-2001-venter-hgs.pdf]] {{t1000Cite{[[pct. 8527|http://scholar.google.de/scholar?hl=de&lr=&cites=12230377424309310152&um=1&ie=UTF-8&ei=2QT_TYb4Ds6Sswbii7XuDQ&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB8QzgIwAA]]}}}
* [[Particle Creation by Black Holes (1975) - S. W. Hawking|http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103899181]] [[local|papers/particle_creation.pdf]] {{t1000Cite{[[pct. 6089|http://scholar.google.at/scholar?cites=15621263267746567439&as_sdt=5&sciodt=0&hl=de]]}}}
* [[The Electrodynamics of Substances with Simultaneously Negative Values of ε and μ (1968) - V. G. Veselago|http://www.physics-online.ru/PaperLogos/1/files/Full_text_English_version.pdf]] [[local|papers/Full_text_English_version.pdf]] {{t1000Cite{[[pct. 6001|http://scholar.google.com/scholar?hl=de&lr=&cites=10629303077499220600&um=1&ie=UTF-8&ei=2wY-T7DPIITItAbWkIjuBA&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCwQzgIwAA]]}}}
* [[On the Einstein-Podolsky Rosen Paradox (1964) - J. S. Bell|http://www.physics.princeton.edu/~mcdonald/examples/QM/bell_physics_1_195_64.pdf]] [[local|papers/bell_physics_1_195_64.pdf]]  {{t1000Cite{[[pct. 5500|http://scholar.google.com/scholar?hl=de&lr=&cites=5148023125170763884&um=1&ie=UTF-8&ei=q_jRTqLnDtOBhQeZguGxDQ&sa=X&oi=science_links&ct=sl-citedby&resnum=3&ved=0CDUQzgIwAg]]}}}
* [[Statistical-Mechanical Theory of Irreversible Processes. I. (1957) - R. Kubo|http://jpsj.ipap.jp/link?JPSJ/12/570/pdf]] [[local|papers/JPSJ-12-570-1.pdf]] {{t1000Cite{[[pct. 4722|http://scholar.google.de/scholar?hl=de&lr=&cites=14804117649850842010&um=1&ie=UTF-8&ei=GOddTvKhLMbssgbwnKGdDw&sa=X&oi=science_links&ct=sl-citedby&resnum=5&ved=0CEYQzgIwBA]]}}}
* [[Theory of Superconductivity (1957) - J. Bardeen, L. N. Cooper, J. R. Schrieffer|http://prola.aps.org/pdf/PR/v108/i5/p1175_1]] [[local|papers/p1175_1.pdf]] {{t1000Cite{[[pct. 4554|http://scholar.google.de/scholar?cites=17583932236096542852&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Noncommutative Geometry Year 2000 - A. Connes|http://arxiv.org/abs/math/0011193]] [[local|papers/0011193v1.pdf]] {{t1000Cite{[[4440|http://scholar.google.de/scholar?hl=de&lr=&cites=5770210089396315588&um=1&ie=UTF-8&ei=Wlg-TuaBJo-d-waZyOH-Bw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCoQzgIwAA]]}}}
* [[String Theory and Noncommutative Geometry (1999) - N. Seiberg, E. Witten|http://arxiv.org/abs/hep-th/9908142]] [[local|papers/9908142v3.pdf]] {{t1000Cite{[[pct. 3619|http://scholar.google.com/scholar?hl=de&lr=&cites=9510950410324187369&um=1&ie=UTF-8&ei=j3-bT_jYHofIsgbvxPhp&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CDMQzgIwAA]]}}}
* [[Coherent and Incoherent States of the Radiation Field (1963) - R. J. Glauber|http://americanphysicalsociety.org/about/pressreleases/upload/pr2.pdf]] [[local|papers/pr2.pdf]] {{t1000Cite{[[pct. 3318|http://scholar.google.de/scholar?cites=14423751054501270697&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Theory of Bose-Einstein Condensation in Trapped Gases (1998) - F. Dalfovo, S. Giorgini, L. P. Pitaevskii, S. Stringari|http://arxiv.org/pdf/cond--mat/9806038]] [[local|papers/9806038v2.pdf]] {{t1000Cite{[[pct. 3179|http://scholar.google.de/scholar?cites=9965730940296946222&as_sdt=5&sciodt=0&hl=de]]}}}
* [[Unity of all Elementary-particle Forces (1974) - H. Georgi, S. L. Glashaw|ftp://www.phy.pku.edu.cn/pub/Books/%CE%EF%C0%ED/%CE%EF%C0%ED%D1%A7%CA%B7/Phy_Rev_100%C4%EA%BE%AB%D1%A1/pdf/12/091.pdf]] [[local|papers/091.pdf]] {{t1000Cite{[[3128|http://scholar.google.de/scholar?cites=16988246690561935311&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Dynamical Breaking of Supersymmetry (1981) - E. Witten|http://zimp.zju.edu.cn/~hep/LHC/DynamicalSUSYBreaking_Witten.pdf]] [[local|papers/DynamicalSUSYBreaking_Witten.pdf]] {{t1000Cite{[[pct. 3112|http://scholar.google.de/scholar?cites=3169233604024524839&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Black Holes and Entropy (1973) - J. D. Bekenstein|http://www.physics.princeton.edu/~mcdonald/examples/QM/bekenstein_prd_7_2333_73.pdf]] [[local|papers/bekenstein_prd_7_2333_73.pdf]] {{t1000Cite{[[pct. 2783|http://scholar.google.at/scholar?cites=12862885560444035932&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[The Hierarchy Problem and New Dimensions at a Millimeter (1998) - N. Arkani-Hamed, S. Dimopoulos, G. Dvali|http://arxiv.org/PS_cache/hep-ph/pdf/9803/9803315v1.pdf]] [[local|papers/9803315v1.pdf]] {{t1000Cite{[[pct. 2711|http://scholar.google.de/scholar?hl=de&lr=&cites=6297293429578150902&um=1&ie=UTF-8&ei=Idw2TaTGCoiRswaQzqR9&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCIQzgIwAA]]}}}
* [[Cosmological Event Horizons, Thermodynamics, and Particle Creation (1977) - G. W. Gibbons, S. W. Hawking|http://lib.org.by/info/0pre-Library/Gibbons,%20Hawking.%20Cosmological%20event%20horizons,%20thermodynamics,%20particle%20creation%20%28PRD%201977%29%28T%29%2814s%29.djvu]] [[local|papers/CosmologicalEventHorizons.djvu]] {{t1000Cite{[[pct. 2654|http://scholar.google.de/scholar?cites=9870722144608430922&as_sdt=5&sciodt=0&hl=de]]}}}
* [[On the Generators of Quantum Dynamical Semigroups (1976) - G. Lindblad|http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103899849]] [[local|papers/Lindblad.pdf]] {{t1000Cite{[[2610|http://scholar.google.com/scholar?hl=de&lr=&cites=5572434144661184242&um=1&ie=UTF-8&ei=dtN5T-2hHIHJsga2nsm_BA&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CC0QzgIwAA]]}}}
* [[On the Attraction between two Perfectly Conducting Plates (1948) - H. B. G. Casimir|http://www.dwc.knaw.nl/DL/publications/PU00018547.pdf]] [[local|papers/PU00018547.pdf]] {{t1000Cite{[[pct. 2548|http://scholar.google.de/scholar?cites=12754256243235039148&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Evolution of Networks (2001) - S. N. Dorogovtsev, J. F. F. Mendes|http://arxiv.org/PS_cache/cond-mat/pdf/0106/0106144v2.pdf]] [[local|papers/0106144v2.pdf]] {{t1000Cite{[[pct. 2514|http://scholar.google.de/scholar?cites=7757634102586467395&hl=de]]}}}
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* [[Thermodynamics of Black Holes in Anti-de Sitter Space (1983) - S. W. Hawking, D. N. Page|http://srv2.fis.puc.cl/~mbanados/Cursos/TopicosRelatividadAvanzada/HawkingPage.pdf]] [[local|papers/HawkingPage.pdf]] {{t500Cite{[[pct. 845|http://scholar.google.de/scholar?hl=de&lr=&cites=9640743466230609949&um=1&ie=UTF-8&ei=ii2QTs3MBMK6-Aam14XSCg&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCYQzgIwAA]]}}}
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* [[Coherent States for Arbitrary Lie Group (1972) - A. M. Perelomov|http://arxiv.org/PS_cache/math-ph/pdf/0203/0203002v1.pdf]] [[local|papers/0203002v1.pdf]] {{t500Cite{[[pct. 633|http://scholar.google.de/scholar?hl=de&lr=&cites=2967057377917328630&um=1&ie=UTF-8&ei=DSvETcieJNHwsgaGrIGTDw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCUQzgIwAA]]}}}
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* [[On an Algebraic Generalization of the Quantum Mechanical Formalism (1934) - P. Jordan, J. Neumann, E. Wigner|http://books.google.com/books?hl=de&lr=&id=Sc3PRTALFpkC&oi=fnd&pg=PA298&ots=5XWQLR9Xxi&sig=jYXFxN1UPbw184EJYVQURNYTxbA]] [[local|papers/AlgebraicGeneralisationOfQM.pdf]] {{t100Cite{[[pct. 457|http://scholar.google.de/scholar?hl=de&lr=&cites=16470069185731839580&um=1&ie=UTF-8&ei=tYqeTsEfxKv5Bp-c7IMN&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCMQzgIwAA]]}}}
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* [[The Space-time Code (1968) - D. Finkelstein|http://streaming.ictp.trieste.it/preprints/P/68/019.pdf]] [[local|papers/019.pdf]] {{t100Cite{[[pct. 130|http://scholar.google.de/scholar?cites=14750110776107060590&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Complex Kerr-Newman Geometry and Black-Hole Thermodynamics (1991) - J. D. Brown, E. A. Martinez|http://etd.lib.ncsu.edu/publications/bitstream/1840.2/427/1/Brown_1991_Phys_Rev_Letters_2281.pdf]] [[local|papers/Brown_1991_Phys_Rev_Letters_2281.pdf]] {{t100Cite{[[pct. 128|http://scholar.google.com/scholar?hl=de&lr=&cites=7962140990669081154&um=1&ie=UTF-8&ei=yrf9TvapCszAtAbbhtX9Dw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCcQzgIwAA]]}}}
* [[A Conceptual Analysis of Quantum Zeno; Paradox, Measurement, and Experiment (1997) - D. Home, M. A. B. Whitaker|http://www.quniverse.sk/buzek/zaujimave/home.pdf]] [[local|papers/home.pdf]] {{t100Cite{[[pct. 127|http://scholar.google.de/scholar?hl=de&lr=&cites=2840776683454384034&um=1&ie=UTF-8&ei=0tZEToKNOM_CswbS5qirCQ&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCYQzgIwAA]]}}}
* [[The Riemann Zeros and Eigenvalue Asymptotics (1999) - M. V. Berry, J. P. Keating|http://www.phy.bris.ac.uk/people/berry_mv/the_papers/Berry307.pdf]] [[local|papers/Berry307.pdf]] {{t100Cite{[[pct. 157|http://scholar.google.de/scholar?hl=de&lr=&cites=1685401931586229327&um=1&ie=UTF-8&ei=TT6xTZboLcTEswbmm6n9Cw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCcQzgIwAA]]}}}
* [[Repulsive Casimir Forces (2002) - O. Kenneth, I. Klich, A. Mann, M. Revzen|http://arxiv.org/abs/quant-ph/0202114v1]] [[local|papers/0202114v1.pdf]] {{t100Cite{[[pct. 134|http://scholar.google.de/scholar?cites=6514261671745828201&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Trace Anomaly Driven Inflation (2004) - S. W. Hawking, T. Hertog, H. S. Reall|http://arxiv.org/abs/hep-th/0010232]] [[local|papers/0010232v4.pdf]] {{t100Cite{[[pct. 132|http://scholar.google.com/scholar?hl=de&lr=&cites=17454728013373251194&um=1&ie=UTF-8&ei=hrqWT_2lEMXJswb80p2ADg&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CDIQzgIwAA]]}}}
* [[Complex Kerr-Newman Geometry and Black-Hole Thermodynamics (1991) - J. D. Brown, E. A. Martinez|http://etd.lib.ncsu.edu/publications/bitstream/1840.2/427/1/Brown_1991_Phys_Rev_Letters_2281.pdf]] [[local|papers/Brown_1991_Phys_Rev_Letters_2281.pdf]] {{t100Cite{[[pct. 128|http://scholar.google.com/scholar?hl=de&lr=&cites=7962140990669081154&um=1&ie=UTF-8&ei=yrf9TvapCszAtAbbhtX9Dw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCcQzgIwAA]]}}}
* [[The Structure of Alternative Division Rings (1950) - R. H. Bruck, E. Kleinfeld|http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063309/pdf/pnas01563-0022.pdf]] [[local|papers/pnas01563-0022.pdf]] {{t100Cite{[[pct. 127|http://scholar.google.de/scholar?hl=de&lr=&cites=6496268522617739391&um=1&ie=UTF-8&ei=CDJfTpyuN4ufOruQyO8C&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCMQzgIwAA]]}}}
* [[Quantum Computation, Entanglement and State Reduction (2002) - R. Penrose|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.84.7047&rep=rep1&type=pdf]] [[local|papers/10.1.1.84.7047.pdf]] {{t100Cite{[[pct. 125|http://scholar.google.de/scholar?cites=11520020952905339803&as_sdt=2005&sciodt=0,5&hl=de]]
}}}
* [[Deformation Quantization: Twenty Years After (1998) - D. Sternheimer|http://arxiv.org/pdf/math.QA/9809056]] [[local|papers/9809056.pdf]] {{t100Cite{[[pct. 125|http://scholar.google.de/scholar?hl=de&lr=&cites=15503651199554240938&um=1&ie=UTF-8&ei=-bDrTavnINHKsgbFoMHnCg&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCMQzgIwAA]]}}}
* [[Inflation and Squeezed Quantum States (1993) - A. Albrecht, P. Ferreira, M. Joyce, T. Prokopec|http://arxiv.org/pdf/astro-ph/9303001]] [[local|papers/9303001.pdf]] {{t1000Cite{[[pct. 125|http://scholar.google.de/scholar?hl=de&lr=&cites=4184935355379956070&um=1&ie=UTF-8&ei=tgXWTbK2D8_n-gbswOnxBw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCQQzgIwAA]]}}}
* [[Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point (2009) - P. Hořava|http://arxiv.org/pdf/0902.3657v2]] [[local|papers/0902.3657v2.pdf]] {{t100Cite{[[pct. 123|http://scholar.google.de/scholar?hl=de&lr=&cites=16060286333843685335&um=1&ie=UTF-8&ei=VmFnTc_-NYvZsgaK9r3wDA&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB8QzgIwAA]]}}}
* [[Aspects of Holographic Entanglement Entropy (2006) - S. Ryu, T. Takayanagi|http://arxiv.org/abs/hep-th/0605073]] [[local|papers/0605073v3.pdf]] {{t100Cite{[[pct. 123|http://scholar.google.com/scholar?hl=de&lr=&cites=10207317677681579425&um=1&ie=UTF-8&ei=xm_cTqTgGsLf4QTq_ZzrDQ&sa=X&oi=science_links&ct=sl-citedby&resnum=2&ved=0CDcQzgIwAQ]]}}}
* [[Gravity, Non-Commutative Geometry and the Wodzicki Residue (1993) - W. Kalau, M. Walze|http://arxiv.org/PS_cache/gr-qc/pdf/9312/9312031v1.pdf]] [[local|papers/9312031v1.pdf]] {{t100Cite{[[pct. 120|http://scholar.google.de/scholar?cites=10321711436103073209&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Quasigroups II (1944) - A. A. Albert|http://www.ams.org/journals/tran/1944-055-00/S0002-9947-1944-0010597-1/S0002-9947-1944-0010597-1.pdf]] [[local|papers/S0002-9947-1944-0010597-1.pdf]] {{t100Cite{[[pct. 120|http://scholar.google.de/scholar?cites=1972350424515353752&as_sdt=5&sciodt=0&hl=de]]}}}
* [[The Dirac Operator and Gravitation (1995) - D. Kastler|http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1104271706]] [[local|papers/DiracAndGravitation.pdf]] {{t100Cite{[[pct. 119|http://scholar.google.de/scholar?hl=de&lr=&cites=12321777489588579981&um=1&ie=UTF-8&ei=zL4-TufWF4XVsgbF78QT&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCEQzgIwAA]]}}}
* [[Gravity, Gauge Theories and Geometric Algebra (1998) - A. Lasenby, C. Doran, S. Gull|http://www.mrao.cam.ac.uk/~clifford/publications/ps/gravity.pdf]] [[local|papers/gravity.pdf]] {{t100Cite{[[pct. 118|http://scholar.google.de/scholar?cites=16075404841437256002&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Relativistic Models of Nonlinear Quantum Mechanics (1978) - T. W. B. Kibble|http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103904623]] [[local|papers/RelativisticModels.pdf]] {{t100Cite{[[pct. 118|http://scholar.google.de/scholar?hl=de&lr=&cites=12264332489813212966&um=1&ie=UTF-8&ei=666eToeoM8T_-gbxpdWDDQ&sa=X&oi=science_links&ct=sl-citedby&resnum=3&ved=0CDwQzgIwAg]]}}}
* [[How far are we from the Quantum Theory of Gravity? (2003) - L. Smolin|http://arxiv.org/abs/hep-th/0303185]] [[local|papers/0303185.pdf]] {{t100Cite{[[pct. 117|http://scholar.google.de/scholar?hl=de&lr=&cites=2161273417093257598&um=1&ie=UTF-8&ei=TxRiTq2iGIz_-gb-w4iWCg&sa=X&oi=science_links&ct=sl-citedby&resnum=2&ved=0CC4QzgIwAQ]]}}}
* [[Forks in the Road, on the Way to Quantum Gravity (1997) - R. D. Sorkin|http://arxiv.org/pdf/gr-qc/9706002]] [[local|papers/9706002.pdf]] {{t100Cite{[[pct. 116|http://scholar.google.de/scholar?hl=de&lr=&cites=5810580870188055110&um=1&ie=UTF-8&ei=_3M4Te3vCZCaOqWl6O0K&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CBsQzgIwAA]]}}}
* [[Solutions to Yang-Mills Field Equations in Eight Dimensions and the Last Hopf Map (1984) - B. Grossman, T. W. Kephart, J. D. Stasheff|http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103941908]] [[local|papers/hopf.pdf]] {{t100Cite{[[pct. 116|http://scholar.google.com/scholar?hl=de&lr=&cites=11110415183727742931&um=1&ie=UTF-8&ei=XpnKToDZIqiK4gT2l9Fi&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCYQzgIwAA]]}}}
* [[A Simple Construction for the Fischer-Griess Monster Group (1985) - J. H. Conway|http://www.kryakin.com/files/Invent_mat_(2_8)/79/79_07.pdf]] [[local|papers/79_07.pdf]] {{t100Cite{[[pct. 114|http://scholar.google.de/scholar?cites=13479633816913418482&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Causal Sets: Discrete Gravity (2003) - R. D. Sorkin|http://arxiv.org/pdf/gr-qc/0309009v1]] [[local|papers/0309009v1.pdf]] {{t100Cite{[[pct. 112|http://scholar.google.com/scholar?hl=de&lr=&cites=5297585953037780874&um=1&ie=UTF-8&ei=2RNWT_62CZHEswac7vT5Bg&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CDMQzgIwAA]]}}}
* [[Noncommutative Gauge Field Theories: A No-Go Theorem (2001) - M. Chaichian, P. Prešnajder, M. M. Sheikh-Jabbari, A. Tureanu|http://arxiv.org/PS_cache/hep-th/pdf/0107/0107037v1.pdf]] [[local|papers/0107037v1.pdf]] {{t100Cite{[[pct. 112|http://scholar.google.de/scholar?hl=de&lr=&cites=11085766719936217009&um=1&ie=UTF-8&ei=LEU-TsTEG4W8-QbEw6GBCA&sa=X&oi=science_links&ct=sl-citedby&resnum=2&ved=0CCIQzgIwAQ]]}}}
* [[Bekenstein Bounds in De Sitter and Flat Space (2000) - R. Bousso|http://arxiv.org/abs/hep-th/0012052]] [[local|papers/0012052v1.pdf]] {{t100Cite{[[pct.  112|http://scholar.google.com/scholar?hl=de&lr=&cites=7443048502559268791&um=1&ie=UTF-8&ei=4kzFTtuZG8T24QTPvsCMDQ&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB0QzgIwAA]]}}}
* [[A New Perspective on Cosmic Coincidence Problems (2000) - N. Arkani-Hamed, L. J. Hall, C. Kolda, H. Murayama|http://arxiv.org/pdf/astro-ph/0005111]] [[local|papers/0005111.pdf]] {{t100Cite{[[pct.112|http://scholar.google.de/scholar?hl=de&lr=&cites=8135023466789952340&um=1&ie=UTF-8&ei=IBZZTve8HMua-ga0pfSqDA&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB8QzgIwAA]]}}}
* [[Fractal Spacetime Structure in Asymptotically Safe Gravity (2005) - O. Lauscher, M. Reuter|http://arxiv.org/pdf/hep-th/0508202v1]] [[local|papers/0508202v1.pdf]] {{t100Cite{[[pct. 109|http://scholar.google.de/scholar?hl=de&lr=&cites=11082239721408893419&um=1&ie=UTF-8&ei=UE0PTq6kBsmKswbgwtnJDg&sa=X&oi=science_links&ct=sl-citedby&resnum=2&ved=0CCUQzgIwAQ]]}}}
* [[Pre-bangian Origin of our Entropy and Time Arrow (1999) - G. Veneziano|http://arxiv.org/pdf/hep-th/9902126v1]] [[local|papers/9902126v1.pdf]] {{t100Cite{[[pct. 109|http://scholar.google.de/scholar?hl=de&lr=&cites=10673949858861273388&um=1&ie=UTF-8&ei=sDxXTp2kGuXd4QT5ttCfDA&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CBoQzgIwAA]]}}}
* [[Feynman Path Integration in Quantum Dynamics (1991) - N. Makri|http://www.scs.illinois.edu/~makri/New-Web-Site/preprints/paper13.pdf]] [[local|papers/paper13.pdf]] {{t100Cite{[[pct. 107|http://scholar.google.at/scholar?cites=12242040296865329255&as_sdt=2005&sciodt=0,5&hl=de]]}}} - Path Integrals -
* [[Decoherence in Quantum Walks - A Review (2006) - V. Kendon|http://arxiv.org/PS_cache/quant-ph/pdf/0606/0606016v3.pdf]] [[local|papers/0606016v3.pdf]] {{t100Cite{[[pct. 106|http://scholar.google.de/scholar?cites=44267111197376751&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Modern Tests of Lorentz Invariance - D. Mattingly|http://www.emis.de/journals/LRG/Articles/lrr-2005-5/download/lrr-2005-5BW.pdf]] {{t100Cite{[[pct. 104|http://scholar.google.de/scholar?hl=de&lr=&cites=9625894262048047594]]}}}
* [[Ground State of Hydrogen as a Zero-point-fluctuation-determined State (1987) - H. E. Puthoff|http://www.earthtech.org/publications/PRDv35_3266.pdf]] [[local|papers/PRDv35_3266.pdf]] {{t100Cite{[[pct. 104|http://scholar.google.com/scholar?hl=de&lr=&cites=6583244378725371338&um=1&ie=UTF-8&ei=pLw8T6KdOMWGswbvrojOBA&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCgQzgIwAA]]}}}
* [[Notes for a Brief History of Quantum Gravity (2001) - C. Rovelli|http://arxiv.org/abs/gr-qc/0006061]] [[local|papers/0006061v3.pdf]] {{t100Cite{[[pct. 102|http://scholar.google.com/scholar?hl=de&lr=&cites=11283646896446693333&um=1&ie=UTF-8&ei=CVNWT5G2BYnptQbP1diVBw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CDQQzgIwAA]]}}}

Abstracts:
* [[Internal Structure of Black Holes (1990) - E. Poisson, W. Israel|http://prd.aps.org/abstract/PRD/v41/i6/p1796_1]] {{t100Cite{[[pct. 377|http://scholar.google.com/scholar?cites=13874594085268993278&as_sdt=2005&sciodt=0,5&hl=de]]}}}

Theses:
* [[NONLINEAR MODELS IN-2 + ε DIMENSIONS (1980) - D. H. Friedan|http://escholarship.org/uc/item/9rz1x6p4.pdf]] [[local|theses/eScholarship UC item 9rz1x6p4.pdf]] {{t100Cite{[[tct. 328|http://scholar.google.de/scholar?hl=de&lr=&cites=18178038255429659217&um=1&ie=UTF-8&ei=LZcFTtWgH8Oe-waiibGFCQ&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB8QzgIwAA]]}}}

Lectures:
* [[The Development of the Space-time View of Quantum Electrodynamics (1965) - R. P. Feynman|http://eurserveur.insa-lyon.fr/LesCours/physique/AppPhysique/approphys/AMERINSA/Enseign_Thema_Program/Physique/Dick_Feyman/pdf/NobelLecture.pdf]] [[local|lectures/NobelLecture.pdf]] {{t100Cite{[[lct. 119|http://scholar.google.de/scholar?hl=de&lr=&cites=2134183645244350910&um=1&ie=UTF-8&ei=vMPjTYr0C47MswbX6OWSBg&sa=X&oi=science_links&ct=sl-citedby&resnum=7&ved=0CGIQzgIwBg]]}}}

Magazines:
* [[There's Plenty of Room at the Bottom (1960) - R. P. Feynman|http://nanoparticles.org/pdf/Feynman.pdf]] [[local|magazines/Feynman.pdf]] {{t1000Cite{[[mct. 1297|http://scholar.google.de/scholar?cites=6255046937502302937&as_sdt=2005&sciodt=0,5&hl=de]]}}}

Links:
* [[The Cosmological Constants (1937) - P. A. M. Dirac|http://www.nature.com/nature/journal/v139/n3512/abs/139323a0.html]] {{t500Cite{[[lct. 828|http://scholar.google.de/scholar?cites=11928574682593484206&as_sdt=2005&sciodt=0,5&hl=de]]}}}

Google Books:
* [[Designs and their Codes (1992) - E. F. Assmus, J. D. Key|http://books.google.com/books?id=OE5lHAZuKUQC&pg=PA209&lpg=PA209&dq=%22%2816,11,4%29%22+%22hamming+code%22+projective&source=bl&ots=5Uh99R2nBD&sig=DbnE2C5JBy8Ar9pQ3t7t7Dw2TWY&hl=de&ei=7gTaSqmNIZSC_QbggozRDA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAgQ6AEwAA#v=onepage&q=%22%2816%2C11%2C4%29%22%20%22hamming%20code%22%20projective&f=false]] [[local|google_books/DesignsAndTheirCodes.pdf]] {{t100Cite{[[bct. 413|http://scholar.google.de/scholar?hl=de&lr=&cites=13520286188419732263&um=1&ie=UTF-8&ei=_pAmTrWEO8_rsgaR7cGtCQ&sa=X&oi=science_links&ct=sl-citedby&resnum=5&ved=0CEsQzgIwBA]]}}}
* [[Introduction to Superstrings and M-theory - M. Kaku|http://books.google.com/books?id=9n-2O7wHnZ4C&dq=%22Introduction+to+Superstrings+and+M-Theory%22+download&printsec=frontcover&source=bn&hl=de&ei=VohrTIa9IaSTOJj76WE&sa=X&oi=book_result&ct=result&resnum=4&ved=0CCwQ6AEwAw#v=onepage&q&f=false]] {{t100Cite{[[bct. 349|http://scholar.google.de/scholar?cites=11923859208152693671&as_sdt=2005&sciodt=2000&hl=de]]}}}
* [[Stochastic Quantization - P. H. Damgaard, H. Hüffel|http://books.google.com/books?id=nxZB4BfDhzcC&pg=PA337&lpg=PA337&dq=%22stochastic+quantization%22&source=bl&ots=XaGMEKRyqu&sig=evveQ9_qk_HH4u1ydSLHVDCfA9s&hl=de&ei=YDe7SuO1BYyW_QaKrqiVDQ&sa=X&oi=book_result&ct=result&resnum=6]] [[local|google_books/Stochastic Quantization.pdf]] {{t100Cite{[[bct. 231|http://scholar.google.de/scholar?cites=2435139135058331786&hl=de]]}}}
* [[Coherence for Tricategories - R. Gordon, A. J. Power, R. Street|http://books.google.com/books?id=m0eIjLi23uEC&pg=PA7&lpg=PA7&dq=Coherence+for+Tricategories&source=bl&ots=_IsoEaQWNw&sig=BYan6HAPnhwcpPzFhaIvCIuaBRU&hl=de&ei=vHe0StDXJ8aksAao7uDuDA&sa=X&oi=book_result&ct=result&resnum=1]] [[local|google_books/tricategories.pdf]] {{t100Cite{[[pct. 177|http://scholar.google.de/scholar?cites=2767671965559650589&hl=de]]}}}
* [[Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics (1994) - G. M. Dixon|http://books.google.com/books?id=_pJB0mMAy1QC&printsec=frontcover&dq=octonions&hl=de]] [[local|google_books/DivisionAlgebras.pdf]] {{t100Cite{[[bct. 134|http://scholar.google.de/scholar?cites=8213752134902241001&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Clifford (Geometric) Algebras with Applications in Physics, Mathematics, and Engineering (1996) - W. E. Baylis|http://books.google.de/books?hl=de&lr=&id=0Nji78YQKfQC&oi=fnd&pg=PA1&dq=%22Clifford+(Geometric)+Algebras%22&ots=C4RTHKB0k-&sig=hN8dJm23R4zzNuqQW1OqlfSPWkY]] {{t100Cite{[[local|google_books/CliffordGeometricAlgebra.pdf]] [[bct. 133|http://scholar.google.de/scholar?cites=4086703771484514773&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [['Nonclassical' States in Quantum Optics: A 'Squeezed' Review of the first 75 Years (2002) - V. V. Dodonov|http://www.quantum.df.ufscar.br/publications/ob21r1.pdf]] [[local|papers/ob21r1.pdf]] {{t100Cite{[[pct. 103|http://scholar.google.de/scholar?hl=de&lr=&cites=956505626697351774&um=1&ie=UTF-8&ei=mivhTfbNNsvwsgbSn5WGBg&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCAQzgIwAA]]}}}
* [[Analysis of Dirac Systems and Computational Algebra (2004) - F. Colombo|http://books.google.de/books?id=gtaWbRwkkkgC&pg=PA298&lpg=PA298&dq=%22complex+octonions%22&source=bl&ots=hvIiJee8Ck&sig=HCAhGtFyAJ_hYZPIQoFSNZM_r2A&hl=de&ei=I-vlTvDcFY_RsgbumLTFCQ&sa=X&oi=book_result&ct=result&redir_esc=y#v=onepage&q=%22complex%20octonions%22&f=false]] {{t100Cite{[[bct. 101|http://scholar.google.de/scholar?cites=16681597676421262134&as_sdt=2005&sciodt=0,5&hl=de]]}}}

<<tiddler [[include_tiddlers/Top Relevance.html#"Top Relevance"]]>>

<<tiddler [[include_tiddlers/Topological Invariant.html#"Topological Invariant"]]>>

<<tiddler [[include_tiddlers/Topology Change.html#"Topology Change"]]>>

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The quotient group $\mathbb R / \mathbb Z$ is a circle.
An n-dimensional ''Torus'' is defined as a direct product of n circles.

The (conventional) ''Total Differential $df$ (Of First Order)'' of a scalar function $f$ of $n$ variables $x_i$ is given by:
\[
df(x) = \sum_{i=1}^n \frac{\partial f(x)}{\partial x_i}dx_i
\]
In the case that $n=1$ one gets
\[
df(x) = \frac{\partial f(x)}{\partial x}dx = f(x+dx) - f(x)
\]
The ''Total Differential (Of First Order)'' therefore describes the first order (or linear) change of a function.

If the $n$ coordinates $x_i$ are functions of $m$ coordinates $y_j$, i.e. $x_i = x_i(y_j)$, we get
\[
df(x(y)) = \sum_{i=1}^{n} \sum_{j=1}^{m} \frac{\partial f(x(y))}{\partial x_i} \frac{\partial x_i (y)}{\partial y_j} dy_j = \sum_{i=1}^{n} \sum_{j=1}^{m} \frac{\partial f(x(y))}{\partial x_i} J_x (y) dy_j
\]
with $J_x(y)$ the [[Jacobi matrix|Jacobi Matrix]].

Videos:
* [[Advanced Differential Calculus 4: The Total Differential|http://www.youtube.com/watch?v=450ac8pQ8dE]] - A very energetic presentation ...

''Transitivity'' means that for any two elements $\mb A$ and $\mb B$ of a space (or a set) there exists an element $\mb g$ of a group $\mathcal G$ such that
\[
\mb B = \mb{g\mb A}
\]
More generally one sais that $\mathcal G$ acts ''n-transitively'' on a set $X$ with $ord(X) \ge n$, if for any pairwise distinct $x_1, \ldots , x_n$ and any pairwise distinct $y_1,\ldots ,y_n$ there is a $g \in \mathcal G$, such that $g x_i = y_i$, $\forall i =1,\ldots,n$.
Transitive defined above thus is equal to 1-transitive.

<<tiddler [[include_tiddlers/Tri-bimaximal Mixing.html#"Tri-bimaximal Mixing"]]>>

<<tiddler [[include_tiddlers/Tricategory.html#"Tricategory"]]>>

<<tiddler [[include_tiddlers/Trigintaduonion.html#"Trigintaduonion"]]>>

<<tiddler [[include_tiddlers/Trigintaduonion Multiplication Tables.html#"Trigintaduonion Multiplication Tables"]]>>

<br><<tiddler [[include_tiddlers/Trotter Formula.html#"Trotter Formula"]]>>

<<tiddler [[include_tiddlers/Tsallis Entropy.html#"Tsallis Entropy"]]>>

In [1] D. Chesley describes what he calls ''Twisted [[Cayley-Dickson Algebras|Cayley-Dickson Algebra]]'', using either one of the following two modified [[Cayley-Dickson doubling|Cayley-Dickson Doubling]] formulas
\begin{eqnarray}
(\mb A_1, \mb A_2)(\mb B_1, \mb B_2) & = &(\mb A_1 \mb B_1 - \mb A_2^*\mb B_2, \mb B_2\mb A_1^* + \mb A_2 \mb B_1) \\
(\mb A_1, \mb A_2)(\mb B_1, \mb B_2) & = & (\mb A_1\mb B_1 - \mb A_2\mb B_2^*, \mb A_1\mb B_2 + \mb B_1^*\mb A_2)
\end{eqnarray}
which are equivalent in the same way as are the two variants of the classical Cayley\-Dickson doubling formulas.

These formulas contain the same elements as the classical Cayley Dickson formulas, however they differ from the latter in that $\mb A_2$ and $\mb B_2$ are exchanged in the first component of the doubled algebra and conjugation is exchanged between $\mb A_1$ and $\mb B_1$ in the second component.

They are the only alternatives to the classical doubling formulas if one requires
* $(\mb A  \mb B)^* = \mb B^* \mb A^*$
* and the existence of $7$ quaternionic triads
for the doubled algebra in case of dimension $8$.

The relevance of twisted CD algebras is given at least by the fact, that twisted octonions naturally occur as [[subalgebras of (untwisted) classical Cayley Dickson sedenion algebras|Sedenion Subalgebras]].

The differences between twisted Cayley\-Dickson algebras and classical Cayley\-Dickson algebras are solely due to their associated [[sign tables|Sign Tables]],  their underlying [[Fano spaces|Fano Spaces]] are the same.

See also: [[twisted octonions|Twisted Octonion]].

Papers:
* [[[1] Superparity and Curvature of Twisted Octonionic Manifolds Embedded in Higher Dimensional Spaces - D. Chesley|http://captaincomputersensor.net/Superparity.html]] pct. 0  prl. 10

Links:
* [[Detailed Derivation and Generalization of Cayley-Dickson Construction - D. Chesley|http://raritan.dl.stevens-tech.edu/personal/dchesley/newCD.html]]

<html><center><img src="images/twistedOctonion.jpg" style="width: 255px; "/></center></html>
''Twisted octonions'' (also refered to as ''Quasi\-Octonions'' in literature) are $8$-dimensional [[twisted Cayley-Dickson algebras|Twisted Cayley-Dickson Algebra]]. They are a variant of the classical [[octonions|Octonion]].

In the following we will restrict ourselves to the description of the [[non-split|Split Algebra]] twisted octonions.

!!!!Comparison of twisted and classical octonions
Characteristic of twisted octonions in comparison to classical octonions are the following properties:
* Loss of [[multiplicative norm|Composition Algebra]].
* Appearance of [[zero divisors|Zero Divisor]].
* Different [[sign tables|Sign Tables]].
* There are $7\cdot480$ different multiplication tables for the twisted octonions compared to $480$ for the classical octonions. Hence all in all one has $8\cdot480 = 3.840$ multiplication tables.
* Their [[automorphism group|Automorphism]] is [[SU(2)]]x[[SU(2)]] rather than [[G2]].

Some properties that are the same:
* The underlying [[projective geometry|Projective Geometry]] of twisted octonions (i.e. the respective [[Fano plane|Fano Plane]]) is the same. (This also applies to the split versions).

!!!! Further properties
For the twisted octonions some quite remarkable properties are known which primarily have to do with their embedding in the [[sedenions|Sedenion]]:
* Doubling non-twisted octonions with either the classical doubling formulas or the "twisted" ones leads to the same sets of sedenions.
* Even the classical doubling formulas applied to the (untwisted) octonions produce sedenions that inevitably contain twisted octonions as subalgebras.
* All the zero divisors of the sedenions are contained in the twisted octonion subalgebras.

Papers:
* [[Twisted Octonions and their Symmetry Groups - S. Catto, D. Chesley|http://captaincomputersensor.net/twisted.html]] pct. 0
* [[Superparity and Curvature of Twisted Octonionic Manifolds Embedded in Higher Dimensional Spaces - D. Chesley|http://captaincomputersensor.net/Superparity.html]] pct. 0

Links:
* [[Standard Cayley-Dickson Construction of Octonions - D. Chesley|http://captaincomputersensor.net/CDocts.html]]
* [[Moreno Variant of Cayley-Dickson Construction of Octonions - D. Chesley|http://captaincomputersensor.net/Morenocts.html]]

Videos:
* [[Michio Kaku on UFOs|http://www.youtube.com/watch?v=zCe4WvdMgzM&feature=related]]
* [[Interview: Illobrand von Ludwiger über wissenschaftliche Beweise für UFOs|http://video.google.com/videoplay?docid=-7486657960591754798#]]

<<tiddler [[include_tiddlers/Ultrafinitism.html#"Ultrafinitism"]]>>

<<tiddler [[include_tiddlers/Uncertainty Relations.html#"Uncertainty Relations"]]>>

<<tiddler [[include_tiddlers/Unification.html#"Unification"]]>>

<<tiddler [[include_tiddlers/Unimodular Gravity.html#"Unimodular Gravity"]]>>

<<tiddler [[include_tiddlers/Universal Enveloping Algebra.html#"Universal Enveloping Algebra"]]>>

<<tiddler [[include_tiddlers/Universal Theory.html#"Universal Theory"]]>>

An observer in an accelerated frame experiences a temperature
\[
T = \frac{\hbar} {2\pi k c} a
\]
where $a$ denotes the acceleration.

This is known as ''(Fulling\-Davies\-)Unruh Effect''. The spectrum is exactly thermal.

!!!!Horizons
One can associate a temperature with any null surface that can act as horizon for a class of observers in any spacetime (including flat spacetime). I.e. the connection of temperature and horizons is quite generic.

This __temperature__ is determined by the behaviour of the [[metric|Metric Tensor]] close to the horizon and __is independent of the field equations__ (if any) which are obeyed by the metric.
The simplest situation is that of Rindler observers in flat spacetime with acceleration $a$ who will attribute an Unruh temperature to the Rindler horizon - which in the flat spacetime has no special significance to the inertial observers.

While this result is usually proved for an eternally accelerating observer, it also holds in the (appropriately) approximate sense for an observer with variable acceleration.
It can be used to show that the vacuum state in a freely falling (local) frame will appear to be a thermal state for high frequency modes if $a^{-1}$ is smaller than the local radius of spacetime curvature.


Papers:
* [[Scalar Particle Production in Schwarzschild and Rindler Metrics (1975) - P. C. W. Davies|http://cosmos.asu.edu/publications/papers/ScalarParticleProductionInSchwarzchild%2015.pdf]] [[local|papers/ScalarParticleProductionInSchwarzchild 15.pdf]] {{t100Cite{[[pct. 329|http://scholar.google.de/scholar?cites=6088379019958548653&as_sdt=2005&sciodt=2000&hl=de]]}}}
* [[Acceleration Radiation and the Second Law of Thermodynamics (1982) - W. G. Unruh, R. M. Wald|http://laplace.physics.ubc.ca/People/benjamin/wald_papers/1982-Unruh-Wald-prd-v25.pdf]] [[local|papers/1982-Unruh-Wald-prd-v25.pdf]] {{t100Cite{[[pct. 180|http://scholar.google.de/scholar?cites=5081230214456944182&as_sdt=2005&sciodt=2000&hl=de]]}}}

Links:
* [[WIKIPEDIA - Unruh Effect|http://en.wikipedia.org/wiki/Unruh_effect]]
* [[WIKIPEDIA - Gibbons-Hawking Effect|http://de.wikipedia.org/wiki/Gibbons-Hawking-Effekt]]

Videos:
* [[Talk by Bill Unruh|http://pirsa.org/10090106]]

<<tiddler [[include_tiddlers/Vacuum.html#"Vacuum"]]>>

<<tiddler [[include_tiddlers/Vertex Operator Algebra.html#"Vertex Operator Algebra"]]>>

Links:
*[[The Ultimate Video Feedback Page|http://www.videofeedback.dk/World/]]

<<tiddler [[include_tiddlers/Videos.html#"Videos"]]>>

<!--{{{-->
<div class='toolbar' macro='toolbar -closeTiddler closeOthers +editTiddler permalink references jump'></div>
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<div class='tagging' macro='tagging'></div>
<div class='tagged'><div macro='tags'></div><div macro='whatLinksHere "back reference(s): <br> "'></div></div>
<div class='viewer' macro='view text wikified'></div>
<div class='tagClear'></div>
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<<tiddler [[include_tiddlers/Virasoro Algebra.html#"Virasoro Algebra"]]>>

<<tiddler [[include_tiddlers/Viscoelasticity.html#"Viscoelasticity"]]>>

<<tiddler [[include_tiddlers/Volume Derivative.html#"Volume Derivative"]]>>

<<tiddler [[include_tiddlers/Von Neumann Entropy.html#"Von Neumann Entropy"]]>>

The ''Von Neumann Equation'' which is the quantum analogue of the Liouville equation is given by
\[
\dot \rho = - \frac i \hbar [H, \rho]
\]
where $H$ is the Hamilton operator.

Links:
* [[Von Neumann Integer Generator|http://www.uni-bonn.de/~manfear/von-neumann-integer-gen.php]]

<<tiddler [[include_tiddlers/Voronin Universality Theorem.html#"Voronin Universality Theorem"]]>>

Given a [[lattice|Lattice]] $L$ in $\mathbb R^n$, a ''Voronoi Polytope'' or ''(Dirichlet-)Voronoi Cell'' $\mathcal P_n$ is defined by
\[
\mathcal P_n \equiv \{x \in \mathbb R^n : \langle x|v \rangle \le \frac 12 ||v||^2 \; \text{for} \; v \in \tilde L \subset L \setminus \{0\}\}
\]
$\tilde L$ are those vectors that define a ''Facet'' (for details see [1]).
<html><center><img src="images/VoronoiConstruction.jpg" style="width: 236px; "/></center></html>
Voronoi polytopes carry many synonymous names in different fields of science. Some examples are given as follows:
* $2$-dimensional lattice theory: ''Dirichlet Domain''.
* $n$-dimensional lattice theory, computational geometry: ''Voronoi Polytope''.
* Solid state physics, crystallography: ''Wigner\-Seitz Cell'', ''First Brillouin Zone'', ''Bernal Polytope'', ''Nearest Neighbour Region''.
* Geography: ''Thiessen Polygon''.
* Politics: ''Domain of Influence''.

$n$-dimensional Voronoi polytopes allow for a [[tesselation|Tiling]] of $\mathbb R^n$. Alternatively one can do a tesselation via Delaunay polytopes.
Both cases are shown in the following picture for $n=2$. The case of a Voronoi tesselation in $\mathbb R^2$ is also called a ''Dirichet Tesselation''.
<html><center><img src="images/Delauny.jpg" style="width: 446px; "/></center></html>

The Voronoi polytope is a remarkably useful tool for investigating lattice geometry.

In $\mathbb R^n$ one has the following number of combinatorial types of parallelohedra which tile $\mathbb R^n$:
* $\mathbb R^3$: $5$
<html><center><img src="images/parallelohedra.jpg" style="width: 255px; "/></center></html>
* $\mathbb R^4$: $52$
* $\mathbb R^5$: Unknown, but estimated to be about $75.000$.

Presentations:
* [[[1] Voronoi L-types and Hypermetrics - M. Deza, M. Dutour|http://www.liga.ens.fr/~deza/Sem-lattices/GeomNumberHyp/Voronoitalk.pdf]] [[local|presentations/Voronoitalk.pdf]]

<<tiddler [[include_tiddlers/W and Z Bosons.html#"W and Z Bosons"]]>>

<<tiddler [[include_tiddlers/WKB Approximation.html#"WKB Approximation"]]>>

<<tiddler [[include_tiddlers/Wavefunction of the Universe.html#"Wavefunction of the Universe"]]>>

<<tiddler [[include_tiddlers/Weak Hypercharge.html#"Weak Hypercharge"]]>>

<<tiddler [[include_tiddlers/Weak Isospin.html#"Weak Isospin"]]>>

<<tiddler [[include_tiddlers/Web.html#"Web"]]>>

<<tiddler [[include_tiddlers/Wedge Product.html#"Wedge Product"]]>>

''Weighing Matrices'' are a generalization of [[Hadamard matrices|Hadamard Matrix]]. Their properties are quite similar to those of Hadamard matrices, and they are important for some construction methods of the latter.

A weighing matrix $W(m, n)$ is defined as a $m \times n$ matrix $H$ with entries $?1$, $0$ and $1$ such that
\[
HH^T = n I
\]
for some nonnegative integer $n$.
$n$ is called the ''Weight'' of $H$.

Given a [[Lie Algebra]] and a representation thereof, ''weights are'' the ''eigenvalues'' of the (maximal) subset of mutually commuting elements of the representation (representation of the [[Cartan subalgebra|Cartan Subalgebra]]).
The weight vectors span a space that is called "Weight Space".

!!!!Physical Applications
* Weight vectors correspond to particles
* Weights vectors correspond to quantum numbers

The ''Weil Conjectures'' (due to André Weil) are a series of theorems about the zeta function of a variety over a finite field.

They were proved by way of constructing a suitable [[cohomology theory|Cohomology]].

What is astounding about them is that they provide a link between the discrete (the number of points on a variety over a finite field) and the continuous (topological notions such as Betti numbers).

The proof of the conjectures can be regarded as one of the great triumphs of algebraic geometry of the 20${}^{th}$ century.

<<tiddler [[include_tiddlers/Weinberg-Witten Theorem.html#"Weinberg-Witten Theorem"]]>>

A ''Weitzenböck Space'' (or ''Teleparallel Space''), also designated $\mathbb T_4$, is a manifold having vanishing [[Riemann curvature|Riemann Tensor]]. It appears as a limiting case of the more general concept of a [[Riemann-Cartan space|Riemann-Cartan Space]].

A Weitzenböck space is characterized by a nontrivial [[tetrad field|Tetrad]] as the fundamental structure which gives rise to the [[metric|Metric Tensor]] as a by-product.

The connection (a.k.a. ''Weizenböck Connection'') is given by
\[
 (\Gamma_{W})_{\mu\nu}^\rho = h^\rho_c (\mb x) \partial_\mu h^c_\nu (\mb x)
\]
Using the more general formula for a connection
\[
\Gamma^\rho_{\mu\nu} (\mb x) =  h^\rho_c (\mb x) \partial_\mu h^c_\nu (\mb x) + h^\rho_c (\mb x) h^a_\nu (\mb x) \omega^c_{\mu a} (\mb x)
\]
demonstrates, that a Weitzenböck connection implies the vanishing of the [[spin connection|Spin Connection]].
This can be interpreted such that, when moving across a Weitzenböck manifold, the (co-)frame is not Lorentz-rotated, which is equivalent to saying that the manifold is "teleparallel" or possesses "distant parallelism".

Compared to the [[Levi-Civita connection|Levi-Civita Connection]] which is uniquely determined by the metric, the teleparallel connection is uniquely determined by the (co-)frame.

~~Version: 2012_05_13, with 1039 out of 1908 (54%) tiddlers online. ~~

<<cloud systemConfig excludeMissing bookmarklet systemConfigX content DiscoveryPackage NavigationPackage excludeLists search systemNotes systemTiddlers transclusion>>

''Welcome'' to this physics, mathematics & philosophy notebook.

Since a while I am maintaining a physics and mathematics notebook locally where I collect facts that seem to be relevant to me and where I write down own thoughts, ideas, results, etc. Some of the things are probably not interesting for the general public as they are too personal, unfinished or preliminary. Some might be and I will contain them in this Wiki. Furthermore I have some results that took me quite some effort to work out over the years and I want to hold them back as I consider publishing some of them one day.
I should really stress that I have not made a distinction between things that are part of general knowledge and things that I have "invented" - therefore be warned! If you are looking for an encyclopedia, you'd probably better consult some other site. (In particular [[P-theory|P-Theory]], the theory of [[polyvector spaces|Polyvector Space]] the way represented here is due to myself. If you don't like it, just ignore the [[tiddlers|What is a Tiddler]] tagged with "polyvector" and the WIKI may still be of some value to you).

As this notebook is based on a [[TiddlyWiki|http://www.tiddlywiki.com]] the information is nicely chunked. So for each "Tiddler" I can decide if I dare to put it on the web or not.

{{center{[img(684px+, )[images/trajectory.jpg]]}}}<html><center><small>Comet Comet <a href="http://www.markus-maute.de/P9385935.JPG">Hale-Bopp </a>1995<center></left></html>
Disclaimer: A lot in this Wiki is second hand information and it is prone to be erroneous. I take no guarantee whatsoever. (I encourage you to do kind of a "peer reviewing" in that you write [[comments|Comments]] - this would be very helpful). Furthermore I am quite lazy painting pictures or reformulating every adequate text found somewhere, so I often just copy the information (occasionally mutatis mutandis) into this Wiki. Therefore if you think there is a problem with violations of copyrights, please let me know and I'll remove the respective content immediately.

Some food for thought: Suppose this Wiki is part of the trajectory of the universe. So then, is it possible to include in it an equation that describes that trajectory that includes the description of the content of this Wiki ?

{{center{<<matchTags popup "label:Some highlights of this WIKI" highlight>>}}}
E\-Mail: <html>
<a href="mailto:trajectory@markus-maute.de">trajectory@markus-maute.de</a> </html>

Comments :
<html><center> <br><iframe name="content" src="http://www.markus-maute.de/trajectory/comments/index.php?do=add_form&page=1" width=65% height=450></iframe></center>
</html>
<html><center>
<p>This Wiki presents independent research not based on any third party funding. If you like it and click a lot through it, consider also clicking on one of the banners at times (or following the donation button).
This way you help to finance the tons of coffee and chocolate required for boosting the author's brain to do all the writings (in particular in a polished style, allowing to make them available to you - for free !) <br><br>

Thank YOU </p>

<form action="https://www.paypal.com/cgi-bin/webscr" method="post">
<input type="hidden" name="cmd" value="_s-xclick">
<input type="hidden" name="hosted_button_id" value="RZXREBYY7AERC">
<input type="image" src="https://www.paypal.com/en_US/i/btn/btn_donateCC_LG.gif" border="0" name="submit" alt="PayPal - The safer, easier way to pay online!">
<img alt="" border="0" src="https://www.paypal.com/de_DE/i/scr/pixel.gif" width="1" height="1">
</form>
<br>
... or just do your next shopping at Amazon via this link: <br>
<iframe name="content" src="amazon_books.html" width=690 height=223 frameborder  = 0></iframe></center></html>
<<tiddler RefreshPageDisplay">>

<<tiddler [[include_tiddlers/Wess-Zumino Model.html#"Wess-Zumino Model"]]>>

In quantum mechanics the ''Weyl Algebra'' is derived from differential calculus via the following correspondence principle:
Let $u$ be the operator $\mb x\cdot$ of multiplication by the coordinate function $x$ on $\mathbb R$ acting on the space of all $C^\infty$ functions on $\mathbb R$ and $v$ the differential operator $i \hbar \partial_x$ (in short: the correspondence principle says $u \leftrightarrow x \cdot$ and $v \leftrightarrow  i \partial_x$), then the Weyl algebra is an associative algebra generated over $C^\infty$ by $u$ and $v$ by the fundamental relation
\[
[u,v] = i \hbar\,
\]

Papers:
* [[Fermi-Walker Transport and the Weyl Connection - B. M. Barbashov, A. B. Pestov|http://www.springerlink.com/content/n3215176w3p1837j/fulltext.pdf]] pct. 0

<<tiddler [[include_tiddlers/Weyl Curvature Hypothesis.html#"Weyl Curvature Hypothesis"]]>>

Given the [[simple roots|Simple Root]] of a [[root system|Root Vector]], all other roots can be constructed from them by means of ''Weyl Reflections''. The set of all such reflections forms a group known as [[Weyl group|Weyl Group]].

In $\mathbb R^l$ each root $r_i$ defines an $(l ? 1)$-dimensional hyperplane which contains the origin and is orthogonal to $r_i$. A Weyl reflection for $r_i$ reflects each of the other roots $r_j$ across this hyperplane, producing the root $r_k$ defined by
\[
r_k = r_j ? 2 \frac{\langle r_j |r_i \rangle} {\langle r_i| r_i\rangle} r_i
\]
!!!!Example
The $14$ roots of [[G2]] can be generated by Weyl reflections starting from the two simple roots of the group.

<html><center><img src="images/Weyl_reflection.jpg" style="width: 590px; "/></center></html>

A ''Weyl Space'' is a non-[[metric compatible|Metric Compatibility]], [[metric affine space|Metric Affine Space]], satisfying
\[
D_\alpha g_{\mu\nu} (\mb x)  =  2 g_{\mu\nu} (\mb x) \phi_\alpha (\mb x)
\]
$\phi_\alpha$ is called ''Weyl Covector'' which is related to scale transformations (and thus extends the [[Lorentz-|Lorentz Transformation]] to the [[conformal group|Conformal Transformation]]).

A Weyl space possesses only a "weak" form of [[non-metricity|Non-Metricity Tensor]], thus it is also referred to as a ''Semi\-Metrical Space''.
Most importantly, the light cone is preserved by parallel transport and causality is not violated in such a space.

<<tiddler [[include_tiddlers/Weyl Tensor.html#"Weyl Tensor"]]>>

''"Tiddler"'' is [[TiddlyWiki]] lingo and means sth. like topic, entry, ...

$2$ examples:
* "What is a Tiddler"
* [[Comments|Comments]]
are tiddlers.
<html><center><iframe name="content" src="http://www.markus-maute.de/trajectory/advert.html" width=51% height=86></iframe></center></html>

/***
|Name|WhatLinksHerePlugin|
|Source|http://rumkin.com/tools/tiddlywiki/#WhatLinksHerePlugin|
|Version|1.0.0|
|Author|Tyler Akins|
|License|Public Domain|
|~CoreVersion|2.1|
|Type|plugin|
|Requires||
|Overrides||
|Description|Displays a list of pages that link to the current page.  Something like a lightweight version of RelatedTiddlersPlugin.  The links[] data is scanned for all tiddlers that link to the current tiddler.|
!Usage
{{{
<<whatLinksHere txtIfList txtIfNoList>>
}}}
* txtIfList: String to print at the top of the list if there is at least one page that links in to the current one.
* txtIfNoList:  String to print at the top if there are no pages that link to the current tiddler.

!Configuration
Do not list these tiddlers:
{{wideInput{<<option txtWhatLinksHereExclude 40>>}}}

!Examples

This is a live example of what links to this page:
|<<whatLinksHere "List of pages that link to me: <br>" "Sorry, nobody links to me.">>|
I like to put it in my ViewTemplate.  Replace the line that says
{{{
<div class='tagged' macro='tags'></div>
}}}
with this
{{{
<div class='tagged'><div macro='tags'></div><div macro='whatLinksHere "<br>What Links Here: <br> "'></div></div>
}}}

!Installation
# Import the WhatLinksHerePlugin tiddler.
# Modify a tiddler or template to use the whatLinksHere macro.  See above for a sample modification to ViewTemplate

!Revision History
* 1.0.0 (2007-09-30)
** Initial version.

!Credits
The RelatedTiddlersPlugin by Eric L. Shulman was the base for this one.  Even though I didn't use 95% of the code, I did use it for inspiration.

!Code
***/
//{{{
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if (config.optionsDesc)
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		if (results.length && params[0])
			out = params[0] + out;
		if (! results.length && params[1])
			out = params[1] + out;
		out = "{{whatLinksHere{" + out + "}}}";
		wikify(out, place);
	}
}

//}}}

<<tiddler [[include_tiddlers/Why the World is neither Deterministic nor Random.html#"Why the World is neither Deterministic nor Random"]]>>

<<tiddler [[include_tiddlers/Wick Rotation.html#"Wick Rotation"]]>>

<<tiddler [[include_tiddlers/Wigner's Theorem.html#"Wigner's Theorem"]]>>

As the [[''Trajectory of the Universe''|Welcome]] is neither solely a Wiki nor solely a Blog, I'll just call it a ''Wiki\-Blog''.

<html><center><img src="images/clifford.jpg" style="width: 455px; "/></center></html>
>I hold in fact:
>1. That small portions of space are in fact of a nature analogous to
>little hills on a surface which is on the average flat; namely, that
>the ordinary laws of geometry are not valid in them.
>2. That this property of being curved or distorted is continually
>being passed on from one portion of space to another after the
>manner of a wave.
>3. That this variation of the curvature of space is what really
>happens in that phenomenon which we call the motion of
>matter, whether ponderable or etherial
>4. That in the physical world nothing else takes place but this
>variation, subject (possibly) to the law of continuity.
>
> - On the Space\-Theory of Matter” Proceedings of the Cambridge Philosophical Society (1876) -

!!!!!A personal remark:
The idea that space is curved is even older and goes back to Riemann who already tried to construct a physical theory under this assumption. It is known that Clifford was highly impressed by Riemann's work and probably was influenced by his thinking.
Both however didn't realize that time has to be included in the overall description of curvature as well. It was Albert Einstein who did this.
On the other hand the statements above go further, namely that matter is also curvature. This idea is at the heart of the classical [[unification|Unification]] idea, also pursued by Einstein, however it has not been realized satisfactorily until today.

Papers:
* [[Non-Abelian BF Theories with Sources and 2-D Gravity - J. P. Lupi , A. Restuccia, J. Stephany|http://arxiv.org/PS_cache/hep-th/pdf/9603/9603013v2.pdf]]

<<tiddler [[include_tiddlers/World Crystal.html#"World Crystal"]]>>

<<tiddler [[include_tiddlers/World Line.html#"World Line"]]>>

<<tiddler [[include_tiddlers/World Polyvector.html#"World Polyvector"]]>>

<<tiddler [[include_tiddlers/World Polyvector Action.html#"World Polyvector Action"]]>>

<<tiddler [[include_tiddlers/World Sheet.html#"World Sheet"]]>>

<<tiddler [[include_tiddlers/World Volume.html#"World Volume"]]>>

<<tiddler [[include_tiddlers/Wormhole.html#"Wormhole"]]>>

The ''Wreath Product "$\wr$"'' is a special [[semidirect product|Semi Direct Product]] between [[groups|Group]].

<<tiddler [[include_tiddlers/X-Product.html#"X-Product"]]>>

Truth table for the ''Exclusive "Not Or"''- (''XNOR''-) operation:

| | !0 | !1|
|!0| 1 | 0|
|!1| 0 |1|

See also:  [[XOR tables|XOR Tables]].

<<tiddler [[include_tiddlers/XOR Tables.html#"XOR Tables"]]>>

|>|!YourSearch Options|
|>|<<option chkUseYourSearch>> Use 'Your Search'|
|!|<<option chkPreviewText>> Show Text Preview|
|!|<<option chkSearchAsYouType>> 'Search As You Type' Mode (No RETURN required to start search)|
|!|Default Search Filter:<<option chkSearchInTitle>>Title ('!')     <<option chkSearchInText>>Text ('%')     <<option chkSearchInTags>>Tags ('#')    <<option chkSearchExtendedFields>>Extended Fields<html><br><font size="-2">The fields of a tiddlers that are searched when you don't explicitly specify a filter in the search text <br>(Explictly specify fields using one or more '!', '%', '#' or 'fieldname:' prefix before the word/text to find).</font></html>|
|!|Number of items on search result page: <<option txtItemsPerPage>>|
|!|Number of items on search result page with preview text: <<option txtItemsPerPageWithPreview>>|

/***
|''Name:''|YourSearchPlugin|
|''Version:''|2.1.5 (2010-02-16)|
|''Source:''|http://tiddlywiki.abego-software.de/#YourSearchPlugin|
|''Author:''|UdoBorkowski (ub [at] abego-software [dot] de)|
|''Licence:''|[[BSD open source license (abego Software)|http://www.abego-software.de/legal/apl-v10.html]]|
|''Copyright:''|&copy; 2005-2010 [[abego Software|http://www.abego-software.de]]|
|''~CoreVersion:''|2.1.0|
|''Community:''|[[del.icio.us|http://del.icio.us/post?url=http://tiddlywiki.abego-software.de/index.html%23YourSearchPlugin]]|
|''Browser:''|Firefox 1.0.4+; Firefox 1.5; ~InternetExplorer 6.0|
!About YourSearch
YourSearch gives you a bunch of new features to simplify and speed up your daily searches in TiddlyWiki. It seamlessly integrates into the standard TiddlyWiki search: just start typing into the 'search' field and explore!

For more information see [[Help|YourSearch Help]].
!Compatibility
This plugin requires TiddlyWiki 2.1.
Check the [[archive|http://tiddlywiki.abego-software.de/archive]] for ~YourSearchPlugins supporting older versions of TiddlyWiki.
!Source Code
***/
/***
This plugin's source code is compressed (and hidden). Use this [[link|http://tiddlywiki.abego-software.de/archive/YourSearchPlugin/Plugin-YourSearch-src.2.1.5.js]] to get the readable source code.
***/
///%
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len=Math.min(_aa.end-_aa.start,_a9);_a1(_9d,s,_9e,_aa.start,_aa.start+len);_a9-=len;}};this.render=function(_ab,s,_ac,_ad){if(s.length<_ac){_ac=s.length;}var _ae=_7b(s,_ad);var _af=_8d(_ae,s,_ac);_94(s,_af,_ac);_9c(_ab,s,_ae,_af,_ac);};};(function(){function _b0(msg){alert(msg);throw msg;};if(version.major<2||(version.major==2&&version.minor<1)){_b0("YourSearchPlugin requires TiddlyWiki 2.1 or newer.\n\nCheck the archive for YourSearch plugins\nsupporting older versions of TiddlyWiki.\n\nArchive: http://tiddlywiki.abego-software.de/archive");}abego.YourSearch={};var _b1;var _b2;var _b3=function(_b4){_b1=_b4;};var _b5=function(){return _b1?_b1:[];};var _b6=function(){return _b1?_b1.length:0;};var _b7=4;var _b8=10;var _b9=2;var _ba=function(s,re){var m=s.match(re);return m?m.length:0;};var _bb=function(_bc,_bd){var _be=_bd.getMarkRegExp();if(!_be){return 1;}var _bf=_bc.title.match(_be);var _c0=_bf?_bf.length:0;var _c1=_ba(_bc.getTags(),_be);var _c2=_bf?_bf.join("").length:0;var _c3=_bc.title.length>0?_c2/_bc.title.length:0;var _c4=_c0*_b7+_c1*_b9+_c3*_b8+1;return _c4;};var _c5=function(_c6,_c7,_c8,_c9,_ca,_cb){_b2=null;var _cc=_c6.reverseLookup("tags",_cb,false);try{var _cd=[];if(config.options.chkSearchInTitle){_cd.push("title");}if(config.options.chkSearchInText){_cd.push("text");}if(config.options.chkSearchInTags){_cd.push("tags");}_b2=new abego.TiddlerQuery(_c7,_c8,_c9,_cd,config.options.chkSearchExtendedFields);}catch(e){return [];}var _ce=_b2.filter(_cc);var _cf=abego.YourSearch.getRankFunction();for(var i=0;i<_ce.length;i++){var _d0=_ce[i];var _d1=_cf(_d0,_b2);_d0.searchRank=_d1;}if(!_ca){_ca="title";}var _d2=function(a,b){var _d3=a.searchRank-b.searchRank;if(_d3==0){if(a[_ca]==b[_ca]){return (0);}else{return (a[_ca]<b[_ca])?-1:+1;}}else{return (_d3>0)?-1:+1;}};_ce.sort(_d2);return _ce;};var _d4=80;var _d5=50;var _d6=250;var _d7=50;var _d8=25;var _d9=10;var _da="yourSearchResult";var _db="yourSearchResultItems";var _dc;var _dd;var _de;var _df;var _e0;var _e1=function(){if(version.extensions.YourSearchPlugin.styleSheetInited){return;}version.extensions.YourSearchPlugin.styleSheetInited=true;setStylesheet(store.getTiddlerText("YourSearchStyleSheet"),"yourSearch");};var _e2=function(){return _dd!=null&&_dd.parentNode==document.body;};var _e3=function(){if(_e2()){document.body.removeChild(_dd);}};var _e4=function(e){_e3();var _e5=this.getAttribute("tiddlyLink");if(_e5){var _e6=this.getAttribute("withHilite");var _e7=highlightHack;if(_e6&&_e6=="true"&&_b2){highlightHack=_b2.getMarkRegExp();}story.displayTiddler(this,_e5);highlightHack=_e7;}return (false);};var _e8=function(){if(!_de){return;}var _e9=_de;var _ea=findPosX(_e9);var _eb=findPosY(_e9);var _ec=_e9.offsetHeight;var _ed=_ea;var _ee=_eb+_ec;var _ef=findWindowWidth();if(_ef<_dd.offsetWidth){_dd.style.width=(_ef-100)+"px";_ef=findWindowWidth();}var _f0=_dd.offsetWidth;if(_ed+_f0>_ef){_ed=_ef-_f0-30;}if(_ed<0){_ed=0;}_dd.style.left=_ed+"px";_dd.style.top=_ee+"px";_dd.style.display="block";};var _f1=function(){if(_dd){window.scrollTo(0,ensureVisible(_dd));}if(_de){window.scrollTo(0,ensureVisible(_de));}};var _f2=function(){_e8();_f1();};var _f3;var _f4;var _f5=new abego.PageWiseRenderer();var _f6=function(_f7){this.itemHtml=store.getTiddlerText("YourSearchItemTemplate");if(!this.itemHtml){_b0("YourSearchItemTemplate not found");}this.place=document.getElementById(_db);if(!this.place){this.place=createTiddlyElement(_f7,"div",_db);}};merge(_f6.prototype,{render:function(_f8,_f9,_fa,_fb){_f3=_fb;_f4=_f9;var _fc=createTiddlyElement(this.place,"div",null,"yourSearchItem");_fc.innerHTML=this.itemHtml;applyHtmlMacros(_fc,null);refreshElements(_fc,null);},endRendering:function(_fd){_f4=null;}});var _fe=function(){if(!_dd||!_de){return;}var _ff=store.getTiddlerText("YourSearchResultTemplate");if(!_ff){_ff="<b>Tiddler YourSearchResultTemplate not found</b>";}_dd.innerHTML=_ff;applyHtmlMacros(_dd,null);refreshElements(_dd,null);var _100=new _f6(_dd);_f5.renderPage(_100);_f2();};_f5.getItemsPerPage=function(){var n=(config.options.chkPreviewText)?abego.toInt(config.options.txtItemsPerPageWithPreview,_d9):abego.toInt(config.options.txtItemsPerPage,_d8);return (n>0)?n:1;};_f5.onPageChanged=function(){_fe();};var _101=function(){if(_de==null||!config.options.chkUseYourSearch){return;}if((_de.value==_dc)&&_dc&&!_e2()){if(_dd&&(_dd.parentNode!=document.body)){document.body.appendChild(_dd);_f2();}else{abego.YourSearch.onShowResult(true);}}};var _102=function(){_e3();_dd=null;_dc=null;};var _103=function(self,e){while(e!=null){if(self==e){return true;}e=e.parentNode;}return false;};var _104=function(e){if(e.target==_de){return;}if(e.target==_df){return;}if(_dd&&_103(_dd,e.target)){return;}_e3();};var _105=function(e){if(e.keyCode==27){_e3();}};addEvent(document,"click",_104);addEvent(document,"keyup",_105);var _106=function(text,_107,_108){_dc=text;_b3(_c5(store,text,_107,_108,"title","excludeSearch"));abego.YourSearch.onShowResult();};var _109=function(_10a,_10b,_10c,_10d,_10e,_10f){_e1();_dc="";var _110=null;var _111=function(txt){if(config.options.chkUseYourSearch){_106(txt.value,config.options.chkCaseSensitiveSearch,config.options.chkRegExpSearch);}else{story.search(txt.value,config.options.chkCaseSensitiveSearch,config.options.chkRegExpSearch);}_dc=txt.value;};var _112=function(e){_111(_de);return false;};var _113=function(e){if(!e){var e=window.event;}_de=this;switch(e.keyCode){case 13:if(e.ctrlKey&&_e0&&_e2()){_e0.onclick.apply(_e0,[e]);}else{_111(this);}break;case 27:if(_e2()){_e3();}else{this.value="";clearMessage();}break;}if(String.fromCharCode(e.keyCode)==this.accessKey||e.altKey){_101();}if(this.value.length<3&&_110){clearTimeout(_110);}if(this.value.length>2){if(this.value!=_dc){if(!config.options.chkUseYourSearch||config.options.chkSearchAsYouType){if(_110){clearTimeout(_110);}var txt=this;_110=setTimeout(function(){_111(txt);},500);}}else{if(_110){clearTimeout(_110);}}}if(this.value.length==0){_e3();}};var _114=function(e){this.select();clearMessage();_101();};var args=_10e.parseParams("list",null,true);var _115=getFlag(args,"buttonAtRight");var _116=getParam(args,"sizeTextbox",this.sizeTextbox);var btn;if(!_115){btn=createTiddlyButton(_10a,this.label,this.prompt,_112);}var txt=createTiddlyElement(null,"input",null,"txtOptionInput searchField",null);if(_10c[0]){txt.value=_10c[0];}txt.onkeyup=_113;txt.onfocus=_114;txt.setAttribute("size",_116);txt.setAttribute("accessKey",this.accessKey);txt.setAttribute("autocomplete","off");if(config.browser.isSafari){txt.setAttribute("type","search");txt.setAttribute("results","5");}else{txt.setAttribute("type","text");}if(_10a){_10a.appendChild(txt);}if(_115){btn=createTiddlyButton(_10a,this.label,this.prompt,_112);}_de=txt;_df=btn;};var _117=function(){_e3();var _118=_b5();var n=_118.length;if(n){var _119=[];for(var i=0;i<n;i++){_119.push(_118[i].title);}story.displayTiddlers(null,_119);}};var _11a=function(_11b,_11c,_11d,_11e){invokeMacro(_11b,"option",_11c,_11d,_11e);var elem=_11b.lastChild;var _11f=elem.onclick;elem.onclick=function(e){var _120=_11f.apply(this,arguments);_fe();return _120;};return elem;};var _121=function(s){var _122=["''","{{{","}}}","//","<<<","/***","***/"];var _123="";for(var i=0;i<_122.length;i++){if(i!=0){_123+="|";}_123+="("+_122[i].escapeRegExp()+")";}return s.replace(new RegExp(_123,"mg"),"").trim();};var _124=function(){var i=_f3;return (i>=0&&i<=9)?(i<9?(i+1):0):-1;};var _125=new abego.LimitedTextRenderer();var _126=function(_127,s,_128){_125.render(_127,s,_128,_b2.getMarkRegExp());};var _129=TiddlyWiki.prototype.saveTiddler;TiddlyWiki.prototype.saveTiddler=function(_12a,_12b,_12c,_12d,_12e,tags,_12f){_129.apply(this,arguments);_102();};var _130=TiddlyWiki.prototype.removeTiddler;TiddlyWiki.prototype.removeTiddler=function(_131){_130.apply(this,arguments);_102();};config.macros.yourSearch={label:"yourSearch",prompt:"Gives access to the current/last YourSearch result",handler:function(_132,_133,_134,_135,_136,_137){if(_134.length==0){return;}var name=_134[0];var func=config.macros.yourSearch.funcs[name];if(func){func(_132,_133,_134,_135,_136,_137);}},tests:{"true":function(){return true;},"false":function(){return false;},"found":function(){return _b6()>0;},"previewText":function(){return config.options.chkPreviewText;}},funcs:{itemRange:function(_138){if(_b6()){var _139=_f5.getLastIndexOnPage();var s="%0 - %1".format([_f5.getFirstIndexOnPage()+1,_139+1]);createTiddlyText(_138,s);}},count:function(_13a){createTiddlyText(_13a,_b6().toString());},query:function(_13b){if(_b2){createTiddlyText(_13b,_b2.toString());}},version:function(_13c){var t="YourSearch %0.%1.%2".format([version.extensions.YourSearchPlugin.major,version.extensions.YourSearchPlugin.minor,version.extensions.YourSearchPlugin.revision]);var e=createTiddlyElement(_13c,"a");e.setAttribute("href","http://tiddlywiki.abego-software.de/#YourSearchPlugin");e.innerHTML="<font color=\"black\" face=\"Arial, Helvetica, sans-serif\">"+t+"<font>";},copyright:function(_13d){var e=createTiddlyElement(_13d,"a");e.setAttribute("href","http://www.abego-software.de");e.innerHTML="<font color=\"black\" face=\"Arial, Helvetica, sans-serif\">&copy; 2005-2008 <b><font color=\"red\">abego</font></b> Software<font>";},newTiddlerButton:function(_13e){if(_b2){var r=abego.parseNewTiddlerCommandLine(_b2.getQueryText());var btn=config.macros.newTiddler.createNewTiddlerButton(_13e,r.title,r.params,"new tiddler","Create a new tiddler based on search text. (Shortcut: Ctrl-Enter; Separators: '.', '#')",null,"text");var _13f=btn.onclick;btn.onclick=function(){_e3();_13f.apply(this,arguments);};_e0=btn;}},linkButton:function(_140,_141,_142,_143,_144,_145){if(_142<2){return;}var _146=_142[1];var text=_142<3?_146:_142[2];var _147=_142<4?text:_142[3];var _148=_142<5?null:_142[4];var btn=createTiddlyButton(_140,text,_147,_e4,null,null,_148);btn.setAttribute("tiddlyLink",_146);},closeButton:function(_149,_14a,_14b,_14c,_14d,_14e){var _14f=createTiddlyButton(_149,"close","Close the Search Results (Shortcut: ESC)",_e3);},openAllButton:function(_150,_151,_152,_153,_154,_155){var n=_b6();if(n==0){return;}var _156=n==1?"open tiddler":"open all %0 tiddlers".format([n]);var _157=createTiddlyButton(_150,_156,"Open all found tiddlers (Shortcut: Alt-O)",_117);_157.setAttribute("accessKey","O");},naviBar:function(_158,_159,_15a,_15b,_15c,_15d){_f5.addPageNavigation(_158);},"if":function(_15e,_15f,_160,_161,_162,_163){if(_160.length<2){return;}var _164=_160[1];var _165=(_164=="not");if(_165){if(_160.length<3){return;}_164=_160[2];}var test=config.macros.yourSearch.tests[_164];var _166=false;try{if(test){_166=test(_15e,_15f,_160,_161,_162,_163)!=_165;}else{_166=(!eval(_164))==_165;}}catch(ex){}if(!_166){_15e.style.display="none";}},chkPreviewText:function(_167,_168,_169,_16a,_16b,_16c){var _16d=_169.slice(1).join(" ");var elem=_11a(_167,"chkPreviewText",_16a,_16c);elem.setAttribute("accessKey","P");elem.title="Show text preview of found tiddlers (Shortcut: Alt-P)";return elem;}}};config.macros.foundTiddler={label:"foundTiddler",prompt:"Provides information on the tiddler currently processed on the YourSearch result page",handler:function(_16e,_16f,_170,_171,_172,_173){var name=_170[0];var func=config.macros.foundTiddler.funcs[name];if(func){func(_16e,_16f,_170,_171,_172,_173);}},funcs:{title:function(_174,_175,_176,_177,_178,_179){if(!_f4){return;}var _17a=_124();var _17b=_17a>=0?"Open tiddler (Shortcut: Alt-%0)".format([_17a.toString()]):"Open tiddler";var btn=createTiddlyButton(_174,null,_17b,_e4,null);btn.setAttribute("tiddlyLink",_f4.title);btn.setAttribute("withHilite","true");_126(btn,_f4.title,_d4);if(_17a>=0){btn.setAttribute("accessKey",_17a.toString());}},tags:function(_17c,_17d,_17e,_17f,_180,_181){if(!_f4){return;}_126(_17c,_f4.getTags(),_d5);},text:function(_182,_183,_184,_185,_186,_187){if(!_f4){return;}_126(_182,_121(_f4.text),_d6);},field:function(_188,_189,_18a,_18b,_18c,_18d){if(!_f4){return;}var name=_18a[1];var len=_18a.length>2?abego.toInt(_18a[2],_d7):_d7;var v=store.getValue(_f4,name);if(v){_126(_188,_121(v),len);}},number:function(_18e,_18f,_190,_191,_192,_193){var _194=_124();if(_194>=0){var text="%0)".format([_194.toString()]);createTiddlyElement(_18e,"span",null,"shortcutNumber",text);}}}};var opts={chkUseYourSearch:true,chkPreviewText:true,chkSearchAsYouType:true,chkSearchInTitle:true,chkSearchInText:true,chkSearchInTags:true,chkSearchExtendedFields:true,txtItemsPerPage:_d8,txtItemsPerPageWithPreview:_d9};for(var n in opts){if(config.options[n]==undefined){config.options[n]=opts[n];}}config.shadowTiddlers.AdvancedOptions+="\n<<option chkUseYourSearch>> Use 'Your Search' //([[more options|YourSearch Options]]) ([[help|YourSearch Help]])// ";config.shadowTiddlers["YourSearch Help"]="!Field Search\nWith the Field Search you can restrict your search to certain fields of a tiddler, e.g"+" only search the tags or only the titles. The general form is //fieldname//'':''//textToSearch// (e."+"g. {{{title:intro}}}). In addition one-character shortcuts are also supported for the standard field"+"s {{{title}}}, {{{text}}} and {{{tags}}}:\n|!What you want|!What you type|!Example|\n|Search ''titles "+"only''|start word with ''!''|{{{!jonny}}} (shortcut for {{{title:jonny}}})|\n|Search ''contents/text "+"only''|start word with ''%''|{{{%football}}} (shortcut for {{{text:football}}})|\n|Search ''tags only"+"''|start word with ''#''|{{{#Plugin}}} (shortcut for {{{tags:Plugin}}})|\n\nUsing this feature you may"+" also search the extended fields (\"Metadata\") introduced with TiddlyWiki 2.1, e.g. use {{{priority:1"+"}}} to find all tiddlers with the priority field set to \"1\".\n\nYou may search a word in more than one"+" field. E.g. {{{!#Plugin}}} (or {{{title:tags:Plugin}}} in the \"long form\") finds tiddlers containin"+"g \"Plugin\" either in the title or in the tags (but does not look for \"Plugin\" in the text). \n\n!Boole"+"an Search\nThe Boolean Search is useful when searching for multiple words.\n|!What you want|!What you "+"type|!Example|\n|''All words'' must exist|List of words|{{{jonny jeremy}}} (or {{{jonny and jeremy}}}"+")|\n|''At least one word'' must exist|Separate words by ''or''|{{{jonny or jeremy}}}|\n|A word ''must "+"not exist''|Start word with ''-''|{{{-jonny}}} (or {{{not jonny}}})|\n\n''Note:'' When you specify two"+" words, separated with a space, YourSearch finds all tiddlers that contain both words, but not neces"+"sarily next to each other. If you want to find a sequence of word, e.g. '{{{John Brown}}}', you need"+" to put the words into quotes. I.e. you type: {{{\"john brown\"}}}.\n\nUsing parenthesis you may change "+"the default \"left to right\" evaluation of the boolean search. E.g. {{{not (jonny or jeremy)}}} finds"+" all tiddlers that contain neither \"jonny\" nor \"jeremy. In contrast to this {{{not jonny or jeremy}}"+"} (i.e. without parenthesis) finds all tiddlers that either don't contain \"jonny\" or that contain \"j"+"eremy\".\n\n!'Exact Word' Search\nBy default a search result all matches that 'contain' the searched tex"+"t. E.g. if you search for {{{Task}}} you will get all tiddlers containing 'Task', but also '~Complet"+"edTask', '~TaskForce' etc.\n\nIf you only want to get the tiddlers that contain 'exactly the word' you"+" need to prefix it with a '='. E.g. typing '=Task' will find the tiddlers that contain the word 'Tas"+"k', ignoring words that just contain 'Task' as a substring.\n\n!~CaseSensitiveSearch and ~RegExpSearch"+"\nThe standard search options ~CaseSensitiveSearch and ~RegExpSearch are fully supported by YourSearc"+"h. However when ''~RegExpSearch'' is on Filtered and Boolean Search are disabled.\n\nIn addition you m"+"ay do a \"regular expression\" search even with the ''~RegExpSearch'' set to false by directly enterin"+"g the regular expression into the search field, framed with {{{/.../}}}. \n\nExample: {{{/m[ae][iy]er/"+"}}} will find all tiddlers that contain either \"maier\", \"mayer\", \"meier\" or \"meyer\".\n\n!~JavaScript E"+"xpression Filtering\nIf you are familiar with JavaScript programming and know some TiddlyWiki interna"+"ls you may also use JavaScript expression for the search. Just enter a JavaScript boolean expression"+" into the search field, framed with {{{ { ... } }}}. In the code refer to the variable tiddler and e"+"valuate to {{{true}}} when the given tiddler should be included in the result. \n\nExample: {{{ { tidd"+"ler.modified > new Date(\"Jul 4, 2005\")} }}} returns all tiddler modified after July 4th, 2005.\n\n!Com"+"bined Search\nYou are free to combine the various search options. \n\n''Examples''\n|!What you type|!Res"+"ult|\n|{{{!jonny !jeremy -%football}}}|all tiddlers with both {{{jonny}}} and {{{jeremy}}} in its tit"+"les, but no {{{football}}} in content.|\n|{{{#=Task}}}|All tiddlers tagged with 'Task' (the exact wor"+"d). Tags named '~CompletedTask', '~TaskForce' etc. are not considered.|\n\n!Access Keys\nYou are encour"+"aged to use the access keys (also called \"shortcut\" keys) for the most frequently used operations. F"+"or quick reference these shortcuts are also mentioned in the tooltip for the various buttons etc.\n\n|"+"!Key|!Operation|\n|{{{Alt-F}}}|''The most important keystroke'': It moves the cursor to the search in"+"put field so you can directly start typing your query. Pressing {{{Alt-F}}} will also display the pr"+"evious search result. This way you can quickly display multiple tiddlers using \"Press {{{Alt-F}}}. S"+"elect tiddler.\" sequences.|\n|{{{ESC}}}|Closes the [[YourSearch Result]]. When the [[YourSearch Resul"+"t]] is already closed and the cursor is in the search input field the field's content is cleared so "+"you start a new query.|\n|{{{Alt-1}}}, {{{Alt-2}}},... |Pressing these keys opens the first, second e"+"tc. tiddler from the result list.|\n|{{{Alt-O}}}|Opens all found tiddlers.|\n|{{{Alt-P}}}|Toggles the "+"'Preview Text' mode.|\n|{{{Alt-'<'}}}, {{{Alt-'>'}}}|Displays the previous or next page in the [[Your"+"Search Result]].|\n|{{{Return}}}|When you have turned off the 'as you type' search mode pressing the "+"{{{Return}}} key actually starts the search (as does pressing the 'search' button).|\n\n//If some of t"+"hese shortcuts don't work for you check your browser if you have other extensions installed that alr"+"eady \"use\" these shortcuts.//";config.shadowTiddlers["YourSearch Options"]="|>|!YourSearch Options|\n|>|<<option chkUseYourSearch>> Use 'Your Search'|\n|!|<<option chkPreviewText"+">> Show Text Preview|\n|!|<<option chkSearchAsYouType>> 'Search As You Type' Mode (No RETURN required"+" to start search)|\n|!|Default Search Filter:<<option chkSearchInTitle>>Title ('!')     <<option chk"+"SearchInText>>Text ('%')     <<option chkSearchInTags>>Tags ('#')    <<option chkSearchExtendedFiel"+"ds>>Extended Fields<html><br><font size=\"-2\">The fields of a tiddlers that are searched when you don"+"'t explicitly specify a filter in the search text <br>(Explictly specify fields using one or more '!"+"', '%', '#' or 'fieldname:' prefix before the word/text to find).</font></html>|\n|!|Number of items "+"on search result page: <<option txtItemsPerPage>>|\n|!|Number of items on search result page with pre"+"view text: <<option txtItemsPerPageWithPreview>>|\n";config.shadowTiddlers["YourSearchStyleSheet"]="/***\n!~YourSearchResult Stylesheet\n***/\n/*{{{*/\n.yourSearchResult {\n\tposition: absolute;\n\twidth: 800"+"px;\n\n\tpadding: 0.2em;\n\tlist-style: none;\n\tmargin: 0;\n\n\tbackground: #ffd;\n\tborder: 1px solid DarkGra"+"y;\n}\n\n/*}}}*/\n/***\n!!Summary Section\n***/\n/*{{{*/\n.yourSearchResult .summary {\n\tborder-bottom-width:"+" thin;\n\tborder-bottom-style: solid;\n\tborder-bottom-color: #999999;\n\tpadding-bottom: 4px;\n}\n\n.yourSea"+"rchRange, .yourSearchCount, .yourSearchQuery   {\n\tfont-weight: bold;\n}\n\n.yourSearchResult .summary ."+"button {\n\tfont-size: 10px;\n\n\tpadding-left: 0.3em;\n\tpadding-right: 0.3em;\n}\n\n.yourSearchResult .summa"+"ry .chkBoxLabel {\n\tfont-size: 10px;\n\n\tpadding-right: 0.3em;\n}\n\n/*}}}*/\n/***\n!!Items Area\n***/\n/*{{{*"+"/\n.yourSearchResult .marked {\n\tbackground: none;\n\tfont-weight: bold;\n}\n\n.yourSearchItem {\n\tmargin-to"+"p: 2px;\n}\n\n.yourSearchNumber {\n\tcolor: #808080;\n}\n\n\n.yourSearchTags {\n\tcolor: #008000;\n}\n\n.yourSearc"+"hText {\n\tcolor: #808080;\n\tmargin-bottom: 6px;\n}\n\n/*}}}*/\n/***\n!!Footer\n***/\n/*{{{*/\n.yourSearchFoote"+"r {\n\tmargin-top: 8px;\n\tborder-top-width: thin;\n\tborder-top-style: solid;\n\tborder-top-color: #999999;"+"\n}\n\n.yourSearchFooter a:hover{\n\tbackground: none;\n\tcolor: none;\n}\n/*}}}*/\n/***\n!!Navigation Bar\n***/"+"\n/*{{{*/\n.yourSearchNaviBar a {\n\tfont-size: 16px;\n\tmargin-left: 4px;\n\tmargin-right: 4px;\n\tcolor: bla"+"ck;\n\ttext-decoration: underline;\n}\n\n.yourSearchNaviBar a:hover {\n\tbackground-color: none;\n}\n\n.yourSe"+"archNaviBar .prev {\n\tfont-weight: bold;\n\tcolor: blue;\n}\n\n.yourSearchNaviBar .currentPage {\n\tcolor: #"+"FF0000;\n\tfont-weight: bold;\n\ttext-decoration: none;\n}\n\n.yourSearchNaviBar .next {\n\tfont-weight: bold"+";\n\tcolor: blue;\n}\n/*}}}*/\n";config.shadowTiddlers["YourSearchResultTemplate"]="<!--\n{{{\n-->\n<span macro=\"yourSearch if found\">\n<!-- The Summary Header ============================"+"================ -->\n<table class=\"summary\" border=\"0\" width=\"100%\" cellspacing=\"0\" cellpadding=\"0\">"+"<tbody>\n  <tr>\n\t<td align=\"left\">\n\t\tYourSearch Result <span 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<<tiddler [[include_tiddlers/Zero Divisor.html#"Zero Divisor"]]>>

See also:
* [[Vacuum]]

Papers:
* [[Inertia as a Zero-point-field Lorentz Force (1994) - B. Haisch, A. Rueda, H. E. Puthoff|http://www.ovalecotech.ca/pdfss/Inertia_as_a_Zero-Point-Field_Lorentz_force.pdf]] [[local|papers/Inertia_as_a_Zero-Point-Field_Lorentz_force.pdf]] {{t100Cite{[[pct. 171|http://scholar.google.de/scholar?cites=9082272909610620817&as_sdt=5&sciodt=0&hl=de]]}}}
* [[Ground State of Hydrogen as a Zero-point-fluctuation-determined State (1987) - H. E. Puthoff|http://www.earthtech.org/publications/PRDv35_3266.pdf]] [[local|papers/PRDv35_3266.pdf]] [[pct. 76|http://scholar.google.de/scholar?cites=13355097493390776833&as_sdt=2005&sciodt=0,5&hl=de]]

Documents:
* [[Energy Extraction from the Vacuum Field|http://www.borderlands.de/Links/SPEEVF.pdf]] [[local|documents/SPEEVF.pdf]]

Links:
* [[WIKIPEDIA - Zero-point Energy|http://en.wikipedia.org/wiki/Zero-point_energy]]

Videos:
* [[Indirect Extraction of Zero-Point Energy from the Quantum Vacuum: Patent 7,379,286 - G. Haisch, B. Moddel|http://www.youtube.com/watch?v=K5IAugkmNso]]

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