Trajectory of the Universe - - A Mathematics, Physics and Philosophy NotebookThe ordinary man wonders at marvellous things; the wise man wonders at the commonplace.

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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-(15,7,3) Design'' is a [[Hadamard 2-Design|Hadamard Design]]. There are five nonisomorphic such designs $D_i$, $i = 1,\ldots, 5$, with full [[automorphism groups|Automorphism]] of order $20.160$, $576$, $96$, $168$ and $168$ respectively.
Hence the number of isomorphic but distinguished $2-(15,7,3)$ designs is
\[
15!\sum_{i=1}^5 \frac{1}{\operatorname{ord}(Aut(D_i))} = 64.864.800 + 2.270.268.000 + 13.621.608.000 + 2\cdot 7.783.776.000  = 31.524.292.800
\]
(Notice, that there also exist $5$ nonisomorphic [[Hadamard-matrices|Hadamard Matrix]] in dimension $16$ with automorphism groups having orders $10.321.920$, $294.912$, $49.152$, $86.016$ and $86.016$ respectively. The orders of the automorphism groups of the designs divide the orders of the automorphism groups of the Hadamard matrices resulting always in the same value $512$. However MAGMA prefers to establish another relationship between the automorphism groups of designs and Hadamard matrices (for details see example below)).

The five $2-(15, 7, 3)$-designs have $1$, $2$, $3$, $2$ and $2$ orbits respectively.

Also see: [[2-(31,15,7) design|2-(31,15,7) Design]].

!!!![[MAGMA|http://magma.maths.usyd.edu.au/calc/]]^^[[Help|MAGMA]]^^ examples
{{{
// Design derived from Hadamard matrix with largest automorphism group by taking +1's as incidences.
K := Design< 2, 15 |
{2,4,7,8,11,13,14},
{3,4,5,8,9,14,15},
{1,4,6,8,10,13,15},
{5,6,7,8,9,10,11},
{1,3,7,8,10,12,14},
{1,2,5,8,11,12,15},
{2,3,6,8,9,12,13},
{9,10,11,12,13,14,15},
{1,3,5,6,11,13,14},
{1,2,6,7,9,14,15},
{2,3,5,7,10,13,15},
{1,2,3,4,9,10,11},
{2,4,5,6,10,12,14},
{3,4,6,7,11,12,15},
{1,4,5,7,9,12,13}>;
AutomorphismGroup(K);
}}}

Papers:
* [[Doubles of Hadamard 2-(15,7,3) Designs - Z. Mateva|http://www.moi.math.bas.bg/acct2008/b36.pdf]] pct. 0
* [[A Method for Construction of Blocks of PG2 (3, 2) and PG3 (4, 2) - V. Mudrinski|http://www.emis.de/journals/NSJOM/Papers/24_2/NSJOM_24_2_095_099.pdf]] pct. 0

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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[[Quantization]] can spoil classical symmetries. As a consequence, symmetry currents, whose classical conservation is assured by [[Noether's theorem|Noether Theorem]], cease to be conserved. Such currents are called ''anomalous''. They possess an anomalous divergence, and the coupling of gauge fields to this current becomes problematical.
The problem afflicts:
* Continuous chiral symmetries in any even-dimensional space-time
* Gravitational symmetries of massless (Weyl) fermions  in space-times with dimensionality $4k + 2, k = 0,1,2,...$
* Discrete symmetries (P,T) in odd dimensions
* Scale/conformal symmetries in any dimension: ''Weyl anomaly'' (also called ''Trace Anomaly'' or ''Conformal Anomaly''), that is, the breakdown of conformal invariance upon quantization. Classically, this invariance leads to the vanishing of the energy–momentum tensor, while its breakdown in the quantum theory leads to a nonvanishing value.
The mathematical connection has come to a sharper focus in the characterization of an anomalous gauge theory by the fact that commutators of gauge transformation generators are anomalous and do not follow the Lie algebra of the gauge group.
For the gauge fields one has
\[
D_\mu D_\nu G^{\mu\nu} = D_\mu J^\mu \propto  G_{\mu\nu} \tilde{G}^{\mu\nu} \ne 0 = \text{ anomaly}
\]
with
\[
G_{\mu\nu}\tilde{G}_{\mu\nu} = \partial_\mu K_\mu
\]
and
\[
K_\mu = 2\epsilon_{\mu\nu\alpha\beta} \left( A_\nu \partial_\alpha A_\beta + \frac{2}{3} i g A_\nu A_\alpha A_\beta \right)
\]

Papers:
* [[Twenty Years of the Weyl Anomaly - M. J. Duff|http://scholar.google.de/scholar?hl=de&lr=&cites=2182335914955876668]] {{t100Cite{[[pct. 183|http://scholar.google.de/scholar?hl=de&lr=&cites=2182335914955876668]]}}}
* [[Topological Invariants, Instantons and Chiral Anomaly on Spaces with Torsion - O. Chandia, J. Zanellia|http://arxiv.org/PS_cache/hep-th/pdf/9702/9702025v1.pdf]] [[pct. 66|http://scholar.google.de/scholar?hl=de&lr=&cites=13079915837245870961]]
* [[On the Chiral Anomaly in Non-Riemannian Spacetimes - Y. N. Obukhov, E. W. Mielke, J. Budczies, F. W. Hehl|http://www.citebase.org/fulltext?format=application/pdf&identifier=oai:arXiv.org:gr-qc/9702011]] [[pct. 28|http://scholar.google.de/scholar?hl=de&lr=&cites=215116949989095649]]
* [[Non-abelian Chiral Anomalies and Wess-Zumino Effective Actions - J. L. Petersen|http://th-www.if.uj.edu.pl/acta/vol16/pdf/v16p0271.pdf]] [[pct. 21|http://scholar.google.de/scholar?hl=de&lr=&cites=16546363659818067930]]
* [[The Axial Anomaly Revisited - P. Federbush|http://www.ma.utexas.edu/mp_arc/c/96/96-316.ps.gz]] [[pct.9|http://scholar.google.de/scholar?hl=de&lr=&cites=5532133868412867919]]
* [[What’s Wrong with Anomalous Chiral Gauge Theory? - T. D. Kieu|http://psroc.phys.ntu.edu.tw/cjp/download.php?d=1&pid=834]] [[pct. 6|http://scholar.google.de/scholar?hl=de&lr=&cites=10696231753488158395]]
* [[Effective Action, Conformal Anomaly and the Issue of Quadratic Divergences - K. A. Meissner, H. Nicolai|http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.2840v2.pdf]] [[pct. 6|http://scholar.google.de/scholar?hl=de&lr=&cites=13813838645528182828]]
* [[Algebraic versus Topologic Anomalies - V. Aldaya, M. Calixto, J. Guerrero|http://repositorio.bib.upct.es/dspace/bitstream/10317/519/6/avt.pdf]] pct. 0
Links:
* [[Schoolarpedia - Axial anomaly|http://www.scholarpedia.org/article/Axial_anomaly]]

> 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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<<tiddler [[include_tiddlers/Antibracket Formalism.html#"Antibracket Formalism"]]>>

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

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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]]

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

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

<<tiddler [[include_tiddlers/Association Type Identities.html#"Association Type Identities"]]>>

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''
\[
[\sum_i \lambda_i \mb A_i,\sum_j  \mu_j \mb B_j,\sum_k  \nu_k \mb C_k ] = \sum_{i,j,k} \lambda_i  \mu_j  \nu_k \mb [ \mb A_i,\mb B_j, \mb C_k]
\]
@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/Albert.jpg][Comments]]}}}@@

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

<<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]]

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

<<tiddler [[include_tiddlers/Atom Laser.html#"Atom Laser"]]>>

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

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

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

<<tiddler [[include_tiddlers/Axial Torsion.html#"Axial Torsion"]]>>

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]]

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

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

<<tiddler [[include_tiddlers/BTZ Black Hole.html#"BTZ Black Hole"]]>>

The ''Baby Monster $B$'' is the second largest [[sporadic simple group|Sporadic Group]] and has the order $2^{41}\cdot 3^{13}\cdot 5^6 \cdot 7^2 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 31 \cdot 47$.

It was discovered by B. Fischer in 1973 (unpublished) and a computer proof of its existence and uniqueness was given by J. S. Leon and C. C. Sims in 1977. An independent computer free construction was given by [[R. Griess|People]] during his construction of the [[Monster group|Monster Group]].

The direct product of the Baby Monster and a cyclic group of order $2$ is the finite automorphism group of the so called ''Shorter [[Moonshine module|Monstrous Moonshine]]'', denoted $VB^\natural$, which is a [[vertex operator superalgebra|Vertex Operator Algebra]] with central charge $23 \frac 12$.
$B$ itself acts as automorphism group on the subspace of $VB^\natural$ spanned by the vectors of conformal weight $2$.

Papers:
* [[The Group of Symmetries of the Shorter Moonshine Module - G. Höhn|http://arxiv.org/PS_cache/math/pdf/0210/0210076v1.pdf]] [[pct. 9|http://scholar.google.de/scholar?cites=3398396966836677972&hl=de&as_sdt=2000]]

''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/Baker-Campbell-Hausdorff Formula.html#"Baker-Campbell-Hausdorff Formula"]]>>

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

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

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]]

<<tiddler [[include_tiddlers/Big Bang.html#"Big Bang"]]>>

<<tiddler [[include_tiddlers/Big Bounce.html#"Big Bounce"]]>>

The ''"Big Desert" Hypothesis'' asserts that, apart from the [[Higgs boson|Higgs Mechanism]], all particles relevant at the [[grand unification|GUT]] scale are already discovered.

* 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"]]>>

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

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

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

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]]

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

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

<<tiddler [[include_tiddlers/Biological Cell.html#"Biological Cell"]]>>

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

<<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]]

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

<<tiddler [[include_tiddlers/Black Hole.html#"Black Hole"]]>>

<<tiddler [[include_tiddlers/Black Holes at the LHC.html#"Black Holes at the LHC"]]>>

<<tiddler [[include_tiddlers/Blaschke Conjecture.html#"Blaschke Conjecture"]]>>

>The objective world simply is, it does not happen.
> - Hermann Weyl -

The ''Block Universe'' view of the universe affords equal (ontological) status to all points in space-time, thus regarding temporality as an illusory human construct with no reference to reality.

This view may have come about as a consequence of the usual way of modelling the mathematics of general relativity as a theory about the curvature of an eternally existing arena of space-time.

Papers:
* [[How Time Passes (2003) - G. Franck|http://www.iemar.tuwien.ac.at/publications/GF_2003c.pdf]] [[local|papers/GF_2003c.pdf]] [[pct. 2|http://scholar.google.com/scholar?hl=de&lr=&cites=9298222414182727390&um=1&ie=UTF-8&ei=fRPCSo39GZXsmwPO-qCxBg&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]

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

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

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]]

<<tiddler [[include_tiddlers/Bose-Einstein Condensate.html#"Bose-Einstein Condensate"]]>>

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

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

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

<<tiddler [[include_tiddlers/Boundary Condition.html#"Boundary Condition"]]>>

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> To look at our brain as 100 billion simple switches - to look at a neuron as a switch or gate - it's an insult to neurons. It's just not that simple.
> - Stuart Hameroff -

The human ''Brain'' consists of roughly $10^{11}$ neurons. (Which is about the number of stars in a typical galaxy).

It consumes around $10\,\rm W$ of energy. Hence the the upper limit on the processing rate, invoking [[Landauer's principle|Landauer's Principle]], is $\sim 10^{21} \,\rm s^{-1}$, or an average of $\sim 10^{10} \, \rm s^{-1}$ per neuron.

Most estimates of the actual processing power of the human brain lie somewhere in the range of $10^{12}\, \rm s^{-1}$ and $10^{18}\, \rm s^{-1}$ and are based on the assumption that a single neuron more or less acts like a "switch". Contrary to this, Stuart Hameroff claims that an individual neuron is capable of carrying out about $10^{17}$ operations per second, leading to an estimate of the overall processing power of the human brain of about $10^{28} \, \rm s^{-1}$ [1]. But this is in conflict with Landauer's principle, taking into account the energy consumption of the human brain.
There seem to be at least three ways out to save the conjecture:
* Not all neurons operate at their limit at the same time, leading to a processing power which de facto is only about $10^{21} \,\rm s^{-1}$ or less on the average. <br><br>
* A large fraction of the computations are reversible. However this implies that they cannot be classical but rather have to be quantum mechanical. Invoking the [[Orch-OR model|Orch-OR Model]] which is based on computations taking place in microtubili, this seems to be justifiable. <br><br>
* The most speculative one is that the [[generalized second law of thermodynamics|Generalized Second Law of Thermodynamics]] needs to be considered, as in the case of neurons/[[cells|Biological Cell]] and consequently brains it leads to non-negligible effects. That is, instead of [[entropy|Entropy]] production merely in the form of classical heat, the bulk of it is produced via horizon states. The question then is, what are these horizons relevant for the functioning of a brain ? In fact, every observer possesses an [[individual|Apparent Horizon]] [[cosmic|Cosmic Horizon]] [[de Sitter|De Sitter Space]] past horizon which could serve as an "extended heat reservoir". More concretely, the [[dark energy|Dark Energy]] modes derived from it could do the job. (Interestingly, Hameroff speaks of "global states" related to [[consciousness|Consciousness]] which one could identify with states of the cosmic horizon, indicating that consciousness could be a global/[[holographic|Holographic Principle]] phenomenon. Moreover this hints to a possible connection between the [[information paradox|Information Loss Paradox]] related with event horizons and the elusive nature of consciousness).

Neuronal responses to environmentally driven demands account for less than 5% of the brain's energy budget, leaving the majority devoted to intrinsic neuronal signalling.

Intrinsic neuronal activity manifests itself as spontaneous fluctuations in the blood oxygen level-dependent functional MRI (fMRI) signal and exhibits synchrony within neuroanatomically and functionally related brain regions.


Papers:
* [[The Cost of Cortical Computation (2003) - P. Lennie|http://www.cns.nyu.edu/~pl/pubs/Lennie03a.pdf]] [[local|papers/Lennie03a.pdf]] {{t100Cite{[[pct. 259|http://scholar.google.de/scholar?hl=de&lr=&cites=1415747101904820378&um=1&ie=UTF-8&ei=0aiRTtnGO4zs-gb04smABQ&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCgQzgIwAA]]}}}
* [[Single Neuron Activity in Human Hippocampus and Amygdala during Recognition of Faces and Objects (1997) - I. Fried, K. A. MacDonald, C. L. Wilson|http://www.cns.nyu.edu/~wendy/class/2006sp/reading6/Fried_etal_Neuron_1997.pdf]] [[local|papers/Fried_etal_Neuron_1997.pdf]] {{t100Cite{[[pct. 211|http://scholar.google.de/scholar?hl=de&lr=&cites=10406762603195009116&um=1&ie=UTF-8&ei=HUaDTu_eEYyb1AXhgumuAQ&sa=X&oi=science_links&ct=sl-citedby&resnum=4&ved=0CEEQzgIwAw]]}}}
* [[When will Computer Hardware Match the Human Brain (1998) - H. Moravec|http://www.jetpress.org/volume1/moravec.pdf]] [[local|papers/moravec.pdf]] {{t100Cite{[[pct. 131|http://scholar.google.de/scholar?cites=3744062543108548637&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Disease and the Brain’s Dark Energy (2010) - D. Zhang, M. E. Raichle|http://www.sciencenet.cn/upload/blog/file/2010/3/2010329101048569788.pdf]] [[local|papers/2010329101048569788.pdf]] [[pct. 40|http://scholar.google.de/scholar?hl=de&lr=&cites=15899758176548449977&um=1&ie=UTF-8&ei=g3VoTqjdBfLZ4QT-9PXcDA&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB8QzgIwAA]]

Magazines:
* [[The Brain's Dark Energy (2010) - M. E. Raichle|http://www.braininnovations.nl/Dark-Energy.pdf]] [[local|magazines/Dark-Energy.pdf]] [[mct. 22|http://scholar.google.de/scholar?cites=11420529491091438792&as_sdt=2005&sciodt=0,5&hl=de]]

Links:
* [[WIKIPEDIA - Brain|http://en.wikipedia.org/wiki/Human_brain]]
* [[WIKIPEDIA - Grandmother Cell|http://en.wikipedia.org/wiki/Grandmother_cell]]
* [[Energy Limits to the Computational Power of the Human Brain (1989) - R. C. Merkle|http://www.merkle.com/brainLimits.html]]
* [[20 Petaflops: New Supercomputer for Oak Ridge Facility to Regain Speed Lead Over the Chinese (2011)|http://www.physorg.com/news/2011-03-petaflops-supercomputer-oak-ridge-facility.html]]

Google books:
* [[Quantifying Human Information Processing (2005) - D. K. McBride, D. Schmorrow|http://books.google.de/books?hl=de&lr=&id=GBVgjK2-Xa4C&oi=fnd&pg=PA1&dq=%22Quantifying+human+information+processing%22&ots=P90L3N_oK8&sig=cHj1QhylvE9rotaT9GpIgHoL1y0#v=onepage&q&f=false]] [[bct. 1|http://scholar.google.de/scholar?cites=10216773155019288992&as_sdt=2005&sciodt=0,5&hl=de]]

Videos:
* [[[1] Google Tech Talks - A New Marriage of Brain and Computer (2007) - S. Hameroff|http://video.google.com/videoplay?docid=-2069501759514424839]]
* [[Supercomputing the Brain's Secrets - H. Markram|http://www.youtube.com/user/TEDtalksDirector#p/u/91/LS3wMC2BpxU]]  [[local|videos/Henry Markram Supercomputing the brain's secrets.wmv]]
* [[Stephen Wiltshire: The Human Camera|http://www.youtube.com/watch?v=a8YXZTlwTAU]]
* [[Karl Pribram: The Holographic Brain|http://www.youtube.com/watch?v=vHpTYs6GJhQ&feature=related]]

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

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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. $\operatorname {card}(\{2, 4, 6\}) = 3$.

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<<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

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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.

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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]].

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"]]>>

Papers:
* [[Space-Time Structure of Weak and Electromagnetic Interactions - D. Hestenes|http://geocalc.clas.asu.edu/pdf-preAdobe8/ST&EW.pdf]] [[pct. 34|http://scholar.google.de/scholar?cites=14346070686986909264&hl=de&as_sdt=2000]]
* [[Gauge Gravity and Electroweak Theory - D. Hestenes|http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.0060v1.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=18168197432238270851&hl=de&as_sdt=2000]]
* [[The Glashow-Salam-Weinberg Electroweak Theory in the Real Algebra of Spacetime - R. Boudet|http://clifford-algebras.org/v7/v7%28S%29/BOUDET95.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=7818588189175864172&hl=de&as_sdt=2000]]
* [[Electroweak Theory - A. Lewis|http://cosmologist.info/notes/old/electroweak.ps]] pct. 0
* [[Electroweak Fields and Duality in the STA-Dirac Equation - F.M.C. Witte|http://www.phys.uu.nl/~witte/dualitySTA.pdf]] pct. 0

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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"]]>>

Papers:
* [[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]]}}}
* [[Complexified Gravity in Noncommutative Spaces (2000) - A. H. Chamseddine|http://arxiv.org/abs/hep-th/0005222]] [[local|papers/0005222v2.pdf]] [[pct. 97|http://scholar.google.com/scholar?hl=de&lr=&cites=16330489882272139813&um=1&ie=UTF-8&ei=_uP9TsypJpHMsgbwkbnXDw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCQQzgIwAA]]
* [[Deformed Reissner-Nordstrom Solutions in Noncommutative Gravity (2007) - P. Mukherjee, A. Saha|http://arxiv.org/abs/0710.5847]] [[local|papers/0710.5847v3.pdf]] [[pct. 26|http://scholar.google.com/scholar?hl=de&lr=&cites=7552024664768814267&um=1&ie=UTF-8&ei=7Wf8TtnHLsLfsgarlrjSDw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB8QzgIwAA]]
* [[Construction of Space-Time from Linear Connections (1990) - P. K. Smrz|http://www.actaphys.uj.edu.pl/vol21/pdf/v21p0603.pdf]] [[local|papers/v21p0603.pdf]] pct. 0

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 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]]

A group $\mathcal G$ can be partitioned into ''Conjugacy Classes'' $C(\mb  X_i)$, $i = 1, \ldots, N$, which are formed by elements $\mb X_i \in \mathcal G$. The conjugacy classes are defined by
\begin{equation}
C(\mb X_i) = \{\mb X \in \mathcal G: \mb X = \mb A^{-1} \mb X_i \mb A\}
\end{equation}
with $\mb A$ any element of $\mathcal G$. The map $\mb X \mapsto \mb A^{-1} \mb X \mb A$ is called ''Conjugacy Map'', ''Conjugation'' or ''Similarity Transformation''. Elements in a conjugacy class said to be ''conjugate'' to one another. The operation of conjugation is an equivalence relation. Therefore every element of $\mb A \in \mathcal G$ is contained in exactly one conjugacy class.

Remark: The notation used here is somewhat unusual, however it is intended to point out the analogies with the [[X-product|X-Product]].

The importance of the notion of conjugation lies in the fact that it is an [[isomorphism|Homomorphism]] called an ''Inner Automorphism''. I.e. conjugation respects the multiplication, since $\mb A^{?1}(\mb X_i \mb X_j)\mb A = (\mb A^{?1}\mb X_i \mb A)(\mb A^{?1} \mb X_j \mb A)$. Moreover, if $\mb A^{?1}\mb X_i \mb A = \mb 1$, then $\mb X_i = \mb A\mb A^{?1} = \mb 1$, hence only the identity is mapped to the identity.

The conjugation $ \mb X = \mb A^{-1} \mb X_i \mb A$ is equivalent to $\mb{AX} = \mb X_i \mb A$. Therefore the existence and number of [[inner automorphisms|Automorphism]] that are not the identity mapping is a kind of measure of the failure of the commutative law in the group.
Consequently conjugacy classes play an important role in the classification of non-commutative groups.

!!!!Generalizations
If the underlying algebraic structure is non-associative (and hence not a group), conjugation doesn't necessarily yield automorphisms any more (e.g. in the [[octonions|Octonion]]). This is due to fact that to prove the isomorphism property (see above) one needs to do rebracketings.

In this case the following generalization might be of interest:
The condition for associativity is:
\begin{equation}
(\mb A \mb X_i) \mb B = \mb A (\mb X_i \mb B)
\end{equation}
If we assume that the [[left-and right inverse properties|Inverse Properties]] still hold, this can be rewritten as:
\begin{equation}
\mb X_i = \mb A^{-1}((\mb A((\mb X_i) \mb B)) \mb B^{-1})
\end{equation}
This can be regarded as the non-associative analog to the conventional conjugation
\begin{equation}
\mb A   = \mb B^{-1}\mb A \mb B
\end{equation}
!!!!Example
The symmetric group $S_3$, consisting of 6 permutations of three elements, has the following 3 conjugacy classes:
\begin{eqnarray}
C (\mb{ABC} \rightarrow \mb{ABC}) & =& \{(\mb{ABC} \rightarrow \mb{ABC}) \} \quad && \text{Identity} \\
C (\mb{ABC} \rightarrow \mb{ACB}) &= & \{(\mb{ABC} \rightarrow \mb{ACB}), (\mb{ABC} \rightarrow \mb{CBA}), (\mb{ABC} \rightarrow \mb{BAC}) \} \quad && \text{Interchanges} \\
C (\mb{ABC} \rightarrow \mb{BCD}) &= & \{(\mb{ABC} \rightarrow \mb{BCD}), (\mb{ABC} \rightarrow \mb{CAB}) \} \quad && \text{Cyclic permutations}
\end{eqnarray}

Lectures:
* [[Group Theory applied to Crystallography - B. Souvignier|http://www.crystallography.fr/mathcryst/pdf/Gargnano/Souvignier_Gargnano_text.pdf]]

<<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"]]>>

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.

/***
|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]]

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 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 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-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]]

> One has to find a possibility to avoid the continuum (together with space and time) altogether. But I have not the slightest idea what kind of elementary concepts could be used in such a theory.
> - Letter from Albert Einstein to David Bohm October 28, 1954

Papers:
* [[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=12701785254001423165&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Quantization of Discrete Deterministic Theories by Hilbert Space (1990) - G. 't Hooft|http://igitur-archive.library.uu.nl/phys/2005-0622-153937/14765.pdf]] [[local|papers/14765.pdf]] [[pct. 36|http://scholar.google.de/scholar?hl=de&lr=&cites=11394643065170924514&um=1&ie=UTF-8&ei=lfT1TISZBcH5sgaVnfHiBA&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCYQzgIwAA]]
* [[Noncommutativity and Discrete Physics (1998) - L. H. Kauffman|http://www2.math.uic.edu/~kauffman/NCDP.pdf]] [[local|papers/NCDP.pdf]] [[pct. 14|http://scholar.google.de/scholar?cites=3342187083863786167&hl=de]]
* [[Scale Free Small World Networks and the Structure of Quantum Space-Time (2003) - M. Requardt|http://arxiv.org/pdf/gr-qc/0308089v1]] [[local|papers/0308089v1.pdf]] [[pct. 6|http://scholar.google.de/scholar?hl=de&lr=&cites=508496736609555692&um=1&ie=UTF-8&ei=3iEoTdu_JoWb8QOuwIn5Ag&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCYQzgIwAA]]
* [[Three Possible Implications of Spacetime Discreteness - S. Gao|http://philpapers.org/archive/GAOTIO.1.pdf]] [[local|papers/GAOTIO.1.pdf]] pct. 0

Abstracts:
* [[Quantum Computation and Combinatorial Spacetime - D. Madina|http://www.qci.jst.go.jp/eqis02/program/abstract/poster36.pdf]] [[local|abstracts/poster36.pdf]]

Journals:
* [[Can Time be a Discrete Dynamical Variable? (1992) - T. D. Lee|journals/DiscreteTime.djvu]] {{t100Cite{[[jct. 100|http://scholar.google.de/scholar?cites=5111132648460527022&as_sdt=2005&sciodt=2000&hl=de]]}}} TRD

Videos:
* [[New Discovery about the Fabric of Space-Time|http://www.youtube.com/watch?v=nByekIx7XXw&feature=related]]

See also:
* [[World crystal|World Crystal]]
* [[Spin networks|Spin Network]]
* [[Causal dynamical triangulation|Causal Dynamical Triangulation]]
* [[Spacetime entropy|Spacetime Entropy]]

''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]]

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.

''E\-Infinity Theory'' assumes that [[spacetime is fractal|Fractal Spacetime]] in nature and its description is based on [[Cantorian sets|Cantor Set]].

Because some dimensions are much smaller than others, the space in which the so-called Cantorian spacetime theory is formulated is hierarchical and consequently, viewed from afar it appears as if it were continuous, diﬀerentiable and of a ﬁnite dimensionality.

This space is called E-inﬁnity and it possesses several kinds of dimensions: Formally speaking it is infinite dimensional, but the expectation value of its Hausdorﬀ dimension is exactly four plus the Golden Mean to the power of three, while its topological dimension is exactly four.

Latest Fermilab experiments with the Tevatron indicate that the precise prediction of the [[Higgs|Higgs Mechanism]]-mass of $161.8033989$ \GeV by E-infinity theory can be excluded [1].

Papers:
* [[A Review of E Infinity Theory and the Mass Spectrum of High Energy Particle Physics (2004) - M.S. El Naschie|http://www.complexity.ru/papers/science25.pdf]] [[local|papers/science25.pdf]] {{t100Cite{ [[pct. 331|http://scholar.google.de/scholar?cites=14121921044845368187&hl=de]]}}}
* [[The VAK of Vacuum Fluctuation, Spontaneous Self-Organization and Complexity Theory Interpretation of High Energy Particle Physics and the Mass Spectrum (2003) - M.S. El Naschie|http://www.el-naschie.net/bilder/file/7.%20The%20VAK%20of%20vacuum%20fluctuation,%20spontaneous.pdf]] [[local|papers/7. The VAK of vacuum fluctuation, spontaneous.pdf]] [[pct. 41|http://scholar.google.de/scholar?cites=16744730194865314043&hl=de]]
* [[On John Nash's Crumpled Surface (2003) - M. S. El Naschie|http://www.scribd.com/doc/35388639/On-John-Nash-s-crumpled-surface]] [[pct. 9|http://scholar.google.com/scholar?hl=de&lr=&cites=14772261508872687800&um=1&ie=UTF-8&ei=N3RWTY2nOJHAtAaGxrGlCw&sa=X&oi=science_links&ct=sl-citedby&resnum=2&ved=0CCgQzgIwAQ]]
* [[From Arthur Cayley via Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan, Emmy Noether and Superstrings to Cantorian Space-Time (2008) - L. Marek-Crnjac|http://www.el-naschie.net/bilder/file/Crnjac_From_Arthur_Cayley.pdf]] [[local|papers/Crnjac_From_Arthur_Cayley.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=10159834154703874053&hl=de]]
* [[Exceptional Lie Groups, E-infinity Theory and Higgs Boson (2007) - A. A. El-Okaby|http://arxiv.org/ftp/arxiv/papers/0709/0709.2394.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=16105425813544884648&hl=de]]

Links:
* [[[1] Fermilab Experiments Constrain Higgs Mass|http://www.fnal.gov/pub/presspass/press_releases/Higgs-mass-constraints-20090313.html]]
* [[RATIONALWIKI - Mohamed El Naschie|http://rationalwiki.org/wiki/Mohamed_El_Naschie]]

''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"]]>>

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 Space.html#"Einstein Space"]]>>

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

The ''Einstein\-Hilbert Action'' is given by:
\[
S[\mb g] = \int \, \mathrm{d}^4 x \sqrt{-\det(\mb g)} \left (  {1 \over 2\kappa} \, R + \mathcal{L}_\mathrm{M} \right)
\]
with the constant $\kappa = \frac{8 \pi G}{c^4}$, $\mathcal{L}_\mathrm{M}$ an arbitrary matter Lagrangian and $g$ the determinant of the [[metric tensor|Metric Tensor]] and $R$ the [[Ricci scalar|Ricci Scalar]].

The Einstein\-Hilbert action is of second-order since $R$ contains second derivatives of $g_{\mu\nu}$.
Strictly speaking the variation of the action functional with respect to $g_{\mu\nu}$ does not yield the [[Einstein equations|Einstein Field Equations]] as an additional surface integral shows up. This seems to pose a potential problem, but it can be handled by simply adding to the action a boundary term which (upon variation) exactly cancels the surface integral. This boundary term is also known as [[Gibbons-Hawking-York boundary term|Gibbons-Hawking-York Boundary Term]].

Alternatively one may revise the variational principle so that the metric and the connection are varied independently ([[Palatini principle|Palatini Principle]]). Thus the functional is to be replaced according to $S[\mb g] \rightarrow S[\mb g, \bs \Gamma]$.
@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/Hilbert.jpg][Comments]]}}}@@

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

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

Some conversions:
\begin{eqnarray}
1~\mathrm{eV} &=& 1{,}602\,176\,487(40)\cdot 10^{-19}~\mathrm J \\
                       &\approx & 1,783·10^{−36}~\mathrm {kg}
\end{eqnarray}
Links:
* [[WIKIPEDIA - Elektronvolt|http://de.wikipedia.org/wiki/Elektronvolt#Verwendung]]

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/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]]

''F4'' is the second smallest of the $5$ exceptional [[Lie groups|Lie Group]] with dimension $52$. Its compact form is the [[automorphism group|Automorphism]] of the [[Albert algebra|Albert Algebra]], the biggest exceptional [[Jordan Algebra|Jordan Algebra]] $\mathfrak{H}$${}_3 (\mathbb {\tilde O})$.
It furthermore is the group of [[isometries|Isometry]] of the [[octonionic projective plane|Octonionic Projective Plane]].
$F_4$ has $52$ roots ($48$ of them non-negative) which can be represented by means of [[Hurwitz integers|Hurwitz Integer]].

Papers:
* [[Quaternionic Root Systems and Subgroups of the Aut(F4) - M. Koca, M. Al-Barwani|http://arxiv.org/PS_cache/hep-th/pdf/0510/0510172v2.pdf]] [[local|papers/0510172v2.pdf]]  [[pct. 2|http://scholar.google.de/scholar?cites=16836134822806176296&hl=de]]

<<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"]]>>

!!!!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"]]>>

Arguments against ''Fine Funing'':
* David Gross pointed out that [[quantum chromodynamics|QCD]] is fixed, complete, and not tunable, and it just so happens that the carbon level is there, Hoyle or no Hoyle.  <br><br>
* The description of the [[vacuum|Vacuum]] by means of [[chaotic quantization|Chaotic Quantization]] demonstrates that at least it is conceivable that fundamental parameters can be calculated from first principles and are thus not any numbers (as is the case for instance for $\pi$).

See also:
* [[Organic universe|Organic Universe]]
* [[Anthropic principle|Anthropic Principle]]
* [[Intelligent design|Intelligent Design]]

Papers:
* [[Why the Universe is just so (2000) - C. J. Hogan|http://llacolen.ciencias.uchile.cl/~vmunoz/download/papers/h00.pdf]] [[local|papers/h00.pdf]] {{t100Cite{[[pct. 111|http://scholar.google.de/scholar?cites=2011791294939170025&as_sdt=2005&sciodt=2000&hl=de]]}}}
* [[Evidence Against Fine Tuning for Life (2011) - D. N. Page|http://arxiv.org/PS_cache/arxiv/pdf/1101/1101.2444v1.pdf]] [[local|papers/1101.2444v1.pdf]] pct. 0

Links:
* [[WIKIPEDIA - Feinabstimmung der Naturkonstanten|http://de.wikipedia.org/wiki/Feinabstimmung_der_Naturkonstanten]]

Videos:
* [[Alpha-Centauri: Was ist die Beryllium-Barriere?|http://www.br-online.de/br-alpha/alpha-centauri/alpha-centauri-beryllium-barriere-2005-ID1207917317644.xml]]

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"]]>>

<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/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]]

The ''(Dirac) Gamma Matrices'' $\bs \gamma_\mu$ form a [[Clifford algebra|Clifford Algebra]], i.e. they satisfy
\begin{equation}
\{\bs{\gamma}_\mu,\bs{\gamma}_\nu\} = 2 \eta_{\mu\nu} \bs{1}
\end{equation}
''Dirac representation''
\begin{equation}
\bs \gamma_{0}=\left(\begin{matrix}\bs 1 & \bs 0\\
\bs 0 & -\bs 1\end{matrix}\right),\quad \bs \gamma_{m}=\left(\begin{matrix}\bs{0} & \bs \sigma_{m}\\
-\bs \sigma_{m} & \bs 0 \end{matrix}\right) \quad m= 1,2,3
\end{equation}
with $\bs\sigma_{m}$ the [[Pauli matrices|Pauli Matrices]].

Explicitely this reads
\begin{equation}
\bs \gamma_{0} = \begin{pmatrix}\mb 1 &\mb  0 & \mb  0 & \mb 0\\
\mb 0 & \mb 1 &\mb  0 &\mb  0 \\
\mb 0 & \mb 0 &- \mb 1 &\mb  0\\
\mb 0 & \mb 0 & \mb 0 &-\mb 1\end{pmatrix}
\end{equation}
\begin{eqnarray}
\bs \gamma_{1} =  \begin{pmatrix}\mb  0 & \mb  0 &\mb  0 & \mb  1 \\
\mb 0 & \mb 0 & \mb 1 &\mb  0\\
\mb 0 &-\mb 1 &\mb  0 &\mb  0\\
\mb -1 & \mb 0 & \mb 0 & \mb 0\end{pmatrix}, \quad

\bs \gamma_{2} &= & \begin{pmatrix}\mb 0 & \mb 0 & \mb 0 &-\mb i\\
\mb 0 & \mb 0 & \mb i & \mb 0\\
\mb 0 &\mb  i & \mb 0 & \mb 0\\
-\mb i & \mb 0 &\mb  0 &\mb  0\end{pmatrix}, \quad

\bs \gamma_{3}= \begin{pmatrix}\mb 0 & \mb 0 &\mb  1 & \mb 0\\
\mb 0 & \mb 0 &\mb  0 &-\mb  1\\
-\mb 1 & \mb 0 & \mb 0 &\mb  0\\
\mb 0 & \mb 1 &\mb  0 & \mb  0\end{pmatrix}
\end{eqnarray}
furthermore one defines a matrix by means the product of all the 4 matrices above
\begin{equation}
\bs \gamma_{5}= i  \bs \gamma_{0} \bs \gamma_{1} \bs \gamma_{2} \bs \gamma_{3}
\end{equation}
''Weyl (chiral) representation''
\begin{equation}
\bs{\gamma}_{0}= -i \left(\begin{matrix}\mb 0 & \mb 1\\
\mb 1 & \mb 0\end{matrix}\right),\quad\bs{\gamma}_{m}=\left(\begin{matrix}\mb 0 & \bs{\sigma}_{m}\\
-\bs{\sigma}_{m} & \mb 0\end{matrix}\right) \quad m= 1,2,3
\end{equation}
or explicitely
\begin{equation}
\bs \gamma_{0} = \begin{pmatrix}\mb 1 &\mb  0 & \mb  0 & \mb 0\\
\mb 0 & \mb 1 &\mb  0 &\mb  0 \\
\mb 0 & \mb 0 &- \mb 1 &\mb  0\\
\mb 0 & \mb 0 & \mb 0 &-\mb 1\end{pmatrix}
\end{equation}
\begin{eqnarray}
\bs \gamma_{1} =  \begin{pmatrix}\mb  0 & \mb  0 &\mb  0 & \mb  1 \\
\mb 0 & \mb 0 & \mb 1 &\mb  0\\
\mb 0 &-\mb 1 &\mb  0 &\mb  0\\
\mb -1 & \mb 0 & \mb 0 & \mb 0\end{pmatrix}, \quad

\bs \gamma_{2} &= & \begin{pmatrix}\mb 0 & \mb 0 & \mb 0 &-\mb i\\
\mb 0 & \mb 0 & \mb i & \mb 0\\
\mb 0 &\mb  i & \mb 0 & \mb 0\\
-\mb i & \mb 0 &\mb  0 &\mb  0\end{pmatrix}, \quad

\bs \gamma_{3}= \begin{pmatrix}\mb 0 & \mb 0 &\mb  1 & \mb 0\\
\mb 0 & \mb 0 &\mb  0 &-\mb  1\\
-\mb 1 & \mb 0 & \mb 0 &\mb  0\\
\mb 0 & \mb 1 &\mb  0 & \mb  0\end{pmatrix}
\end{eqnarray}

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

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

<<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]]

The ''Gelfand\-Naimark Theorem'' asserts that an arbitrary abstractly defined [[C*-algebra|C*-Algebra]] is isometrically isomorphic to a norm-closed self-adjoint algebra of bounded operators on a complex [[Hilbert space|Hilbert Space]].

This result was proven by Israel Gelfand and Mark Naimark in 1943 and was a significant point in the development of the theory of C${}^*$-algebras since it established the possibility of considering a C${}^*$-algebra as an abstract algebraic entity without reference to particular realizations as an algebra of operators.

It may be seen as quite an astounding result, as the definition of a C${}^*$-algebra neither involves the notion of a locally compact Hausdorff-space nor that of a Hilbert space.

!!!! Generalisations
The Gelfand\-Naimark axioms applied to a general nonassociative unital algebra imply [[alternativity|Alternative Algebra]] and the algebras obtained are [[alternative C*-algebras|Noncommutative Jordan C*-Algebra]].

Papers:
* [[A Gelfand-Neumark Theorem for C*-Alternative Algebras - R. B. Braun|http://www.springerlink.com/content/jp0267109575141m/fulltext.pdf]] [[pct. 11|http://scholar.google.de/scholar?cites=8477055504176070159&as_sdt=2005&sciodt=2000&hl=de]]
* [[New Associative and Nonassociative Gelfand-Naimark Theorems - M. C. García, A. R. Palacios|http://www.springerlink.com/content/j4p7603l68774951/]] [[pct. 3|http://scholar.google.de/scholar?cites=10088134754760090978&as_sdt=2005&sciodt=2000&hl=de]]
@@display:block;text-align:right;[img[My comments ...|images/Gelfand.jpg][Comments]]&nbsp;@@

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"]]>>

When quantizing gauge theories, ''Ghost Fields'' are used to compensate for the effects of the gauge degrees of freedom, so that [[unitarity|Unitarity]] is preserved.

In electrodynamics in the linear gauges, ghosts decouple and can be ignored. In non-abelian gauge theories, convenient gauges generically involve interacting ghosts.
A major step towards an understanding of ghost fields was the introduction of the [[Faddeev-Popov quantization|Quantization]] procedure.

<<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"]]>>

/***
|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/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]]

See:
* [[Propagator]]

Documents:
* [[Green’s Functions in Physics (2003) - M. Baker, S. Sutlief|http://www.phys.washington.edu/users/baker/green.pdf]] [[local|documents/green.pdf]]

>... 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]]

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]]

A ''Hadamard 3-Design'' is a symmetric ''$(4m, 2m, m-1)$''-[[block design|Design]] which is equivalent to a [[Hadamard matrix|Hadamard Matrix]] of order $4m$.

A ''Hadamard 2-Design'' is a symmetric ''$(4m-1, 2m-1,m-1)$''-[[balanced incomplete block design (SBIBD)|Design]] which is again equivalent to a Hadamard matrix of order $4m$. It is the [[derived design|Design]] of a Hadamard $3$-Design $3-(4m, 2m, m?1)$.

It is conjectured that Hadamard designs exist for all integers $m>0$, which is one of the most important unsolved problems in combinatorics. (This problem can equivalently be formulated in terms of the unresolved problem of the general existence of related Hadamard matrices).

!!!! Examples:
Some examples of Hadamard $2$-designs are:
* $m = 1$: $(3,1,0)$
* $m = 2$: $(7,3,1)$, which is related to the [[projective plane|Fano Plane]] $PG(2,2)$.
* $m = 4$: $(15,7,3)$, which is related to the projective space [[PG(3,2)]].
* $m = 6$: $(23,11,5)$
* $m = 8$: $(31,15,7)$

Papers:
* [[Skew Hadamard Designs and Their Codes - J. L. Kim, Patrick Sol|http://www.math.louisville.edu/~jlkim/wcc07_dcc_6.pdf]] [[pct. 7|http://scholar.google.de/scholar?cites=11229682409815488230&hl=de]]
* [[On Affine Designs and Hadamard Designs with Line Spreads - V. C. Mavron, T. P. McDonough, V. D. Tonchev|http://users.aber.ac.uk/tpd/papers/spreads_04_07_01.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=15248201958044827840&hl=de]]
* [[Doubles of Hadamard 2-(15,7,3) Designs - Z. Mateva|http://www.moi.math.bas.bg/acct2008/b36.pdf]] pct. 0

!!!![[SAGE|http://www.sagenb.org/]]^^[[Help|Sage]]^^ examples
{{{
Had = HadamardDesign(31);
Had.is_block_design();
Had.blocks();
Had.incidence_matrix();
AUT = Had.automorphism_group();
AUT;
AUT.order();
}}}

!!!![[MAGMA|http://magma.maths.usyd.edu.au/calc/]]^^[[Help|MAGMA]]^^ examples
{{{
R := MatrixRing(Integers(), 4);
H := R ! [1,1,1,-1, 1,1,-1,1, 1,-1,1,1, -1,1,1,1];
L := TensorProduct(H, -H);
Had := HadamardRowDesign(L, 1);
Had;
Aut := AutomorphismGroup(Had);
Aut;
}}}

<<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"]]>>

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.

A ''Homomorphism'' is a map from one algebraic structure to another of the same type that preserves all the relevant structure; i.e. properties like identity elements, inverse elements, and binary operations.
If an algebraic structure includes more than one operation, homomorphisms are required to preserve each operation.

Types of homomorphisms:
* ''Endomorphism'' - homomorphism from an object to itself.
* ''Monomorphism'' (also sometimes called an extension) - injective homomorphism.
* ''Epimorphism'' - surjective homomorphism.
* [[Isomorphism]].
* [[Automorphism|Automorphism]] - endomorphism which is also an isomorphism.
From the point of view of [[category theory|Category Theory]], a homomorphism is a ''Morphism''.

!!!!Example
For a ring which consists of addition and multiplication a homomorphism $\Phi$ must satisfy:
\begin{eqnarray}
\Phi(A+B)& =& \Phi(A) + \Phi(B) \\
\Phi(AB) &= &\Phi(A)\Phi(B)
\end{eqnarray}
for any element $A$, $B$ of the ring.

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"]]>>

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"]]>>

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]]
[<img( 5% , )[images/meow.gif]]
[<img( 3% , )[images/meow.gif]]
[<img( 2% , )[images/meow.gif]]
[img(  1%+,+)[images/meow.gif]]
{{tagClear{
}}}
<<<
!!!!!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)};
//}}}
//{{{
var f=config.formatters[config.formatters.findByField("name","image")];
f.match="\\[[<>]?[Ii][Mm][Gg](?:\\([^,]*,[^\\)]*\\))?\\[";
f.lookaheadRegExp=/\[([<]?)(>?)[Ii][Mm][Gg](?:\(([^,]*),([^\)]*)\))?\[(?:([^\|\]]+)\|)?([^\[\]\|]+)\](?:\[([^\]]*)\])?\]/mg;
f.handler=function(w) {
	this.lookaheadRegExp.lastIndex = w.matchStart;
	var lookaheadMatch = this.lookaheadRegExp.exec(w.source)
	if(lookaheadMatch && lookaheadMatch.index == w.matchStart) {
		var floatLeft=lookaheadMatch[1];
		var floatRight=lookaheadMatch[2];
		var width=lookaheadMatch[3];
		var height=lookaheadMatch[4];
		var tooltip=lookaheadMatch[5];
		var src=lookaheadMatch[6];
		var link=lookaheadMatch[7];

		// Simple bracketted link
		var e = w.output;
		if(link) { // LINKED IMAGE
			if (config.formatterHelpers.isExternalLink(link)) {
				if (config.macros.attach && config.macros.attach.isAttachment(link)) {
					// see [[AttachFilePluginFormatters]]
					e = createExternalLink(w.output,link);
					e.href=config.macros.attach.getAttachment(link);
					e.title = config.macros.attach.linkTooltip + link;
				} else
					e = createExternalLink(w.output,link);
			} else
				e = createTiddlyLink(w.output,link,false,null,w.isStatic);
			addClass(e,"imageLink");
		}

		var img = createTiddlyElement(e,"img");
		if(floatLeft) img.align="left"; else if(floatRight) img.align="right";
		if(width||height) {
			var x=width.trim(); var y=height.trim();
			var stretchW=(x.substr(x.length-1,1)=='+'); if (stretchW) x=x.substr(0,x.length-1);
			var stretchH=(y.substr(y.length-1,1)=='+'); if (stretchH) y=y.substr(0,y.length-1);
			if (x.substr(0,2)=="{{")
				{ try{x=eval(x.substr(2,x.length-4))} catch(e){displayMessage(e.description||e.toString())} }
			if (y.substr(0,2)=="{{")
				{ try{y=eval(y.substr(2,y.length-4))} catch(e){displayMessage(e.description||e.toString())} }
			img.style.width=x.trim(); img.style.height=y.trim();
			config.formatterHelpers.addStretchHandlers(img,stretchW,stretchH);
		}
		if(tooltip) img.title = tooltip;

		// GET IMAGE SOURCE
		if (config.macros.attach && config.macros.attach.isAttachment(src))
			src=config.macros.attach.getAttachment(src); // see [[AttachFilePluginFormatters]]
		else if (config.formatterHelpers.resolvePath) { // see [[ImagePathPlugin]]
			if (config.browser.isIE || config.browser.isSafari) {
				img.onerror=(function(){
					this.src=config.formatterHelpers.resolvePath(this.src,false);
					return false;
				});
			} else
				src=config.formatterHelpers.resolvePath(src,true);
		}
		img.src=src;
		w.nextMatch = this.lookaheadRegExp.lastIndex;
	}
}

config.formatterHelpers.addStretchHandlers=function(e,stretchW,stretchH) {
	e.title=((stretchW||stretchH)?'DRAG=stretch/shrink, ':'')
		+'SHIFT-CLICK=show full size, CTRL-CLICK=restore initial size';
	e.statusMsg='width=%0, height=%1';
	e.style.cursor='move';
	e.originalW=e.style.width;
	e.originalH=e.style.height;
	e.minW=Math.max(e.offsetWidth/20,10);
	e.minH=Math.max(e.offsetHeight/20,10);
	e.stretchW=stretchW;
	e.stretchH=stretchH;
	e.onmousedown=function(ev) { var ev=ev||window.event;
		this.sizing=true;
		this.startX=!config.browser.isIE?ev.pageX:(ev.clientX+findScrollX());
		this.startY=!config.browser.isIE?ev.pageY:(ev.clientY+findScrollY());
		this.startW=this.offsetWidth;
		this.startH=this.offsetHeight;
		return false;
	};
	e.onmousemove=function(ev) { var ev=ev||window.event;
		if (this.sizing) {
			var s=this.style;
			var currX=!config.browser.isIE?ev.pageX:(ev.clientX+findScrollX());
			var currY=!config.browser.isIE?ev.pageY:(ev.clientY+findScrollY());
			var newW=(currX-this.offsetLeft)/(this.startX-this.offsetLeft)*this.startW;
			var newH=(currY-this.offsetTop )/(this.startY-this.offsetTop )*this.startH;
			if (this.stretchW) s.width =Math.floor(Math.max(newW,this.minW))+'px';
			if (this.stretchH) s.height=Math.floor(Math.max(newH,this.minH))+'px';
			clearMessage(); displayMessage(this.statusMsg.format([s.width,s.height]));
		}
		return false;
	};
	e.onmouseup=function(ev) { var ev=ev||window.event;
		if (ev.shiftKey) { this.style.width=this.style.height=''; }
		if (ev.ctrlKey)  { this.style.width=this.originalW; this.style.height=this.originalH; }
		this.sizing=false;
		clearMessage();
		return false;
	};
	e.onmouseout=function(ev) { var ev=ev||window.event;
		this.sizing=false;
		clearMessage();
		return false;
	};
}
//}}}

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

>Researchers studying the theory of error-correcting codes have discovered, in recent years, that finite geometries and designs can provide the basis for excellent communications schemes. The basic idea is to take the linear span (over some finite field) of the rows of the incidence matrix of such a structure as the allowable messages.
> - Eric S. Lander - Symmetric Designs: An Algebraic Approach

One way to represent a design is in terms of its ''Incidence Matrix M''.

An incidence matrix is a $n \times b$-matrix with the rows indexed by the $n$ points of the design and the columns by its $b$ blocks. In a given row, $1$'s are placed in the columns with blocks that contain the point of that row and $0$'s in the remaining cells.

!!!!Relationship with [[codes|Blockcode]]
A binary incidence matrix of a [[Steiner quadruple system|Steiner Quadruple System]] $SQS(n)$ is a constant weight-$4$ $(n, 4, \frac{n(n-1)(n-2)}{24})$-code which is strongly optimal. E.g.
* SQS(8): (8,4,14)
* SQS(16): (16,4,140)

!!!!Examples
A $(4,4,3,3,2)$-design with points $\{1,2,3,4\}$ and blocks $\{\{1,2,3\}, \{2,3,4\}, \{3,4,1\}, \{4,1,2\}\}$ defines the following incidence matrix:
| |!{1,2,3}|!{2,3,4}|!{3,4,1}|!{4,1,2}|
|!1|1|0|1|1|
|!2|1|1|0|1|
|!3|1|1|1|0|
|!4|0|1|1|1|

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( {
	name: "inlineJavascript",
	match: "\\<script",
	lookahead: "\\<script(?: src=\\\"((?:.|\\n)*?)\\\")?(?: label=\\\"((?:.|\\n)*?)\\\")?(?: title=\\\"((?:.|\\n)*?)\\\")?(?: key=\\\"((?:.|\\n)*?)\\\")?( show)?\\>((?:.|\\n)*?)\\</script\\>",

	handler: function(w) {
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			var src=lookaheadMatch[1];
			var label=lookaheadMatch[2];
			var tip=lookaheadMatch[3];
			var key=lookaheadMatch[4];
			var show=lookaheadMatch[5];
			var code=lookaheadMatch[6];
			if (src) { // external script library
				var script = document.createElement("script"); script.src = src;
				document.body.appendChild(script); document.body.removeChild(script);
			}
			if (code) { // inline code
				if (show) // display source in tiddler
					wikify("{{{\n"+lookaheadMatch[0]+"\n}}}\n",w.output);
				if (label) { // create 'onclick' command link
					var link=createTiddlyElement(w.output,"a",null,"tiddlyLinkExisting",wikifyPlainText(label));
					var fixup=code.replace(/document.write\s*\(/gi,'place.bufferedHTML+=(');
					link.code="function _out(place,tiddler){"+fixup+"\n};_out(this,this.tiddler);"
					link.tiddler=w.tiddler;
					link.onclick=function(){
						this.bufferedHTML="";
						try{ var r=eval(this.code);
							if(this.bufferedHTML.length || (typeof(r)==="string")&&r.length)
								var s=this.parentNode.insertBefore(document.createElement("span"),this.nextSibling);
							if(this.bufferedHTML.length)
								s.innerHTML=this.bufferedHTML;
							if((typeof(r)==="string")&&r.length) {
								wikify(r,s,null,this.tiddler);
								return false;
							} else return r!==undefined?r:false;
						} catch(e){alert(e.description||e.toString());return false;}
					};
					link.setAttribute("title",tip||"");
					var URIcode='javascript:void(eval(decodeURIComponent(%22(function(){try{';
					URIcode+=encodeURIComponent(encodeURIComponent(code.replace(/\n/g,' ')));
					URIcode+='}catch(e){alert(e.description||e.toString())}})()%22)))';
					link.setAttribute("href",URIcode);
					link.style.cursor="pointer";
					if (key) link.accessKey=key.substr(0,1); // single character only
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				else { // run script immediately
					var fixup=code.replace(/document.write\s*\(/gi,'place.innerHTML+=(');
					var c="function _out(place,tiddler){"+fixup+"\n};_out(w.output,w.tiddler);";
					try	 { var out=eval(c); }
					catch(e) { out=e.description?e.description:e.toString(); }
					if (out && out.length) wikify(out,w.output,w.highlightRegExp,w.tiddler);
				}
			}
			w.nextMatch = lookaheadMatch.index + lookaheadMatch[0].length;
		}
	}
} )
//}}}

// // Backward-compatibility for TW2.1.x and earlier
//{{{
if (typeof(wikifyPlainText)=="undefined") window.wikifyPlainText=function(text,limit,tiddler) {
	if(limit > 0) text = text.substr(0,limit);
	var wikifier = new Wikifier(text,formatter,null,tiddler);
	return wikifier.wikifyPlain();
}
//}}}

// // 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"]]>>

A map $\tau : \mathcal {A} \to \mathcal{A}$ is called an ''Involution'' or ''Involutive Antiautomorphism'' on an algebra $\mathcal{A}$ if it is an [[antiautomorphism|Automorphism]] of period $2$, i.e. if
\begin{eqnarray}
\tau (\mb A + \mb B) & = & \tau (\mb A) + \tau (\mb B) \\
\tau (\mb{AB})  & = & \tau (\mb B)  \tau (\mb A) \\
\tau (\tau(\mb A)) & = & \mb A
\end{eqnarray}
$\forall \mb A, \mb B \in \mathcal{A}$.

The algebra is called scalar if for the [[norm|Norm]] defined by $\mathcal N (\mb A) \equiv \tau (\mb A) \mb A$ one has $\mathcal N (\mb A) \in \mathbb R \;\, \forall \mb \in \mathcal A$.

!!!!Examples
[[Partity reversions P|CPT Transformations]] and [[time reversions T|CPT Transformations]] in Minkowski space
\begin{eqnarray}
P: (t, \vec x) \mapsto (t, -\vec x) \\
T: (t, \vec x) \mapsto (-t, \vec x)
\end{eqnarray}
Papers:
* [[Nonassociative Algebras with Scalar Involution - K. McCrimmon|http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102707250]] [[pct. 24|http://scholar.google.de/scholar?cites=13727272288851598106&hl=de]]
* [[Involution on Composition Algebras - S. Pumplün|http://homepage.uibk.ac.at/~c70202/jordan/archive/unitams/unitams.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=2718813721862090480&hl=de&as_sdt=2000]]
* [[The Hermitian Level of Composition Algebras - S. Pumplün, T. Unger|http://www.math.uni-bielefeld.de/LAG/man/086.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=4086996909278571255&hl=de&as_sdt=2000]]

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

A ''Isomorphism'' is a bijective [[homomorphism|Homomorphism]]. Isomorphic objects are completely indistinguishable as far as the structure in question is concerned. A generalisation of isomorphism are [[isotopies|Isotopy]].

Papers:
* [[Isospin and Local Space-Time Rotations - J. G. Valatin|http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103841070]]

<<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"]]>>

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}

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"]]>>

{{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"]]>>

''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"]]>>

The ''M\-Algebra'' is the maximal extension of the $\mathcal{N}=1$ super-Poincaré algebra in eleven dimensions.
It is spanned by the set $G_{A}=\{J_{ab},P_a,Q_\alpha,Z_{ab},Z_{abcde}\}$, where $J_{ab}$ and $P_a$ are the generators of the [[Poincaré group|Poincaré Transformation]] and $Q_\alpha$ is a Majorana spinor supercharge with anticommutator
\begin{equation}
\{Q_{\alpha },Q_{\beta }\}=\left( C\Gamma ^{a}\right) _{\alpha \beta
}P_{a}+(C\Gamma ^{ab})_{\alpha \beta }Z_{ab}+(C\Gamma ^{abcde})_{\alpha
\beta }Z_{abcde}
\end{equation}
The charge conjugation matrix $C$ is antisymmetric, and the central charges $Z_{ab}$ and $Z_{abcde}$ are tensors under Lorentz rotations but otherwise Abelian generators. In standard eleven-dimensional supergravity, these generators correspond to the "electric" and "magnetic" charges of the $M2$ and $M5$ branes, respectively.

Papers:
* [[Poincaré Invariant Gravity with Local Supersymmetry as a Gauge Theory for the M-algebra - M. Hassaine, R. Troncoso, J. Zanelli|http://arxiv.org/PS_cache/hep-th/pdf/0306/0306258v2.pdf]] [[pct. 16|http://scholar.google.de/scholar?cites=10560539088242203663&hl=de]]
* [[On the Octonionic Superconformal M-algebra - F. Toppan|ftp://ftp2.biblioteca.cbpf.br/pub/apub/2002/nf/nf_zip/nf04502.pdf]] pct. 0

Papers:
* [[Topics in M-theory - E. Sezegin|http://arxiv.org/PS_cache/hep-th/pdf/9809/9809204v2.pdf]] [[pct. 16|http://scholar.google.de/scholar?cites=15550580083989394648&hl=de]]

!!!![[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

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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,