Trajectory of the Universe - Mathematics and Physics Notebook of Markus MauteWhen I hear of Schrödinger's cat, I reach for my gun.

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

''16 Vectors'' are a generalization of Lorentz 4 vectors. A 16 vector is understood as a $2^4$-dimensional vector in [[P-space|Polyvector Space]]. A description by means of 16 vectors therefore implies that space-time is 4-dimensional.

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]]

Let $\mathcal Q$ be a local analytic [[quasigroup|Quasigroup]] with the multiplication $z = x\cdot y$ and $dim \mathcal Q \equiv r \ge 1$.
Three foliations of the ($2r$-dimensional) manifold $\mathcal M = \mathcal Q \times \mathcal Q$ given by $\lambda_1: x = const.$, $\lambda_2 : y = const$ and $\lambda_3 : z = const.$ form a ''$3$''-[[Web]] (or ''$3$-Net'') $W = \{X, \lambda_\alpha\}$, corresponding to the quasigroup $\mathcal Q$.

Conversely, let $W = \{X, \lambda_{\alpha}\}$ be a $3$-web formed by $3$ $r$-dimensional foliations $\lambda_\alpha$ on a manifold $\mathcal M$ with $\operatorname{dim} (\mathcal M) = 2r$. Then a local quasigroup $Q$ is defined in a neighbourhood $U$ of a point of $\mathcal M$. The relation $z = x\cdot y$ means that the leaves $x \in \lambda_1$, $y \in \lambda_2$ and $z \in \lambda_3$ pass through a point of $U$. The quasigroup $\mathcal Q$ is called the ''Coordinate Quasigroup'' of the $3$-web $W$.

Two $3$-webs $W = \{X, \lambda_\alpha\}$ and $\tilde W = \{\tilde X, \tilde \lambda_\alpha\}$ are said to be equivalent if there exists a local diffeomorphism $f: X \rightarrow \tilde X$ such that $f(\lambda_\alpha) = \tilde \lambda_\alpha$ for $\alpha = 1,2,3$.

A ''Three\-Web'' can be regarded as a geometrical interpretation of a [[quasigroup|Quasigroup]]. Three webs with closed [[G-structure|G-Structure]] are a natural and far-reaching generalization of [[Lie groups|Lie Group]].

With any [[loop|Loop]] one can associate a three-net. Conversely, every three-net leads to a class of [[isotopic|Isotopy]] loops called ''Coordinate Loops'' of the web.

The original classification of three-webs was in respect to closure conditions which are denoted $T$ (G. Thomsen), $R$ (K. Reidemeister), $M$ (R. Moufang), $B_l$ and $B_r$ (G. Bol) and $H$ (hexagonal) respectively. They correspond to the properties of commutativity, associativity and various forms of "relaxed" associativity like [[right- and left-alternativity|Alternative Algebra]] and [[monoassociativity|Monoassociativity]], which hold in the coordinate loops of a three-web.
[[Hexagonal three-webs|Hexagonal Three Web]] form the widest class of webs characterized by known closure conditions and include all the other types mentioned.

The following table lists the $3$-webs with some of their properties:
<html><center><img src="images/3Webs.jpg" style="width: 390px; "/></center></html>

M. A. Akivis pointed out that a [[G-structure|G-Structure]] defined by a multidimensional $3$-web of the classes $T$, $R$, $M$ and $B$ is closed. Shelekov showed [1] that this is also true for $H$-webs, however requiring a $4^{th}$-order prolongation of the [[tangent space|Tangent Algebra]].

See also: [[Hexagonal three-webs|Hexagonal Three Web]].

Papers:
* [[Classification of Multidimensional Three-webs According to Closure Conditions - A. M. Shelekhov (Russian)|http://www.mathnet.ru/php/getFT.phtml?jrnid=intg&paperid=181&volume=21&year=1989&issue=&fpage=109&what=fullt&option_lang=eng]] [[local|papers/getFT.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=6396154066767780326&hl=de&as_sdt=2000]]
* [[[1] On the Higher-Order Differential-Geometric Objects of a Multidimensional Three-Web - A. M. Shelekhov|http://www.springerlink.com/content/l844mjp21074j410/fulltext.pdf]] [[original (Russian)|http://www.kazan.mathnet.ru/php/getFT.phtml?jrnid=intg&paperid=168&volume=19&year=1987&issue=&fpage=101&what=fullt&option_lang=eng]] [[local|papers/Shelekhov.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=17328826859130719070&hl=de&as_sdt=2000]] prl. 10
* [[A Classification and Examples of Four-Dimensional Isoclinic Three-Webs - V. V. Goldberg|http://arxiv.org/PS_cache/math/pdf/0010/0010176v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=9503186043654802454&hl=de]]
* [[On an Algebraical Computation of the Tensor and the Curvature for 3-Webs - T. B. Bouetou|http://arxiv.org/PS_cache/math/pdf/0310/0310097v1.pdf]] pct. 0 prl. 6

Google books:
* [[Geometry and Algebra of Multidimensional Three-webs - M. A. Akivis, A. M. Shelekhov|http://books.google.com/books?id=xsIhSTzheh8C&dq=akivis&printsec=frontcover&source=bl&ots=-a0RWOgKaO&sig=DtXSNcvK8SYDqWdkH-21tQnHPOM&hl=de&sa=X&oi=book_result&resnum=7&ct=result#PPP1,M1]] [[bct. 60|http://scholar.google.de/scholar?hl=de&lr=&cites=14458088870284223965]] [[local|google_books/GeometryAndAlgebra.pdf]]

Videos:
* [[Edward Witten Lecture - Dimensional Gravity Revisited|http://www.youtube.com/view_play_list?p=36BF70A0707857EF]]

A ''4-Cube'' is the 4 dimensional counterpart of the conventional 3-dimensional cube. Other designations are: ''Tesseract'', ''Octachoron'', ''4-Dimensional Hypercube''.

<html><center><img src="images/tesseract.gif" style="width: 210px; "/></center></html>
<html><center><img src="images/4-cube.jpg" style="width: 150px; "/></center></html>
!!!!Properties
Vertices: 16 = 2$\cdot$8
Edges: 32 = 2$\cdot$12+8
Faces: 24
Cells: 8

A ''5-Cube'' is the $5$-dimensional counterpart of the conventional $3$-dimensional cube. Other designations are: ''Pentaract'', ''Octachoron'', ''5-dimensional Hypercube''.

<html><center><img src="images/5-cube.jpg" style="width: 240px; "/></center></html>
!!!!Properties
* Vertices: $32$
* Edges: $80$
* Faces: $80$ squares
* Cells: $40$ cubes 
* Hypercells: $10$ [[tesseracts|4-Cube]]

The ''Seven\-Dimensional Sphere'' or ''7-Sphere'' $S^7$ is given by:
\begin{eqnarray}
S^7 &=& \{x \in \mathbb R^8 : \langle x|x\rangle = 1\} \\
& = & \{(\mathbf A_1, \mathbf A_2) \in \mathbb H \times \mathbb H : ||\mathbf A_1||^2 + ||\mathbf A_2||^2 = 1\}
\end{eqnarray}

The $7$-sphere is the unique [[parallelizable|Parallelizability]] manifold but it is not a [[group|Group]] manifold.

It is the only compact [[Riemannian manifold|Riemann Space]] which shares with [[Lie groups|Lie Group]] the property of [[absolute parallelism|Parallelizability]].

The $7$-sphere has the structure of an analytic [[Moufang loop|Moufang Loop]], inherited from the multiplication in the real division [[octonion algebra|Octonion]] $\mathbb O$. Hence, if we identify $S^7$ with $\{\mathbf A \in \mathbb O: ||\mathbf A||^2 = 1\}$, with $||\,.\,|| $ the [[norm|Norm]] in $\mathbb O$, then the product in $\mathbb O$ of two elements in $S^7$ belongs to $S^7$.
!!!!Properties
* The structure group of the tangent bundle over $S^7$ can be either [[SO(7)]] or [[G2]]
* $ S^7 \cong SO(8)/SO (7)$
* $ S^7 \times \mathbb R^7 \cong SO(8, \mathbb C)/SO (7, \mathbb C)$

Papers:
* [[Seven-Spheres from Octonions - J. Lukierski, P. Minnaert|http://streaming.ictp.trieste.it/preprints/P/83/189.pdf]] [[pct. 8|http://scholar.google.de/scholar?cites=10021972343777246011&hl=de]]
* [[S7 and cS7 - M. Cederwall, C. R. Preitschopf|http://arxiv.org/PS_cache/hep-th/pdf/9309/9309030v1.pdf]] [[pct. 7|http://scholar.google.de/scholar?hl=de&lr=&cites=2101453315758664030]] [[local|papers/9309030v1.pdf]] prl. 9

''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]]

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

''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 $\mathbf A$ and $\mathbf X$ of an algebra $\mathcal A$, the ''Adjoint'' $ad_{\mathbf A}$ is defined as a linear map $ad_{\mathbf A}: \mathcal A \rightarrow \mathcal A$ given by the [[commutator|Commutator]] product:
\begin{equation}
ad_{\mathbf A}(\mathbf X) = [\mathbf A, \mathbf X]
\end{equation}
If one replaces every element of an algera 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 $[\mathbf A, \mathbf B] = \mathbf C$ it follows $[ad_{\mathbf A}, ad_{\mathbf B}] = ad_{\mathbf C}$.

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]]

A $d$-dimensional manifold equipped with an affine connection is sometimes called an affinely connected space and is denoted by $L_d$.
The category of spaces with affine connection is equivalent to the category of spaces with a geo-odular structure, where an algebraic system with a nonassociative binary operation ([[geodesic multiplication|Geodesic Loop]]) is given in a neighbourhood of each  point of the space.

An ''Akivis Algebra'' is a vector space endowed with a skew-symmetric bilinear product $a$ and a trilinear product $b$ satisfying the identity
\begin{eqnarray}
&&a(\mathbf A,a(\mathbf B,\mathbf C)) + a(\mathbf B,a(\mathbf C,\mathbf A)) + b(\mathbf C,b(\mathbf A, \mathbf B)) = \\
&& b(\mathbf A,\mathbf B,\mathbf C) + b(\mathbf B,\mathbf C,\mathbf A) + b(\mathbf C,\mathbf A,\mathbf B) - b(\mathbf A,\mathbf C,\mathbf B) - b(\mathbf C,\mathbf B,\mathbf A) - b(\mathbf B,\mathbf A,\mathbf C)
\end{eqnarray}
a.k.a. ''Akivis Identity'' or ''Generalized'' [[Jacobi Identity|Jacobian]].

For any (nonassociative) algebra one obtains an Akivis algebra by identifying $a$ with the [[commutator|Commutator]] and $b$ with the [[associator|Associator]]. The equation is then an identity if one resolves the double commutators and the associators.

Akivis Algebras were introduced in 1976 by [[M. A. Akivis|http://d-omega.org/index.php?action=show&post_id=55&PHPSESSID=eb69f92142fc5664dc7becfb633d4d0c]] as local algebras of [[three-webs|3-Web]].

''Theorem''
A local algebra ([[tangent algebra|Tangent Algebra]]) of a differentiable [[quasigroup|Quasigroup]] is an Akivis algebra. (This theorem is a generalization of [[Lie's first theorem|Lie's Theorems]] for differentiable quasigroups).

''Theorem (Shestakov)''
Any Akivis algebra can be isomorphically embedded in the algebra of commutators and associators of a certain nonassociative algebra. (This generalizes the corresponding theorem for Lie algebras which says that every Lie algebra is isomorphic to a subalgebra of commutators of a certain associative algebra).
However an Akivis algebra does not in general uniquely determine a differentiable quasigroup. (See next theorem).

''Theorem''
Local Akivis algebras associated with [[Moufang-|Moufang Loop]] and [[Bol-quasigroups|Bol Loop]] determine these quasigroups in a unique way. Monoassociative quasigroups however are only determined uniquely by prolonged Akivis algebras (which are [[hyperalgebras|Sabinin Algebra]]).

Akivis algebras are so called binary-ternary algebras and are a generalization of Lie algebras which are binary algebras only. Akivis algebras can be regarded as [[prolonged Lie algebras (hyperalgebras)|Sabinin Algebra]] and can describe tangent algebras of quasigroups up to third order. In case that higher orders are relevant, a generalization is required, which are called prolonged Akivis algebras.
Every Lie algebra, Akivis algebra or prolonged Akivis algebra is in fact a special case of a [[Sabinin algebra|Sabinin Algebra]].
Akivis algebras therefore allow for a straightforward and natural generalization of Lie theory.

The relation between Akivis algebras and [[Sabinin algebras|Sabinin Algebra]] was clarified by Shestakov and Umirbaev (2002).They showed that free nonassociative algebras are the universal enveloping algebras of free Akivis algebras, just as free associative algebras are the universal enveloping algebras of free Lie algebras.

!!!!Examples
i) [[Lie algebras|Lie Algebra]]: As these are associative the right hand side is zero and the expression reduces to the [[Jacobi identity|Jacobian]].
ii)  [[Alternative Algebras|Alternative Algebra]] ([[Octonions|Octonion]]): As the associator is antisymmtrical, the right hand side collapses to 6 times the associator of $\mathbf A$, $\mathbf B$ and $\mathbf C$.

Papers:
* [[Every Akivis Algebra is Linear ? - I. P. Shestakov|http://www.springerlink.com/content/k637348274094870/fulltext.pdf]] [[local|papers/AkivisAlgebra.pdf]] [[pct. 15|http://scholar.google.de/scholar?hl=de&lr=&cites=3943685980755657013]]

''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]]

Papers:
* [[Encyclopedia of Types of Algebras - J.-L. Loday|http://www-irma.u-strasbg.fr/~loday/PAPERS/EncyclopALG(root).pdf]]

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]].

''Analytic Loops'' are a generalization of [[Lie groups|Lie Group]] first considered by Maltsev.

Papers:
* [[On Anti-Commuative Algebras and Analytic Loops - A. A. Sagle|http://books.google.com/books?hl=de&lr=&id=px2Qr7kHgcUC&oi=fnd&pg=PA550&ots=TT4BtuDc1r&sig=Qwt3XmMCsvsICz3L6fZoPaMaVyQ#v=onepage&q=&f=false]] [[local|journals/Sagle.pdf]] [[pct. 3|http://scholar.google.com/scholar?cites=17448209514636332778&hl=de]]

Links:
* [[Springer Encyclopaedia of Mathematics - Loop, analytic|http://eom.springer.de/L/l060840.htm]]

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 ''anamolous''. 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]]

The ''Anti-Commutator'' of two elements $\mathbf A$, $\mathbf B$ of an algebra is defined as:
\[
\{\mathbf A,\mathbf B\}= \mathbf{AB}+\mathbf{BA}
\]
An algebra that is obtained from an algebra $\mathcal A$ by replacing the product $\mathbf{AB}$ with the anti-commutator $\{\mathbf A, \mathbf B\}$ is denoted $\mathcal A^+$.
The anti-commutator can also be expressed in terms of the [[Jordan product|Jordan Algebra]]:
\[
\{\mathbf A,\mathbf B\} = 2 \mathbf A \circ \mathbf B
\]
$\mathcal A^+$ is called a ''Commutative Algebra''.

$\mathcal A$ is called anticommutative if it satisfies the identity
\[
\mathbf A^2 = 0 \quad \forall A \in \mathcal A
\]
This implies that $\{\mathbf A, \mathbf B \} = 0$ and the converse holds in characteristic $\ne 2$.

The ''Antibracket Formalism'' is used to [[quantize|Quantization]] a [[gauge theory|Gauge Transformation]] and appears to be the most powerful method to do this.
Contrary to the Faddeev\-Popov  functional integration procedure it does not fail for:
* So called "open gauge algebras" which only close on-shell. Such algebras occur in gravity and [[supergravity|Supergravity]].  
* "Reducible theories" where the gauge generators are all not independent, 
* [[Yang-Mills theories|Yang-Mills Theory]], if one considers exotic gauge-fixing procedures for which "extraghosts" appear.

For each field and ghost one introduces an antifield, thereby doubling the total number of original fields. The antibracket is an odd non-degenerate symplectic form on the space of fields and antifields and plays the role of the Poisson bracket. As a consequence, Hamiltonian concepts, such as canonical transformations, can be formulated and used. The original classical [[action|Action Principle]] is extended to a new action, in an essentially unique way, to arrive at a theory with manifest [[BRST symmetry|BRST Quantization]]. The antibracket formalism proceeds via the functional integral, hence the powerful techniques of functional integration are available.

Papers: 
* [[Antibracket, Antifields and Gauge-Theory Quantization - J. Gomis, J. Paris, S. Samuel|http://arxiv.org/PS_cache/hep-th/pdf/9412/9412228v1.pdf]] {{t100Cite{[[pct. 212|http://scholar.google.de/scholar?cites=8635365789647031556&hl=de&as_sdt=2000]]}}}

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

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]]

An ''Association Type'' of degree $n$ is a way to put parentheses in a product of degree $n$.
The number of association types of degree $n$ is given by the [[Catalan number|Catalan Numbers]] $\operatorname{Cat}[n]$.

!!!!Examples
| !n | !Bracket Combinations |
| $3$ | $\mathbf A(\mathbf{BC})$, $(\mathbf{AB})\mathbf C$ |
| $4$ | $\mathbf A(\mathbf B(\mathbf{CD}))$, $\mathbf A((\mathbf{BC})\mathbf D)$, $(\mathbf A(\mathbf{BC}))\mathbf D$, $(\mathbf{AB})(\mathbf{CD})$, $((\mathbf{AB})\mathbf C)\mathbf D$ |
| $5$ | $\mathbf A(\mathbf B(\mathbf C(\mathbf{DE})))$, $\mathbf A(\mathbf B((\mathbf{CD})\mathbf E))$, $\mathbf A((\mathbf{BC})(\mathbf{DE}))$, $\mathbf A((\mathbf B(\mathbf{CD}))\mathbf E)$, $\mathbf A(((\mathbf{BC})\mathbf D)\mathbf E)$, $(\mathbf{AB})(\mathbf C(\mathbf{DE}))$, $(\mathbf{AB})((\mathbf{CD})\mathbf E)$, $(\mathbf A(\mathbf{BC}))(\mathbf{DE})$, $(\mathbf A(\mathbf B(\mathbf{CD})))\mathbf E$, $(\mathbf A((\mathbf{BC})\mathbf D))\mathbf E$, $((\mathbf{AB})\mathbf C)(\mathbf{DE})$, $((\mathbf{AB})(\mathbf{CD}))\mathbf E$, $((\mathbf A(\mathbf{BC}))\mathbf D)\mathbf E$, $(((\mathbf{AB})\mathbf C)\mathbf D) \mathbf E$ |

Papers:
* [[The Nucleus of the Free Alternative Algebra - I. R. Hentzel, L. A. Peresi |http://www.expmath.org/expmath/volumes/15/15.4/Peresi.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=69275940962992650&hl=de]]
* [[Non-Associative Operations - N. J. Lord|http://www.jstor.org/pss/2689567]] pct. 0

A curve $x^\sigma = x^\sigma(\tau)$ is said to be ''autoparallel'' (with respect to the [[connection|Connection]] $\Gamma^\sigma_{\mu\nu}$) if the covariant derivative $ \frac{D}{D\tau} \equiv \frac{d}{d\tau} + \Gamma^\sigma_{\mu\nu}$  along the curve of the tangent vector $u^\sigma(\tau) = \frac{dx^\sigma(\tau)}{d\tau}$ is zero at every point:
\begin{eqnarray}
&&\boxed{\frac{Du^\sigma }{D\tau}  = \frac{d u^\sigma}{d\tau} + \Gamma^\sigma_{\mu\nu} u^\mu u^\nu \equiv  0}
\end{eqnarray}
Using $\frac{d}{d\tau} = \frac{d x^\mu(\tau)}{d\tau} \frac{\partial}{\partial x^\mu} = u^\mu(\tau) \frac{\partial}{\partial x^\mu} $ this can also be expressed as
\begin{eqnarray}
\frac{D u^\sigma}{D\tau}  & = & u^\mu \frac{\partial u^\sigma}{\partial x^\mu} + \Gamma^\sigma_{\mu\nu} u^\mu u^\nu  \\
& = & u^\mu \left (\frac{\partial u^\sigma}{\partial x^\mu}+ \Gamma^\sigma_{\mu\nu}  u^\nu \right ) \\
\end{eqnarray}
Hence the autoparallelity condition reads
\begin{eqnarray}
&&\boxed{u^\mu D_\mu u^\sigma = u^\mu \frac{Du^\sigma}{D x^\mu} = 0}
\end{eqnarray}
with $D^\mu$ the [[covariant derivative|Covariant Derivative]] in respect to the coordinates $x^\mu$.

Only if [[torsion|Torsion]] is totally antisymmetric ($T_{ijk} = -T_{jik} = -T_{ikj}$) are autoparallels identical to [[geodesics|Geodesic Equation]] (i.e. the connection reduces to a [[Levi-Civita connection|Levi-Civita-Connection]]). That is to say autoparallels are a ''generalization of geodesics''.

!!!!Physical interpretation
An autoparallel describes a trajectory of an intertial system, that is one, in which no forces act. Therefore the momenta in this system are conserved. Locally this is a system with an orthonormal basis $\{\mathbf e_a\}$.
Therefore
\begin{eqnarray}
0 & = &\mathbf a = \frac{d}{d\tau} \mathbf p = \frac{d  p^a}{d\tau}\mathbf e_a = \frac{d}{d\tau} ( p^a \mathbf e_a) \\
   & \equiv & \frac{d}{d\tau} (p^\mu \mathbf e_\mu) \\
   & = & \frac{D  p^\mu}{D\tau} \mathbf e_\mu \\
\end{eqnarray}
Hence
\[
\frac{D  p^\mu}{D\tau} = 0
\]
!!!!Generalizations
See [[Polyvector Autoparallelity]].

Papers:
* [[New Action Principle for Classical Particle Trajectories in Spaces with Torsion - P. Fiziev, H. Kleinert|http://arxiv.org/PS_cache/hep-th/pdf/9503/9503074v1.pdf]] [[pct. 36|http://scholar.google.de/scholar?cites=1374319893785598759&hl=de]]
* [[Autoparallels From a New Action Principle - H. Kleinert and A. Pelster|http://arxiv.org/PS_cache/gr-qc/pdf/9605/9605028v2.pdf]] [[pct. 13|http://scholar.google.de/scholar?cites=15539175244601932971&hl=de]]

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]]

''BRST Quantization'' (or the ''BRST Formalism'') is a differential geometric approach to performing consistent, [[anomaly|Anomaly]]-free perturbative calculations in a non-abelian gauge theory. It is due to C. M. Becchi, A. Rouet, R. Stora and I. V. Tyutin.
In the BRST approach, one selects a perturbation-friendly __gauge fixing procedure__ for the action principle of a gauge theory using the differential geometry of the gauge bundle on which the field theory lives. One then quantizes the theory to obtain a Hamiltonian system in the interaction picture in such a way that the "unphysical" fields introduced by the gauge fixing procedure resolve gauge anomalies without appearing in the asymptotic states of the theory. The result is a set of Feynman rules for use in a Dyson series perturbative expansion of the S-matrix which guarantee that it is unitary and renormalizable at each loop order—in short, a coherent approximation technique for making physical predictions about the results of scattering experiments.

After quantization there remains a nilpotent, odd, global symmetry involving transformations of both fields and [[ghosts|Ghost Field]] which is called ''Becchi\-Rouet\-Stora\-Tyutin (BRST) Symmetry''.

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''
\[
T_{(ij)} = \frac1{2!}(T_{ij} +T_{ji})
\]
\[
T_{(ijk)} = \frac1{3!}(T_{ijk} +T_{ikj}+T_{jki} +T_{jik}+T_{kij}+T_{kji})
\]

''Anti\-Symmetrisation''
\[
T_{[ij]} = \frac{1}{2!}(T_{ij} -T_{ji})
\]
\[
T_{[ijk]} = \frac{1}{3!}(T_{ijk} -T_{ikj}+T_{jki}-T_{jik}+T_{kij}-T_{kji})
\]
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]]:
\[
S_{[a}T_{b]} = \frac{1}{2}[S_{a} , T_{b}]
\]

The ''Barnes\-Wall Lattices $\Lambda_n \equiv BW_n$'' define an infinite sequence of sphere packings in dimensions $n = 2^m$ with $m \in \mathbb N$, which include the densest packings known in dimensions $1, 4, 8$ and $16$. In dimensions $32$ and higher they are less dense than other known packings.
The [[BW8-lattice|E8 Lattice]] is [[1-modular|Lattice]] whereas all the other BW\-lattices are [[2-modular|Lattice]].

Barnes\-Wall lattices are closely related to [[Reed-Muller codes|Reed-Muller Code]].

!!!!Automorphism groups
$Aut(BW_n) = \mathcal C_m \cong 2_+^{1+2m} O^+(2m, 2)$ which is a  subgroup of the [[orthogonal group|Orthogonal Group]] $O(2^m, \mathbb R)$ that preserves $BW_n$. This group and its complex analogue have also arisen in connection with the construction of orthogonal spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. It has also been used in the classification of finite simple groups.

$C_m$ has a normal subgroup $E$ which is an extra-special $2$-group of order $2^{1+2m}$
$C_m$ is a maximal finite subgroup of $GL(2^m,\mathbb R)$

The [[order|Order]] of the [[automorphism group|Automorphism]] is given by
\[
\operatorname{ord}(Aut(BW_{2^m})) = \begin{cases}
  696.729.600  & \text{for}\;m = 3\\
  2^{m^2+m+1}(2^m-1)\prod_{i=1}^{m-1}(2^{2i}-1) & \text{otherwise}
\end{cases}
\]
or
\[
\operatorname{ord}(Aut(BW_{2^{m+1}}) = \frac{2^{2m+2}(2^{2m}-1)(2^{m+1} -1)}{(2^m-1)} \operatorname{ord}(Aut(BW_{2^{m}})
\]
for $m = 1$ and $m > 3$.

For the first few lattices one gets the orders: $2$, $8$, $1.152$, $696.729.600$, $89.181.388.800$, $48.126.558.103.142.400$, ... (see: [[Sloane's A014116|http://www.research.att.com/~njas/sequences/A014116]]).

The [[kissing number|Kissing Number]] of a Barnes Wall lattice $BW_{2^m}$ is given by:
\[
K(BW_{2^m}) = \prod_{i = 1}^m (2^i +2) = (2^m + 2)K(BW_{2^{m-1}})
\]
So for example:
\begin{eqnarray}
K(BW_2) &=& 4 \\
K(BW_4) &=& 6\cdot 4 = 24 \\
K(BW_8) &=& 10 \cdot 24 = 240 \\
K(BW_{16}) &=& 18 \cdot 240 = 4.320 \\
K(BW_{32}) &=& 34 \cdot 4.320 = 146.880 \\
\end{eqnarray}

!!!! Equivalences
$BW_2\cong \mathbb Z^2$, which is the $2$-dimensional [[integer lattice|Integer Lattice]].

$BW_4\cong \mathbb D_4$, which is the $4$-dimensional [[checkerboard lattice|Checkerboard Lattice]] (a.k.a. Schläfli lattice).

$BW_8$ is the famous [[E8 lattice|E8 Lattice]].

$BW_{16}$, see: [[Lambda 16 lattice|Lambda 16 Lattice]].


Papers:
* [[The Invariants of the Clifford Groups - G. Nebe, E. M. Rains, N. J. A. Sloane|http://arxiv.org/PS_cache/math/pdf/0001/0001038v2.pdf]] [[pct. 45|http://scholar.google.de/scholar?cites=15209257309795139415&hl=de&as_sdt=2000]]
* [[A Family of Optimal Packings in Grassmannian Manifolds - P. W. Shor and N. J. A. Sloane|http://arxiv.org/PS_cache/math/pdf/0208/0208003v1.pdf]] [[pct. 18|http://scholar.google.de/scholar?cites=7133739350584600065&hl=de&as_sdt=2000]]
* [[The Genus of the Barnes-Wall Lattice - R. Scharlau, B. B. Venko|http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=B57E13057DCD1983FB0DFFA7A9EE2CF6?doi=10.1.1.29.9284&rep=rep1&type=pdf ]] [[local|papers/10.1.1.29.9284-1.pdf]] [[pct. 15|http://scholar.google.de/scholar?cites=3128810909691478320&hl=de]]
* [[A Simple Construction for the Barnes-Wall Lattices - G. Nebe|http://www.math.rwth-aachen.de/~Gabriele.Nebe/papers/bw.pdf]] [[pct. 8|http://scholar.google.de/scholar?cites=15240790726815424632&hl=de]]
* [[A Very Intuitive Geometric Picture of the 24-cell, E8 and Lambda-16 Lattices Given by Using the Hopf Maps - E. Lewin Altschuler, A. Perez–Garrido|http://arxiv.org/PS_cache/math/pdf/0612/0612728v1.pdf]] pct. 0

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

Papers:
* [[The Berry Phase of D0-Branes - C. Pedder, J. Sonner, D. Tong|http://aps.arxiv.org/PS_cache/arxiv/pdf/0801/0801.1813v3.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.

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
\[
\sum_{k=0}^{n} {n \choose k} = 2^n
\]
\[
 {n \choose k} = {n-1 \choose k-1} + {n-1 \choose k}
\]

!!!!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]]

''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.

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]]

The ''Blaschke Conjecture'' (made in 1920) is related to the theory of [[webs|Web]]. It states that conditions of [[linearizability|Linearizability]]
* for a 3-web should consist of four relations for the ninth order web invariants (four \PDEs of ninth order) and
* for a 4-web of two relations for the fourth order web invariants (two \PDEs of fourth order).

Papers:
* [[On the Blaschke Conjecture for 3-Webs - V. V. Goldberg, V. V. Lychagin|http://m.njit.edu/CAMS/Technical_Reports/CAMS04_05/report6c.pdf]] [[pct. 14|http://scholar.google.de/scholar?cites=820694232038145569&hl=de]]

>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 - G. Franck|http://www.iemar.tuwien.ac.at/publications/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]]

Given a set of $q$ symbols $A$ (a.k.a. letters of an alphabet) and a set of $n$-tuples $A^n \equiv \{(a_0, a_1, \ldots , a_{n?1}) : a_i \in A\}$ over $A$ (a.k.a. ''(Code)-Words''), a ''Blockcode'' $C \equiv (n,M,d)_q$ of length $n$ is a subset $C \subseteq A^n$.
$M$ denotes the size of the code, i.e. the number of codewords it contains and $d$ the minimal distance of the words, its minimal [[Hamming distance|Hamming Distance]].

A blockcode is a special form of a code in that it is assumed that all the words are of equal length. (Often the word "code" is used, although it is assumed that in fact it is a  blockcode).

If the code contains only $2$ letters one speaks of a ''Binary (Block-)Code'' which can also be denoted $C \equiv (n,M,d)$ for short.

A common setting is to take a [[Galois field|Galois Field]] for $A$, i.e. $A \equiv \mathbb F_q = GF(q)$. In this case the subset is a vector space and the code is called a [[Linear (block-)code|Linear Blockcode]].

The goal of [[coding theory|Coding Theory]] in engineering is to construct codes with small $n$, large $M$ and large $d$. These constraints are however incompatible. Therefore in practical applications a compromise has to be found.

!!!!Notation
Some authors use the notation $(\, ... \, , x,\, ...\,)_q$ for linear codes instead of $[\, ... \, , x,\, ...\,]_q$ where $x$ stands for the rank of the code.
This should not be confused with the similar notation used for non-linear codes where $x$ stands for the number of code words. (See also examples below).

!!!!Automorphism group
The (permutation) [[automorphism group|Automorphism]] of a code $C$, denoted $Aut(C)$, is given by $Aut(C) \subseteq S^n$ where $S^n$ is the symmetric group. It acts on the code by permuting coordinates. $Aut(C)$ sends a code $C$ to itself, i.e. every codeword of $C$ is sent to another codeword of $C$.
The number of equivalent codes $N(Aut(C))$ under automorphisms is given by
\[
N(Aut(C)) = \frac{(q-1)^n n!}{\operatorname{ord}(Aut(C))}
\]
!!!!Examples
*$(4,8,2) = (4,2^3,2) $ describes a binary code with $2^3 = 8$ codewords of length $4$ each and a minimum [[Hamming distance|Hamming Distance]] between the closest two words of $2$, i.e. differing in $2$ letters.
*$(16, 2048, 6) = (2^4, 2^{11}, 4) = [16, 11, 4]$ [[Reed-Muller code|Reed-Muller Code]] $\operatorname{RM}(2,4)$ (a linear binary code).
* $(16, 256, 6) = (2^4, 2^8, 6)$ [[Nordstrom-Robinson code|Nordstrom-Robinson Code]] (a non-linear binary code).

Links:
* [[WIKIPEDIA: Blockcode|http://de.wikipedia.org/wiki/Blockcode]]

Papers:
* [[Coset Codes-Part I: Introduction and Geometrical Classification - G. D. Forney, J. R.|http://www.ensc.sfu.ca/people/faculty/cavers/ENSC805/readings/34it05-forney-a.pdf]] {{t100Cite{[[pct. 303|http://scholar.google.de/scholar?cites=673769417590896897&hl=de]]}}}

Lectures:
* [[Introduction to Coding Theory - J. Bierbrauer|http://www.math.mtu.edu/~jbierbra/HOMEZEUGS/codesintro.ps]]
* [[Gitter und Codes, 2.1 Codes: Einige Grundbegriffe - R. Scharlau|http://www.mathematik.tu-dortmund.de/~scharlau/SoSe09/GC_Mat/gc2_1und2.pdf]]
* [[Introduction to Modern Coding Theory - G. David Forney, Jr.|http://www.ima.umn.edu/talks/workshops/PISG6.8-26.04/forney/NDSlides.pdf]]
* [[Informationstheorie Codierung und Kryptologie - M. Kutylowski, W.-B. Strothmann|http://inf.uweb.bbzs.ch/Unterrichtsmaterialien_Schuljahr_200607/1_Lehrjahr/Stuff/Informationstheorie%20Codierung%20Und%20Kryptographie.pdf]]

Theses:
* [[Error Correcting Codes on Algebraic Surfaces - C. C. Lomont|http://www.lomont.org/Math/Papers/2003/Thesis.pdf]]

A ''Left Bol Loop'' satisfies the identity
\begin{equation}
 (\mathbf A(\mathbf{BA}))\mathbf C=\mathbf A(\mathbf B(\mathbf{AC}))
\end{equation}
a ''Right Bol Loop'' the identity
\begin{equation}
\mathbf A((\mathbf{BC})\mathbf B)=((\mathbf{AB})\mathbf C)\mathbf B
\end{equation}
A loop is both left- and right- Bol if and only if it is a [[Moufang loop|Moufang Loop]].
A Bol loop is [[diassociative|Diassociativity]] if and only if it is a [[Moufang loop|Moufang Loop]].
Related to Bol Loops are [[Bol algebras|Bol Algebra]].

Links:
* [[WIKIPEDIA - Boltzmann Brain|http://en.wikipedia.org/wiki/Boltzmann_brain]]

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]]

Papers:
*[[Quantum Physics, Semester 1 2008 - J. D. Cresser|http://www.physics.mq.edu.au/~jcresser/Phys301/LectureSlides/Phys301Lectures.pdf]] - (download very slow)
* [[Quantum Physics Notes, Chapter 9: Operations on States - J. D.  Cresser|http://www.physics.mq.edu.au/~jcresser/Phys301/Chapters/OperationsOnStates.pdf]]

Unlike bosonic [[p-branes|P-Brane]] which can be formulated in arbitrary spacetime dimensions $D$, [[supersymmetric|Supersymmetry]] $p$-branes only can be formulated for certain combinations of $d = p + 1$ and $D$. This restriction, enforced by supersymmetry, gives rise to the so called ''Brane Scan''.

<html><center><img src="images/brane_scan.jpg" style="width: 350px; "/></center></html>
The brane scan only tells us which branes are not forbidden by supersymmetry. If these branes actually exist as solutions to any supersymmetric field theory is another question.

In 11 space-time dimensions one only has 2 possible $p$-branes, therefore in case of [[M-theory|M-Theory]] supersymmetry is quite restrictive in respect to possible p-branes.

In so called ''Braneworld 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

Papers:
* [[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

Lectures:
* [[Grundgedanken einer einheitlichen Feldtheorie der Materie und Gravitation|http://www.engon.de/protosimplex/downloads/02%20heim%20-%20mbb%201.2.pdf]] - The basic idea is that the gravitational connection is composed of the classical symmetric Christoffel part and an antisymmetric part. Consequently the stress energy tensor also contains an antisymmetrical part. The latter represents matter and is quantized.

A ''C''${}^*$''-Algebra'' $\mathcal A$ is a Banach algebra over the field of complex numbers supplemented with an [[involution|Involution]] ${}^*$ which satisfies the so called $C^*$-identity:
\begin{equation}
\|\mathbf A^* \mathbf A\| = \|\mathbf A\|^2
\end{equation}
or equivalently
\begin{equation}
\|\mathbf A \mathbf A^* \| = \|\mathbf A\|^2
\end{equation}
$\forall \mathbf A \in \mathcal A$.
Every $C^*$-algebra is automatically a Banach *-algebra, however the converse is not true in general.

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

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

Given a [[Riemann Manifold|Riemann Space]] $\mathcal M$, a ''Calibrated Manifold'' is a k-dimensional submanifold of $\mathcal M$ defined by a closed k-form, called the ''Calibration''. Calibrated k-submanifolds minimize volume within their homology class.
Calibrated geometry was introduced by Harvey and Lawson.

Papers:
* [[Calibrated Geometries - R. Harvey, H. B. Lawson, Jr. (pct. 759)|http://www.springerlink.com/content/8451j84w08j28432/fulltext.pdf]] [[local|papers/CalibratedGeometries.pdf]] - The paper which started off the subject of calibrated geometry. prl. 10

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

The ''Cartan Matrix'' $g_{ij}$ of a rank $r$ [[root system|Root Lattice]] is a $r \times r$ matrix given by
\[
g_{ij} = 2\frac{\langle\mathbf e_i|\mathbf e_j\rangle}{\langle \mathbf e_i|\mathbf e_i \rangle}
\]
where $\mathbf e_i$ 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 $\mathbf T_i$, a ''Cartan Subalgebra'' is generated by a maximal subset of $N$ commuting [[generators|Generator]] $\mathbf T_j$, i.e. for which
\[
[\mathbf T_j, \mathbf T_k] = 0\text{,}   \quad  \forall \, j,k \in \{1,\ldots N\}
\]
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.

A ''Catalan Number'' $\operatorname{Cat}[N]$ defined as:
\[
\operatorname{Cat}[N] = \frac{1}{N!} {2N-2 \choose N-1} = \frac{(2N-2)!}{N!(N-1)!}
\]
Examples:
| !N | !Cat[N]|
|0| 1   |
|1| 1   |
|2| 1   |
|3| 2   |
|4| 5   |
|5| 42  |
|6| 132 |
|7| 429 |

See also: [[Sloane's A000108|http://www.research.att.com/~njas/sequences/A000108]].

Links:
* [[Catalan Numbers - R. M. Dickau|http://mathforum.org/advanced/robertd/catalan.html]]

Papers:
* [[Catalan Numbers - T. Davis|http://www.geometer.org/mathcircles/catalan.pdf]]

!!!!''Monoidal Category'' (a.k.a ''Bicategory'' or ''2-Category'')
A monoidal category $\mathcal C$ is a category with elements $\mathbf A, \mathbf B, \mathbf C \in \mathcal C$, a functor $\otimes : C \times C \mapsto C $ and a collection
of functorial isomorphisms  $\Phi_{\mathbf A, \mathbf  B, \mathbf  C}  : (\mathbf A \otimes \mathbf B) \otimes \mathbf C \mapsto \mathbf A \otimes (\mathbf B \otimes
\mathbf C) $ called the rebracketting associators between any three objects, that satisfy the [[pentagon identity|Pentagon Identity]].

!!!! [[Tricategory (a.k.a. 3-Category)|Tricategory]]
!!!! [[Tetracategory|Tetracategory]]

!!!! Historical
Categories, functors and natural transformations were introduced by Eilenberg and Mac Lane in their 1945 paper [1]. The language they introduced transformed modern mathematics. Their focus was not on categories and functors, but on natural transformations, which are maps between functors. Implicitly, they were introducing the $2$-category "Cat" of categories, functors and natural transformations.

Papers:
* [[[1] General Theory of natural equivalences - S. Eilenberg, S. MacLane|http://killingbuddha.altervista.org/FILOSOFIA/GToNe.pdf]] {{t100Cite{[[pct. 329|http://scholar.google.de/scholar?cites=12446028328535803067&hl=de]]}}}
* [[The Algebra of Oriented Simplexes - R. Street|http://www.math.mq.edu.au/~street/aos.pdf]] {{t100Cite{[[pct. 172|http://scholar.google.de/scholar?cites=18277909809131901522&hl=de]]}}}
* [[Higher Operads, Higher Categories - Tom Leinster|http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3Amath%2F0305049]] {{t100Cite{[[pct. 119|http://scholar.google.de/scholar?cites=11867756092802353709&hl=de]]}}}
* [[An Introduction to n-Categories - John C. Baez|http://arxiv.org/PS_cache/q-alg/pdf/9705/9705009v1.pdf]] [[pct. 57|http://scholar.google.de/scholar?cites=13588647644112862672&hl=de]]
* [[A Prehistory of n-Categorical Physics -J. C. Baez, A. Lauday|http://www.math.ucr.edu/home/baez/history.pdf]] [[local|papers/history.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=4772089887807811478&hl=de]]

Links:
* [[Quantization, Deformations, and new Homological and Categorical Methods in Mathematical Physics - J. Jones|http://www.maths.manchester.ac.uk/~tv/LMS/jones.html]]

Google Books:
* [[Higher Homotopy Structures in Topology and Mathematical Physics - J. D. Stasheff, J. McCleary|http://books.google.com/books?id=vMaRoWCRFHoC&pg=PA235&lpg=PA235&dq=stasheff+polyhedron&source=bl&ots=D56uqzTxfs&sig=FhuG-_kWqbxQGfkGARY_waK9wbQ&hl=de&sa=X&oi=book_result&resnum=2&ct=result#PPA227,M1]] [[local|google_books/HigherHomotopyStructures.pdf]] [[bct. 5|http://scholar.google.de/scholar?cites=14783958081477946542&hl=de]]
* [[Categories in Algebra, Geometry and Mathematical Physics - A. Davydov, R. Street|http://books.google.com/books?id=MZuFR_CU-ykC&printsec=frontcover&dq=Categories+in+Algebra,+Geometry+and+Mathematical+Physics+Davydov&source=bl&ots=muA-F_qjPh&sig=cdT5gf4CfuIgUrDee6UjNav4C-k&hl=de&ei=xsNpS9zSGY-j_gau8LHWCQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAcQ6AEwAA]] [[bct. 2|http://scholar.google.de/scholar?cites=16469916136609465560&hl=de&as_sdt=2000]]

''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.

>The Cayley\-Dickson algebras are just Clifford algebras with some signs in their multiplication tables changed, to make the elements all mutually anticommute. The two lines of algebras both agree on the reals, the complex numbers and the quaternions, but then they part company: Clifford goes next to an 8-dimensional matrix algebra while Cayley\-Dickson produces the 8-dimensional octonions.
>- D. R. Finkelstein - Time, Quantum, and Information

!!!! Construction
A ''Cayley\-Dickson (CD) Algebra'' can be generated by means of the (classical) [[Cayley-Dickson doubling process|Cayley-Dickson Doubling]].
The sequence $\mathbb A_n$ of algebras generated is: $\mathbb A_0 = \mathbb{R}$, $\mathbb A_1 = \mathbb{C}$, $\mathbb A_2 = \mathbb{H}$, $\mathbb A_3 = \mathbb{O}$, $\mathbb A_4 = \mathbb{S}$, $\mathbb A_5 = \mathbb{T}, \ldots$ namely the real numbers, complex numbers, [[quaternions|Quaternion]], [[octonions|Octonion]], [[sedenions|Sedenion]], [[trigintaduonions|Trigintaduonion]],....

All Cayley\-Dickson- and [[Clifford-algebras|Clifford Algebra]] can be obtained by so called proper twists on $\mathbb Z_2^N$ (for details see [2]).

!!!! General Properties
$\mathbb A_0$ and $\mathbb  A_1$ are commutative.
$\mathbb A_0$, $\mathbb  A_1$ and $\mathbb  A_2$ are associative.
$\mathbb A_0, \ldots, \mathbb  A_3$ are [[alternative|Alternative Algebra]] and [[normed|Norm]].
$\mathbb A_n$ with $n \ge 4$  is [[flexible|Flexible Algebra]] and has [[zero divisors|Zero Divisor]].

Given a basis $\{\mathbf e_1, \ldots, \mathbf e_{2^n}\}$ of $\mathbb A_n$, one has $\prod_{i=1}^{2^n} \mathbf e_i = \pm 1$. The result being $+1$ or $-1$ depends on the order of the multiplications and the [[association type|Association Type]] of the $2^n-1$-fold product.

!!!! [[Automorphism groups|Automorphism]]
$Aut(\mathbb A_1)$ = $\mathbb Z_2$ = {Identity, Conjugation}.
$Aut(\mathbb A_2$) = $SO(3)$ the rotation group in $\mathbb R^3$.
$Aut(\mathbb A_3$) = [[G2]], the exceptional Lie group.
$Aut(\mathbb A_n$) = $Aut(\mathbb A_{n?1}) \times S_3$ for $n \ge 4$ (showed by P. Eakin and A. Sathaye) where $S_3$ is the symmetric group of order 6.
Therefore $Aut(\mathbb A_4)$ = [[G2]] $\times S_3$.

For all Cayley Dickson algebras one has the following
Adjoint properties:
\begin{eqnarray}
\langle \mathbf A \mathbf B| \mathbf C \rangle& =& \langle \mathbf A | \mathbf B^* \mathbf C \rangle \\
\langle \mathbf A |\mathbf B \mathbf C \rangle& =& \langle\mathbf A \mathbf C^*| \mathbf B \rangle
\end{eqnarray}
[[Norm|Norm]] properties:
\[
|| \mathbf{AB} || = || \mathbf{A^*B}|| = ||\mathbf{AB^*} || = || \mathbf{BA}||
\]
[[Associator|Associator]]-identities:
\[
- [\mathbf A, \mathbf B, \mathbf C] = [\mathbf A^*, \mathbf B, \mathbf C] = [\mathbf A, \mathbf B^*, \mathbf C] =  [\mathbf A, \mathbf B, \mathbf C^*]
\]

!!!! Rotations
The multiplication map $R_{\mathbf B} : \mathbb A_n \rightarrow \mathbb A_n : \mathbf A \mapsto \mathbf{AB}$ with $||\mathbf B|| = 1$ belongs to $SO(2^n)$ for $n \le 3$ as these algebras are [[composition algebras|Composition Algebra]].
Therefore:
($R_{\mathbf B \in \mathbb C}: \mathbb A_1 \rightarrow \mathbb A_1) \in SO(2)$
($R_{\mathbf B \in \mathbb H}: \mathbb A_2 \rightarrow \mathbb A_2) \in$ [[SO(4)]]
($R_{\mathbf B \in \mathbb O}: \mathbb A_3 \rightarrow \mathbb A_3) \in$ [[SO(8)]]

!!!![[Zero Divisors|Zero Divisor]]
* Every zero-divisor in $\mathbb A_n$ is imaginary.
* For the dimension $d$ of zero divisor subspaces in  $\mathbb A_n$ one has: $d \le 2^n - 4n+4$. $d$ is always a multiple of 4 and for any possible dimension, zero divisors do in fact occur. Therefore in the [[sedenions|Sedenion]] one has only zero divisors for $d = 0,4$, whereas in the [[trigintaduonions|Trigintaduonion]] they occur for $d = 0,4,8,16$.
* For $n \ge 4$ the largest zero divisor subspace of $\mathbb A_n$ is [[homeomorphic|Homeomorphism]] to a disjoint union of $2^{n-4}$ copies of the [[Stiefel variety|Stiefel Manifold]] $V_2(\mathbb R^7)$, i.e. the space of ordered pairs of orthonormal vectors in $\mathbb R^7$. For the sedenions one therefore gets one copy and for the trigintaduonions $2$ copies.

!!!!Alternative Subspaces
''Theorem'' The greatest possible dimension $d$ of an alternative subspace of $\mathbb A_n$ is $d = 2(n + 1)$, given $n \ge 3$. Hence $d_{\mathbb S} = 10\,$ and $d_{\mathbb T} = 12$.

!!!![[Multiplication Tables]]
For the number of inequivalent multiplication tables $N(\mathbb A_n)$ of $\mathbb A_n$ one has (this is a conjecture !)
\begin{eqnarray}
N(\mathbb A_n) &= & N_{Fano}(\mathbb A_n) \cdot N_{signs} \\
& =& \frac{(2^n -1)!}{\operatorname{ord}(PGL(n,2))} \cdot 2^{2^n-n-1}
\end{eqnarray}
where $N_{Fano}(A_n)$ is the number of inequivalent [[Fano Spaces]] associated with the algebra and $N_{signs}$ the number of sign permutations of base vectors leading to inequivalent multiplication tables. $\operatorname{ord}(PGL(n,2))$ is the order of the [[projective linear group|Projective General Linear Group]] $PGL(n,2)$.

Thence:
\begin{eqnarray}
N(\mathbb H) & = & N_{Fano\, lines}(\mathbb H)\cdot N_{signs}(\mathbb H)
=1 \cdot 2 = 2 \\

N(\mathbb O) &=&N_{Fano\, planes}(\mathbb O)\cdot N_{signs}(\mathbb O)
= \frac{7!}{168} \cdot 2^4 = 30 \cdot 16 = 480 \\

N(\mathbb S) &=& N_{Fano\, tetrahedra}(\mathbb S)\cdot N_{signs}(\mathbb S)
=  \frac{15!}{20.160} \cdot 2^{11} = 64.864.800 \cdot 2.048 = 132.843.110.400 \approx 1.3\cdot 10^{11} \\

N(\mathbb T) & = & N_{Fano\, hyper-tetrahedra}(\mathbb T) \cdot N_{signs}(\mathbb T)
= \frac{31!}{9.999.360} \cdot 2^{26} = 822.336.494.953.469.303.808.000.000 \cdot 67.108.864 \approx 5.5\cdot 10^{34}  \\

N(\mathbb A_6) & = & N_{Fano\, hyper^2-tetrahedra}(\mathbb A_6) \cdot N_{signs}(\mathbb A_6)
= \frac{63!}{20.158.709.760} \cdot 2^{57} \approx 1.4\cdot 10^{94}
\end{eqnarray}

Papers:
* [[[1] On Generalized Cayley-Dickson Algebras - R. B. Brown|http://www.projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102992693]] [[local|papers/Brown_generalizedCDAlgebras.pdf]] [[pct. 20|http://scholar.google.de/scholar?cites=3336004767924200788&hl=de]]
* [[Alternative Elements in the Cayley–Dickson Algebras - G. Moreno|http://arxiv.org/PS_cache/math/pdf/0404/0404395v1.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=4013125457504762682&hl=de]]
* [[[2] Properly Twisted Groups and their Algebras - John W. Bales|http://jwbales.us/ptgal.pdf]] [[local|papers/ptgal.pdf]] pct. 0
* [[A Tree for Computing the Cayley-Dickson Twist J. W. Bales|http://jwbales.us/cdtree.pdf]] [[local|papers/cdtree.pdf]] pct. 0
* [[Cayley-Dickson and Clifford Algebras as Twisted Group Algebras (2003) J. W. Bales|http://jwbales.home.mindspring.com/combined.pdf]] pct. 0

Links:
* [[The Cayley-Dickson Calculator|http://jwbales.us/]]

The ''Cayley\-Dickson Doubling (Process)'' (a.k.a. ''Cayley\-Dickson Construction'') doubles a [[Cayley-Dickson algebra|Cayley-Dickson Algebra]], extending its multiplication, [[involution|Involution]] ${}^*$ and [[norm|Norm]] $||\,.\,||$  to the new Cayley\-Dickson algebra. The resulting algebra contains the originial algebra as a subalgebra.

There are two possible formulas for the ''extended multiplication'', which are equivalent. The first one is given by
\[
(\mathbf A_1, \mathbf A_2)(\mathbf B_1, \mathbf B_2) = (\mathbf A_1\mathbf B_1 - \lambda \mathbf B_2^* \mathbf A_2, \mathbf B_2\mathbf A_1+ \mathbf A_2 \mathbf B_1^*)
\]
the second one by
\[
(\mathbf A_1, \mathbf A_2)(\mathbf B_1, \mathbf B_2) = (\mathbf A_1 \mathbf B_1 - \lambda \mathbf B_2 \mathbf A^*_2,  \mathbf A_2 \mathbf B_1 + \mathbf A^*_1 \mathbf B_2)
\]
with $ \lambda = \pm 1$. (Notice that the sign of $\lambda$ is a matter of convention and formulas with a "+$\lambda$"-term are also found in literature).
We will stick to the first formula given above.

A Cayley\-Dickson algebra that results from $n$ doubling steps will be designated $CD (\lambda_1,\lambda_2, \ldots,\lambda_n)$ with $\lambda_i = \pm 1$.
If $\lambda_i = 1$, $\forall i$ one gets a (standard/canonical) ''Non-split (Circular) Cayley\-Dickson Algebra''. Else the algebras are called ''[[Split (Hyperbolic)|Split Algebra]] Cayley Dickson Algebras''.
In case of $n \le 3$ one has the theorem, that any two split [[composition algebras|Composition Algebra]] of the same dimension over a given field are isomorphic. This can be proofed using the [[Moufang identity|Moufang Loop]].

The multiplication tables resulting from doublings with different $\lambda$'s differ only in the signs of their structure constants (e.g. see multiplication table under [[split octonions|Split Octonion]]). In particular, the structure of the multiplication tables of split and non-split algebras differs only in respect to their associated [[sign tables|Sign Tables]] and not to their underlying [[Fano spaces|Fano Spaces]].

A $N = 2^n$-dimensional split-algebra has signature $(\frac{N}{2}-1, \frac{N}{2}+1)$, where the first number is the number of "-" signs and the second one the number of "+" signs.
For the split versions of the following Cayley\-Dickson algebras one therefore has the following signatures:
* Complex numbers: $(0,2)$
* [[Quaternions|Split Quaternion]]: $(1,3)$
* [[Octonions|Split Octonion]]: $(3,5)$
* [[Sedenions|Split Sedenion]]: $(7,9)$
* Trigintaduonions: $(15,17)$
The signs of the $N$ terms of the square of the norm of these algebras consists of an equal number of "+" and "-" signs (i.e. $N'$ of each). This might be the reason why these algebras are called split algebras.

For the ''extended involution'' one has
\[
(\mathbf{A}, \mathbf{B})^{*} = (\mathbf{A}^{*}, -\mathbf{B})
\]
and for the ''extended norm'' $||\,.\,||$
\[
 ||(\mathbf{A},\mathbf{B})||^2 = ||\mathbf{A}||^2 + \lambda ||\mathbf{B}||^2
\]

!!!!Alternative Doubling Formulas
Computer searches by Warrent Smith have yielded over 100 different doubling formulas beyond the octonions with all kind of different algebraic properties, most of them however "unpleasant".

Some examples are:
!!!!![[2n-ons|2n-Ons]]
!!!!![[Twisted Cayley-Dickson algebras|Twisted Cayley-Dickson Algebras]]
!!!!!Ternary Doubling Formula
In [1] a modified Cayley\-Dickson doubling process is described which results in a $16$-dimensional ternary algebra.

!!!!![[Clifford algebras|Clifford Algebra]]
A doubling process for Clifford algebras similar to the Cayley\-Dickson process is described in [2].

Papers:
* [[[2] From Clifford Algebras to Cayley Algebras - H. Albuquerque|http://www.cim.pt/files/FromLieAlgebrasToQuantumGroups.pdf#page=7]]  [[local|papers/FromCliffordToCayley.pdf]] pct. 0
* [[Algorithms for Programmers, Ideas and Source Code - J. Arndt|http://pds11.egloos.com/pds/200903/02/10/Algorithms_for_Programmers.pdf]] [[local|papers/Algorithms_for_Programmers.pdf]] - in particular: "37.14.1 The Cayley\-Dickson Construction"

Google books:
* [[[1] Ternary Sedenions and their Representations - J. Lõhmus, K. Sorgsepp|http://books.google.com/books?id=jVGE7UWXaGcC&pg=PA283&lpg=PA283&dq=%22ternary+sedenion%22&source=bl&ots=5SWm76FYh6&sig=F51fjabHGOdzXMsZYF55HvQJdIc&hl=de&ei=oJWXSvaoD87K_gbxu9S7BQ&sa=X&oi=book_result&ct=result&resnum=2#v=onepage&q=%22ternary%20sedenion%22&f=false]] [[local|journals/TernarySedenions.pdf]]

>It's probable, given a large enough Life space, initially in a random state,  that after a long time, intelligent self-reproducing animals will emerge and populate some parts of the space.
>- John H. Conway

The concept of a ''Cellular Automaton'' (''CA'') was initiated in the early 1950's by by John Von Neumann and Stan Ulam. Von Neumann showed that a cellular automaton can be universal.

The nine-cell neighbourhood CA (Moore neighbourhood), with two states per cell and appropriate rules, has been shown to be capable of universal computation. This structure has been utilized with a specified set of local rules to create the ''Game Of Life'' CA.
<html><center><img src="images/neighbourhood.jpg" style="width:310px; "/></center></html>

Relevant parameters of cellular automata are:
* Dimension (of the underlying grid)
* Number of neighbouring cells
* Boundary conditions (e.g. "null", periodic or "intermediate")
* Next state function (local rule): deterministic vs. probabilistic/"fuzzy", constant vs. time-/state dependent

Despite the simple construction of cellular automata, they are capable of highly complex behavior. For most cellular automata models, the only general method to determine the qualitative (average) dynamics of the system is to run simulations on a computer for various initial global configurations.
Wolfram classified \CAs into four broad categories:
* class 1: \CAs which evolve to a homogeneous state,
* class 2: \CAs displaying simple separated periodic structures,
* class 3: \CAs exhibiting chaotic or pseudo-random behaviour,
* class 4: \CAs which yield complex patterns of localized structures and are capable of universal computation.

!!!!Conway's "Game of Life"
<html><center><img src="images/Coway_life.jpg" style="width:440px; "/></center></html>
Remarkable features of this rule are:
* Its ability to give rise to complex ordered patterns out of an initially disordered state, or primordial soup.
* It has been proven to be capable of universal computation. I.e. by a proper selection of initial conditions, "Life" can be made to carry out arbitrary algorithmic procedures making it a general purpose computer.
* Since there are fundamental limits on the predictability of universal computers as for examples the famous Halting theorem, the same limits must apply to the general evolution of "Life".

!!!! [[Probabilistic cellular automata|Probabilistic Cellular Automaton]]
!!!!Applications
Cellular automata ...
* have been proposed as an alternative to [[differential equations|Differential Geometry]] in modelling laws of physics.
* allow for the the description of systems exhibiting non-equilibrium dynamics.

Links:
* [[Stephen Wolfram: Articles on Cellular Automata|http://www.stephenwolfram.com/publications/articles/ca/]]
* [[CelLab by Rudy Rucker and John Walker|http://www.fourmilab.ch/cellab/]] <<LaunchApplicationButton "launch locally" "simulation" "programs\CellLab\CELLAB.exe">>
* [[Quantum Computation Archive maintained by Tom Toffoli and Zac Walton|http://web.archive.org/web/20030501081438/http://pm1.bu.edu/~tt/qcl/]] - Interesting list of articles; links are dead, however.

Papers:
* [[A Survey on Cellular Automata - N. Ganguly, B. K. Sikdar,  A. Deutsch, G. Canright, P. P. Chaudhuri|http://polaris.ing.unimo.it/didattica/cas/L8/CAsurvey.pdf]] [[pct. 51|http://scholar.google.de/scholar?cites=6810868859815360340&hl=de]]
* [[Occam, Turing, von Neumann, Jaynes: How Much Can You Get For How Little?(A Conceptual Introduction to Cellular Automata - T. Toffoli|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.40.7699&rep=rep1&type=pdf]] [[local|papers/10.1.1.40.7699-2.pdf]] [[pct. 21|http://scholar.google.de/scholar?cites=8072421601376078705&hl=de]]
* [[Evolution of Fault-tolerant Self-replicating Structures - L. Righetti, S. Shokur, M. S. Capcarrere|http://birg2.epfl.ch/users/righetti/ecal03.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=5630930575514864362&hl=de]]
* [[Turing Machine Engineering and Immortality - W. D. Smith|http://www.math.temple.edu/~wds/homepage/immortal.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=8046887322951118560&hl=de]]

Theses:
* [[Spiking Cellular Associative Neural Networks for Pattern Recognition - G. Brewer|http://www.cs.york.ac.uk/ftpdir/reports/2008/YCST/03/YCST-2008-03.pdf]]

Presentations:
* [[Cellular Automata: Overview and Classical Results - S. Capobianco|http://www2.ru.is/kennarar/luca/CENTRE/Conferences/Seminars/SLIDES/silvio1.pdf]]

Videos:
* [[The H. Paul Rockwood Memorial Lecture: A New Kind of Science - Stephen Wolfram|http://www.uctv.tv/search-details.aspx?showID=7153]]

The ''Center'' $Z(\mathcal G)$ of a [[group|Group]] $\mathcal G$ is the subset of all elements of the group that commute with all other elements of the group.
\[
Z(\mathcal G) = \{z \in \mathcal G \ | gz = zg, \, \forall g \in \mathcal G \}
\]
$Z(G)$ is an abelian subgroup of $G$.
The quotient of a group and its center is isomorphic to the group of [[inner automorphisms|Automorphism]] of the group, thus
\[
\mathcal G/Z(\mathcal G) \simeq Inn (\mathcal G)
\]
!!!!Generalisations
The center of a [[non-associative algebra|Nonassociative Algebra]] is defined as the intersection of the center - defined in the same way as for a group - and the [[nucleus|Nucleus]].

Links:
* [[PlanetMath.org - Centralizers in Algebra|http://planetmath.org/?method=l2h&from=objects&name=AdditiveCommutator&op=getobj]]

Given a $n \times n$-matrix $\mathbf M$ the ''Characteristic Polynomial'' $p_{\mathbf M} (\lambda)$ is defined by
\begin{equation}
p_{\mathbf M} (\lambda) \equiv \det(\lambda \mathbf I_n- \mathbf M)
\end{equation}
It is the solution to the [[eigenvalue problem|Eigenvalue Theory]] $ \mathbf M \mathbf A = \lambda \mathbf A$.
The equation
\begin{equation}
p_{\mathbf 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_{\mathbf M} (\lambda) &=&  \lambda^2 ? \lambda (m_{11} + m_{22}) + (m_{11}m_{22} - m_{12}m_{21}) \\
& =& \lambda^2 ? \operatorname{Tr} (\mathbf M) \lambda + \det (\mathbf M)
\end{eqnarray}

An $n$-dimensional ''Checkerboard Lattice $\mathbb D_n$'' ($n \ge 3$) is the [[even sublattice|Lattice]] of the [[integer lattice|Integer Lattice]] $\mathbb Z^n$, given by:
\[
\mathbb D_n = \{(x_1, \ldots, x_n) \in \mathbb Z^n: \, \sum_{i = 1}^n x_i \in 2\mathbb Z^n\}
\]
In other words, $\mathbb D_n$ is obtained by coloring the points of an integer lattice alternately blue and white with a checkerboard coloring and taking the blue points for example.

<html><center><img src="images/D2_lattice.jpg" style="width: 255px; "/></center></html>
The [[kissing number|Kissing Number]] is given by $K(\mathbb D_n) = 2n(n-1) = 4 \large {n \choose 2} $. E.g. it is $4, 24, 112, 480, 1.104, 1.984$ for dimensions $2,4,8,16,24$ and $32$ respectively.

A possible basis $B$ of $\mathbb D_n$ is given by
\[
B= \{-\mathbf e_1 - \mathbf e_2,  \mathbf e_1 -\mathbf e_2 , \mathbf e_2 - \mathbf e_3, \ldots, \mathbf e_{n-1} - \mathbf e_n\}
\]
or, equivalently in terms of a [[generator matrix|Generator Matrix]], by
\[
M = \begin{pmatrix} -1 & -1  & 0 & \ldots & 0 & 0 \\
                     1 & -1  & 0 & \ldots & 0 & 0 \\
                     0 & 1  & -1 & \ldots & 0 & 0 \\
                     . & .  & . & \ldots & . & . \\
                     0 & 0  & 0 & \ldots & 1 & -1
 \end{pmatrix}
\]
Its associated [[Dynkin Diagram]] is depicted as follows:

<html><center><img src="images/DnDynkin.jpg" style="width: 425px; "/></center></html>

$\mathbb D_4$ represents the densest lattice packing in $\mathbb R^4$.

!!!! $\mathbb D_n^+$-lattices
A $\mathbb D_n^+$-packing is obtained if one "doubles" a $\mathbb D_n$-lattice. However, if and only if $n$ is even it is a lattice. For $n\, mod \; 4 = 0$ it is an integral lattice.

Some particular lattices are:
* $\mathbb D^+_4 \cong \mathbb Z^4$
* $\mathbb D^+_8 \cong \mathbb E_8$
A particular non-lattice packing, also known as tetrahedral or diamond packing, is $\mathbb D^+_3$. This is how carbon atoms are arranged in a diamond, the hardest known substance.

The existence of two inequivalent lattices [[E8xE8|E8xE8 Lattice]] and $\mathbb D^+_{16}$ that cannot be distinguished by their [[theta functions|Theta Series]] has a nice consequence in differential geometry: as Milnor observed, the tori $\mathbb R^{16}/\mathbb E^2_8$ and $\mathbb R^{16}/\mathbb D^+_{16}$ are non-isometric but [["isospectral"|Spectral Geometry]], in that their Laplacians have the same eigenvalues with the same multiplicities. In the language of Mark Kac: One cannot hear the difference between these two $16$-dimensional drums.

!!!!Applications
The compactification of the $16$ left-moving coordinates of the [[heterotic string|Heterotic String]] requires an even, self-dual lattice in $16$ dimensions. The only possible lattices are therefore the $\mathbb E^2_8$- and the $16$-dimensional $\mathbb D^+_{16}$-lattice. This gives rise to the $2$ heterotic string theories.

Google books:
* [[Coding Theory and Number Theory - T. Hiramatsu, G. Köhler|http://books.google.com/books?id=rujHMF_cHKcC&pg=PA89&lpg=PA89&dq=%22checkerboard+lattice%22+%22dynkin+diagram%22&source=bl&ots=RQn0lVXfNi&sig=Xxp1NSpnhQp86M5lXAq7XaS3e_Q&hl=de&ei=BnBRS8myB46nsQbyzLnYCw&sa=X&oi=book_result&ct=result&resnum=8&ved=0CC0Q6AEwBw#v=onepage&q=%22checkerboard%20lattice%22%20%22dynkin%20diagram%22&f=false]] [[bct. 6|http://scholar.google.de/scholar?cites=7435121675051430839&hl=de&as_sdt=2000]]

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

The ''Church\-Turing Hypothesis'' states that there are no computational concepts beyond that of a Turing machine.
This including humans that, to be in accordance with the hypothesis, also operate algorithmically and hence are not able to calculate functions beyond those that can be calculated by a Turing machine.

Papers:
* [[Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer - D. Deutsch|http://physics.princeton.edu/~mcdonald/examples/QM/deutsch_prsl_a400_97_85.pdf]] {{t1000Cite{[[pct. 2218|http://scholar.google.de/scholar?cites=11455492807395938722&hl=de&as_sdt=2000]]}}}

>That I have been able to accomplish anything in mathematics is really due to the fact that I have always found it so difficult. When I read, or when I am told about something, it nearly always seems so difficult, and practically impossible to understand, and then I cannot help wondering if it might not be simpler. And on several occasions it has turned out that it really was more simple!
>- David Hilbert -

>The ordinary man wonders at marvellous things;
>the wise man wonders at the commonplace
>- Confucius -

Links:
* [[Verman University Mathematical Quotations Server|http://math.furman.edu/~mwoodard/mqs/mquot.shtml]]

''Coding Theory'' is concerned with transmitting data across noisy channels and recovering the message. Coding theory is about making messages easy to read which should not be confused with cryptography which is the art of making messages hard to read.

The simplest method for detecting errors in binary data is the ''Parity Code'' which transmits an extra "parity" bit after every $7$ bits from the source message. However, this method can only detect errors, the only way to correct them is to ask for the data to be transmitted again.

A simple way to correct as well as detect errors can be accomplished by means of a ''Repetition Code'' for which each bit set is repeated a number of times. The recipient sees which value, "$0$" or "$1$", occurs more often and assumes that that was the intended bit. The scheme can tolerate error rates up to $1$ error in every $2$ bits.

The disadvantage of the repetition scheme is that it multiplies the number of bits transmitted by a factor which may prove unacceptably high.
In 1948, Claude Shannon inaugurated the whole subject of coding theory by showing that it was possible to encode messages in such a way that the number of extra bits transmitted was as small as possible. Unfortunately his proof did not give any explicit recipes for these optimal codes.

However two years later, [[Hamming|Hamming Code]] published details of his work on explicit error-correcting codes with information transmission rates more efficient than simple repetition.

Papers:
* [[Channel Coding: The Road to Channel Capacity - D. J. Costello, G. D. Forney|http://arxiv.org/PS_cache/cs/pdf/0611/0611112v1.pdf]] [[pct. 13|http://scholar.google.de/scholar?cites=817826004769441746&hl=de&as_sdt=2000]]
* [[The Quest for Error Correction in Biology Recent Developments in Codes and Biology - M. K. Gupta|http://magazine.embs.org/Past_Issues/2006January/Gupta.pdf]] [[pct. 5|http://scholar.google.com/scholar?cites=7535388237544424643&hl=de]]

Lectures:
* [[Codierungstheorie - P. Hauck|http://www-dm.informatik.uni-tuebingen.de/skripte/Codierungstheorie/CodierungstheorieSS2004.pdf]]

Videos:
* [[MITOPENCOURSEWARE: Principles of Digital Communication I - R. Gallager, L. Zheng|http://ocw.mit.edu/OcwWeb/Electrical-Engineering-and-Computer-Science/6-450Fall-2006/CourseHome/index.htm]]
* [[MITOPENCOURSEWARE: Principles of Digital Communication II - D. Forney|http://ocw.mit.edu/OcwWeb/Electrical-Engineering-and-Computer-Science/6-451Spring-2005/CourseHome/index.htm]]
* [[Coding Theory - MSRI (Mathematical Sciences Institute)|http://www.msri.org/calendar/workshops/WorkshopInfo/489/show_workshop]]

A ''Coherence Law'' (of degree $n$) roughly speaking says that given an algebra of dimension $n$ with basis $\{\mathbf e_1, \ldots, \mathbf e_n\}$, the way $n-1$ products between base elements have to be carried out is specified by $1, \ldots, n-2$-fold products of the algebra. 

!!!!Examples: 
$n = 3$, e.g. [[quaternions|Quaternion]]: 
The [[multiplication table|Multiplication Tables]] of such an algebra makes no statement as to how to put brackets in case of $2$ consecutive products (the algebra is $2$-graded). Therefore the expressions $(\mathbf e_i \mathbf e_j)\mathbf e_k$ and $\mathbf e_i (\mathbf e_j\mathbf e_k)$ must be transformable into one another by (a sequence) of $1$-fold products of the algebra. Put it differently, on a fundamental level the algebra is "blind" when it comes to distinguishing (generic) $2$-fold products, i.e. it is associative.
 
$n = 4$, e.g. [[octonions|Octonion]]: 
The [[multiplication table of such an algebra|Octonion Multiplication Tables]] makes no statement as to how to put brackets in case of $3$ consecutive products (the algebra is $3$-graded). Therefore the expressions $\mathbf e_i(\mathbf e_j(\mathbf e_k \mathbf e_l))$, $\mathbf e_i((\mathbf e_j \mathbf e_k)\mathbf e_l)$, $(\mathbf e_i(\mathbf e_j \mathbf e_k))\mathbf e_l$, $(\mathbf e_i \mathbf e_j)(\mathbf e_k \mathbf e_l)$, $((\mathbf e_i \mathbf e_j)\mathbf e_k)\mathbf e_l$ must be transformable into one another by (a sequence) of at maximum $2$-fold products of the algebra. Put it differently on a fundamental level the algebra is "blind" when it comes to distinguishing (generic) $3$-fold products.
This coherence law is known as [[Pentagon identity|Pentagon Identity]] and related to the [[Stasheff pentagon|Associahedron]] $\mathcal K(4)$. 

$n = 5$, e.g. [[sedenions|Sedenion]]: 
The [[multiplication table of such an algebra|Sedenion Multiplication Tables]] makes no statement as to how to put brackets in case of $4$ consecutive products (the algebra is $4$-graded). Therefore the expressions corresponding with [[association types|Association Type]] of degree $5$ (e.g. $(\mathbf e_i(\mathbf e_j(\mathbf e_k \mathbf e_l))\mathbf e_m$) must be transformable into one another by (a sequence) of at maximum $3$-fold products of the algebra. Put it differently on a fundamental level the algebra is "blind" when it comes to distinguishing (generic) $4$-fold products.
This coherence law is related to the [[Stasheff polytope|Stasheff Polytope]] $\mathcal K(5)$. 

Note, that a multiplication table of an algebra that specifies $n$-fold products, doesn't imply that this is essential for determining a coherence relation of degree $n+1$. This is the case for [[Clifford algebras|Clifford Algebra]], where due to the fact that they are in general associative, coherence laws for degree $n$ are completely determined by binary products. Therefore these algebras code less information as [[Cayley-Dickson algebras|Cayley-Dickson Algebra]] for example, i.e. they are the more "trivial" algebras in comparison and the information contained in their multiplication tables is (more) compressible (i.e. the multiplication tables possess a certain degree of redundancy).

According to M. Gertenhaber, “every restricted deformation theory generates its proper ''Cohomology'' theory”. The deformation theory of associative algebras (resp. [[Lie algebras|Lie Algebra]]) and their modules involves the ''Hochschild Cohomology'' theory of those algebras and modules (which is related to [[non-commutative geometry|Noncommutative Geometry]]). Today there doesn’t exist any standard way to associate a proper cohomology theory to every given category of algebras and modules.

Papers:
* [[Loop Cohomology - K. W. Johnson, C. R. Leedham-Green|http://www.dml.cz/bitstream/handle/10338.dmlcz/102372/CzechMathJ_40-1990-2_2.pdf]] [[local|papers/CzechMathJ_40-1990-2_2.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=18243030275162540099&hl=de]]

The obstruction imposed by the ''Coleman Mandula Theorem'' can be avoided by invoking supersymmetry or by using of dual numbers. The latter has been shown by Wald.

Papers:
* [[Universal Enveloping Algebras and Some Applications in Physics - Xavier Bekaert|http://www.ulb.ac.be/sciences/ptm/pmif/Rencontres/ModaveI/Xavier.pdf]]

The ''Coleman\-Mandula Theorem'' states that under certain natural hypotheses the symmetry group of the S-matrix must be a direct product of the [[Poincaré group|Poincaré Transformation]] and an internal symmetry. It is usually interpreted as a no-go theorem, forbidding a nontrivial mixing of spacetime and internal symmetries.

[[Supersymmetry]] ([[Haag-Łopuszański-Sohnius theorem|Haag-Łopuszański-Sohnius Theorem]]) and certain quantum groups famously manage to avoid the theorem: in these cases the symmetry is not an ordinary [[Lie group|Lie Group]], as assumed by the theorem. 
Apart from supersymmetry, [[dual numbers|Dual Number]] offer another possibility to avoid the obstruction imposed by the Coleman\-Mandula theorem (shown by Wald).

Papers:
* [[All Possible Symmetries of the S Matrix - S. Coleman, J. Mandula|http://hep.phy.tu-dresden.de/Lehre/SS2009/SUSY/literatur/coleman_madula_p1251_1.pdf]]  {{t500Cite{[[pct. 566|http://scholar.google.de/scholar?cites=2608624804188340434&hl=de&as_sdt=2000]]}}} - The original paper on the topic.
* [[Universal Enveloping Algebras and Some Applications in Physics - X. Bekaert|http://www.ulb.ac.be/sciences/ptm/pmif/Rencontres/ModaveI/Xavier.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=3148727214875758285&hl=de]]

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

The ''Commutator'' (a.k.a. ''Lie Bracket'') of two elements $\mathbf A$, $\mathbf B$ of an algebra $\mathcal A$ is defined as:
\[
[\mathbf A,\mathbf  B] = \mathbf{AB} - \mathbf{BA}
\]
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 $\mathbf{AB}$ with the commutator $[\mathbf A, \mathbf B]$ is denoted $\mathcal A^−$.

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

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

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]]}}}

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

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'' $C(\mathbf  X_i)$, $i = 1, \ldots, N$, which are formed by elements $\mathbf X_i \in \mathcal G$. The conjugacy classes are defined by
\begin{equation}
C(\mathbf X_i) = \{\mathbf X \in \mathcal G: \mathbf X = \mathbf A^{-1} \mathbf X_i \mathbf A\}
\end{equation}
with $\mathbf A$ any element of $\mathcal G$. The map $\mathbf X \mapsto \mathbf A^{-1} \mathbf X \mathbf A$ 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 $\mathbf A \in \mathcal G$ 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 $\mathbf A^{?1}(\mathbf X_i \mathbf X_j)\mathbf A = (\mathbf A^{?1}\mathbf X_i \mathbf A)(\mathbf A^{?1} \mathbf X_j \mathbf A)$. Moreover, if $\mathbf A^{?1}\mathbf X_i \mathbf A = \mathbf 1$, then $\mathbf X_i = \mathbf A\mathbf A^{?1} = \mathbf 1$, hence only the identity is mapped to the identity.

The conjugation $ \mathbf X = \mathbf A^{-1} \mathbf X_i \mathbf A$ is equivalent to $\mathbf{AX} = \mathbf X_i \mathbf A$. 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}
(\mathbf A \mathbf X_i) \mathbf B = \mathbf A (\mathbf X_i \mathbf B)
\end{equation}
If we assume that the [[left-and right inverse properties|Inverse Properties]] still hold, this can be rewritten as:
\begin{equation}
\mathbf X_i = \mathbf A^{-1}((\mathbf A((\mathbf X_i) \mathbf B)) \mathbf B^{-1})
\end{equation}
This can be regarded as the non-associative analog to the conventional conjugation
\begin{equation}
\mathbf A   = \mathbf B^{-1}\mathbf A \mathbf 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 (\mathbf{ABC} \rightarrow \mathbf{ABC}) & =& \{(\mathbf{ABC} \rightarrow \mathbf{ABC}) \} \quad && \text{Identity} \\
C (\mathbf{ABC} \rightarrow \mathbf{ACB}) &= & \{(\mathbf{ABC} \rightarrow \mathbf{ACB}), (\mathbf{ABC} \rightarrow \mathbf{CBA}), (\mathbf{ABC} \rightarrow \mathbf{BAC}) \} \quad && \text{Interchanges} \\
C (\mathbf{ABC} \rightarrow \mathbf{BCD}) &= & \{(\mathbf{ABC} \rightarrow \mathbf{BCD}), (\mathbf{ABC} \rightarrow \mathbf{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]]

Papers:
* [[Forces from Connes' Geometry - T. Schücker|http://www.esi.ac.at/preprints/esi1237.pdf]]

Links:
* [[Quantum Consciousness - Stuart Hameroff|http://www.quantumconsciousness.org/]]

Papers:
* [[Quantum Computation in Brain Microtubules|http://www.cs.indiana.edu/classes/b629-sabr/QuantumComputationInBrainMicrotubules.pdf]] {{t100Cite{[[pct. 119|http://scholar.google.de/scholar?cites=17233805516325330389&hl=de]]}}}
* [[Theory of Brain Function, Quantum Mechanics and Superstrings - D. Nanopoulos|http://arxiv.org/PS_cache/hep-ph/pdf/9505/9505374v1.pdf]] [[pct. 27|http://scholar.google.de/scholar?cites=15335628828834998581&hl=de]]
* [[Non-Computability of Consciousness - D. Song|http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.1617v1.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=4241235277389295173&hl=de]]

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 $\pi : X ? \mathbb R$.
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]]

See [[Covariance]].

The ''Cosmological Constant'' $\Lambda$ can be introduced via the [[Einstein equations|Einstein Field Equations]] as a global curvature term that adds to the [[Ricci curvature|Ricci Tensor]]. Modern field theory associates this term with the energy density of the vacuum. The measured value is negative which means that in the (near) absence of gravity spacetime is curved negatively.

''Cosmological Constant Problem:''
If the universe is described by an effective local quantum field theory down to the Planck scale, one would expect a cosmological constant of the order of $m_{\rm pl}^4$. In fact the measured value is smaller than that by a factor of $10^{120}$.

''Experimental status:''
Current experimental findings are consistent with the idea of [[dark energy|Dark Energy]] behaving like Einstein's cosmological constant,  i.e. it describes a density and pressure associated with "empty" space.
The latest Hubble data contradict theories that postulate that dark energy behaved differently billions of years ago to how it does today. The observations also confirmed that the expansion rate of the cosmos began speeding up about five to six billion years ago. This is when astronomers believe that dark energy's repulsive force starts dominating over the gravitational force.

Papers:
*[[New Hubble Space Telescope Discoveries of Type Ia Supernovae at z ? 1: Narrowing Constraints on the Early Behavior of Dark Energy - A. G. Riess, L.-G. Strolger, S. Casertano, H. C. Ferguson, B. Mobasher, B. Gold, P. J. Challis, A. V. Filippenko, S. Jha, W. Li, J. Tonry, R. Foley, R. P. Kirshner, M. Dickinson, E. MacDonald, D. Eisenstein, M. Livio, J. Younger, C. Xu, T. Dahlen, D. Stern|http://xxx.lanl.gov/PS_cache/astro-ph/pdf/0611/0611572v2.pdf]]  {{t100Cite{[[pct. 221|http://scholar.google.de/scholar?cites=14144653772503374306&hl=de]]}}}

''Covariance'' and ''Contravariance'' refer to the way tensors transform under a change of the coordinate system $x \rightarrow x' = x'(x)$.

Covariant transformation:
\begin{equation}
T_\mu \rightarrow T'_\mu = {\partial x^\nu \over \partial x'^\mu} {T}_\nu
\end{equation}
A tensor $T_\mu$ that transforms this way is called a covariant tensor.

Contravariant transformation:
\begin{equation}
T^\mu \rightarrow T'^\mu = {\partial x'^\mu \over \partial x^\nu} {T}^\nu
\end{equation}
A tensor $T^\mu$ that transforms this way is called a contravariant tensor.

Mixed covariant and contravariant transformation:
\begin{equation}
T^\mu_\nu \rightarrow T'^\mu_\nu = {\partial x'^\mu \over \partial x^\rho} {\partial x^\sigma \over \partial x'^\nu} {T}^\rho_\sigma
\end{equation}

Example vectors:
\begin{equation}
\vec v =x'^\mu \mathbf{\mathbf{e'}}_\mu= {\partial x'^\mu \over \partial x^\nu} x^\nu {\partial x^\rho \over \partial x'^\mu} \mathbf{e}_\rho  = x_\mu \mathbf{e}^\mu
\end{equation}
Therefore components and basis vectors transform with opposite variance.

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]]

''Curie's Principle'' states that the symmetry of a cause is always preserved in its effects. It was published in 1894 by Pierre Curie.

Papers:
* [[Curie's Principle - J. Ismael|http://www.usyd.edu.au/time/ismael/papers/curies_principle.pdf]] [[pct. 11|http://scholar.google.com/scholar?hl=de&lr=&cites=1122593746217649839&um=1&ie=UTF-8&ei=IQeuSqC_N8HDsgauzvjsBw&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]

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]]

The ''D'Alembert Equation'' is given by
\[
\partial_\mu\partial^\mu \Phi (\mathbf x) \equiv  \square  \Phi (\mathbf x) = 0
\]
with $ \Phi (\mathbf x)$ a scalar field.

In [[Superstring Theory]] a string can be either closed or open. For a closed string one has periodic, for an open string Dirichlet- or Von Neumann-boundary conditions. The latter were considered unphysical in the beginning, as they break Lorentz invariance. However when [[p-branes|P-Brane]] were discovered it was realized, that open strings can end on them and satisfying the Dirichlet boundary condition. Such special p-branes are called Dp-branes, with the "D" standing for "Dirichlet" or short ''D-branes''.

Papers:
* [[Dark Eneregy and 3-Manifold Topology - T. Asselmeyer-Maluga, H. Rosé|http://th-www.if.uj.edu.pl/acta/vol38/pdf/v38p3633.pdf]]

''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.

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.

!!!!De Sitter Space-time
De Sitter Spacetime is the most symmetric spacetime, namely a homogeneous constant-curvature spacetime with cosmological constant $\Lambda$.
For a $n$-dimensional space-time the relevant symmetry groups are:
\begin{equation}
    G = \left\{
      \begin{array}{cll}
          SO(n,1) & \Lambda > 0  &  \text { (de Sitter)}\\
          ISO(n-1,1)&  \Lambda = 0 & \text { (Minkowski)} \\
          SO(n-1,2) &  \Lambda < 0 &\text { (anti de Sitter)}
      \end{array}
      \right.
\end{equation}
The metric is given by
\begin{equation}
d\tau^2 = dt^2 - e^{-2\Lambda t} dr^2
\end{equation}
Papers:
* [[The de Sitter and Anti-de Sitter Sightseeing Tour - U. Moschella|http://www.bourbaphy.fr/moschella.ps]] [[pct. 3|http://scholar.google.de/scholar?hl=de&lr=&cites=8046863754320787269]] [[local|papers/Moschella.ps]]

[[Welcome]]

!!!!T\-Design
A ''T\-Design $t-(v, k, \lambda)$'' is a set of $v$ points together with a collection of subsets of size $k$ (called ''Blocks'') such that each set of $t$ points is contained in precisely $\lambda$ blocks.

Associated with a $t-(v, k, \lambda)$ design is a $(t?1)-(v ?1, k ?1, \lambda)$-design which is called a ''Derived Design''.

<html><center><img src="images/t-designs.jpg" style="width: 545px; "/></center></html>

Until the early 80s it was believed that t-designs only exist for $t\le 5$. Then S. Magliveras und D. W. Leavitt discovered a $6-(33,8,36)$ design and subsequently L. Teirlinck showed by a famous theorem, that t-designs exist for all $t$.

An important subclass of t-designs are [[resolvable t-designs|Resolvable Design]].

!!!! Balanced Incomplete Design
A ''Balanced Incomplete Block Design $(v, k, \lambda, r, b)$'' or ''BIBD'' is a special t-designs, namely a $2-(v, k, \lambda)$ t-design and can be described as follows:
Given a set of $v \ge 2$ elements called ''Varieties'', ''Treatments'' or ''Points'' and a collection of $b > 0$ subsets, called ''Blocks'', the following conditions hold:
* __each block__ consists of exactly $k$ points with $v > k > 0$, i.e. the block-size is $k$,
* __each point__ appears in exactly $r$ blocks with $r > 0$,
* __each pair of points__ appear simultaneously in exactly $\lambda$ blocks with $\lambda > 0$.
The designation "Incomplete" refers to the condition $v > k$. If one allowed for $v = k$, the design would be trivial. Furthermore "Balanced" refers to the constancy of the parameter $\lambda$.

''Theorem:''
Given a $(v,b,r,k,\lambda)$-design, necessary conditions that its parameters must satisfy are
\begin{eqnarray}
bk &=& vr \\
r(k-1)& =&  \lambda(v-1)
\end{eqnarray}
Since only three of the five parameters are independent, the shorter notation ''$(v, k, \lambda)$-BIBD'' is also used to represents a BIBD on $v$ points, with block size $k$ and index $\lambda$.

A $(v, k, \lambda)$-BIBD with $k = 3$ and $\lambda = 1$ is called a [[Steiner triple system|Steiner Triple System]].

!!!!!Configuration
A $(v', k')$-configuration in a BIBD is a subset of $k'$ blocks whose union is an $v'$-element subset of points of the design. A ''Pasch Configuration'' or ''Quadrilateral'' is a ''$(6, 4)$-configuration'' in a [[Steiner triple system|Steiner Triple System]]. (For an example see there).

!!!! Symmetric block design
For a symmetric $(v, k, \lambda, r, b)$-block design, a.k.a. ''SBIBD'', __the number of blocks equals the number of points__, $b=v$. Equivalently: $r=k$. It follows that the [[incidence matrix|Incidence Matrix]] of such a design is a square matrix.
All \SBIBDs satisfy $4m - 1\le v \le m^2 + m + 1$. [[Projective planes|Projective Plane]] attain the upper bound. [[Hadamard designs|Hadamard Design]] satisfy the lower one.
A ''Dual Design'' of a symmetric design is obtained by exchanging blocks and points. This construction however fails for nonsymmetric designs. A special and well known case of such a duality is related to the exchange of points and lines of a [[projective plane|Projective Plane]].

''Theorem:''
There exists a [[Hadamard matrix|Hadamard Matrix]] of order $4m$ if and only if there exists a (symmetric) $(4m-1,2m-1,m-1)$-BIBD. Such a design is a [[Hadamard 2-Design|Hadamard Design]].

!!!!Example
Let $P = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ and $B = \{\{(1, 2, 3) , (4, 5, 6) , (7, 8, 9) , (1, 4, 7) , (2, 5, 8) , (3, 6, 9), (1, 5, 9), (2, 6, 7), (3, 4, 8), (1, 6, 8), (2, 4, 9), (3, 5, 7)\}\}$,
then $(P,B)$ is a $(9,3,1)$-BIBD which is also a Steiner triple system. (One can take any pair of points in a block and will not find this pair in any other block).

Papers:
* [[Isomorph-free Exhaustive Generation of Combinatorial Designs - P. Kaski|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.74.7345&rep=rep1&type=pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=17864799794830258171&hl=de&as_sdt=2000]]
* [[A Short Course in Combinatorial Designs - I. Anderson, I. Honkala|http://users.utu.fi/honkala/designs.ps]] [[pct. 2|http://scholar.google.com/scholar?hl=de&lr=&cites=16950545933888252384&um=1&ie=UTF-8&ei=kQ4-S5G1MZLInAPRz4zPBA&sa=X&oi=science_links&resnum=9&ct=sl-citedby&ved=0CEIQzgIwCA]]
* [[Partitioning of Hypercubes by Resolution of Combinatorial Designs - H. C. Burg|https://www.fz-juelich.de/zam/files/docs/ib/ib-96/ib-9619.ps]] [[local|papers/PartitioningHypercubes.pdf]] pct. 0

Theses:
* [[Aufbau einer SQL-Datenbank für t-Design Parametersätze unter Einbezug von Regeln - M. Weiß|ftp://ftp.mathe2.uni-bayreuth.de/DIPLOM/ba_weiss.pdf]] [[local|theses/ba_weiss.pdf]] trl. 8
* [[A new Polyhedral Approach to Combinatorial Designs - A. Mercado|http://txspace.tamu.edu/bitstream/handle/1969.1/358/etd-tamu-2004A-INEN-Arambula-1.pdf?sequence=1]]

Lectures:
* [[Research in Designs & Codes - K. Martin|http://www.math.ou.edu/~kmartin/papers/designs-codes2.pdf]]

Presentations:
* [[Combinatorial Designs: Balanced Incomplete Block Designs|http://www-math.cudenver.edu/~wcherowi/courses/m6409/Blockdesigns.pdf]]

Given two manifolds $\mathcal M$ and $\mathcal M'$, a ''Diffeomorphism'' is a smooth, bijective and differentiable function, say $f$, from $\mathcal M$ to $\mathcal M'$ which has an inverse $f^{-1}$ which is also smooth and differentiable.

''Properties:''
A diffeomorphism is always a [[homeomorphism|Homeomorphism]] whereas the converse is not true in general (see [[exotic geometries|Exotic Geometry]]).

>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) -

Papers:
* [[Discrete Differential Calculus, Graphs, Topologies and Gauge Theory - A. Dimakis, F. Müller-Hoissen|http://arxiv.org/PS_cache/hep-th/pdf/9404/9404112v2.pdf]] [[pct. 72|http://scholar.google.de/scholar?cites=15140035106911831270&hl=de&as_sdt=2000]]
* [[Discrete Riemannian Geometry - A. Dimakis, F. Müller-Hoissen|http://arxiv.org/PS_cache/gr-qc/pdf/9808/9808023v1.pdf]] [[pct. 53|http://scholar.google.de/scholar?cites=15200090245903738717&hl=de&as_sdt=2000]]

Links:
* [[Zuse's Thesis: The Universe is a Computer|http://www.idsia.ch/~juergen/digitalphysics.html]]
* [[Digital Philosophy.org|http://www.digitalphilosophy.org/]]

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 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|Burkhard Heim]])
* Ons (Goertzel)

See also:
* [[Cellular automaton|Cellular Automaton]]
* [[Process physics|Process Physics]]
* [[Discrete spacetime|Discrete Spacetime]]
* [[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/]]

Papers:
* [[Algorithmic Theories of Everything - J. Schmidhuber|http://arxiv.org/PS_cache/quant-ph/pdf/0011/0011122v2.pdf]] [[pct. 46|http://scholar.google.de/scholar?cites=7282820845356865291&hl=de]]

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.

Papers:
* [[Noncommutativity and Discrete Physics - L. H. Kauffman|http://www2.math.uic.edu/~kauffman/NCDP.pdf]] [[pct. 14|http://scholar.google.de/scholar?cites=3342187083863786167&hl=de]]

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]]

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

See also:
* [[World crystal|World Crystal]]
* [[Spin networks|Spin Network]]

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

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

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

Papers:
* [[Hyperbolic Weyl Groups and the four Normed Division Algebras - A. J. Feingold, A. Kleinschmidt, H. Nicolai|http://aps.arxiv.org/PS_cache/arxiv/pdf/0805/0805.3018v1.pdf]]

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]]

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!}$

''Dual Numbers'' are a variant of complex numbers, having a basis $\{\mathbf e, \mathbf e_1\}$ with a nilpotent "imaginary unit", i.e. $\mathbf e_1^2 = 0$.

In [[composition algebras|Composition Algebra]] there is a one to one correspondence between non-standard [[involutions|Involution]] (of the first kind) and composition subalgebras of half rank.

It can be shown that every such involution is a map $D : \mathbb A_n = \mathbb A_{n-1} \times \mathbb A_{n-1} \rightarrow  \mathbb A_n$, given by
\[
D ({\mathbf A}) \equiv \tilde {\mathbf A} = \widetilde {(\mathbf A_1,\mathbf A_2)} = (\mathbf A_2^{*},  \mathbf A_1)
\]
with ${}^{*}$ the standard involution of $\mathbb A_{n-1}$.

$D$ is algebraically identical with the [[duality|Duality]] operation ${ }^\sim$ which is widely used in physics and mathematics. It furthermore coincides with what is known as ''Reciprocity'' or ''Reciprocal Transformation''. The duality involution is also used in the context of non-composition algebras (e.g. for $\mathbb A_n$ with $n \ge 4$, see [1]).

We will also use the duality involution in a wider context than just with composition-algebras and often simply refer to it as "duality".

!!!!Properties
!!!!!1.
\begin{eqnarray}
D (\mathbf A_1, \mathbf A_2) & = & (\mathbf A_2^*, \mathbf A_1) \\
D^2 (\mathbf A_1, \mathbf A_2)  & = & (\mathbf A_1^*, \mathbf A_2^*) = (\mathbf A_1, \mathbf A_2)^* \\
D^3 (\mathbf A_1, \mathbf A_2) & = & (\mathbf A_2, \mathbf A_1^*) \\
D^4 (\mathbf A_1, \mathbf A_2) & = & (\mathbf A_1, \mathbf A_2)^{**} =  (\mathbf A_1, \mathbf A_2)
\end{eqnarray}
The duality involution therefore is an involution of period $4$. Furthermore one has the relationship $D^2 = {}^{*}$ between the duality- and the standard involution. (The latter being an involution of period $2$).

!!!!!2.
\begin{eqnarray}
D(\mathbf {AB}) &=& D((\mathbf A_1, \mathbf A_2)(\mathbf B_1, \mathbf B_2))  \\
                             &=&D(\mathbf A_1\mathbf B_1 - \lambda \mathbf B_2^* \mathbf A_2, \mathbf B_2\mathbf A_1+ \mathbf A_2 \mathbf B_1^*)\\
&=& (\mathbf B_1 \mathbf A_2^* + \mathbf A_1^*\mathbf B_2^*,\mathbf A_1\mathbf B_1 - \lambda \mathbf B_2^* \mathbf A_2 )
\end{eqnarray}

!!!!!3.
\[
||\mathbf A||^2 = ||\tilde {\mathbf A}||^2 = ||\mathbf A_1||^2 + ||\mathbf A_2||^2
\]

!!!!! 4.
If $\mathbf B$ is [[doubly pure|Pure Element]] then $\widetilde{\mathbf {AB}} = - \tilde {\mathbf A} \mathbf B$.

If both $\mathbf A$ and $\mathbf B$ are doubly pure then
\begin{eqnarray}
\tilde {\mathbf A} \mathbf B + \tilde {\mathbf B} \mathbf A &=& 0 \;\;\text{iff} \;\; \mathbf A ? \mathbf B \\
\mathbf A \mathbf B - \tilde{\mathbf B}\tilde{\mathbf A} &=& 0 \;\;\text{iff} \;\; \tilde{\mathbf A} ? \mathbf B \\
\tilde{\mathbf A} \mathbf B - \mathbf A \tilde{\mathbf B} &=& 0 \;\;\text{iff} \;\;  \mathbf A  \perp \mathbf B \; \text {and} \; \tilde{\mathbf A } \perp \mathbf B  \\
\tilde{\mathbf A} \mathbf B &=& 0  \;\;\text{iff} \;\;  \mathbf {AB} = \mathbf 0
\end{eqnarray}
!!!!Examples
!!!!!Electrodynamics
In [[electrodynamics|Electrodynamics]] the duality involution (a.k.a. ''Electric/Magnetic Reciprocity'') represents a [[duality rotation|Duality Rotation]] of the electromagnetic field by an angle of $90^{\circ}$:
\[
\begin{pmatrix} \vec E  \\ \vec B \end{pmatrix} \xrightarrow D \begin{pmatrix} - \vec B \\  \vec E \end{pmatrix} \xrightarrow D
\begin{pmatrix} -\vec E \\  -\vec B \end{pmatrix}  \xrightarrow D \begin{pmatrix} \vec B  \\ -\vec E  \end{pmatrix}  \xrightarrow D  \begin{pmatrix} \vec E  \\ \vec B  \end{pmatrix}
\]
A duality involution can therefore be interpreted as a special case of a duality rotation.

!!!!!Phase Space
In the context of phase space, duality also goes under the name ''Phase Space Reciprocity'' or ''Born Reciprocity''. The laws of physics are invariant under the reciprocal transformation $(q, p) \rightarrow (-p, q)$, where $q$, $p$ are the position and momentum respectively.
Examples:

Bosonic harmonic oscillator:
Hamiltonian: $H = \frac{1}{2} (q^2 + p^2)$
Commutation relations: $[q, p] = i,\, [q, q] = [p, p] = 0$

Fermionic harmonic oscillator:
Hamiltonian: $H = \frac{i}{2} (qp ? pq)$
Anti-commutation relations: $\{q, p\} = 0, \, \{q, q\} = \{p, p\} = 1$

All these identities are invariant under the given map.

Papers:
* [[Alternative Elements in the Cayley–Dickson Algebras - G. Moreno|http://arxiv.org/PS_cache/math/pdf/0404/0404395v1.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=4013125457504762682&hl=de]]
* [[Space, Phase Space and Quantum Numbers of Elementary Particles - P. ?enczykowski|http://th-www.if.uj.edu.pl/acta/vol38/pdf/v38p2053.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=11380504504379261666&hl=de&as_sdt=2000]]
* [[Inductive Multiplication in Dickson Algebras - F. Chaitin-Chatelin|http://www.umcs.maine.edu/~chaitin/f11.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=18436882527483780969&hl=de&as_sdt=2000]]
* [[Involutions on Composition Algebras - S. Pümplin|http://homepage.uibk.ac.at/~c70202/jordan/archive/unitams/unitams.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=301299799629836786&hl=de]]
* [[[1] Alternativity and Reciprocity in the Cayley-Dickson Algebra - S. Kuwata, H. Fujii, A. Nakashima|http://arxiv.org/PS_cache/hep-th/pdf/0601/0601033v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=5646731258760761994&hl=de&as_sdt=2000]]

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.

Papers:
* [[A Review of E Infinity Theory and the Mass Spectrum of High Energy Particle Physics - M.S. El Naschie|http://www.complexity.ru/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 - M.S. El Naschie|http://www.el-naschie.net/bilder/file/7.%20The%20VAK%20of%20vacuum%20fluctuation,%20spontaneous.pdf]] [[pct. 41|http://scholar.google.de/scholar?cites=16744730194865314043&hl=de]]
* [[From Arthur Cayley via Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan, Emmy Noether and Superstrings to Cantorian Space–Time - L. Marek-Crnjac|http://www.el-naschie.net/bilder/file/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 - 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]]

''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]]

> $E_8$ is as complicated as symmetry can get. Mathematics can almost always offer another example that's harder than the one you're looking at now, but for Lie groups, $E_8$ is the hardest one. The literature on this subject is very dense and very difficult to understand.
> - David Vogan -
<html><center><a href="http://www.flickr.com/photos/reactiongrid/show/with/3254287800/"><img src="images/E8Polytope.png" style="width: 465px; "/></a></center></html>
''$E_8$'' is the largest of the $5$ exceptional [[Lie groups|Lie Group]]. It has a rank $8$ [[root system|E8 Lattice]] consisting of $240$ [[root vectors|Root Vector]] spanning $\mathbb R^8$.

There exist 4 forms (3 real and 1 complex):
!!!!Real
* $E_{8\left(-248\right)}$, ''compact'', (which is usually the one meant if no other information is given), simply connected, trivial [[outer automorphism group|Automorphism]].
* [[E8(8)]], ''split'', maximal compact subgroup $Spin(16)/\mathbb Z_2\,$, double cover, trivial outer automorphism group.
* $E_{8\left(-24\right)}$ (quaternionic $E_8$), maximal compact subgroup $E7×SU(2)/(?1×?1)$, double cover, trivial outer automorphism group.
!!!!Complex
* $E_8$, dimension $248 =744/3$ (= real dimension $496$), simply connected, maximal compact subgroup is the compact form of $E_8$, outer automorphism group of order $2$ generated by complex conjugation.

$E_8$ is the group of [[isometries|Isometry]] of the [[octooctonionic projective plane|Octooctonionic Projective Plane]]. "Defining" the octooctonionic projective plane is however difficult and as yet has not been managed without explicit reference to $E_8$, which makes this relationship somewhat circular.

!!!!$E_8$ and Physics
$E_8$ arises in [[heterotic string theory|Superstring Theory]] because in order for the initial reduction from $26$ to $10$ dimensions to proceed consistently, one needs to endow a $16$-dimensional subspace of the original $26$-dimensional space with an [[even, unimodular lattice|Lattice]]. It turns out that there are exactly two such lattices in $16$ dimensions, one of which is the [[root lattice|Root Lattice]] of $E_8 \oplus E_8$.

Papers:
* [[Should E8 SUSY Yang-Mills be Reconsidered as a Family Unification Model? - S. L. Adler|http://arxiv.org/PS_cache/hep-ph/pdf/0201/0201009v3.pdf]] [[pct. 9|http://scholar.google.de/scholar?cites=10661975116370373957&hl=de&as_sdt=2000]]
* [[The Octic E8 Invariant - M. Cederwall, J. Palmkvist|http://arxiv.org/PS_cache/hep-th/pdf/0702/0702024v1.pdf]] [[pct. 6|http://scholar.google.de/scholar?cites=16361402885191875124&hl=de&as_sdt=2000]]
* [[Further Thoughts on Supersymmetric E8 as a Family and Grand Unification Theory - S. L. Adler|http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3Ahep-ph%2F0401212]] [[pct. 5|http://scholar.google.de/scholar?cites=16880445888517271289&hl=de&as_sdt=2000]]
* [[A Novel View on the Physical Origin of E8 - Matej Pavsic|http://arxiv.org/PS_cache/arxiv/pdf/0806/0806.4365v1.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=10420560560647573279&hl=de]]
* [[A Geometric Construction of the Exceptional Lie Algebras F4 and E8 - J. F.-O’Farill|http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.2829v1.pdf]] pct. 0

Presentations:
* [[Special Geometries in Mathematical Physics The E8 challenge - I. Agricola|http://www-irm.mathematik.hu-berlin.de/~agricola/material/bms-E8-final.pdf]]

Links:
* [[Mathematicians Map E8|http://www.aimath.org/E8/]]

> There are innumerable ways to make E8.
> - Robert Wilson [1]

The ''$E_8$ Lattice'' (short ''$\Gamma_8$'' or ''$\mathbb E_8$''), which also goes under the name ''Gosset Lattice'', is identical with the $8$-dimensional [[Barnes-Wall lattices|Barnes-Wall Lattice]] $BW_8$.

It is the unique [[even, unimodular lattice|Lattice]] in less than $16$ dimensions and as no other lattice in $8$ dimensions has higher density, it is an important example in [[coding theory|Coding Theory]] and in respect to the general [[sphere-packing problem|Kissing Number]].

$\mathbb E_8$ is the [[root lattice|Root Vector]] of the group [[E8]].

The $240$ elements of the inner shell of $\mathbb E_8$ can be represented by means of the $240$ [[integral octonions|Integral Octonion]], which are elements of the unit octonion [[7-sphere|7-Sphere]] (i.e. $\mathbb E_8 \subset S^7$) and which form a closed non-associative discrete algebra. This implies various [[Hopf fibrations|Hopf Map]] $H_2:S^{7}  \xrightarrow{S^3} \ S^4$.
On the other hand the inner shell of the $E_8$-lattice can be seen as one of the Hopf-fibrations $H_3:S^{15}  \xrightarrow{S^7} \ S^8$.

One way to classify the $E_8$-roots is as follows:
i) $2$ real roots $\{\pm \mathbf e_0\}$ of [[SU(2)]]
ii) $126 = 112 + 7\cdot2 = 2\cdot 7\cdot 2^4 + 7\cdot2$ imaginary roots of [[E7]]
iii) $112 = 2\cdot 7\cdot 2^4$  roots with non-zero scalar parts of $E_8/(E_7 \times SU(2))$

Using icosians it is also possible to give an embedding of the $E_8$-root system into $\mathbb H$, however at the cost of introducing a new inner product (for details see [2], [3]).

The [[Weyl group|Weyl Group]] $W(E_8)$ of the $E_8$-lattice is the group generated by reflections in the hyperplanes orthogonal to the $240$ roots of the lattice. Its [[order|Order]] is
\[
    \operatorname{ord}(W(\mathrm{E}_8)) = 696.729.600 = 4! \cdot 6! \cdot 8!
\]
For the formula of the order see: [[Barnes-Wall lattice|Barnes-Wall Lattice]].
!!!!Sublattices
$\mathbb E_8$ contains the $A_8$- and $D_8$-lattices.

!!!!Applications
* The $E_8$-lattice has made its appearance in the theory of [[quasicrystals|Quasicrystal]] where phenomena can be described by certain natural projections of the lattice down to $2$, $3$ and $4$ dimensions.

!!!![[MAGMA|http://magma.maths.usyd.edu.au/calc/]]^^[[Help|MAGMA]]^^ examples
{{{
L := Lattice("Lambda", 8);
L2 := Lattice("E", 8);
IsIsomorphic(L, L2);
IsIsomorphic(L, Dual(L2));
AutomorphismGroup(L)
IsEven(L)
}}}

Papers:
* [[Octonion X-Product and E8 Lattices - G. Dixon|http://arxiv.org/PS_cache/hep-th/pdf/9411/9411063v1.pdf]] [[pct. 8|http://scholar.google.de/scholar?cites=4817530804157203118&hl=de]] prl. 10
* [[[2] Hyperbolic Weyl Groups and the four Normed Division Algebras - A. J. Feingold, A. Kleinschmidt, H. Nicolai|http://aps.arxiv.org/PS_cache/arxiv/pdf/0805/0805.3018v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=6176913252844757445&hl=de]]
* [[Symmetries of the Octonionic Root System of E8 - M. Koca|http://streaming.ictp.trieste.it/preprints/P/90/275.pdf]] [[local|papers/275.pdf]] pct. 0 prl. 10
* [[Gosset’s Figure in a Clifford Algebra - D. A. Richter|http://homepages.wmich.edu/~drichter/papersrichter/gossetfigurecliffordalgebra2004.pdf]] [[local|papers/gossetfigurecliffordalgebra2004.pdf]] pct. 0
* [[Quaternionic Roots of E8 Related Coxeter Graphs and Quasicrystals - M. Koca, N. Ö. Koca|http://journals.tubitak.gov.tr/physics/issues/fiz-98-22-5/fiz-22-5-8-97022.pdf]] pct. 0
* [[[3] E8 Lattice with Octonions and Icosians - M. Koca|http://cdsweb.cern.ch/record/197019/files/198906327.pdf?version=1]] pct. 0

Lectures:
* [[[1] E8 - R. A. Wilson|http://www.maths.qmul.ac.uk/~raw/talks_files/E8.pdf]]

$E_{8\left(8\right)}$ 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) - M. Günaydin, K. Koepsell, H. Nicolai|http://arxiv.org/PS_cache/hep-th/pdf/0109/0109005v2.pdf]] [[pct. 39|http://scholar.google.de/scholar?cites=13037309818640150601&hl=de&as_sdt=2000]]
* [[An Exceptional Geometry for d = 11 Supergravity? - K. Koepsell, H. Nicolai, H. Samtleben|http://arxiv.org/PS_cache/hep-th/pdf/0006/0006034v1.pdf]] [[pct. 30|http://scholar.google.de/scholar?cites=6784179047447449427&hl=de&as_sdt=2000]]

>I believe there are
>$15.747.724.136.275.002.577.605.653.961.181.555.468.044.717.914.527.116.709.366.231.425.076.185.631.031.296 $
>protons in the universe and the same number of electrons.
> - Arthur Eddington, Mathematical Theory of Relativity (1923) -
Eddington arrived at the outrageous conclusion of the citation above after a series of convoluted (and wrong!) calculations in which he first "proved" that the value of the so-called fine-structure constant was exactly $1/136$. This value appears as a factor in his prescription for the number of particles (protons + electrons; neutrons were not discovered until 1930) in the universe: $2 \cdot 136 \cdot 2^{256} = 17 × 2^{260} = 3.149544\ldots \cdot 10^{79}$ (double the number written out in full in the quote above). This is the Eddington number, notable for being the largest specific integer (as opposed to an estimate or approximation) ever thought to have a unique and tangible relationship to the physical world. However, experimental data gave a slightly lower value for the fine-structure constant, closer to $1/137$. Unfazed, Eddington simply amended his "proof" to show that the value had to be exactly $1/137$, prompting the satirical magazine Punch to dub him "Sir Arthur Adding\-One."

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]]

An ''Einstein Space'' or ''Einstein Manifold'' is a [[Riemannian or pseudo-Riemannian space|Riemann Space]] whose [[Ricci tensor|Ricci Tensor]] is proportional to the [[metric|Metric Tensor]], i.e.
\begin{equation}
R_{\mu\nu} = k g_{\mu\nu}
\end{equation}
with $k$ a constant.
If $k=0$ the space is called ''Ricci-flat''.

Dimension $4$ ist the smallest dimension where non-trivial Einstein-metrics occur.

''Einstein–Cartan Theory'' extends general relativity to allow for the description of spin.
The description of spacetime includes [[torsion|Torsion]] and the energy-momentum tensor is generally nonsymmetric.
The Einstein–Cartan theory differs only very slightly from the Einstein theory. The effects of spin and torsion are significant only at very high densities of matter, however still   much smaller than the Planck density at which quantum gravitational effects are believed to dominate.

Papers:
* [[The Einstein–Cartan Theory - A. Trautman|http://www.fuw.edu.pl/~amt/ect.pdf]]
*[[On a Completely Antisymmetric Cartan Tensor - L. Fabbri|http://arxiv.org/PS_cache/gr-qc/pdf/0608/0608090v2.pdf]]

The ''Weak Equivalence Principle'' states that all particles follow the same path in a gravitational field independent of their mass.

The ''Strong Equivalence Principle'' states that an accelerated reference frame is equivalent to gravitation, or that mass curves space, 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.

''Exact Sequences'' play a crucial role in [[(co)homological algebra|(Co)homology]].

An exact sequence is defined as a sequence of objects $A_n$ (e.g. groups, modules, chain complexes, sheaves) and [[morphisms|Homomorphism]] (called ''Boundary Operators'') $\delta_n$ between them
\[
\ldots \to A_{n-1} \overset{\delta_n}{\to}A_n\overset{\delta_{n+1}}{\to}A_{n+1}\to \ldots
\]
such that the image of one morphism equals the kernel of the next,
\[
\text{Im} (\delta_{n}) =\text{Ker} (\delta_{n+1})
\]

A special case is a ''Short Exact Sequence'' which is of the form
\[
0 \overset{\delta_1} \to A_1 \overset{\delta_2}{\to}A_2\overset{\delta_3}{\to}A_3\overset{\delta_4}{\to}  0
\]
!!!!Examples
!!!!!Exact sequence
$n$-chain- and $n$-cochain complexes (see [[homology/cohomology|(Co)homology]]).

!!!!!Short exact sequence
The following facts from classical vector analysis
\begin{eqnarray}
\operatorname{curl}\,(\operatorname{grad}\,f) &=& \nabla \times (\nabla f) = 0 \\
\operatorname{div}\,(\operatorname{curl}\,\vec v) &= &\nabla \cdot \nabla \times \vec{v} = 0
\end{eqnarray}
constitute a short exact sequence
\[
\Bbb R \ \xrightarrow{\mbox{grad}}\ \Bbb{R}^3  \xrightarrow{\mbox{curl}}\ \Bbb R^3\ \xrightarrow{\mbox{div}}\ \Bbb R
\]

Papers:
* [[A Proof of the De Rham Theorem using Induction on Open Sets - M. Price|http://www.cs.bath.ac.uk/MPrice/downloads/DeRham.pdf]] pct. 0

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]]

The ''Exterior Derivative'' $\mathbf{d}$, which goes back to Élie Joseph Cartan, generalises the classical Leibniz differential known from analysis and the differential operators $\operatorname{grad}$, $\operatorname{div}$ and $\operatorname{curl}$ known from vector analysis to [[differential forms|Differential Form]].  

The exterior derivative transforms [[p-forms|Differential Form]] $\mathbf{\omega}$ into $(p+1)$-forms $\mathbf{d\omega }$, with the main property ([[Poincaré lemma|Poincaré Lemma]]) that
\[
\mathbf{dd}=\mathbf{d}^{2}=0
\]
Its explicit form is given by 
\[
\mathrm d\omega (\mathbf x)=\sum_{1\leq i_1<\ldots<i_k\leq n} \sum_{i=1}^n \frac{\partial \omega_{i_1\ldots i_k} (\mathbf x)}{\partial x_{i}} \mathrm d x_{i}\wedge\mathrm dx_{i_1}\wedge\ldots\wedge\mathrm dx_{i_k}
\]
The operation of exterior differentiation does not depend on the choice of coordinates on the manifold. 
!!!!Examples 
In $\mathbb{R}^{3}$ one has that 

1. any scalar function $f=f(x,y,z)$ is a $0$-form,

2. the gradient $\mathbf{d}f=\mathbf{\omega }$ of any smooth function $f$ is a $1$-form
\[
\mathbf{d}f = \mathbf{\omega}=\frac{\partial f}{\partial x}dx+\frac{\partial f}{%
\partial y}dy+\frac{\partial f}{\partial z}dz \text{,}
\]

3. the curl $\mathbf{\alpha =d\omega}$ of any smooth $1$-form $\mathbf \omega$ is a $2$-form
\begin{eqnarray}
&&\mathbf \alpha =d\omega =\left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) dydz +
       \left( \frac{\partial P}{\partial z} -\frac{\partial R}{\partial x}\right) dzdx +
       \left( \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right) dxdy \\
\end{eqnarray}
If $\mathbf \omega$ is [[exact|Exact Sequence]], i.e. $\mathbf{\omega =d}f$, then $\mathbf{\alpha =dd}f=0$.

4. the divergence $\mathbf{\beta =d\alpha}$ of any smooth $2$-form $%\mathbf{\alpha} $ is a $3$-form
\[
\mathbf{\beta =d\alpha} =\left( \frac{\partial A}{\partial x}+\frac{\partial
B}{\partial y}+\frac{\partial C}{\partial z}\right) dxdydz
\]
and if $\alpha$ is exact with $\mathbf{\alpha =d\omega}$, then $\mathbf{\beta =dd\omega}=0$.

In general, for any two smooth functions $f=f(x,y,z)$ and $g=g(x,y,z)$, the exterior derivative $\mathbf{d}$\ obeys the Leibniz rule
\begin{equation}
\mathbf{d}(fg)=g\,\mathbf{d}f+f\,\mathbf{d}g
\end{equation}
and the chain rule
\begin{equation}
\mathbf{d}\left(g(f)\right) =g^{\prime }(f)\,\mathbf{d}f
\end{equation}
Papers: 
* [[Geometrical Bioelectrodynamics - V. G. Ivancevic, T. T. Ivancevic|http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.4014v3.pdf]] pct. 0

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]]

Letting the symmetric group $S_7$ act on the [[Fano plane|Fano Plane]] one gets $\operatorname{ord}(S_7) = 7!$ different labellings.
Equivalently one can describe the Fano plane as a [[Steiner triple system|Steiner Triple System]] on seven points, i.e. a $STS(7)$, given for example by:
\begin{eqnarray}
STS(7) &= &\{(\mathbf e_1,\mathbf e_2,\mathbf e_3),(\mathbf e_1,\mathbf e_4,\mathbf e_5),(\mathbf e_1,\mathbf  e_6,\mathbf  e_7),(\mathbf  e_2,\mathbf  e_4,\mathbf  e_6),\\
&&\;\;(\mathbf  e_2,\mathbf  e_5,\mathbf  e_7),(\mathbf e_3,\mathbf e_4,\mathbf e_7),(\mathbf e_3,\mathbf  e_5,\mathbf  e_6) \}
\end{eqnarray}
The action of a permutation $\sigma \in S_7$ on it is given by
\begin{eqnarray}
\sigma(STS(7))& =&\{(\mathbf e_{\sigma(1)},\mathbf e_{\sigma(2)},\mathbf e_{\sigma(3)}),(\mathbf e_{\sigma(1)},\mathbf e_{\sigma(4)},\mathbf e_{\sigma(5)}),(\mathbf e_{\sigma(1)},\mathbf  e_{\sigma(6)},\mathbf e_{\sigma(7)}),(\mathbf  e_{\sigma(2)},\mathbf  e_{\sigma(4)},\mathbf  e_{\sigma(6)}), \\
&& \;\;(\mathbf  e_{\sigma(2)},\mathbf  e_{\sigma(5)},\mathbf  e_{\sigma(7)}),(\mathbf e_{\sigma(3)},\mathbf e_{\sigma(4)},\mathbf e_{\sigma(7)}),(\mathbf e_{\sigma(3)},\mathbf  e_{\sigma(5)},\mathbf  e_{\sigma(6)}) \}
\end{eqnarray}
[[Automorphisms|Automorphism]] are those permutations the preserve the groupings of the triples (blocks). These are the elements of the automorphism group [[PSL(2,7)]].

Example: $\sigma(1) = 1,\, \sigma(2) = 4,\, \sigma(3) = 5, \, \sigma(4) = 2, \, \sigma(5) = 3,\, \sigma(6) = 6,\, \sigma(7) = 7$
\begin{eqnarray}
\Rightarrow \, \sigma(STS(7)) &=&\{(\mathbf e_1,\mathbf e_4,\mathbf e_5),(\mathbf e_1,\mathbf e_2,\mathbf e_3),(\mathbf e_1,\mathbf  e_6,\mathbf e_7),(\mathbf  e_4,\mathbf  e_2,\mathbf  e_6),(\mathbf  e_4,\mathbf  e_3,\mathbf  e_7),(\mathbf e_5,\mathbf e_2,\mathbf e_7),(\mathbf e_5,\mathbf  e_3,\mathbf  e_6) \} \\
&=& STS(7)
\end{eqnarray}

$PSL(2,7)$ divides the order of $S_7$ by $168$ such that one is left with $30$ [[cosets|Coset]] (which are still isomorphic to one another). These $30$ cosets are considered in the following, picking one representant of each, resulting in a set of $30$ Fano planes, which will be refered to as "different Fano planes".

The $30$ differentFano planes form one [[orbit|Orbit]] under the symmetric group $S_7$ and two orbits of length $15$ each under the [[alternating group|Alternating Group]] $A_7$.

Alternatively they can be partitioned into $6$ [[orbits|Orbit]] of orders $7,7,7,7,7,1,1$ respectively under the action of a cyclic shift. (See example below).

$2$ among the the $30$ labellings have either $0$, $1$ or $3$ lines (=triples of labels) in common.
There is a unique partition of the $30$ different Fano planes into $2$ sets of $15$ planes each, such that any $2$ Fano planes in one of the sets have exactly $1$ line in common. Both sets allow for the construction of the Fano tetrahedron of the projective geometry [[PG(3,2)]] or the [[Hoffman-Singleton graph|Hoffman-Singleton Graph]].

!!!!Class 1:
The following $2$ equivalent pictures show the $15$ Fano planes of the first class mentioned above:

<html><center><img src="images/FanoPlanes.gif" style="width: 750px; "/></center></html>
<html><center><img src="images/15planes1.jpg" style="width: 420px; "/></center></html>
The associated $15$ [[Steiner triple systems|Steiner Triple System]] (STS(7)) ordered lexicographically are given by:
\begin{eqnarray}
O_1: \{1,2,3\}, \{1,4,5\}, \{1,6,7\}, \{2,4,7\}, \{2,5,6\}, \{3,4,6\}, \{3,5,7\}\\
\{1,2,3\}, \{1,4,6\}, \{1,5,7\}, \{2,4,5\}, \{2,6,7\}, \{3,4,7\}, \{3,5,6\}\\
\{1,2,3\}, \{1,4,7\}, \{1,5,6\}, \{2,4,6\}, \{2,5,7\}, \{3,4,5\}, \{3,6,7\}\\
\{1,2,4\}, \{1,3,5\}, \{1,6,7\}, \{2,3,6\}, \{2,5,7\}, \{3,4,7\}, \{4,5,6\}\\
\{1,2,4\}, \{1,3,6\}, \{1,5,7\}, \{2,3,7\}, \{2,5,6\}, \{3,4,5\}, \{4,6,7\}\\
\{1,2,4\}, \{1,3,7\}, \{1,5,6\}, \{2,3,5\}, \{2,6,7\}, \{3,4,6\}, \{4,5,7\}\\
\{1,2,5\}, \{1,3,4\}, \{1,6,7\}, \{2,3,7\}, \{2,4,6\}, \{3,5,6\}, \{4,5,7\}\\
\{1,2,5\}, \{1,3,6\}, \{1,4,7\}, \{2,3,4\}, \{2,6,7\}, \{3,5,7\}, \{4,5,6\}\\
\{1,2,5\}, \{1,3,7\}, \{1,4,6\}, \{2,3,6\}, \{2,4,7\}, \{3,4,5\}, \{5,6,7\}\\
\{1,2,6\}, \{1,3,4\}, \{1,5,7\}, \{2,3,5\}, \{2,4,7\}, \{3,6,7\}, \{4,5,6\}\\
\{1,2,6\}, \{1,3,5\}, \{1,4,7\}, \{2,3,7\}, \{2,4,5\}, \{3,4,6\}, \{5,6,7\}\\
\{1,2,6\}, \{1,3,7\}, \{1,4,5\}, \{2,3,4\}, \{2,5,7\}, \{3,5,6\}, \{4,6,7\}\\
\{1,2,7\}, \{1,3,4\}, \{1,5,6\}, \{2,3,6\}, \{2,4,5\}, \{3,5,7\}, \{4,6,7\}\\
O_1: \{1,2,7\}, \{1,3,5\}, \{1,4,6\}, \{2,3,4\}, \{2,5,6\}, \{3,6,7\}, \{4,5,7\}\\
\{1,2,7\}, \{1,3,6\}, \{1,4,5\}, \{2,3,5\}, \{2,4,6\}, \{3,4,7\}, \{5,6,7\}
\end{eqnarray}
An example of a cyclic shift is the following: We take the first STS and get:
\begin{eqnarray}
\{1,2,3\}, \{1,4,5\}, \{1,6,7\}, \{2,4,7\}, \{2,5,6\}, \{3,4,6\}, \{3,5,7\} \rightarrow\\
\{2,3,4\}, \{2,5,6\}, \{2,7,1\}, \{3,5,1\}, \{3,6,7\}, \{4,5,7\}, \{4,6,1\} = \\
\{2,3,4\}, \{2,5,6\}, \{1,2,7\}, \{1,3,5\}, \{3,6,7\}, \{4,5,7\}, \{1,4,6\} = \\
\{1,2,7\}, \{1,3,5\}, \{1,4,6\}, \{2,3,4\}, \{2,5,6\}, \{3,6,7\}, \{4,5,7\}\\
\end{eqnarray}
which is the second last of the STS listed above, denoted $O_1$.

!!!!Class 2:
The class set of $15$ Fano planes is depicted in the following:

<html><center><img src="images/15planes2.jpg" style="width: 420px; "/></center></html>
The associated $15$ Steiner triple systems ordered lexicographically are:
\begin{eqnarray}
\{1,2,3\},\{1,4,5\},\{1,6,7\},\{2,4,6\},\{2,5,7\},\{3,4,7\},\{3,5,6\}\\
\{1,2,3\},\{1,4,6\},\{1,5,7\},\{2,4,7\},\{2,5,6\},\{3,4,5\},\{3,6,7\}\\
\{1,2,3\},\{1,4,7\},\{1,5,6\},\{2,4,5\},\{2,6,7\},\{3,4,6\},\{3,5,7\}\\
\{1,2,4\},\{1,3,5\},\{1,6,7\},\{2,3,7\},\{2,5,6\},\{3,4,6\},\{4,5,7\}\\
\{1,2,4\},\{1,3,6\},\{1,5,7\},\{2,3,5\},\{2,6,7\},\{3,4,7\},\{4,5,6\}\\
\{1,2,4\},\{1,3,7\},\{1,5,6\},\{2,3,6\},\{2,5,7\},\{3,4,5\},\{4,6,7\}\\
\{1,2,5\},\{1,3,4\},\{1,6,7\},\{2,3,6\},\{2,4,7\},\{3,5,7\},\{4,5,6\}\\
\{1,2,5\},\{1,3,6\},\{1,4,7\},\{2,3,7\},\{2,4,6\},\{3,4,5\},\{5,6,7\}\\
\{1,2,5\},\{1,3,7\},\{1,4,6\},\{2,3,4\},\{2,6,7\},\{3,5,6\},\{4,5,7\}\\
\{1,2,6\},\{1,3,4\},\{1,5,7\},\{2,3,7\},\{2,4,5\},\{3,5,6\},\{4,6,7\}\\
\{1,2,6\},\{1,3,5\},\{1,4,7\},\{2,3,4\},\{2,5,7\},\{3,6,7\},\{4,5,6\}\\
\{1,2,6\},\{1,3,7\},\{1,4,5\},\{2,3,5\},\{2,4,7\},\{3,4,6\},\{5,6,7\}\\
\{1,2,7\},\{1,3,4\},\{1,5,6\},\{2,3,5\},\{2,4,6\},\{3,6,7\},\{4,5,7\}\\
\{1,2,7\},\{1,3,5\},\{1,4,6\},\{2,3,6\},\{2,4,5\},\{3,4,7\},\{5,6,7\}\\
\{1,2,7\},\{1,3,6\},\{1,4,5\},\{2,3,4\},\{2,5,6\},\{3,5,7\},\{4,6,7\}
\end{eqnarray}
!!!![[MAGMA|http://magma.maths.usyd.edu.au/calc/]]^^[[Help|MAGMA]]^^ examples
* [[Code File|code/MAGMAFanoPlanes.txt]]

Papers:
* [[A Note on the Covering of all Triples on 7 Points with Steiner Triple Systems - A. E. Brouwer|http://www.win.tue.nl/~aeb/preprints/zn63.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=15007749364926377386&hl=de&as_sdt=2000]]
* [[YEA WHY TRY HER RAW WET HAT A Tour of the Smallest Projective Space - B. Polster| http://www.qedcat.com/articles/yea.pdf]] [[local|papers/yea.pdf]] [[pct. 3|http://scholar.google.com/scholar?cites=11018116709271941363&hl=de&as_sdt=2000]]

Presentations:
* [[Some New and Old Results Regarding Room Squares and Related Designs - J. Dinitz|http://www.emba.uvm.edu/~dinitz/mcccc.09.ppt]] [[local|presentations/mcccc.09.ppt]]

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)]].

<html><center><img src="images/fano_tetrahedron.jpg" style="width: 420px; "/></center></html>

[[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".

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]]

!!!!Properties
* For flat spaces the path integral formalism is completely equivalent to the [[Schwinger action principle|Action Principle]]. This equivalence holds equally well for quantum field theory, which is usually regarded as involving a flat configuration space.

Papers:
*[[The Schwinger Action Principle and the Feynman Path Integral for Quantum Mechanics in Curved Space - D. J. Toms|http://arxiv.org/PS_cache/hep-th/pdf/0411/0411233v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=12337870966204431579&hl=de]]
* [[Challenges to Path Integral Formulations of Quantum Theories - R. Jackiw|http://arxiv.org/PS_cache/arxiv/pdf/0711/0711.1514v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=7114550071588992508&hl=de]]

>We are not to think of M as being a part of B (i.e. M is not inside B); instead, B is to be viewed as a separate space from M, which we tend to regard as standing, in some sense, above the base space M. There are many copies of the fibre F in the bundle B, one entire copy of F standing above each point of M. The copies of the fibres are all disjoint (i.e. no two intersect), and together they make up the entire bundle B. The way to think of M in relation to B is as a factor space of the bundle B by the family of fibres F. That is to say, each point of M corresponds precisely to a separate individual copy of F. There is a continuous map from B down to M, called the canonical projection from B to M, which collapses each entire fibre F down to that particular point of M which it stands above.
> - R. Penrose - Road to Reality
<html><center><img src="images/bundle.jpg"  "/></center></html>

A ''Fiber Bundle $\mathcal B$'' is a space which locally looks like a product space, i.e. $\mathcal B = \mathcal M \times \mathcal F$ with $\mathcal F$ the so called ''Fiber of the bundle'' and $\mathcal M$ the ''Base Space'' or ''Base Manifold''. It consists of a continuous surjective map
\[
\pi : \mathcal B \to \mathcal M
\]
called ''Canonical Projection'' or ''Submersion'', regarded as part of the bundle. $\pi$ may be viewed as the collapsing of each fiber $\mathcal F$ down to a single point.

''Trivial Bundle''
<html><center><img src="images/trivial.jpg"  "/></center></html>
If the whole space is globally the product of the base space and the fiber space, the fiber bundle is called a ''Trivial Bundle''.

''Twisted Bundle''
<html><center><img src="images/twist.jpg"  "/></center></html>
A ''Twisted Bundle'' is a non-trivial bundle. Examples: Möbius strip, Klein bottle.
The dimension of a twisted bundle is always the sum of the dimensions of $\mathcal M$ and $\mathcal F$.

''Section''
A ''Section or Cross Section of a bundle'' is a so called ''Lift'' of the base space $\mathcal M$ into the bundle.

<html><center><img src="images/section.jpg"  "/></center></html>

The map is the inverse of canonical projection, i.e. $\pi^{-1}$.

The section can be regarded as a generalization of the notion of a graph $\Gamma$ of a function $f$ which is the map $ X → X \times f(X)$. Both are equivalent for the trivial bundle.
<html><center><img src="images/graph.jpg"  "/></center></html>
For a general bundle the notion of a graph translates into: $ \Gamma: M → M \times \pi^{-1}(M)$.

[[Tangent Bundle]]
The fibers are tangent spaces $T(\mathcal M)$ of the base space.

''Cotangent Bundle''
The fibers are the [[cotangent vector spaces|Covector]] $T^∗(\mathcal M)$ of the base space.

''Principle Bundle''
A principal bundle is a special case of a fiber bundle where the fiber is a [[group|Group]], usually a [[Lie group|Lie Group]].

!!!!Examples
* [[Hopf maps|Hopf Map]]

!!!!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

A long range ''Fifth Force'' is predicted by some extensions of the [[standard model|Standard Model]].
It has been hypothesized to impact large-scale structure formation. If it was attractive and very long range, it would effectively increase the gravitational field strength and thus accelerate structure formation. Previous studies have shown that such a force could reduce discrepancies between observations and predictions, e.g. by increasing the number of galaxy clusters and superclusters and reducing voids, which would agree better with observations.

Papers:
* [[Fifth Force from Fifth Dimension: A Comparison between two Different Approaches - F. Dahia, E. M. Monte, C. Romero|http://arxiv.org/PS_cache/gr-qc/pdf/0303/0303044v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=17847174983934044315&hl=de]]

Papers:
* [[The Natural Selection of Universes Containing Intelligent Life - 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]] [[pct. 29|http://scholar.google.de/scholar?cites=17506788885958317714&hl=de]]

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 is a manifold $\mathcal M$, 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, these 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 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 - S.-S. Chern|http://www.ams.org/notices/199609/chern.pdf]]

The first fundamental form of a manifold is given by:
\begin{equation}
|d\mathbf s|^2 = g_{\mu\nu}(\mathbf x) dx^\mu dx^\nu
\end{equation}
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\mathbf 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 $\mathcal M \in \mathbb R^n$ $m \le n$ parameterized by $\mathbf X(\mathbf x)$.
Furthermore let $\mathbf C(\tau)$ be a 1-parameter curve  in the manifold.
We have:
\begin{equation}
\frac{d\mathbf C(\tau)}{d\tau} = \sum_{i=1}^m \frac{\partial \mathbf X (\mathbf x)}{\partial x_i}  \frac{d x_i}{d\tau} = \sum_{i=1}^m \mathbf e_i (\mathbf x)  \frac{d x_i}{d\tau}
\end{equation}
where the $\mathbf e_i(\mathbf x)$ define a local basis in the manifold.
The path length $l_{\tau_0}(\tau_1)$  of the curve is given by:
\begin{equation}
l_{\tau_0}(\tau_1) = \int_{\tau_0}^{\tau_1} {\sqrt{\langle \frac{d\mathbf C(\tau)}{d\tau} | \frac{d\mathbf C(\tau)}{d\tau}\rangle} d\tau}
\end{equation}
and therefore
\begin{eqnarray}
l_{\tau_0}(\tau_1) & = & \int_{\tau_0}^{\tau_1} {\sqrt{\sum_{i,j} \langle \mathbf e_i (\mathbf x) | \mathbf e_j (\mathbf x) \rangle dx_i dx_j} d\tau} \\
   & = & \int_{\tau_0}^{\tau_1} {\sqrt{\sum_{i,j} g_{ij}(\mathbf x) dx_i dx_j} d\tau} \\
   & = & \int_{\tau_0}^{\tau_1} {\sqrt{|d\mathbf s|^2} d\tau} \\
\end{eqnarray}
with $g_{ij} (\mathbf x)$ 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 $AG(n, q)$.

An algebra is called ''flexible'' if it satisfies the condition:
\[
[\mathbf A,\mathbf B,\mathbf A] = 0
\]
i.e.
\[
(\mathbf{AB})\mathbf A = \mathbf A(\mathbf{BA})
\]
Most of the interesting [[nonassociative algebras|Nonassociative Algebra]] do have this property.

An algebra $\mathcal A$ is flexible if and only if   the identity
\[
[\mathbf D, \mathbf A \circ \mathbf B] =[\mathbf D, \mathbf A] \circ \mathbf B + \mathbf A \circ [\mathbf D, \mathbf B]
\]
holds for $\forall \, \mathbf B, \mathbf C, \mathbf D \ \in \mathcal A$. I.e. the map $ad_{\mathbf D}: \mathcal A \rightarrow \mathcal A$ defined by $ad_{\mathbf D} (\mathbf A) =[\mathbf D, \mathbf A]$ is a [[derivation|Derivation]] of the [[commutative algebra|Anti-Commutator]] $A^+$.


Linearisation leads to
\[
[\mathbf A,\mathbf B,\mathbf C] =  - [\mathbf C,\mathbf B,\mathbf A]
\]
i.e. the [[associator|Associator]] is antisymmetric in the 1${}^{st}$ and 3${}^{rd}$ component.

Due to flexibility the [[Akivis identity|Akivis Algebra]] simplifies to
\[
[[\mathbf A,\mathbf B ],\mathbf C] + [[\mathbf B,\mathbf C],\mathbf A] + [[\mathbf C,\mathbf A], \mathbf B] = 2[\mathbf A,\mathbf B,\mathbf C] + 2[\mathbf B,\mathbf C,\mathbf A] +2 [\mathbf C,\mathbf A,\mathbf B]
\]
The [[Teichmüller identity|Teichmüller Identity]] can be expressed as:
\[
[\mathbf A, [\mathbf B, \mathbf C, \mathbf D]]  + [[\mathbf A, \mathbf B, \mathbf C], \mathbf D] = [\mathbf A, \mathbf B, [\mathbf C, \mathbf D]]  - [\mathbf A, [\mathbf B, \mathbf C], \mathbf D] + [[\mathbf A,\mathbf B], \mathbf C, \mathbf D]
\]
Any flexible quadratic algebra is a [[noncommutative Jordan algebra|Jordan Algebra]].

''Further identities:''
\begin{eqnarray}
& [\mathbf A, [\mathbf B, \mathbf C], \mathbf D] = [[\mathbf B, \mathbf C],\mathbf D, \mathbf A] \\
& \langle \mathbf A | \mathbf{BC} \rangle =  \langle \mathbf{B}^* \mathbf A | \mathbf C \rangle =  \langle \mathbf A  \mathbf C^* | \mathbf{B} \rangle \\
& (\mathbf B \mathbf A^*)\mathbf A = \mathbf A^*(\mathbf{AB})
\end{eqnarray}

Papers:
* [[Loops with Universal Elasticity - P. N. Syrbu|http://www.quasigroups.eu/contents/download/1994/1_6.pdf]] [[pct. 1|http://scholar.google.com/scholar?hl=de&lr=&cites=14547103960313458104&um=1&ie=UTF-8&ei=0L5NSpTsG5yPsAaT5e3xBw&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]

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]]

!!!!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]]

<html><center><img src="images/p_mannheim.jpg" style="width: 680px; "/></center></html>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]]

Links:
* [[VisWIKI|http://viswiki.com/en/Freudenthal_magic_square]]

Papers:
* [[Magic Squares and Matrix Models of Lie Algebras - C. H. Barton, A. Sudbery|http://arxiv.org/PS_cache/math/pdf/0203/0203010v2.pdf]] [[local|papers/0203010v2.pdf]]

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]].

Links:
* [[WIKIPEDIA - Mathematical Joke|http://en.wikipedia.org/wiki/Mathematical_joke]]

Videos:
* [[Big Bang Theory Full Episodes|http://www.dipity.com/timeline/Big-Bang-Theory-Full-Episodes]]
* [[Math Doesn't Suck|http://www.amazon.com/Math-Doesnt-Suck-Survive-Breaking/dp/0452289491/ref=sr_1_1?ie=UTF8&s=books&qid=1249646448&sr=8-1]] - watch these videos !

!!!! Papers
Reading papers, one finds this or that peculiarity at times. Here's a selection of some of them ...

{{center{[img(489px+, )[images/fun1.jpg]]}}}
... how hard it is to become famous ! Want more ? Then click [[here|http://www.springerlink.com/content/vj150j6542650wkl/fulltext.pdf]].


{{center{[img(405px+, )[images/name.jpg]]}}}
Why abbreviated e-mail adresses make sense in some cases. (Try to type this name on your computer if you don't believe it).


Space-time supposedly is not continuous, but why not the introduction of a [[paper|http://arxiv.org/PS_cache/gr-qc/pdf/0512/0512002v1.pdf]] ... 
{{center{[img(489px+, )[images/continuous.jpg]]}}} 

A little competition in LaTeX:

{{center{[img(481px+, )[images/ConnesSM.jpg]]}}}

{{center{[img(457px+, )[images/big_formula.jpg]]}}}
Some more such awesome LaTeX and the complete formula can be found [[here|http://www.math.uni-muenster.de/u/raimar/physics/thesis/]]. 

O.K., do you still believe that WORD can do better than \LeTeX ?

!!!!Miscellanea 
!!!!! Women and money
{{center{[img(378px+, )[images/women_and_algebra.jpg]]}}} 
!!!!! A true story ?
The following concerns a question in a physics degree exam at the University of Copenhagen:

"Describe how to determine the height of a skyscraper with a barometer."

One student replied: "You tie a long piece of string to the neck of the barometer, then lower the barometer from the roof of the skyscraper to the ground. The length of the string plus the length of the barometer will equal the height of the building."

This highly original answer so incensed the examiner that the student was failed immediately. He appealed on the grounds that his answer was indisputably correct, and the university appointed an independent arbiter to decide the case. The arbiter judged that the answer was indeed correct, but did not display any noticeable knowledge of physics.

To resolve the problem, it was decided to call the student in and allow him six minutes in which to provide a verbal answer which showed at least a minimal familiarity with the basic principles of physics. For five minutes the student sat in silence, forehead creased in thought. The arbiter reminded him that time was running out, to which the student replied that he had several extremely relevant answers, but couldn't make up his mind which to use.

On being advised to hurry up the student replied as follows:

"Firstly, you could take the barometer up to the roof of the skyscraper, drop it over the edge, and measure the time it takes to reach the ground. The height of the building can then be worked out from the formula $H =\frac12 g  t^2$. But bad luck on the barometer."

"Or if the sun is shining you could measure the height of the barometer, then set it on end and measure the length of its shadow. Then you measure the length of the skyscraper's shadow, and thereafter it is a simple matter of proportional arithmetic to work out the height of the skyscraper."

"But if you wanted to be highly scientific about it, you could tie a short piece of string to the barometer and swing it like a pendulum, first at ground level and then on the roof of the skyscraper. The height is worked out by the difference in the gravitational restoring force $T = 2 \pi^2 \sqrt{\frac{l}{g}}$."

"Or if the skyscraper has an outside emergency staircase, it would be easier to walk up it and mark off the height of the skyscraper in barometer lengths, then add them up."

"If you merely wanted to be boring and orthodox about it, of course, you could use the barometer to measure the air pressure on the roof of the skyscraper and on the ground, and convert the difference in millibars into feet to give the height of the building."

But since we are constantly being exhorted to exercise independence of mind and apply scientific methods, undoubtedly the best way would be to knock on the janitor's door and say to him 'If you would like a nice new barometer, I will give you this one if you tell me the height of this skyscraper'."

The student was Nils Bohr, the only Dane to win the Nobel prize for Physics.

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

It is defined by a formally completely integrable system of exterior differential equations. Examples of closed $G$-structures are [[Lie groups|Lie Group]], [[symmetric spaces|Symmetric Space]] and some of their generalizations and many known classes of [[3-webs|3-Web]].
The order $k$ of a closed $G$-structure is a measure of how close is is to [[Lie groups|Lie Group]] for which $k = 2$.

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]].

!!!!Example
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 $\frac12 n^2(n ? 1)$.
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.

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.

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]] [[pct. 4|http://scholar.google.de/scholar?cites=4528943419224839982&hl=de]]

A ''G2-manifold'' is a seven-dimensional [[Riemannian manifold|Riemann Space]] with holonomy group [[G2]]. A compact $G2$-manifold is also known as ''Joyce 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|Homomorphism]] 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 $SL(4,2)$.

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]]

>A final word about Clifford algebras in general: they are, like Lie groups, profligate. There are too many of them, an infinite number of both Lie groups and Clifford algebras that are physically irrelevant, not a part of the design of reality. This is and always was the problem with GUT theories based on a unifying large Lie group, and it is and always will be the problem with unification theories based on large Clifford algebras. In both cases it is the principal of the educated guess that leads to the choice of unifying algebraic object. This is unsatisfactory. Nature can not be so arbitrarily ugly.
>- Geoffrey M. Dixon - Division algebras: octonions, quaternions, complex numbers, and the Algebraic Design of Physics -

Group embeddings of the standard model:
[[E8]] $\supset$ [[E7]] $\supset$ [[E6]] $\supset SO(10) \supset SU(5) \supset SU(3)_c \times SU(2)_L  \times U(1)_Y$
However the irreducible representations of a GUT should be [[chiral|Chirality]]. The inclusions [[E8]] $\supset$ [[E7]] $\supset$ [[E6]] must therefore be ruled out. Instead it is conjectured that the gauge group is $E_8 \times E_8$ or [[SO(32)]] which one has to somehow break down to e.g. $E_6$.

!!!! Unification with Clifford Algebras:
<html><center><img src="images/clifford_unification.jpg" style="width: 420px; "/></center></html>
Papers:
* [[Aspects of Grand Unification in Higher Dimensions - A. A. Wingerter|http://deposit.ddb.de/cgi-bin/dokserv?idn=976474522&dok_var=d1&dok_ext=pdf&filename=976474522.pdf]] [[local|papers/dokserv.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=6613607189809313311&hl=de&as_sdt=2000]]
* [[The Algebra of Grand Unified Theories - J. Baez, J. Huerta|http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.1556v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=1321160442974930747&hl=de&as_sdt=2000]]

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

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

\boldsymbol \gamma_{2} &= & \begin{pmatrix}\mathbf 0 & \mathbf 0 & \mathbf 0 &-\mathbf i\\
\mathbf 0 & \mathbf 0 & \mathbf i & \mathbf 0\\
\mathbf 0 &\mathbf  i & \mathbf 0 & \mathbf 0\\
-\mathbf i & \mathbf 0 &\mathbf  0 &\mathbf  0\end{pmatrix}, \quad

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

\boldsymbol \gamma_{2} &= & \begin{pmatrix}\mathbf 0 & \mathbf 0 & \mathbf 0 &-\mathbf i\\
\mathbf 0 & \mathbf 0 & \mathbf i & \mathbf 0\\
\mathbf 0 &\mathbf  i & \mathbf 0 & \mathbf 0\\
-\mathbf i & \mathbf 0 &\mathbf  0 &\mathbf  0\end{pmatrix}, \quad

\boldsymbol \gamma_{3}= \begin{pmatrix}\mathbf 0 & \mathbf 0 &\mathbf  1 & \mathbf 0\\
\mathbf 0 & \mathbf 0 &\mathbf  0 &-\mathbf  1\\
-\mathbf 1 & \mathbf 0 & \mathbf 0 &\mathbf  0\\
\mathbf 0 & \mathbf 1 &\mathbf  0 & \mathbf  0\end{pmatrix}
\end{eqnarray}

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 - M. Braun|http://www.ieja.net/papers/2009/V6/2-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]]

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]]) ${h_\mu}^a$.

Videos:
* [[Einstein's Theory (lecture 1 - 12) - L. Susskind|http://www.youtube.com/view_play_list?p=6C8BDEEBA6BDC78D]]
* [[Caltech's Physics: Gravitational Waves - A Web-Based Course|http://elmer.tapir.caltech.edu/ph237/]]

!!!!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:
\[
\mathbf{G} := \begin{pmatrix} 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1\\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1\\ 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \end{pmatrix}
\]
!!!![[Lattices|Lattice]]
Given a lattice $L$ with basis vectors $\{\mathbf e_1, \ldots, \mathbf e_n$\}, its ''Generator Matrix'' (or ''Basis Matrix'') ''$B$'' is defined by $B_{ij} \equiv (\mathbf e_i)_j$. 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:
\[
L = \{\vec x' : \vec x' = B \vec x = \sum_{i=1}^n x_i\mathbf e_i, \, x_i \in \mathbb Z\}
\]

* [[The Sequence of the Human Genome - J. C. Venter et al.|http://www.upch.edu.pe/facien/dbmbqf/gorjeda/cursos/geneticaavanzada%202007/articulos/Science-2001-venter-hgs.pdf]] {{t1000Cite{[[pct. 7321|http://scholar.google.de/scholar?cites=7202406481006653037&hl=de]]}}}

The ''Geodesic Equation'' describes the trajectory $\boldsymbol \gamma$ with the shortest path length between two points $a$ and $b$.
It can be attained by varying the path length of possible paths from $a$ to $b$ and finding the minimum.
The path length $l(\boldsymbol \gamma)$ of a curve $\boldsymbol \gamma$ is given by
\[
l(\boldsymbol \gamma) = \int_a^b ds = \int_a^b \sqrt {g_{\mu\nu}(\boldsymbol \gamma(s)) \dot {\mathbf \gamma}^\mu (s) \dot {\mathbf{\gamma}}^\nu(s)} ds
\]
This can be expressed in terms of an [[action|Action Principle]] $S$ as
\[
S(\boldsymbol \gamma, \dot{\boldsymbol \gamma}) = \int_a^b L (\boldsymbol \gamma , \dot{\boldsymbol \gamma}) ds
\]
with
\[
L(\boldsymbol \gamma, \dot{\boldsymbol \gamma}) = \int_a^b \sqrt {g_{\mu\nu}(\boldsymbol \gamma (s)) \dot \gamma^\mu(s) \dot \gamma^\nu(s)} ds
\]

Finding the path with minimum length therefore is equivalent to minimizing the action, which results in the Lagrange equations, i.e.
\[
{\partial L\over\partial \gamma^\rho} - {d\over ds}{\partial L\over\partial \dot{\gamma}^\rho} = 0
\]
Inserting (the square of) our Lagrangian, we get
\[
\frac{\partial g_{\mu\nu}(\gamma)}{\partial \gamma^\rho} \dot{\gamma}^\mu \dot{\gamma}^\nu - 2 {d\over ds} \left (g_{\rho \nu} (\boldsymbol \gamma) \dot{\gamma}^\nu \right )= 0
\]
assuming that the metric tensor is symmetric.

Carrying out the total differentiation yields:
\[
\frac{\partial g_{\mu\nu}(\boldsymbol \gamma)}{\partial \gamma^\rho} \dot{\gamma}^\mu \dot{\gamma}^\nu - 2 \frac{\partial g_{\rho\nu}(\boldsymbol \gamma)}{\partial \gamma^\mu}\dot{\gamma}^\mu \dot{\gamma}^\nu - 2 g_{\rho\nu} (\boldsymbol \gamma) \ddot{\gamma}^\nu = 0
\]

Splitting up the second term and renaming indices leads to
\[
\left (\frac{\partial g_{\mu\nu}(\boldsymbol\gamma)}{\partial \gamma^\rho} - \frac{\partial g_{\rho\nu}(\mathbf{\gamma})}{\partial \gamma^\mu} - \frac{\partial g_{\rho\mu}(\boldsymbol \gamma)}{\partial \gamma^\nu} \right ) \dot{\gamma}^\mu \dot{\gamma}^\nu - 2 g_{\rho\nu} (\boldsymbol \gamma) \ddot{\gamma}^\nu = 0
\]

!!!!Examples

''Point particle in special relativity:''
\[
S = - m c^2 \int_{C} \, d \tau
\]
This action is invariant under reparametrizations of $\tau$. One can fix this invariance by different gauge fixings (e.g. static gauge, light-front gauge).

''Point particle in general relativity:''
In General Relativity the ''Geodesic Equation'' describes the trajectory $x^\mu(\tau)$ of a point particle (without spin) - the ''Geodesic'' - under the action of gravitation:
\[
a^{\lambda}(\tau) + \Chr{\lambda}{\mu \nu} u^{\mu}(\tau) u^{\nu}(\tau) = 0
\]
with $  \Chr{\lambda}{\mu \nu} $ the [[Christoffel connection|Levi-Civita-Connection]] and $\tau$ proper time.

''Spinning particle in special relativity:''
The geodesic is a straight line which is not influenced by the particle's spin.

''Spinning particle in general relativity:''
If spin is considered (but without spin precession) the equation has to be modified by an additional force term which yields one of the [[Mathisson-Papapetrou equations|Mathisson-Papapetrou Equations]]
\[
a^{\lambda} + \Chr{\lambda}{\mu \nu} u^{\mu}(\tau) u^{\nu} = \frac{1}{2m} R^\lambda_{\rho\sigma\omega} S^{\sigma\omega} u^{\rho}
\]
Hence the spinning particle does not follow a geodesic any more. 

If furthermore a (constant) electromagnetic field $F_{\mu\nu}$ is added one gets the following equation which is a special case of the [[Dixon-Souriau equations|Dixon-Souriau Equations]]:
\[
a^{\lambda} + \Chr{\lambda}{\mu \nu} u^{\mu}(\tau) u^{\nu} (\tau)= (\frac{1}{2} R^\lambda_{\rho\sigma\omega} S^{\sigma\omega}  + e F_{\lambda\rho}) \frac {u^{\rho}}{m}
\]

Given an analytic manifold $\mathcal {M}$ with an analytic linear connection and $e \in \mathcal {M} $ a fixed point, a ''Geodesic Loop'' is defined by the multiplication
\[
a\cdot b = \exp (T_{e \to a} \exp^{-1} (b))
\]
with $T_{e \to a}$ the parallel transport of tangent vectors along a geodesic segment from the point $e \in \mathcal {M}$ to the point $a \in \mathcal {M}$ .
This construction justifies the role of smooth [[quasigroups|Quasigroup]] and [[loops|Loop]] in differential geometry.
Geodesic loops are [[mono-associative|Monoassociativity]].
If the connection is locally symmetric, the geodesic loop is a [[Bol loop|Bol Loop]].
Geodesic loops were introduced by Kikkawa.

A possibility for a physical interpretation of the geo-odular structure of a curved space-time may follow from the fact that the geodesic multiplication is a generalization of a rigid shift $x^\mu \rightarrow x^\mu + a^\mu$ of a flat space-time. So the representation theory for geodesic loops must naturally embrace the representation theory of Poincaré translations.

Papers:
* [[Geodesic Loops - A. Figula|http://www.emis.de/journals/JLT/vol.10_no.2/figulat2e.ps.gz]] [[pct. 4|http://scholar.google.de/scholar?cites=245422329403152310&hl=de]]
* [[Geodesic Multiplication as a Tool for Classical and Quantum Gravity - P. Kuusk, E. Paal|http://arxiv.org/PS_cache/arxiv/pdf/0803/0803.1241v1.pdf]] [[pct. 3|http://scholar.google.de/scholar?hl=de&lr=&cites=17164858885421195931]] [[local|papers/0803.1241v1.pdf]]
* [[Geodesic Multiplication and Quantum Kinematics in a Newtonian Spacetime - P. Kuusk|http://arxiv.org/PS_cache/gr-qc/pdf/9906/9906093v1.pdf]] pct. 0
* [[Geodesic Multiplication, BRST and Dynmical Cohomology of Space-Time - P. Kuusk, J. Örd, E. Paal |http://hexagon.park.tartu.ee/~jyri/lr_brst.ps.gz]] pct. 0

''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.

''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]].

/***
|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 $\vec {\mathbf X} = \{\mathbf A_1, \ldots, \mathbf A_N\}$ of elements of an algebra, having a [[inner product|Scalar Product]] $<.|.>$, the ''Gram Matrix'' is defined as
\[
\mathbf M(\mathbf A_1, \ldots,\mathbf A_N) \equiv \langle \vec {\mathbf X}| \vec {\mathbf X}^t \rangle = \left ( \begin{matrix}  \langle \mathbf  A_1,\mathbf  A_1\rangle & \langle \mathbf  A_1, \mathbf  A_2\rangle & \ldots & \langle \mathbf  A_1,\mathbf  A_N\rangle \\  \langle \mathbf  A_2, \mathbf  A_1\rangle & \langle \mathbf  A_2, \mathbf  A_2\rangle & \ldots & \langle \mathbf  A_2, \mathbf  A_N\rangle \\  \ldots & \ldots & \ldots & \ldots \\  \langle \mathbf  A_N,\mathbf  A_1\rangle & \langle \mathbf  A_N,\mathbf  A_2\rangle & \ldots & \langle \mathbf  A_N, \mathbf A_N\rangle \\  \end{matrix} \right )
\]
or in component form
\[
M_{ij} = \langle \mathbf A_i|\mathbf A_j \rangle
\]
!!!!Properties
* Any Gram matrix is symmetric, since inner products are symmetric.
* Given a Gram matrix the vectors $\vec {\mathbf X}$ 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]]

!!!!Interpretations
* Gravitation represents a "metrical elasticity" of space which is brought about by quantum fluctuations of the vacuum (Andrei Sakharov).
* Space-time is a crystal with dislocations and disclinations ([["world crystal"|World Crystal]]) which has undergone a quantum phase transition to a nematic phase by a condensation of dislocations.
* Space-time is a medium which can be described as a [[Fermi system|Fermi System]].

Papers:
* [[Quantum Phase Transitions and the Breakdown of Classical General Relativity - G. Chapline|http://arxiv.org/PS_cache/gr-qc/pdf/0012/0012094v1.pdf]]  {{t100Cite{[[pct. 104|http://scholar.google.de/scholar?cites=4549488117690173279&hl=de]]}}}
* [[Nonholonomic Mapping Principle for Classical and Quantum Mechanics in Spaces with Curvature and Torsion - H. Kleinert|http://arxiv.org/PS_cache/gr-qc/pdf/0203/0203029v1.pdf]] [[local|papers/0203029v1.pdf]][[pct. 27|http://scholar.google.de/scholar?cites=5964503436918444234&hl=de]]

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]]

>... 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.

Since Griess’ construction of the [[Monster simple group|Monster Group]] as the automorphism group 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''.

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]]

<html><center><img src="images/grouplike_structures.jpg" style="width: 330px; "/></center></html>
Papers:
*[[Group Theory for Unified Model Building - R. Slansky (version 1)|http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?198102061]]  [[local|papers/allpdf2.pdf]]  [[(version 2)|http://bolvan.ph.utexas.edu/~vadim/classes/2004f.homeworks/slansky.pdf]]  [[local|papers/slansky.pdf]] [[pct. 614|http://scholar.google.de/scholar?cites=2139962486156996772&hl=de]]
* [[An Elementary Introduction to Groups and Representations - B. C. Hall|http://arxiv.org/PS_cache/math-ph/pdf/0005/0005032v1.pdf]] [[local|papers/0005032v1.pdf]]  [[pct. 6|http://scholar.google.de/scholar?cites=14279749498558430801&hl=de]]

A remark:
One usually distinguishes between a commtutative (abelian) group and a non-commutative (nonabelian) group, dropping commutativity from the set of group axioms in the latter case. It would only be consequent to define a nonassociative group analogously, dropping the associativity assumption instead. Such an object is however called a [[loop|Loop]] and is in general not regarded as a group any more. This is somewhat awkward, as it obscures the close relationship between the two classes of objects. In particular it tends to hide away the fact that the algebras associated to loops are a consequent generalization of [[Lie algebras|Lie Algebra]], associated to groups.
Another awkward fact is that the name "loop" is also used in other contexts. Therefore an internet search will drown the search results for loops, alluded to above, in the search results for the other kinds of loops.

Lectures:
* [[Discrete Groups - R. Loll|http://www.phys.uu.nl/~sahlmann/teaching/lecture%20notes/discrete%20groups%20lecture%20notes.pdf]] [[local|papers/discrete groups lecture notes.pdf]]

There are at least two definitions of ''Groupoid'' currently in use:

I. $\;\,$  An algebraic structure on a set with a binary operator. The only restriction on the operator is closure (i.e., applying the binary operator to two elements of a given set $S$ returns a value which is itself a member of $S$). [[Associativity, commutativity, etc., are not required. A groupoid can be empty. An associative groupoid is called a semigroup.

II.$\;\,$  An algebraic structure first defined by Brandt (1926) and also known as a virtual group.

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

A ''H-Space'' is a set $M$ with multiplication function $m: M \times M \rightarrow M$, having an identity element $e$.

A variant (also of local groups) is a so called ''Local Analytic H-Space''. It is defined by a triple $(M, E, m)$, provided $M$ is an analytic manifold, $E$ is an open set of $M$ containing $e$, and $m$ is an analytic function $m: E \times E \rightarrow M$ satisfying $m(e, x) = m(x, e) = x, \; \forall x \in E$.

Every topological group is a H-space; however, in the general case, as compared to a topological group, H-spaces may lack associativity and inverses.

Links:
* [[Homepage J. P. May|http://www.math.uchicago.edu/~may/]]

Papers:
* [[Homotopy Associativity of H-spaces. I - J. D. Stasheff|http://www.math.sunysb.edu/~blafard/tex/stash.pdf]] [[local|papers/stash.pdf]] {{t500Cite{[[pct. 768|http://scholar.google.de/scholar?cites=15888580365419436557&hl=de]]}}}

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;
}}}

A ''Hadamard Matrix'' of order $n$, $H_n$, is a square $n\times n$-matrix with entries that are either $+1$ or $?1$. All rows and columns of the matrix are mutually orthogonal, i.e. any two different rows (columns) have matching entries in exactly half of their rows (columns). This can be expressed as
\[
H_n H_n^{T}= n I_n
\]
with $I_n$ the $n\times n$-identity matrix.

The $n$-dimensional parallelepiped spanned by the rows (columns) of an $n\times n$-Hadamard matrix has the maximum possible $n$-dimensional volume among parallelepipeds spanned by vectors whose entries are bounded in absolute value by $1$.
Equivalently, a Hadamard matrix has maximal absolute determinant among matrices with entries of absolute value less than or equal to $1$, which is given by $\det (H_n) = \pm n^{n/2}$. E.g $|\det (H_n)| = 1, 2, 16, 4.096, 4.294.967.296$ for $n = 1,2,4,8,16$. (For details see [[Sloane's A003433|http://www.research.att.com/~njas/sequences/A003433]]).

Hadamard matrices have been completely classified up to order $28$. For higher orders, only partial classifications are known.

!!!!Equivalence/Automorphisms
Two Hadamard matrices $H$ and $H'$ are said to be ''equivalent'' (which is also known as ''Hadamard Equivalence'') if one of them can be obtained from the other one by permuting rows or columns or by multiplying rows or columns by $-1$. This can be expressed as
\[
H' = P^{?1}HQ
\]
where $P$ and $Q$ are monomial matrices (having just one non-zero element in each row or column) with non-zero entries $\pm 1$.
This kind of equivalence defines an equivalence relation.

The [[automorphism group|Automorphism]] of a Hadamard matrix is defined by all pairs $(P,Q)$ with the group operation given by $(P_1,Q_1) (P_2,Q_2) = (P_1P_2,Q_1Q_2)$.
In particular $\{-1_n,-1_n\}$ is an automorphism for any $H_n$.

Example:
All $12 \times 12$ Hadamard matrices are equivalent. $G = Aut(H_{12})/\{-1,-1\}$ is isomorphic to the [[Mathieu group|Mathieu Group]] $M_{12}$.

!!!!Secial types
* A ''Normalized Hadamard Matrix'' is defined as a Hadamard matrix with all entries equal to "1" in the first row and the first column. It is always possible to normalize a Hadamard matrix, however not necessarily in a unique way. Normalized Hadamard matrices make their appearance as [[sign matrices|Sign Tables]] of non-split [[Cayley-Dickson algebras|Cayley-Dickson Algebra]]. The classical [[Cayley-Dickson construction|Cayley-Dickson Doubling]] therefore allows for the generation of Hadamard matrices of this kind in dimensions $2^n$ with $n\in \mathbb N$.
* For a Hadamard matrix $H$, the matrix $B = \frac 12 (H +J)$ is called the ''Binary Hadamard Matrix'' associated with $H$. If $H$ is normalized, $B$ may serve to generate a [[binary blockcode|Blockcode]].
* A ''Skew\-Hadamard Matrix'' $SH$ is a Hadamard matrix satisfying $H = A + I$, where $A^T = -A$ and $I$ is the identity matrix. There is a unique SH matrix of order $8$, whose associated code is the [[binary Hamming code|Hamming Code]] $[8,4,4]$. Similarly there is a unique SH matrix of order $12$, whose associated code is the [[extended ternary Golay code|Golay Code]] $[12, 6, 6]$. There are two inequivalent SH matrices of order $16$. These are associated with extremal type II codes $[16, 8, 4]$. Skew Hadamard matrices are known to exist for for all $n<188$ with $n$ divisible by $4$.
* [[Sylvester matrices|Sylvester Matrix]].
* ''Regular Hadamard matrices''  have constant row and column sums.
* A ''Paley\-Hadamard Matrix'' is a Hadamard matrix that has order $p+1$ with $p$ a prime number.
* ''Walsh Matrices''.

!!!!Properties
* If a Hadamard matrix of order $n$ exists, then ($n = 1$ or $2$) or ($n \operatorname{mod} 4 = 0$ with $n \ge 4$).
* Every Hadamard matrix of order $n$ is equivalent to a normalized Hadamard matrix of this order.
* The matrices of orders $1, 2, 4, 8$ and $12$ are unique (up to isomorphism). For $n = 16, 20, 24, 28$ one has $5, 3, 60, 487$ inequivalent realisations respectively (see [[Sloane's A036297|http://www.research.att.com/~njas/sequences/A036297]]).

!!!!Applications
Hadamard matrices have numerous applications in different fields. Here are some examples:
!!!!!Physics/chemistry
* Hadamard transform spectroscopy
* Chemical design
* Optical multiplexing techniques
* Stereochemistry
* Nuclear magnetic resonance (NMR) spectroscopy
!!!!!Mathematics/computer sciences
* [[Block designs|T-Design]]
* [[Coding theory|Coding Theory]] (error correcting codes)
* Graph designs
* Tournaments
* Orthogonal arrays
* [[Projective planes|Projective Plane]]
* [[Group theory|Group]]
* Terminal networks
* Circuit theory

!!!! Generalizations
* [[Weighing matrices|Weighing Matrix]]

!!!![[Examples|Hadamard Matrix - Examples]]

Papers:
* [[Hadamard Matrices and Their Applications - A. Hedayat, W. D. Wallis|http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aos/1176344370]] [[pct. 98|http://scholar.google.de/scholar?cites=1675355368834419103&hl=de&as_sdt=2000]]
* [[Characterization of Hadamard Matrices - K. Balasubramanian|http://www.mcs.csueastbay.edu/~kbalasub/reprints/283.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=12773239850263391881&hl=de&as_sdt=2000]]
* [[On the Cassification of Hadamard Matrices of Order 32 - H. Kharaghani, B. Tayfeh-Rezaie|http://www.cs.uleth.ca/~hadi/research/h32/h32-classification-fv.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=1793574277052483166&hl=de&as_sdt=2000]]
* [[Group Actions on Hadamard Matrices - P. O. Cathain|http://www.maths.nuigalway.ie/~padraig/Docs/PadraigOCathainMastersThesis.pdf]] pct. 0

Lectures:
* [[Hadamard Matrices - P. J. Cameron|http://designtheory.org/library/encyc/topics/had.pdf]]

Theses:
* [[A Survey of the Hadamard Conjecture - E. Tressler|http://scholar.lib.vt.edu/theses/available/etd-05042004-120929/unrestricted/thesis_revised.pdf]]

Links:
* [[A Library of Hadamard Matrices - N. J. A. Sloane|http://www.research.att.com/~njas/hadamard/]]
* [[Hadamard matrix data by P. Ó. Catháin|http://www.maths.nuigalway.ie/~padraig/DataFiles/had16.txt]]

Google books:
* [[Hadamard Matrices and Their Applications - K. J. Horadam|http://books.google.de/books?hl=de&lr=&id=fZ3quix-ifMC&oi=fnd&pg=PR11&dq=horadam+hadamard+google+books&ots=KNiffWGIwr&sig=CtKmwJbFk3cdqHViDOhBbLNVBK8#v=onepage&q=&f=false]]   [[local|google_books/HadamardMatrices.pdf]] [[pct. 36|http://scholar.google.de/scholar?cites=7256276410987669113&hl=de]]

Given an order $n$, one has the following examples of [[Hadamard matrices|Hadamard Matrix]]:

!!!!!n = 1
~~
|+|
~~
!!!!!n = 2
~~
|+|+|
|+|-|
~~
!!!!!n = 4
~~
|+|+|+|+|
|+|-|+|-|
|+|+|-|-|
|+|-|-|+|
~~
flipping the second and the fourth column one gets:
~~
|+|+|+|+|
|+|-|+|-|
|+|-|-|+|
|+|+|-|-|
~~
which represents the [[sign matrix|Sign Tables]] of an antisymmetric [[multiplication table|Multiplication Tables]] and corresponds to the multiplication table of the right handed [[quaternions|Quaternion]].
If instead on one flips the third and the fourth column one gets the sign matrix of the multiplication table of the left handed quaternions:
~~
|+|+|+|+|
|+|-|-|+|
|+|+|-|-|
|+|-|+|-|
~~
!!!!!n = 8
~~
|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|+|-|
|+|+|-|-|+|+|-|-|
|+|-|-|+|+|-|-|+|
|+|+|+|+|-|-|-|-|
|+|-|+|-|-|+|-|+|
|+|+|-|-|-|-|+|+|
|+|-|-|+|-|+|+|-|
~~
[[Automorphism group|Automorphism]]: order  $2^{10} \cdot 3 \cdot 7 = 21.504$.

With the following map of the columns $(1 \to 1, 2 \to 6, 3 \to 8, 4 \to 7, 5 \to 5, 6 \to 4, 7 \to 3, 8 \to 2)$ one gets the table
~~
|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|-|+|
|+|-|-|+|+|+|-|-|
|+|+|-|-|+|-|+|-|
|+|-|-|-|-|+|+|+|
|+|+|-|+|-|-|-|+|
|+|+|+|-|-|+|-|-|
|+|-|+|+|-|-|+|-|
~~
which is the sign matrix of the multiplication table of the [[octonions|Octonion]] obtained by classical [[Cayley-Dickson doubling|Cayley-Dickson Doubling]].
(Notice however that by flipping the columns we have changed the determinant from $2^{12}$ to $-2^{12}$, which happens if two rows (or columns) are switched).
By appropriately flipping rows and columns one can get all $240$ non-equal sign tables associated with the [[480 different octonion algebras|480 Octonion Multiplication Tables]].
Furthermore, Monte\-Carlo simulations suggest that there are all in all $2.640 = 11\cdot 240$ different possible $8$-dimensional sign tables when requiring
* the number of $+1$- and $-1$-entries is equal for each row and column (except for the border-row and -column). I.e. the matrices are Hadamard matrices,
* they are normalized,
* the diagonal consists of $-1$ entries (except for the one element of the border), i.e. it corresponds to a non-split algebra,
* the matrices are antisymmetric.
(Unfortunately I am lacking any explanation of this result yet).

The algorithm also reproduces the $2$ different sign tables in case of the quaternions, given above.

Furthermore this matrix is Hadamard equivalent to a representation of the [[Fano plane|Fano Plane]].

!!!!!n = 16
The five distinct Hadamard matrices of order $16$ can be taken to be:
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|+|-|+|-|+|-|+|-|+|-|
|+|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|
|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|+|
|+|+|+|+|-|-|-|-|+|+|+|+|-|-|-|-|
|+|-|+|-|-|+|-|+|+|-|+|-|-|+|-|+|
|+|+|-|-|-|-|+|+|+|+|-|-|-|-|+|+|
|+|-|-|+|-|+|+|-|+|-|-|+|-|+|+|-|
|+|+|+|+|+|+|+|+|-|-|-|-|-|-|-|-|
|+|-|+|-|+|-|+|-|-|+|-|+|-|+|-|+|
|+|+|-|-|+|+|-|-|-|-|+|+|-|-|+|+|
|+|-|-|+|+|-|-|+|-|+|+|-|-|+|+|-|
|+|+|+|+|-|-|-|-|-|-|-|-|+|+|+|+|
|+|-|+|-|-|+|-|+|-|+|-|+|+|-|+|-|
|+|+|-|-|-|-|+|+|-|-|+|+|+|+|-|-|
|+|-|-|+|-|+|+|-|-|+|+|-|+|-|-|+|
~~
[[Automorphism group|Automorphism]]: order $ 2^{15} \cdot 3^2 \cdot 5 \cdot 7 = 10.321.920$.
Determinant: $2^{32}$.
Symmetric, contrary to all the other matrices listed in the following.
This matrix is Hadamard equivalent to a representation of the [[Fano tetrahedron|Fano Spaces]].
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|+|-|+|-|+|-|+|-|+|-|
|+|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|
|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|+|
|+|+|+|+|-|-|-|-|+|+|+|+|-|-|-|-|
|+|-|+|-|-|+|-|+|+|-|+|-|-|+|-|+|
|+|+|-|-|-|-|+|+|+|+|-|-|-|-|+|+|
|+|-|-|+|-|+|+|-|+|-|-|+|-|+|+|-|
|+|+|+|+|+|+|+|+|-|-|-|-|-|-|-|-|
|+|-|+|-|+|-|-|+|-|+|-|+|-|+|+|-|
|+|+|-|-|+|+|-|-|-|-|+|+|-|-|+|+|
|+|-|-|+|+|-|+|-|-|+|+|-|-|+|-|+|
|+|+|+|+|-|-|-|-|-|-|-|-|+|+|+|+|
|+|-|+|-|-|+|+|-|-|+|-|+|+|-|-|+|
|+|+|-|-|-|-|+|+|-|-|+|+|+|+|-|-|
|+|-|-|+|-|+|-|+|-|+|+|-|+|-|+|-|
~~
[[Automorphism group|Automorphism]]: order $2^{15}\cdot 3^2 = 294.912$.
Determinant: $-2^{32}$.
Hadamard equivalent to its transpose.

~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|+|-|+|-|+|-|+|-|+|-|
|+|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|
|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|+|
|+|+|+|+|-|-|-|-|+|+|+|+|-|-|-|-|
|+|-|+|-|-|+|-|+|+|-|+|-|-|+|-|+|
|+|+|-|-|-|-|+|+|+|+|-|-|-|-|+|+|
|+|-|-|+|-|+|+|-|+|-|-|+|-|+|+|-|
|+|+|+|+|+|+|+|+|-|-|-|-|-|-|-|-|
|+|+|+|+|-|-|-|-|-|-|-|-|+|+|+|+|
|+|+|-|-|+|-|+|-|-|-|+|+|-|+|-|+|
|+|+|-|-|-|+|-|+|-|-|+|+|+|-|+|-|
|+|-|+|-|+|-|-|+|-|+|-|+|-|+|+|-|
|+|-|+|-|-|+|+|-|-|+|-|+|+|-|-|+|
|+|-|-|+|+|+|-|-|-|+|+|-|-|-|+|+|
|+|-|-|+|-|-|+|+|-|+|+|-|+|+|-|-|
~~
[[Automorphism group|Automorphism]]: order $2^{14} \cdot 3 = 49.152$.
Determinant: $2^{32}$.
Hadamard equivalent to its transpose.
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|+|-|+|-|+|-|+|-|+|-|
|+|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|
|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|+|
|+|+|+|+|-|-|-|-|+|+|+|+|-|-|-|-|
|+|-|+|-|-|+|-|+|+|-|+|-|-|+|-|+|
|+|+|-|-|-|-|+|+|+|+|-|-|-|-|+|+|
|+|-|-|+|-|+|+|-|+|-|-|+|-|+|+|-|
|+|+|+|+|+|+|+|+|-|-|-|-|-|-|-|-|
|+|+|+|-|+|-|-|-|-|-|-|+|-|+|+|+|
|+|+|-|+|-|-|-|+|-|-|+|-|+|+|+|-|
|+|+|-|-|-|+|+|-|-|-|+|+|+|-|-|+|
|+|-|+|+|-|+|-|-|-|+|-|-|+|-|+|+|
|+|-|+|-|-|-|+|+|-|+|-|+|+|+|-|-|
|+|-|-|+|+|-|+|-|-|+|+|-|-|+|-|+|
|+|-|-|-|+|+|-|+|-|+|+|+|-|-|+|-|
~~
[[Automorphism group|Automorphism]]: order $ 2^{12} \cdot 3 \cdot 7 = 86.016$.
Determinant: $-2^{32}$
Transpose equals the next matrix in the list.
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|+|-|+|+|+|+|-|-|-|-|
|+|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|
|+|-|-|+|+|-|-|+|+|-|+|-|+|-|+|-|
|+|+|+|+|-|-|-|-|+|+|-|-|-|-|+|+|
|+|-|+|-|-|+|-|+|+|-|-|+|+|-|-|+|
|+|+|-|-|-|-|+|+|+|-|-|+|-|+|+|-|
|+|-|-|+|-|+|+|-|+|-|+|-|-|+|-|+|
|+|+|+|+|+|+|+|+|-|-|-|-|-|-|-|-|
|+|-|+|-|+|-|+|-|-|-|-|-|+|+|+|+|
|+|+|-|-|+|+|-|-|-|-|+|+|-|-|+|+|
|+|-|-|+|+|-|-|+|-|+|-|+|-|+|-|+|
|+|+|+|+|-|-|-|-|-|-|+|+|+|+|-|-|
|+|-|+|-|-|+|-|+|-|+|+|-|-|+|+|-|
|+|+|-|-|-|-|+|+|-|+|+|-|+|-|-|+|
|+|-|-|+|-|+|+|-|-|+|-|+|+|-|+|-|
~~
[[Automorphism group|Automorphism]]: order $ 2^{12} \cdot 3 \cdot 7 = 86.016$.
Determinant: $-2^{32}$
Transpose equals the former matrix in the list.
This matrix is Hadamard equivalent (but not equal) to the [[sign matrix|Sign Tables]] of the [[sedenions|Sedenion]] (see MAGMA example below).

//Question: Are there any algebraic counterparts for the other three $16$-dimensional Hadamard matrices ?//

!!!![[MAGMA|http://magma.maths.usyd.edu.au/calc/]]^^[[Help|MAGMA]]^^ examples
* [[Code File|code/MAGMAHadamardMatrices.txt]]
* [[Handbook of MAGMA Functions, Chapter 121 - Hadamard Matrices|code/MAGMAHadamardMatrices.pdf]]

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]].

A ''Hamiltonian Loop'' is a (non-associative) [[loop|Loop]] for which its subloops are [[normal subloop|Normal Subloop]].

>... this E8 lattice generates the Hamming-8 code. This is important for the E8 x E8 heterotic superstring theory. The lattice structure in this Lie algebra provides for the "anomaly cancellations" for the heterotic string theory. And it was this even-self dual lattice structure that provides this "anomaly cancellation".
> What physicists call anomaly cancellation is, in effect, what the mathematicians call error-correcting - every even-self dual lattice corresponds to an even self-dual error-correcting code.
> - Jack Sarfatti -

One distinguishes between a ''Hamming Code'' and an ''Extended Hamming Code'' which are described seperately in the following:

!!!!Hamming code
A $q$-ary Hamming code $\operatorname{Ham}(r, q)$ with redundancy $r \ge 2$ is a linear and [[perfect (single error) correcting code|Perfect Code]], given by
\begin{eqnarray}
\operatorname{Ham}(r,q) &=& \left [\frac {q^r - 1} {q - 1}, \frac {q^r - 1}{q - 1} -r, 3 \right]\\
& =& [n, n -r, 3]_q  \\
&= & (n, q^{n -r}, 3)_q
\end{eqnarray}
Hence any received word with at most one error will be decoded correctly and the code has the smallest possible size of any code that does this.

In case of a binary code one has:
\begin{eqnarray}
\operatorname{Ham}(r,2) \equiv \operatorname{Ham}(r) &=& \left [2^r ? 1, 2^r - r - 1, 3 \right] \\
&= &[n, n - r, 3] \\
&= &(n, 2^{n - r}, 3)
\end{eqnarray}
For a given $r$, any [[perfect|Perfect Code]] $1$-error correcting linear code of length $n=2r-1$ and dimension $n-r$ is a Hamming Code.

!!!!Extended Hamming code
A ''Binary Extended Hamming Code $\operatorname{Ham_E}(m, 2) \equiv \operatorname{Ham_E}(m)$'' is a $[n, n - m - 1,4] = [2^m, 2^m - m - 1, 4]$-code which is a  $\operatorname{RM}(m-2,m)$ [[Reed-Muller code|Reed-Muller Code]].
It is obtained by adding a "parity check bit" to a Hamming code, i.e. $\operatorname {Ham}(r) = [2^r ? 1, 2^r ? r - 1, 3] \rightarrow [2^r , 2^r ? r -1, 3 + 1] = \operatorname {Ham_E}(r)$.

(For $\mathbb F_2$) the [[dual code|Dual Code]] of an extended Hamming code is a [[Hadamard code|Hadamard Code]] (i.e. a first order Reed\-Muller code $\operatorname{RM}(1,m)$).

The [[automorphism group|Automorphism]] of the extended Hamming code is isomorphic to [[AGL(r, 2)|Affine General Linear Group]]. (It is the same automorphism as for the simplex code of same length $[2^r, r+ 1, 2^{r?1}]$).

Contrary to the Hamming code, an extended Hamming code is not perfect.

!!!!! Examples
* $\operatorname{Ham}_E(3) = [8,8,4] = \operatorname{RM}(1,3)$. Dual code: $\operatorname {RM}(1,3) = \operatorname{Had}(3) = \operatorname{Ham}_E(3)$, i.e. this Hamming code is self-dual.
* $\operatorname{Ham}_E(4) = [16,11,4] = \operatorname{RM}(2,4)$. Dual code: $\operatorname {RM}(1,4) = \operatorname{Had}(4)$.
* $\operatorname{Ham}_E(5) = [32,26,4] = \operatorname{RM}(3,5)$. Dual code: $\operatorname {RM}(1,5) = \operatorname{Had}(5)$.

Papers:
* [[Error Detecting and Error Correcting Codes - R. W. Hamming|http://vigeland.paradise.caltech.edu/ist4/handouts/IST4_hamming1950.pdf]] [[local|papers/IST4_hamming1950.pdf]] {{t1000Cite{[[pct. 1164|http://scholar.google.de/scholar?cites=15739239885988894465&hl=de]]}}}

Links:
* [[A Library of Linear (and Nonlinear) Codes - N. J. A. Sloane|http://www.research.att.com/~njas/codes/]]
* [[A Venn Diagram of the Hamming Code - D. A. Richter|http://homepages.wmich.edu/~drichter/hammingvenn.htm]]

The ''Hamming Distance $d(c,c?)$'' of two words $c$ and $c'$ of equal length of a code $C$ is defined as the number of letters that are different.

Formally:
\[
d(c,c?) \equiv w(c-c?)
\]
where $w(c - c?)$  is the [[Hamming weight|Hamming Weight]].

If the code is a [[linear blockcode|Linear Blockcode]] the function $d(c,c?) : \mathbb F^n_q \rightarrow  \mathbb F^n_q \times \mathbb F^n_q \rightarrow \mathbb R \ge 0$ defines a metric in the usual topological sense.

In the special case of a binary code ($q = 2$), the comparison of two blocks can be carried out via an XOR\-operation and the Hamming distance is determined by the number of "1's".

The Hamming distance determines the maximal number of errors $t$ that can be corrected, which is
\[
t = \left\lfloor\frac{d-1}{2}\right\rfloor
\]
!!!!Examples
: $001 \underline{1}0$ and $001\underline{0}0 \rightarrow$ Hamming-distance =$1$
: $1\underline{2}34\underline{5}$ and $1\underline{3}34\underline{4} \rightarrow$ Hamming-distance=$2$
: $\underline{h}o\underline{r}se$ and $\underline{m}o\underline{u}se \rightarrow$ Hamming-distance=$2$

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]]: $w(x) \in 4\mathbb Z$.

The ''Heat Kernel'' $U(t)$ is an operator acting on a n-dimensional [[Riemannian manifold|Riemann Space]] $\mathcal M$ without boundary and is given by:
\begin{equation}
U(t) = exp(t\square)
\end{equation}
with $\square$ is the [[Laplace-Beltrami operator|Laplace-Beltrami Operator]].

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:
\[
(\mathbf H_f)_{ij}(\mathbf{x}) = \frac{\partial^2 f(\mathbf{x})}{\partial x_i\partial x_j} = \partial_i \partial_j f(\mathbf{x})\,\!
\]
Written out explicitly it is:
\[
\mathbf H_f(\mathbf{x}) = \begin{pmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1\,\partial x_n} \\  \\ \frac{\partial^2 f}{\partial x_2\,\partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2\,\partial x_n} \\  \\ \vdots & \vdots & \ddots & \vdots \\  \\ \frac{\partial^2 f}{\partial x_n\,\partial x_1} & \frac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{pmatrix}
\]
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:
\[
\Delta f(\mathbf{x}) =f(\mathbf{x}+\Delta\mathbf{x})\approx f(\mathbf{x}) + \mathbf J_f(\mathbf{x})\Delta \mathbf{x} +\frac{1}{2} \Delta\mathbf{x}^\mathrm{T} \mathbf H_f(\mathbf{x}) \Delta\mathbf{x}
\]
The first order change of $f$ is described by the [[Jacobian matrix|Jacobi Matrix]] $\mathbf J_f$.

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

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

!!!! An essay ...
<html><center><img src="images/pyramide-hierarchie.jpg" style="width: 395px; "/></center></html>
For the world to be structured the way it is, a big seperation of the orders of the grades of the world polyvector is required.
If we assume that Planck's constant determines the separation between the vector- and the bivector grades (between relativity and quantum mechanics/spin), it becomes clear, that if this constant were not very small, classical physics and quantum mechanics would be blurred and we would not observe a distinct classical world.
If we assume that the values of elementary charges determine the separation between the second and third grades, then it follows, that if this value were much bigger, the charges were also much bigger and the world would be much more clumped together.
A similar problem arises if one considers the separation of the third and the fourth grade. One might speculate that the cosmological constant - which is of the order $10^{-120}$ - is the relevant value for their separation. If it were much bigger, the universe would not have expanded to a size comparable to what we see today.
Furthermore if the seperations would generally be smaller, higher grades (5 and above) might play a significant role if they existed.
If one assumes that they do in fact exist, given the actual constants of nature, they can be neglected as they are expected to be of order $10^{-160}$ and smaller.

Summarizing the things just said heuristically in terms of a world polyvector $\Phi$, it would look sth. like this:
\begin{eqnarray}
\Phi &=&c \mathbf e_\mu + \hbar \mathbf e_{\mu\nu}  + e \mathbf e_{\mu\nu\rho}  + \Lambda \mathbf e_{\mu\nu\rho\sigma} + \Omega \mathbf e_{\mu\nu\rho\sigma\tau} + \ldots  \\
&\approx& \mathbf e_\mu + 10^{-40} \mathbf e_{\mu\nu} + 10^{-80} \mathbf e_{\mu\nu\rho}  + 10^{-120} \mathbf e_{\mu\nu\rho\sigma} + 10^{-160} \mathbf e_{\mu\nu\rho\sigma\tau} + \ldots \\
&\approx& \mathbf e_\mu + l_P \mathbf e_{\mu\nu} + l_P^2 \mathbf e_{\mu\nu\rho}  + l_p^3 \mathbf e_{\mu\nu\rho\sigma} +  l_p^4 \mathbf e_{\mu\nu\rho\sigma\tau} + \ldots
\end{eqnarray}
with $c$ the speed of light, $\hbar$ Planck's constant, $e$ the elementary charge, $\Lambda$ the cosmological constant and $\Omega$ a constant, which would be a signature of a 5-th dimension. // QUESTION: Could the Bolzmann constant be involved here ? //

In the last step we have assumed that the separations are of the order of a multiple of Planck's length. Due to dimensionality reasons, to be able to add two adjacent grades, one must relate them by a dimension of length. As the natural and most fundamental length scale is the Planck length, it is chosen.
The representation in term of a polyvector also suggest that all the constants in nature can be derived from one fundamental unit which here is assumed to be the [[planck length|Planck Units]].
As we are dealing with ratios of the order $10^{40}$, the [[large number hypothesis|Large Number Hypothesis]] might be related to polyvector physics.

// QUESTION: Could the scalar part which has been omitted here be the mass of the universe which is about $10^{55}g \approx 10^{40}m_P$ (Eddington ?) //

© by Markus Maute, 2009

Papers:
* [[The Hierarchy Problem and New Dimensions at a Millimeter - N. Arkani–Hamed, S. Dimopoulos, G. Dvali|http://arxiv.org/PS_cache/hep-ph/pdf/9803/9803315v1.pdf]]

''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]]

<html><center><img src="images/HoffmanSingletonGraph.jpg" style="width: 325px; "/></center></html>
The ''Hoffman\-Singleton Graph'' is a [[Moore graph|Moore Graph]] where each vertex has degree $7$, and the girth is $5$. It is the unique [[(7,5)-cage graph|Cage]].
It is a symmetric graph, having $50$ vertices and $175$ edges. The [[automorphism group|Automorphism]] of the graph has order $252.000 =  2^5 \cdot 3^2 \cdot 5^3 \cdot 7 = 525 \cdot 480$.
Since it is a symmetric graph its automorphism group acts transitively on its vertices, edges, and arcs.

The Hoffman\-Singleton graph is contained $704$ times in the [[Higman-Sims graph|Higman-Sims Graph]]. On the other hand is has $525$ [[Petersen graph|Moore Graph]] as subgraphs.

One construction is to identify its vertices with the $15$ vertices and $35$ lines of the projective space [[PG(3,2)]].

Links:
* [[Math Games - The Hoffman-Singleton Game|http://www.maa.org/editorial/mathgames/mathgames_11_01_04.html]]
* [[Hoffman-Singleton Graph -  A. E. Brouwer|http://www.win.tue.nl/~aeb/graphs/Hoffman-Singleton.html]]
* [[Hoffman-Singleton Playing Cards|http://www.mathpuzzle.com/facecards.jpg]] - Every vertex of the Hoffman\-Singleton Graph corresponds to a card.

Presentations:
* [[Everything you wanted to know about the Hoffman-Singleton Graph ... but were afraid to draw - R. Beezer|http://buzzard.ups.edu/talks/beezer-2009-portland-maa-hoffman-singleton.pdf]]

Papers:
* [[Rick’s Tricky Six Puzzle: S5 Sits Specially in S6 - A. Fink, R. Guy|http://www.maa.org/pubs/mmapr09.pdf]] pct. 0
* [[Symmetric Drawings of the Hoffman-Singleton Graph - R. A. Litherland|http://www.math.lsu.edu/~lither/hoff-sing/pictures.pdf]] pct. 0

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

A function $f: X ? Y$ 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'' - bijective homomorphism. Isomorphic objects are completely indistinguishable as far as the structure in question is concerned. A generalisation of isomorphism are [[isotopies|Isotopy]].
* [[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]]

''Hopf Algebras'' can be considered as generalizations of [[groups|Group]] in the sense that they are "noncommutative algebras of functions on groups" (Drinfield).
Hence the name ''Quantum Groups''. The coproduct encodes the group structure.

A ''Hopf Algebra'' is a ''Bialgebra'' $(H,\nabla,\eta,\Delta,\epsilon)$ over a field $\mathbb K$ with the extra structure of an ''Antipode'' $S$ (''Co\-Inverse'').

For the bialgebra one has the following maps:
''Product'': $ \nabla: H  \otimes H \mapsto H $ with ''Unit'': $ \eta : \mathbb K \mapsto H$
''Coproduct'': $ \Delta: H \mapsto H  \otimes H $ with ''Counit'': $\epsilon : H \mapsto \mathbb K$

The antipode $S$ is given by:
$S: H \mapsto H$.
The relation between the maps is shown in the following diagram:

<html><center><img src="images/png/HopfAlgebra.png" style="width: 233px; "/></center></html>

A generalisation are so called [[Quasi-Hopf Algebras]].
Papers:
* [[The Hopf Algebra Structure of GL(1,Hq) and the Isomorphism between SPq(1) and SUq(2) - S. Celik|http://arxiv.org/PS_cache/math/pdf/0112/0112118v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=4542758295980108055&hl=de]]

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.

A ''Hurwitz Integer'' (a.k.a ''Hurwitz Quaternion'', ''Integral Quaternion'') is a [[quaternion|Quaternion]] whose components are either all integers or all half-integers. The set $H$ of all Hurwitz integers therefore is given by
\[
H = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z} \;\vee\, a,b,c,d \in \mathbb{Z} + \tfrac{1}{2}\right\}
\]
and forms a lattice in $\mathbb R^4$
$H$ is closed under quaternion-multiplication and -addition and hence forms a subring of the quaternion ring $\mathbb H$.

There are 24 units of norm 1 given by
\[
\{ \pm e, \pm i, \pm j, \pm k, \frac 12 (\pm e \pm i  \pm j  \pm k) \}
\]
These form a nonabelian group of order 24 known as the [[binary tetrahedral group|Binary Tetrahedral Group]]. It has the [[quaternion group|Quaternion Group]] as a [[normal subgroup|Normal Subgroup]].
Furthermore they represent the [[root vectors|Root Vector]] of the Lie algebra [[SO(8)]] which are the most inner shell of the lattice spanned by the Hurwitz integers.

The the vectors defined by the first and the second shell of the lattice (the latter having length $\sqrt 2$) represent the non-zero root vectors of the exceptional [[Lie algebra|Lie Algebra]] [[F4]]. Therefore the lattice formed by the Hurwitz integers is also refered to as ''$F_4$ Lattice''.

The set of Hurwitz integers contains the set of complex [[Gaussian integers|Gaussian Integer]] and the set of complex Eisenstein integers as 2-dimensional sections.

See also:
* [[Gaussian Integers|Gaussian Integer]] (complex integers)
* [[Integral octonions|Integral Octonion]] (octonionic integers)

Papers:
* [[A Critical Analysis of the Hydrino Model - A. Rathke|http://arxiv.org/PS_cache/quant-ph/pdf/0505/0505150v1.pdf]] [[pct. 8|http://scholar.google.de/scholar?cites=13240601268371920337&hl=de]]

Links:
* [[BlackLight Power Inc.|http://www.blacklightpower.com/]]

Considerable effort has been spent to constructing a quaternionic analysis which in respect to its richness of internal properties and applications can be compared to complex analysis. However, in the opinion of the majority of contemporary mathematicians, no safisfactory solution for this problem has been found until now.
The natural definition of a “right”- and "left"-derivative is also unproductive, since the requirement for its existence and the uniqueness of the limit leads to a considerably over-determined system of PDE’s which appears to be compatible only for the trivial case of linear functions.
Nonetheless, numerous attempts to bypass these difficulties have been undertaken, among others by Fueter. All these attempts, however, cannot, perhaps, be considered as a successful version of quaternionic analysis. The same applies to the construction of a more complicated non-associative analysis (e.g. over the algebra of octonions).

Papers:
* [[Quaternionic Analysis and the Algebrodynamics - V. V. Kassandrov|http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.2895v1.pdf]] [[local|papers/algebrodynamics.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=3171071724294877694&hl=de]]
* [[On Regular Functions of a Nonalternative Hypercomplex Variable - K. Imaeda, M. Imaeda |http://nels.nii.ac.jp/els/110000047027.pdf;jsessionid=03F7CEB2823D0DEBC1AE390C5CE879F8?id=ART0000382546&type=pdf&lang=jp&host=cinii&order_no=&ppv_type=0&lang_sw=&no=1222930590&cp=]] [[pct. 1|http://scholar.google.de/scholar?cites=6263168713478112476&hl=de]]

Given an algebra $\mathcal A$, an ''Ideal'' is a special kind of subalgebra $\mathcal A'$ of $\mathcal A$ with the property, that for any $\mathbf A' \in \mathcal A'$ and $\mathbf A \in \mathcal A$, $\mathbf {AA'} \in \mathcal A'$.
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 $\mathbf e$ is also a [[zero divisor|Zero Divisor]] as $\mathbf A^2 = \mathbf A$ implies $\mathbf A (\mathbf A - \mathbf e) = 0$.

/***
|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)};
//}}}
//{{{
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>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 $O(n)$.

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 $\mathcal M$.

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]].

>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

<html><center><img src="images/integralOctonions.jpg" style="width: 257px; "/></center></html>
The ''Integral Octonions'' (a.k.a. ''Integer Cayley Numbers'', ''Integral Octaves'', ''Octavians'') are a discrete non-associative ring based on the [[octonion algebra|Octonion]] $\mathbb O$. They define a [[maximal order|Maximal Order]] within the octonions and can be regarded as a natural generalisation of the ring of integers $\mathbb Z$, as is the case for the [[Gaussian integers|Gaussian Integer]] and the [[Hurwitz integers|Hurwitz Integer]].

The integral octonions form a lattice inside $\mathbb O$ which is a rescaled [[E8 lattice|E8 Lattice]].

A special subset are the ''Octavian Units'' which are [[integral elements|Integral Elements]] having inverses that are again integral octonions. There are $240$ such elements and they correspond to the vertices of the inner shell of the $E_8$-lattice. After rescaling with $\sqrt 2$ they are identical with the $E_8$-roots.

!!!!Construction
Given a non-split octonion algebra obtained by a classical [[Cayley-Dickson doubling|Cayley-Dickson Doubling]], one possible representation of the $240$ octavian units is as follows (see [1]) :
\begin{eqnarray}
&& \mathbf r = \pm \mathbf e_0 \\
&&  \mathbf c_{i} = \pm \mathbf e_i, \;\, i  = 1, \ldots, 7 \\
&&\mathbf q_1 \equiv \frac 12  (\pm \mathbf e_0 \pm \mathbf e_2 \pm \mathbf e_3 \pm \mathbf e_5), \;\;\,  \mathbf q_1^\bot \equiv \frac 12  (\pm \mathbf e_1 \pm \mathbf e_4 \pm \mathbf e_6 \pm \mathbf e_7) \\
&&\mathbf q_2 \equiv \frac 12  (\pm \mathbf e_0 \pm \mathbf e_1 \pm \mathbf e_3 \pm \mathbf e_6), \;\;\,  \mathbf q_2^\bot \equiv \frac 12  (\pm \mathbf e_2 \pm \mathbf e_4 \pm \mathbf e_5 \pm \mathbf e_7) \\
&&\mathbf q_3 \equiv \frac 12  (\pm \mathbf e_0 \pm \mathbf e_1 \pm \mathbf e_2 \pm \mathbf e_7), \;\;\,  \mathbf q_3^\bot \equiv\frac 12  (\pm \mathbf e_3 \pm \mathbf e_4 \pm \mathbf e_5 \pm \mathbf e_6) \\
&&\mathbf q_4 \equiv \frac 12  (\pm \mathbf e_0 \pm \mathbf e_5 \pm \mathbf e_6 \pm \mathbf e_7), \;\;\, \mathbf q_4^\bot \equiv \frac 12  (\pm \mathbf e_1 \pm \mathbf e_2 \pm \mathbf e_3 \pm \mathbf e_4) \\
&&\mathbf q_5 \equiv \frac 12  (\pm \mathbf e_0 \pm \mathbf e_1 \pm \mathbf e_4 \pm \mathbf e_5), \;\;\,  \mathbf q_5^\bot \equiv \frac 12  (\pm \mathbf e_2 \pm \mathbf e_3 \pm \mathbf e_6 \pm \mathbf e_7) \\
&&\mathbf q_6 \equiv \frac 12  (\pm \mathbf e_0 \pm \mathbf e_2 \pm \mathbf e_4 \pm \mathbf e_6), \;\;\,  \mathbf q_6^\bot \equiv \frac 12  (\pm \mathbf e_1 \pm \mathbf e_3 \pm \mathbf e_5 \pm \mathbf e_7) \\
&&\mathbf q_7 \equiv \frac 12  (\pm \mathbf e_0 \pm \mathbf e_3 \pm \mathbf e_4 \pm \mathbf e_7), \;\;\, \mathbf q_7^\bot \equiv \frac 12  (\pm \mathbf e_1 \pm \mathbf e_2 \pm \mathbf e_5 \pm \mathbf e_6) \\
\end{eqnarray}
The set of units therefore decomposes according to $240 = 8\cdot 2 + 14 \cdot 2^4 = 16 + 2\cdot 56 + 2\cdot 56 = 16 + 112 + 112$.

Computer simulations show that $\langle \mathbf q_i|\mathbf q_i^\bot\rangle = 0$ holds irrespective of sign combinations.

(Notice that for a given pair $\mathbf q_i$, $\mathbf q_i^\bot$, every index $0, \ldots, 7$ occurs exactly once. I.e. the indices of $\mathbf q_i$ and $\mathbf q_i^\bot$ are complementary (or "dual") to one another. As $\mathbf e_0$ is contained in all the $\mathbf q_i$'s, one has a duality ${}^*$ between $3$ and $4$ indices, e.g. $(235)^* = (1467)$).

This construction fails in case of the [[split octonion algebras|Split Octonion]].

!!!!! Relationship with [[Steiner systems|Steiner System]]
The elements $\mathbf q_1, \ldots, \mathbf q_7$ can also be obtained from the [[Steiner triple system|Steiner Triple System]] $STS(7) = \{\{1,2,7\},\{1,3,6\},\{1,4,5\},\{2,3,4\},\{2,5,6\},\{3,5,7\},\{4,6,7\}\}$ which is equivalent to the [[Fano plane|Fano Planes - Classification]] $15$ of class $1$ (for details see: [[classification of Fano planes|Fano Planes - Classification]]).
<html><center><img src="images/E8Fano.jpg" style="width: 155px; "/></center></html>
Furthermore the elements $\mathbf q_1, \ldots, \mathbf q_7, \mathbf q_1^\bot, \ldots, \mathbf q_7^\bot$  are in $1:1$-correspondence with a [[SQS(8)-Steiner quadruple system|Steiner Quadruple System]]. (Given the $7$ triples of a $STS$, the $\mathbf q_i^\bot$ are determined by its extension to the respective $SQS$).

There are $30$ inequivalent representations of the Fano plane and hence one has $30$ variants of the construction of the octavian units given above. However only for $7$ of them one obtains a closed algebra. This corresponds to the known fact that there exist $7$ different [[maximal orders|Maximal Order]] of the octonions.
Besides the construction described above, the other $6$ constructions are based on the Fano planes $2,3,4,7$ and $12$ of class $1$, according to the classification mentioned.

The situation is more involved if one considers all [[480 different octonion algebras|480 Octonion Multiplication Tables]]. (We restrict ourselves to the non-split case here):
These fall into $30$ classes, each one represented by a different Fano plane. (The $7$ lines of the respective Fano plane are the $7$ associative triples of the associated algebra).
Among algebras within a class, multiplication tables are equal modulo [[signs|Sign Tables]] of their [[structure constants|Structure Constants]]. It turns out that given a class, the maximal orders are identical for all $16$ algebras therein.
For different classes however they are different. Yet the number of maximal orders is the same for all $30$ classes and hence for all $480$ different octonion algebras.

The following table shows the maximal orders for the $30$ octonion algebras based on the different Fano planes. The notation of the Fano planes is given by ($\langle class\rangle$, $\langle number \rangle$):
~~
|!Algebra |!Maximal orders|
|(1,1) |(2,1), (2,2), (2,3), (2,4), (2,7), (2,12), (2,15)|
|(1,2) |(2,1), (2,2), (2,3), (2,5), (2,9), (2,10), (2,14)|
|(1,3) |(2,1), (2,2), (2,3), (2,6), (2,8), (2,11), (2,13)|
|(1,4) |(2,1), (2,4), (2,5), (2,6), (2,7), (2,11), (2,14)|
|(1,5) |(2,2), (2,4), (2,5), (2,6), (2,8), (2,10), (2,15)|
|(1,6) |(2,3), (2,4), (2,5), (2,6), (2,9), (2,12), (2,13)|
|(1,7) |(2,1), (2,4), (2,7), (2,8), (2,9), (2,10), (2,13)|
|(1,8) |(2,3), (2,5), (2,7), (2,8), (2,9), (2,11), (2,15)|
|(1,9) |(2,2), (2,6), (2,7), (2,8), (2,9), (2,12), (2,14)|
|(1,10)|(2,2), (2,5), (2,7), (2,10), (2,11), (2,12), (2,13)|
|(1,11)|(2,3), (2,4), (2,8), (2,10), (2,11), (2,12), (2,14)|
|(1,12)|(2,1), (2,6), (2,9), (2,10), (2,11), (2,12), (2,15)|
|(1,13)|(2,3), (2,6), (2,7), (2,10), (2,13), (2,14), (2,15)|
|(1,14)|(2,2), (2,4), (2,9), (2,11), (2,13), (2,14), (2,15)|
|(1,15)|(2,1), (2,5), (2,8), (2,12), (2,13), (2,14), (2,15)|
|(2,1) |(1,1), (1,2), (1,3), (1,4), (1,7), (1,12), (1,15)|
|(2,2) |(1,1), (1,2), (1,3), (1,5), (1,9), (1,10), (1,14)|
|(2,3) |(1,1), (1,2), (1,3), (1,6), (1,8), (1,11), (1,13)|
|(2,4) |(1,1), (1,4), (1,5), (1,6), (1,7), (1,11), (1,14)|
|(2,5) |(1,2), (1,4), (1,5), (1,6), (1,8), (1,10), (1,15)|
|(2,6) |(1,3), (1,4), (1,5), (1,6), (1,9), (1,12), (1,13)|
|(2,7) |(1,1), (1,4), (1,7), (1,8), (1,9), (1,10), (1,13)|
|(2,8) |(1,3), (1,5), (1,7), (1,8), (1,9), (1,11), (1,15)|
|(2,9) |(1,2), (1,6), (1,7), (1,8), (1,9), (1,12), (1,14)|
|(2,10)|(1,2), (1,5), (1,7), (1,10), (1,11), (1,12), (1,13)|
|(2,11)|(1,3), (1,4), (1,8), (1,10), (1,11), (1,12), (1,14)|
|(2,12)|(1,1), (1,6), (1,9), (1,10), (1,11), (1,12), (1,15)|
|(2,13)|(1,3), (1,6), (1,7), (1,10), (1,13), (1,14), (1,15)|
|(2,14)|(1,2), (1,4), (1,9), (1,11), (1,13), (1,14), (1,15)|
|(2,15)|(1,1), (1,5), (1,8), (1,12), (1,13), (1,14), (1,15)|
~~
Some observations:
* Given an algebra, the $7$ maximal orders stem exclusively from one of the $2$ classes of Fano planes.
* The class of the Fano plane class on which the algebra is based is always opposite to the ones of its maximal orders. I.e. the construction given above always fails if one chooses the Steiner system based on the the $7$ associative triples ("quaternionic roots") of the algebra.
* All in all there are $210 = 30\cdot 7$ maximal orders for the $30$ Fano planes. A closer inspection of the table reveals that each Fano plane occurs exactly $7$ times. I.e. the Fano planes are equidistributed among the maximal orders. In other words, given a Fano plane, one can always find $7\cdot 16 = 112$ algebras that have it as maximal order. For the other $368$ cases the $240$ integral elements do not close under the respective octonion product.

As a consequence of the above one has $3.360 = 480 \cdot 7 = 30\cdot 112$ different constructions of the octavian units and hence of the $E_8$-root system. (An observation: This number divides the order of the $E_8$-Weyl group. Is this an accident ?)

!!!!!Relationship with [[codes|Blockcode]]
The $\mathbf E_8$-lattice can also be expressed in terms of [[Reed-Muller-|Reed-Muller Code]], [[Hadamard-|Hadamard Code]] or [[Hamming-|Hamming Code]] codes as follows:
\begin{eqnarray}
\mathbb E_8 &= & 2 \mathbb Z^8 + [8,4,4] \\
& =& 2 \mathbb Z^8 + \operatorname{RM}(1,3) \\
& = & 2 \mathbb Z^8 + \operatorname{Had}(3) \\
&=& 2 \mathbb Z^8 + \operatorname{Ham_E}(3)
\end{eqnarray}

If we take the inner shell of this lattice and rescale it appropriately we get the octavian units.
The $r$ and $c_i$ in the construction above correspond with the $\mathbb Z^8$ sublattice and the $q_i$ and $q^\bot_i$ with the sublattice generated by the code.

The  $q_i$ and $q^\bot_i$ can be mapped to the $14$ weight-$4$ codewords as follows:
\begin{eqnarray}
&& q_1 = [10110100], \; \bar q^\bot_1 = [01001011], \\
&& q_2 = [11010010], \; \bar q^\bot_2 = [00101101], \\
&& q_3 = [11100001], \; \bar q^\bot_3 = [00011110], \\
&& q_4 = [10000111], \; \bar q^\bot_4 = [01111000], \\
&& q_5 = [11001100], \; \bar q^\bot_5 = [00110011], \\
&& q_6 = [10101010], \; \bar q^\bot_6 = [01010101], \\
&& q_7 = [10011001], \; \bar q^\bot_7 = [01100110], \\
\end{eqnarray}
For the scalar product in $\mathbb F_2$-space one also has $\langle \mathbf q_i|\mathbf q_i^\bot\rangle = 0$.

Using the $q_i$'s, replacing the $0$'s by $-1$'s and adding the weight-$8$ word of the code one can create the following $8\times 8$-matrix
\[
\begin{pmatrix} q_1\\ q_2\\q_3\\q_4\\q_5\\q_6\\q_7\\q_8\end{pmatrix} = \begin{pmatrix}
1&-1&1&1&-1&1&-1&-1\\
1&1&-1&1&-1&-1&1&-1\\
1&1&1&-1&-1&-1&-1&1\\
1&-1&-1&-1&-1&1&1&1\\
1&1&-1&-1&1&1&-1&-1\\
1&-1&1&-1&1&-1&1&-1\\
1&-1&-1&1&1&-1&-1&1\\
1&1&1&1&1&1&1&1 \end{pmatrix}
\]
which is a [[Hadamard Matrix|Hadamard Matrix]]. (See Magma example below). In case of $8$ dimensions this matrix is unique up to automorphisms of the order $21.504 = 2^{10} \cdot 3 \cdot 7$ which is a divisor of the order of the $E_8$-Weyl group $ \operatorname{ord}(Aut(E_8))/21.504 = 32.400$. (Is this accidential ?)
A permutation of rows or columns leads to an equivalent Hadamard matrix. In terms of the code this means that the three relevant parameters $n$, $k$ and $d$ remain unchanged. Analogous arguments apply to the $\bar{\mathbf q_i}$'s.
Notice that the [[(minimal) Hamming distance|Hamming Distance]] of $4$ translates into the fact that for a given triad of the Fano plane at least two elements must be different. In terms of designs that means that two blocks must be different in at least two points. (In fact they differ in exactly two points).

!!!!!Roots of unity
One can group the integral octonions according to their multiplicative [[order|Order]] $m$:
\begin{eqnarray}
m = 1:&& \mathbf e_0 \\
m = 2:&& -\mathbf e_0 \\
m = 3: &&\frac 12  (\mathbf e_0 \pm \mathbf e_i \pm \mathbf e_j \pm \mathbf e_k) \\
m = 4: && \pm \mathbf e_i\\
           &&   \frac 12  (\pm \mathbf e_i \pm \mathbf e_j \pm \mathbf e_k \pm \mathbf e_l) \\
m = 6: && \frac 12  (-\mathbf e_0 \pm \mathbf e_i \pm \mathbf e_j \pm \mathbf e_k)
\end{eqnarray}
or in other words, given elements are an $m$-th ''Root Of Unity''.
This way the number $240$ splits up according to $240 = 1 + 1 + 56 + 14+ 112 + 56$.

Computer experiments show that for any two integral octonions $\mathbf o_1$ and $\mathbf o_2$ one has
\[
\mathbf o_1 \mathbf o_2^m = \mathbf o_1
\]
with $m \in \{1,2,3,4,6\}$.
This generalizes the multiplicative orders given above, which are reproduced in the case $\mathbf o_1=\mathbf o_2$.

One has $114$ sixth roots of unity given by the $\mathbf r$'s and the $\mathbf q_i$'s. (I.e. the sixth roots of unity are exactly those integral octonions that contain the identity element).
They fall into two classes with $57$ elements each, those with $\mathbf r^3 = \mathbf q_i^3 = 1$ and those with $\mathbf r^3 = \mathbf q_i^3 = -1$.
The $\mathbf q_i$'s are also referred to as ''Brandt Transformers'' in literature due to a theorem by Brandt stating:
The map$ \mathbf X \rightarrow \mathbf A^{-1} \mathbf X \mathbf A$ is an [[automorphism|Automorphism]] if and only if $\mathbf A$ is a sixth root of unity.

!!!!! [[D8|Checkerboard Lattice]] (= [[SO(16)]]) construction
Alternatively one can construct the octavian units building upon the set $R$ of $112$ roots of the group $SO(16)$.
This requires an adequate change of basis of the octonion algebra.
The elements of the new basis $\mathbf e'_i$ (which are [[simple roots|Simple Root]] of $SO(16$)) can be chosen as follows:
\begin{eqnarray}
\mathbf e'_0 \equiv (\mathbf e_0 + \mathbf e_4), \quad \mathbf e'_4 \equiv (\mathbf e_0 - \mathbf e_4) \\
\mathbf e'_1 \equiv (\mathbf e_1 + \mathbf e_5), \quad \mathbf e'_5 \equiv (\mathbf e_1 - \mathbf e_5) \\
\mathbf e'_2 \equiv (\mathbf e_2 + \mathbf e_6), \quad \mathbf e'_6 \equiv (\mathbf e_2 - \mathbf e_6) \\
\mathbf e'_3 \equiv (\mathbf e_3 + \mathbf e_7), \quad \mathbf e'_7 \equiv (\mathbf e_3 - \mathbf e_7) \\
\end{eqnarray}
or, written in a more compact form:
\[
\mathbf e'_i \equiv (\mathbf e_i + \mathbf e_{i+4}), \quad \mathbf e'_{i+4} \equiv (\mathbf e_i - \mathbf e_{i+4}), \quad i = 0,1,2,3
\]
With them the set of roots of $SO(16)$ is given by:
\[
R = \pm \mathbf e'_i \pm \mathbf e'_j,  \quad i, j = 0,\ldots 7, \;\,  i<j
\]
That is, one picks all possible $2$-element sets out of a set of $8$ elements. There are $28 = \large {8 \choose 2}$ possibilities to do this. Combining this with the $4$ sign combinations one gets all the $112$ roots.
Notice that $R$ is not closed under octonion multiplication.

$R$ can be extended to the full set of $E_8$-roots by adjoining those $128$ elements
\[
\sum_{i =0}^7 \pm \mathbf e'_i
\]
which have an odd number of "-" signs.

!!!![[Isotopies|Isotopy]]
The group of isotopies is isomorphic to $Spin^+_8 (\mathbb F_2) = 2^2 O_8^+(2)$ with $O_8^+(2)$ the finite, simple [[orthogonal group|Orthogonal Group]] over $\mathbb F_2$. It is generated by the triple of maps $(\mathbf L_{\mathbf A}, \mathbf R_{\mathbf A}, \mathbf B_{\mathbf A})$ with $\mathbf L_{\mathbf A}$ and $\mathbf R_{\mathbf A}$ [[left- and right translations|Left- and Right Translation]] and $\mathbf B_{\mathbf A}$ a bimultiplication, given by $\mathbf B_{\mathbf A}(\mathbf X) = \mathbf A^{-1} \mathbf X \mathbf A^{-1}$.

!!!![[SAGE|http://www.sagenb.org/]]^^[[Help|Sage]]^^ examples
{{{
RM = gap.ReedMullerCode(1,3)
N = gap.Elements(RM)
gap.Size(N)
Aut = RM.AutomorphismGroup()
gap.Size(Aut)
gap.Elements(RM)
gap.WeightDistribution(RM)
}}}

!!!![[MAGMA|http://magma.maths.usyd.edu.au/calc/]]^^[[Help|MAGMA]]^^ examples
{{{
R := MatrixRing(Integers(), 8);

H := R!
[1,-1,1,1,-1,1,-1,-1,
1,1,-1,1,-1,-1,1,-1,
1,1,1,-1,-1,-1,-1,1,
1,-1,-1,-1,-1,1,1,1,
1,1,-1,-1,1,1,-1,-1,
1,-1,1,-1,1,-1,1,-1,
1,-1,-1,1,1,-1,-1,1,
1,1,1,1,1,1,1,1];

IsHadamard(H);
}}}

Papers:
* [[Ideals in the Intgral Octaves - D. Allcock|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.22.8580&rep=rep1&type=pdf]] [[pct. 6|http://scholar.google.de/scholar?cites=2431900507781036899&hl=de]]
* [[Beyond Ideals in the Dickson Ring of Integral Octonions - F. Chaitin-Chatelin|http://www.umcs.maine.edu/~chaitin/f7.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=10547392125336397789&hl=de]]
* [[Hyperbolic Weyl Groups and the Four Normed Division Algebras - A. J. Feingold, A. Kleinschmidt, H. Nicolai|http://arxiv.org/PS_cache/arxiv/pdf/0805/0805.3018v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?hl=de&lr=&cites=6176913252844757445&um=1&ie=UTF-8&ei=OjV8Stn9Cdi1sgaT6_zYAg&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]
* [[Prime Factorization of Integral Cayley Octaves - P. Rehm|http://archive.numdam.org/ARCHIVE/AFST/AFST_1993_6_2_2/AFST_1993_6_2_2_271_0/AFST_1993_6_2_2_271_0.pdf]] [[pct. 7|http://scholar.google.de/scholar?cites=5391424943379139762&hl=de&as_sdt=2000]]
* [[Cayley Orders - A. M. Cohen, G. Nebe, W. Plesken|http://archive.numdam.org/ARCHIVE/CM/CM_1996__103_1/CM_1996__103_1_63_0/CM_1996__103_1_63_0.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=15837980437847642931&hl=de&as_sdt=2000]]
* [[Hyperbolic Weyl Groups and the four Normed Division Algebras - A. J. Feingold, A. Kleinschmidt, H. Nicolai|http://aps.arxiv.org/PS_cache/arxiv/pdf/0805/0805.3018v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=6176913252844757445&hl=de]]
* [[Integral Octonions and E8 - M. Koca|http://streaming.ictp.trieste.it/preprints/P/86/224.pdf]] pct 0

Theses:
* [[Ganzzahlige Oktonionen - T. Quade|http://www.quadi.de/~thomas/diplom/Diplom.pdf]] [[local|theses/GanzzahligeOktonionen.pdf]] - Related website: [[Thomas Mathe-Seite|http://www.quadi.de/~thomas/mathe.htm]]

Google Books:
* [[[1] The Beauty of Geometry: Twelve Essays - H. S. M. Coxeter (chapt. 2)|http://books.google.com/books?id=beTjmcibCH8C&pg=PA21&lpg=PA21&dq=The+Beauty+of+Geometry+integral+cayley+numbers&source=bl&ots=kRPl6CTZZ1&sig=SIz34DSqcVyLopq-LeyiYwtvv2c&hl=de&ei=fUuOSvzuBJGe_AbOoo32DQ&sa=X&oi=book_result&ct=result&resnum=1#v=onepage&q=&f=false]] [[local|google_books/TheBeautyOfGeometry.pdf]] [[bct. 7|http://scholar.google.de/scholar?cites=13159578209779229825&hl=de]] brl. 10

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 (\mathbf A + \mathbf B) & = & \tau (\mathbf A) + \tau (\mathbf B) \\
\tau (\mathbf{AB})  & = & \tau (\mathbf B)  \tau (\mathbf A) \\
\tau (\tau(\mathbf A)) & = & \mathbf A
\end{eqnarray}
$\forall \mathbf A, \mathbf B \in \mathcal{A}$.

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

!!!!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]]

@@display:block;text-align:center;[img[images/isometry.jpg]]@@
In ''Isometry (= isometric isomorphism or congruence mapping)'' is a distance-preserving isomorphism f between [[metric|Metric Tensor]] spaces.
That is:
\begin{equation}
\mathcal{N}(f(x)-f(y))=\mathcal{N}(x-y)
\end{equation}

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]]

!!!![[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 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 in 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) 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 by it.

There is a lot more that could said about \JHyperComplex. If you have questions, please contact me [[here|Welcome]].

If you don't believe that hypercomplex numbers are interesting, you should check out another piece of software I have written, 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]]:
\[
\det (\mathbf J_{\mathbf f}(\mathbf{x})) = \begin{vmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\  \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_2} \\  \\ \vdots & \vdots & \ddots & \vdots \\  \\ \frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_n} \end{vmatrix}
\]
Its value gives the following information about the behavior of the function $\mathbf f$ in the point $\mathbf x$:
*  $> 0$  orientation preserving
* $< 0$ orientation reversing
* $= 0$ not invertible
* $= 1$ volume preserving

''General Relativity''
For a coordinate transformation $x_\mu \mapsto x'_\nu(\mathbf x) $ in general relativity, the Jacobi determinant $\det (\mathbf J_{\mathbf{x'}}(\mathbf{x}))$ can be written as
\[
\det (J_{\mathbf{\mathbf x'}}(\mathbf{x})) = \epsilon_{\mu\nu\rho\sigma} \frac{\partial x'^\mu}{\partial x^0} \frac{\partial x'^\nu}{\partial x^1} \frac{\partial x'^\rho}{\partial x^2} \frac{\partial x'^\sigma}{\partial x^3}
\]
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 $\mathbf f$ is defined by:
\[
(\mathbf J_{\mathbf f})_{ij}(\mathbf{x}) = \frac{\partial f_i(\mathbf{x})}{\partial x_j} = \partial_j f_i(\mathbf{x})\,\!
\]
Written out explicitely it is:
\[
\mathbf J_{\mathbf{f}}(\mathbf{x}) = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\  \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_2} \\  \\ \vdots & \vdots & \ddots & \vdots \\  \\ \frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_n} \end{pmatrix}
\]
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:
\[
\Delta f_i(\mathbf{x}) = f_i(\mathbf{x}+\Delta\mathbf{x})\approx f_i(\mathbf{x}) + (\mathbf J_{\mathbf{f}})_i(\mathbf{x})\Delta \mathbf{x} + \frac{1}{2} \Delta\mathbf{x}^\mathrm{T} \mathbf H_{f_i}(\mathbf{x}) \Delta\mathbf{x}
\]
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 $ \mathbf x = (x_1, x_2, \ldots, x_n) \to \mathbf x' = (x'_1(\mathbf x), x'_2 (\mathbf x), \ldots, x'_n (\mathbf x))$.
One has, up to first order:
\[
x'^i (\mathbf x + d\mathbf x) \mathbf{e}_i  = x'^i (\mathbf x) \mathbf{e}_i + \frac{\partial x'^i (\mathbf x)}{\partial x^j} dx^j  \mathbf{e}_i = x'^i (\mathbf x) \mathbf{e}_j + J_{\mathbf {x'}}(\mathbf {x})_{ij}  \mathbf{e}_j
\]

<html><center><img src="images/JohnHortonConway.jpg " style="width: 165px; "/></center></html>
Papers:
* [[The Free Will Theorem - J. Conway, S. Kochen|http://arxiv.org/PS_cache/quant-ph/pdf/0604/0604079v1.pdf]] [[pct. 34|http://scholar.google.de/scholar?cites=16215323905036932989&hl=de]]
* [[The Strong Free Will Theorem - J. Conway, S. Kochen|http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.3286v1.pdf]] [[pct. 3|http://scholar.google.com/scholar?hl=de&lr=&cites=10339603623079889885&um=1&ie=UTF-8&ei=BWi_SuPuKtH-_AbssfGCAQ&sa=X&oi=science_links&resnum=3&ct=sl-citedby]]

Videos:
* [[Princeton University: Free Will Lecture Series|http://www.princeton.edu/WebMedia/lectures/]]

Books:
* [[On Numbers and Games|books/JHConway_OnNumbersAndGames.djvu]] {{t500Cite{[[bct. 624|http://scholar.google.de/scholar?cites=11910834121902210311&hl=de]]}}}

A ''Jordan Algebra'' is a [[commutative|Commutator]] algebra (however generally not associative) satisfying the [[Jordan identity|Jordan Identity]].
A Jordan algebra can be constructed from a non-commutative algebra by introducing a symmetric product $\boldsymbol \circ$ called the ''Jordan Product'':
\[
\mathbf A \circ \mathbf B \equiv \frac{1}{2} (\mathbf{AB} + \mathbf{BA}) = \frac{1}{2}\{\mathbf A,\mathbf B \} = \frac{1}{4}[(\mathbf A+\mathbf B)^2 - (\mathbf A-\mathbf B)^2]
\]
Any Jordan algebra (of characteristic $\ne 2$) is [[power-associative|Power Associative Algebra]].

A weaker form of a Jordan algebra is a [[Noncommutative Jordan Algebra]].

''Theorem (Zelmanov)'' 
Every simple Jordan algebra (of any dimension) is isomorphic to one of the following: 
## an algebra of a bilinear form, 
## an algebra of Hermitian type or 
## an [[Albert algebra|Albert Algebra]].
!!!! Historical
* In 1933 Jordan suggested a new formulation of quantum mechanics based on Jordan algebras.

* In 1934 Jordan, von Neumann and Wigner showed that the Jordan algebras are always in one-to-one correspondence with a matrix algebra over the complex numbers with one exception, the exceptional Jordan algebra $\mathfrak H$${}_3 (\mathbb O)$ of $3 \times 3$ matrices over the [[octonions|Octonion]]. Later Albert proved that this is the only exceptional Jordan algebra the so called [[Albert algebra|Albert Algebra]]. The other Jordan algebras are called special.

* In 1978, Günaydin, Piron, and Ruegg showed that it is possible to formulate quantum mechanics based on the exceptional Jordan algebra. It is called octonionic quantum mechanics. (The formulation of quantum mechanics with the special Jordan algebras brings about nothing new as it is equivalent to the Dirac formulation in terms of commutators).

* In 1983 Zelmanov accomplished  the classification of infinite dimensional Jordan algebras. It appears that all infinite dimensional simple Jordan algebras are extensions of special Jordan algebras and that there are no infinite dimensional exceptional Jordan algebras. This implies that no Hilbert space formulation of octonionic quantum mechanics is possible.

Papers:
* [[Quasi-Jordan Algebras - R. Velásquez, R. Felipe|http://www.cimat.mx/reportes/enlinea/I-06-14.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=1481025530681856671&hl=de]]
* [[On Anti-Commutative Algebras with an Invariant Form - A. Sagle|http://books.google.de/books?hl=de&lr=&id=RSNqggY5Q5cC&oi=fnd&pg=PA370&dq=identities+cayley+dickson+process+autor:a-sagle&ots=t70c7xnB43&sig=nRPFGdsMaGGis-6UTDZMKI0k-Kk#PPA370,M1]] [[pct. 4|http://scholar.google.de/scholar?cites=8670043287416005504&hl=de]]

Lectures:
* [[Mini Course on Jordan Algebras - K. McCrimmon|http://www.ime.usp.br/~liejor/2009/Conference_Manaus/MiniCourse-McCrimmon.pdf]]

The ''Jordan Identity'' comes in $4$ different versions, given by:
# \begin{eqnarray}
&&(\mathbf{BA})\mathbf A^2 = (\mathbf {BA}^2)\mathbf A 
\end{eqnarray} 
#  \begin{eqnarray}
&&(\mathbf A \mathbf B ) \mathbf A^2  =  \mathbf A (\mathbf{BA}^2)  \Leftrightarrow   [\mathbf A, \mathbf B, \mathbf A^2]  =   0
\end{eqnarray} 
#\begin{eqnarray}
&&\mathbf A^2 (\mathbf{BA}) =  (\mathbf A^2 \mathbf B ) \mathbf A  \Leftrightarrow  [\mathbf A^2, \mathbf B, \mathbf A] =  0
\end{eqnarray} 
# \begin{eqnarray}
&&\mathbf A^2(\mathbf{AB}) = \mathbf A(\mathbf A^2 \mathbf B)
\end{eqnarray} 
Versions $2$ and $4$ are mentioned by Pascual Jordan in the context of a measurement algebra of quantum mechanics (see [1]).

If the algebra is [[flexible|Flexible Algebra]] all $4$ versions are equivalent. 

The Jordan identity in its various versions establishes a relationship between two different [[association types|Association Type]] of degree $4$, namely the one of the form $(..)(..)$ and one of the four other ones, each one occuring once in one of the identities.
As a rule of thumb: All versions consist of an association type having $\mathbf A$ as outermost element and one having $\mathbf A^2$ as outermost element. Furthermore $\mathbf B$ occurs at any of the $4$ possible positions within an association type. (The ordering of the identities is accordingly). 

Polarizing the identities once, one gets: 
#  \begin{eqnarray}
(\mathbf{BA})(\mathbf{AC}) + (\mathbf{BA})(\mathbf{CA}) + (\mathbf{BC})\mathbf A^2 = (\mathbf {BA}^2)\mathbf C + (\mathbf B (\mathbf {AC}))\mathbf A + (\mathbf B (\mathbf{CA}))\mathbf A
\end{eqnarray} 
# \begin{eqnarray}
&&(\mathbf A \mathbf B ) (\mathbf A \mathbf C) + (\mathbf A \mathbf B )  (\mathbf C \mathbf A) + (\mathbf C \mathbf B ) \mathbf A^2 = \mathbf A (\mathbf B (\mathbf{AC})) + \mathbf A (\mathbf B (\mathbf{CA})) + \mathbf C (\mathbf{BA}^2)  \Leftrightarrow  \\
&&[\mathbf A, \mathbf B, \mathbf{AC}] +  [\mathbf A, \mathbf B, \mathbf{CA}] + [\mathbf C, \mathbf B, \mathbf A^2]  =   0
\end{eqnarray} 
# \begin{eqnarray}
&& \mathbf A^2 (\mathbf{BC}) + (\mathbf{AC}) (\mathbf{BA}) + (\mathbf{CA}) (\mathbf{BA}) = (\mathbf A^2 \mathbf B ) \mathbf C +  ((\mathbf {AC}) \mathbf B ) \mathbf A + ((\mathbf {CA})\mathbf B ) \mathbf A \Leftrightarrow  \\
&& [\mathbf A^2, \mathbf B, \mathbf C] + [\mathbf {AC}, \mathbf B, \mathbf A] + [\mathbf{CA}, \mathbf B, \mathbf A]=  0
\end{eqnarray} 
# \begin{eqnarray}
\mathbf A^2(\mathbf{CB}) + (\mathbf {AC})(\mathbf{AB}) + (\mathbf{CA})(\mathbf{AB})  = \mathbf A( (\mathbf {AC}) \mathbf B) + \mathbf A( (\mathbf {CA})\mathbf B) + \mathbf C(\mathbf A^2 \mathbf B)
\end{eqnarray} 

The fully polarized versions look as follows: 
#  \begin{eqnarray}
(\mathbf{BA}) (\mathbf{CD}) + (\mathbf{BA}) (\mathbf{DC}) +  (\mathbf{BC})(\mathbf{AD}) +  (\mathbf{BC})  (\mathbf{DA}) +  (\mathbf{BD})  (\mathbf{AC}) +  (\mathbf{BD}) (\mathbf{CA}) = \\
(\mathbf B (\mathbf {AC}))\mathbf D + (\mathbf B (\mathbf{AD}))\mathbf C + (\mathbf B (\mathbf{CA}))\mathbf D + (\mathbf B (\mathbf{CD})) \mathbf A + (\mathbf B (\mathbf{DA}))\mathbf C + (\mathbf B ( \mathbf{DC}))\mathbf A  
\end{eqnarray} 
# \begin{eqnarray}
&&(\mathbf{AB}) (\mathbf{CD}) + (\mathbf{AB}) (\mathbf{DC}) + (\mathbf{CB})(\mathbf{AD}) + (\mathbf{CB})(\mathbf{DA}) + (\mathbf{DB})(\mathbf{AC}) +  (\mathbf{DB}) (\mathbf{CA}) = \\
&&\mathbf A (\mathbf B (\mathbf {CD})) + \mathbf A (\mathbf B ( \mathbf {DC})) + \mathbf C (\mathbf B ( \mathbf {AD})) + \mathbf C (\mathbf B ( \mathbf {DA})) +  \mathbf D (\mathbf B (\mathbf {AC})) + \mathbf D (\mathbf B (\mathbf {CA})) \Leftrightarrow  \\
&& \sigma_{\{\mathbf A,\mathbf C, \mathbf D\}} [\mathbf A, \mathbf B, \{\mathbf C, \mathbf D\}] =  0
\end{eqnarray} 
#\begin{eqnarray}
&&(\mathbf{AC}) (\mathbf{BD}) + (\mathbf{AD}) (\mathbf{BC}) +  (\mathbf{CA})(\mathbf{BD}) +  (\mathbf{CD})  (\mathbf{BA}) +  (\mathbf{DA})  (\mathbf{BC}) +  (\mathbf{DC}) (\mathbf{BA}) = \\
&&((\mathbf{AC})\mathbf B)\mathbf D + ( (\mathbf{AD})\mathbf B)\mathbf C  + ( (\mathbf{CA})\mathbf B)\mathbf D + ( (\mathbf{CD})\mathbf B) \mathbf A  + ( (\mathbf{DA})\mathbf B)\mathbf C +  ( (\mathbf{DC})\mathbf B) \mathbf A  \Leftrightarrow  \\
&& \sigma_{\{\mathbf A,\mathbf C, \mathbf D\}} [\{\mathbf C, \mathbf D\}, \mathbf A, \mathbf B] =  0
\end{eqnarray} 
# \begin{eqnarray} 
(\mathbf{AC}) (\mathbf{DB}) + (\mathbf{AD}) (\mathbf{CB}) +  (\mathbf{CA})(\mathbf{DB}) +  (\mathbf{CD})  (\mathbf{AB}) +  (\mathbf{DA})  (\mathbf{CB}) +  (\mathbf{DC}) (\mathbf{AB}) = \\
\mathbf A ((\mathbf {CD})\mathbf B) + \mathbf A ((\mathbf{DC})\mathbf B) + \mathbf C ((\mathbf{AD})\mathbf B) + \mathbf C ((\mathbf{DA}) \mathbf B) + \mathbf D ((\mathbf{AC})\mathbf B) + \mathbf D ((\mathbf{CA})\mathbf B)  
\end{eqnarray} 

Papers:
* [[A Moufang Loop, the Exceptional Jordan Algebra, and a Cubic Form in 27 Variables - R. L. Griess, Jr.|http://deepblue.lib.umich.edu/bitstream/2027.42/28568/1/0000371.pdf]] [[local|papers/0000371.pdf]] [[pct. 18|http://scholar.google.de/scholar?cites=9341850002796726845&hl=de&as_sdt=2000]]
* [[[1] Über das Verhältnis der Theorie der Elementarlänge zur Quantentheorie - P. Jordan|http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103840802]] [[local|papers/PJordan.pdf]] [[pct. 6|http://scholar.google.de/scholar?cites=12918464759599452930&hl=de-]] prl. 10

The the ''Jordan Triple Product'' $\{\mathbf A, \mathbf B,\mathbf C\}_J$ is defined as
\begin{equation}
\{\mathbf A, \mathbf B, \mathbf C \}_J =  (\mathbf A \mathbf B^*) \mathbf C + (\mathbf C \mathbf B^*) \mathbf A ? (\mathbf A \mathbf C) \mathbf B^*
\end{equation}

The (abelian) ''Kalb\-Ramond Field'' or ''Axion Field'' $B_{\mu\nu}$ is a two-form field which appears in the low energy limit of [[string theory|Superstring Theory]], in [[quantum gravity|Quantum Gravity]] and in several other frameworks in particle physics. Most attempts to incorporate mass to gauge field models in four dimensions take into account this object added to a one form gauge field.

(In string theory) the axion field  also shows up in the context of [[3-cocycles|3-Cocycle]] which are related to a violation of the [[Jacobi identity|Jacobian]], leading to a nonassociative algebra.

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]]

> Fifteen young ladies of a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast.
> - Kirkman -
<html><center><img src="images/kirkmanSchoolgirls.gif" style="width: 155px; "/></center></html>
>The Schoolgirl Problem connects block designs, finite projective geometries, algebraic number fields, error-correcting codes, and recreational mathematics. Are there any other connections? Yes, ... the Fundamental Theorem of Galois Theory, one of the most beautiful theorems in mathematics.
>- E. Brown, K. E. Mellinger [1] -

The solution to the ''Kirkman's Schoolgirl Problem'' is an example of a [[resolvable|Resolvable Design]] $(15, 35, 7, 3, 1)$-[[design|Balanced Incomplete Block Design]] (or equivalently a resolvable $2-(15,3,1)$ [[t-design|T-Design]]), a.k.a. [[Kirkmann Triple System of order 15, or KTS(15)|Kirkman Triple System]].

Technically speaking it is a 1-resolution of the design, i.e. an  arrangement into $7$ parallel classes, each consisting of $5 = 15/5$ blocks, such that each variety appears exactly once in each class.

With $15$ girls, there are $455$ different ways to group them in three. A solution to Kirkman's schoolgirl problem is to "pick" $35$ adequate groupings from these.

An example of such a solution is as follows:
<html>
<table border="1" cellspacing="0" cellpadding="3">
    <tbody>
        <tr>
            <th>Sun</th>
            <th>Mon</th>

            <th>Tue</th>
            <th>Wed</th>
            <th>Thu</th>
            <th>Fri</th>
            <th>Sat</th>
        </tr>

        <tr>
            <td>A, F, K</td>
            <td>A, B, E</td>
            <td>B, C, F</td>
            <td>E, F, I</td>
            <td>C, E, K</td>

            <td>E, G, M</td>
            <td>K, M, D</td>
        </tr>
        <tr>
            <td>B, G, L</td>
            <td>C, D, G</td>
            <td>D, E, H</td>

            <td>G, H, K</td>
            <td>D, F, L</td>
            <td>F, H, N</td>
            <td>L, N, E</td>
        </tr>
        <tr>
            <td>C, H, M</td>

            <td>H, I, L</td>
            <td>I, J, M</td>
            <td>L, M, A</td>
            <td>G, I, O</td>
            <td>I, K, B</td>
            <td>O, B, H</td>

        </tr>
        <tr>
            <td>D, I, N</td>
            <td>J, K, N</td>
            <td>K, L, O</td>
            <td>N, O, C</td>
            <td>H, J, A</td>

            <td>J, L, C</td>
            <td>A, C, I</td>
        </tr>
        <tr>
            <td>E, J, O</td>
            <td>M, O, F</td>
            <td>N, A, G</td>

            <td>B, D, J</td>
            <td>M, N, B</td>
            <td>O, A, D</td>
            <td>F, G, J</td>
        </tr>
    </tbody>
</table>
</html>
The question if the $455$ triples can even be partitioned into $13$ disjoint $KTS(15)$ is known as [[Silverster's problem|Silvester's Problem]].

Papers:
* [[[1] Kirkman’s Schoolgirls Wearing Hats and Walking through Fields of Numbers - E. Brown, K. E. Mellinger|http://www.math.vt.edu/people/brown/doc/kirkman.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=17781259118371940981&hl=de&as_sdt=2000]]

Links:
* [[Kirkman's Ladies - A. Papadimoulis|http://thedailywtf.com/Articles/Kirkmans-Ladies.aspx]]

''Kissing Number $K(L)$'' of a [[lattice|Lattice]] $L$  is the maximum number of spheres of radius $1$ that can simultaneously touch the unit sphere in $n$-dimensional Euclidean space.

The exact number is only known for dimensions $1,2,3,8$ and $24$. In these cases the densest sphere packings correspond to known [[lattices|Lattice]].

In most cases the best known lower bound corresponds to a lattice. Known exceptions are dimensions $12$, where the the densest known sphere packings are non-lattices.

<html><center><img src="images/spherePackings.jpg" style="width: 467px; "/></center></html>

The following table lists the highest kissing numbers presently known in dimensions up to $128$.
The entry gives the lattice with the highest kissing number known. If a nonlattice with a higher kissing number is known, it appears in parentheses.
The kissing number in a nonlattice packing may vary from sphere to sphere. The table contains the largest known value for the respective dimension.
<html>
<TABLE border="1" width="100%">
<tr><th>Dim</th><th>Kissing number for a lattice (nonlattice)</th><th>Lattice (resp. nonlattice)</th>
</tr>
<tr><td>1</td><td>2</td><td><a href="http://www.research.att.com/~njas/lattices/A1.html">LAMBDA<sub>1</sub> = A<sub>1</sub> = Z</a></td>

</tr>
<tr><td>2</td><td>6</td><td><a href="http://www.research.att.com/~njas/lattices/A2.html">LAMBDA<sub>2</sub> = A<sub>2</sub></a></td>
</tr>
<tr><td>3</td><td>12</td><td><a href="http://www.research.att.com/~njas/lattices/A3.html">LAMBDA<sub>3</sub> = A<sub>3</sub></a> = <a href="http://www.research.att.com/~njas/lattices/D3.html">D<sub>3</a></td>

</tr>
<tr><td>4</td><td>24</td><td><a href="http://www.research.att.com/~njas/lattices/D4.html">LAMBDA<sub>4</sub> = D<sub>4</sub></a></td>
</tr>
<tr><td>5</td><td>40</td><td><a href="http://www.research.att.com/~njas/lattices/D5.html">LAMBDA<sub>5</sub> = D<sub>5</sub></a></td>

</tr>
<tr><td>6</td><td>72</td><td><a href="http://www.research.att.com/~njas/lattices/E6.html">LAMBDA<sub>6</sub> = E<sub>6</sub></a></td>
</tr>
<tr><td>7</td><td>126</td><td><a href="http://www.research.att.com/~njas/lattices/E7.html">LAMBDA<sub>7</sub> = E<sub>7</sub></a></td>

</tr>
<tr><td>8</td><td>240</td><td><a href="http://www.research.att.com/~njas/lattices/E8.html">LAMBDA<sub>8</sub> = E<sub>8</sub></a></td>
</tr>
<tr><td>9</td><td>272 (306)</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA9.html">LAMBDA<sub>9</sub></a> ( P<sub>9a</sub> )</td>

</tr>
<tr><td>10</td><td>336 (500)</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA10.html">LAMBDA<sub>10</sub></a> ( P<sub>10b</sub> )</td>
</tr>
<tr><td>11</td><td>438 (582)</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA11.html">LAMBDA<sub>11</sub></a> ( P<sub>11c</sub> )</td>

</tr>
<tr><td>12</td><td>756 (840)</td><td><a href="http://www.research.att.com/~njas/lattices/K12.html">KAPPA<sub>12</sub> = K<sub>12</sub> (Coxeter-Todd lattice)</a> ( P<sub>12a</sub> )</td>
</tr>

<tr><td>13</td><td>918 (1130)</td><td><a href="http://www.research.att.com/~njas/lattices/KAPPA13.html">KAPPA<sub>13</sub></a> ( P<sub>13a</sub> )</td>
</tr>
<tr><td>14</td><td>1422 (1582)</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA14.html">LAMBDA<sub>14</sub></a> ( P<sub>14b</sub> )</td>

</tr>
<tr><td>15</td><td>2340 (2564)</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA15.html">LAMBDA<sub>15</sub></a> ( P<sub>15a</sub> )</td>
</tr>
<tr><td>16</td><td>4320</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA16.html">LAMBDA<sub>16</sub></a></td>

</tr>
<tr><td>17</td><td>5346</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA17.html">LAMBDA<sub>17</sub></a></td>
</tr>
<tr><td>18</td><td>7398</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA18.html">LAMBDA<sub>18</sub></a></td>
</tr>
<tr><td>19</td><td>10668</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA19.html">LAMBDA<sub>19</sub></a></td>

</tr>
<tr><td>20</td><td>17400</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA20.html">LAMBDA<sub>20</sub></a></td>
</tr>
<tr><td>21</td><td>27720</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA21.html">LAMBDA<sub>21</sub></a></td>
</tr>
<tr><td>22</td><td>49896</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA22.html">LAMBDA<sub>22</sub></a></td>

</tr>
<tr><td>23</td><td>93150</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA23.html">LAMBDA<sub>23</sub></a></td>
</tr>
<tr><td>24</td><td>196560</td><td><a href="http://www.research.att.com/~njas/lattices/Leech.html">Leech lattice LAMBDA<sub>24</sub></a></td>
</tr>
<tr><td>25</td><td>196656</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA25.html">LAMBDA<sub>25</sub></a></td>

</tr>
<tr><td>26</td><td>196848</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA26.html">LAMBDA<sub>26</sub></a></td>
</tr>
<tr><td>27</td><td>197142</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA27.html">LAMBDA<sub>27</sub></a></td>
</tr>
<tr><td>28</td><td>197736</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA28.html">LAMBDA<sub>28</sub></a></td>

</tr>
<tr><td>29</td><td>198506</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA29.html">LAMBDA<sub>29</sub></a></td>
</tr>
<tr><td>30</td><td>200046</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA30.html">LAMBDA<sub>30</sub></a></td>
</tr>
<tr><td>31</td><td>202692</td><td><a href="http://www.research.att.com/~njas/lattices/LAMBDA31.html">LAMBDA<sub>31</sub></a></td>

</tr>
<tr><td>32</td><td>261120 (276032)</td><td><a href="http://www.research.att.com/~njas/lattices/Q32.html">Q<sub>32</sub></a> and others (<a href="http://www.research.att.com/~njas/doc/edel.ps">Nonlattice </a>)</td>
</tr>
<tr><td>33</td><td>262272 (294592)</td><td>Q<sub>33</sub> (<a href="http://www.research.att.com/~njas/doc/edel.ps">Nonlattice</a>)</td>

</tr>
<tr><td>34</td><td>264576 (318020)</td><td>Q<sub>34</sub> (<a href="http://www.research.att.com/~njas/doc/edel.ps">Nonlattice</a>)</td>
</tr>
<tr><td>35</td><td>268032 (370892)</td><td>Q<sub>35</sub> (<a href="http://www.research.att.com/~njas/doc/edel.ps">Nonlattice</a>)</td>

</tr>
<tr><td>36</td><td>274944 (438872)</td><td>Q<sub>36</sub> (<a href="http://www.research.att.com/~njas/doc/edel.ps">Nonlattice </a>)</td>
</tr>
<tr><td>37</td><td>284160 (439016)</td><td>Q<sub>37</sub> (<a href="http://www.research.att.com/~njas/doc/edel.ps">Nonlattice </a>)</td>

</tr>
<tr><td>38</td><td>302592 (566652)</td><td>Q<sub>38</sub> (<a href="http://www.research.att.com/~njas/doc/edel.ps">Nonlattice </a>)</td>
</tr>
<tr><td>39</td><td>333696 (714184)</td><td>Q<sub>39</sub> (<a href="http://www.research.att.com/~njas/doc/edel.ps">Nonlattice </a>)</td>

</tr>
<tr><td>40</td><td>399360 (991792)</td><td>Q<sub>40</sub> (<a href="http://www.research.att.com/~njas/lattices/doc/edel.ps">Nonlattice</a>)</td>

</tr>
<tr><td>42</td><td>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (1196788)</td><td>&nbsp; &nbsp; &nbsp; (<a href="http://www.research.att.com/~njas/doc/edel.ps">Nonlattice </a>)</td>

</tr>
<tr><td>44</td><td>2708112 (2948552)</td><td><a href="http://www.research.att.com/~njas/lattices/MW44.html">MW<sub>44</sub></a> (<a href="http://www.research.att.com/~njas/doc/edel.ps">Nonlattice </a>)</td>

</tr>
<tr><td>48</td><td>52416000</td><td><a href="http://www.research.att.com/~njas/lattices/CQ48a.html">P<sub>48n</sub></a>, <a href="http://www.research.att.com/~njas/lattices/P48p.html">P<sub>48p</sub></a>, <a href="P48q.html">P<sub>48q</sub></a></td>

</tr>
<tr><td>64</td><td>138458880 (331737984)</td><td><a href="http://www.research.att.com/~njas/lattices/CQ64b.html">Ne<sub>64</sub></a>
</a> (<a href="http://www.research.att.com/~njas/doc/edel.ps">Nonlattice</a>)</td>
</tr>
<tr><td>80</td><td>1250172000 (1368532064)</td><td><a href="http://www.research.att.com/~njas/lattices/L80.html">L<sub>80</sub></a> of Bachoc-Nebe (<a href="http://www.research.att.com/~njas/doc/edel.ps">Nonlattice</a>)</td>

</tr>
<tr><td>128</td><td>218044170240 (8863556495104)</td><td>MW<sub>128</sub> [Elkies98] (<a href="http://www.research.att.com/~njas/doc/edel.ps">Nonlattice</a>)</td>
</tr>
</TABLE>
</html>
(Taken from [[Catalog of Lattices: Table of the Highest Kissing Numbers Presently Known - Nebe, Sloane|http://www.research.att.com/~njas/lattices/kiss.html]] and adapted).

Papers:
* [[The Sphere Packing Problem - N. J. A. Sloane|http://www.emis.ams.org/journals/DMJDMV/xvol-icm/13/Sloane.MAN.ps.gz]] [[pct. 22|http://scholar.google.de/scholar?cites=13177131311783204990&hl=de]]

The multiplication table of the ''Klein Four-group'' is given by
||!e|!i|!j|!k|
|!e|e|i|j|k|
|!i|i|e|k|j|
|!j|j|k|e|i|
|!k|k|j|i|e|
The group can be represented by means of the [[Fano line|Fano Spaces]].

The ''Klein Gordon Equation'' describes the dynamics of a scalar field $\Phi(\mathbf x)$ with mass m and is given by:
\begin{equation}
(\partial_\mu\partial^\mu + m^2) \Phi (\mathbf x) \equiv (\square + m^2) \Phi (\mathbf x) = 0
\end{equation}
In case that $m=0$ one gets the [[D'Alembert equation|D'Alembert Equation]].

The ''Klein Gordon Equation'' describes the dynamics of a scalar field $\Phi(\mathbf x)$ with mass m and is given by:
\[
(\partial_\mu\partial^\mu + m^2) \Phi (\mathbf x) \equiv (\square + m^2) \Phi (\mathbf x) = 0
\]
In case that $m=0$ one gets the [[D'Alembert equation|D'Alembert Equation]].

The ''Kleinfeld Identities'' are given by
\begin{eqnarray}
[\mathbf A, \mathbf B] [[\mathbf A, \mathbf B]^2, \mathbf C, \mathbf D] = [\mathbf A, \mathbf B]\, \mathbf K (\mathbf A, \mathbf B, \mathbf C, \mathbf D) = 0\\
[[\mathbf A, \mathbf B]^2, \mathbf C, \mathbf D] [\mathbf A, \mathbf B] = \mathbf K (\mathbf A, \mathbf B,\mathbf C, \mathbf D) \, [\mathbf A, \mathbf B] = 0\\
[[\mathbf A, \mathbf B]^4, \mathbf C, \mathbf D] = 0
\end{eqnarray}
where $ \mathbf K (\mathbf A, \mathbf B,\mathbf C, \mathbf D)  \equiv ([\mathbf A, \mathbf B]^2, \mathbf C, \mathbf D)$ defines the so called ''Kleinfeld Element''.

The identities are satisfied in any [[alternative algebra|Alternative Algebra]].

Papers:
* [[The Kleinfeld Identities in Generalized Accessible Rings - G. V. Dorofeev|http://www.springerlink.com/content/p77j52r1w5j5434p/fulltext.pdf]] [[local|papers/KleinfeldIdentities.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=1309922877507733707&hl=de]]

The ''Kochen\-Specker Theorem'' (''KS Theorem''), formulated in 1967, states that there cannot be any hidden variables in a non-contextual quantum mechanical model.  Besides Bell's theorem the KS theorem is a fundamental "no go theorem" for hidden variable theories in quantum mechanics.

The proof of the KS theorem is notoriously complex.

Papers:
* [[The Problem of Hidden Variables in Quantum Mechanics - S. Kochen, E. P. Specker|http://www.hep.princeton.edu/~mcdonald/examples/QM/kochen_iumj_17_59_68.pdf]] [[local|papers/kochen_iumj_17_59_68.pdf]]  {{t500Cite{[[pct. 843|http://scholar.google.de/scholar?cites=18441803985553364076&hl=de&as_sdt=2000]]}}} - The original paper on the subject.

Links:
* [[Stanford Encyclopedia of Philosophy - The Kochen-Specker Theorem|http://plato.stanford.edu/entries/kochen-specker/]]
* [[Webpage of Karl Svozil|http://tph.tuwien.ac.at/~svozil/]]

Videos:
* [[The Paradox of Kochen and Specker - John Conway|http://www.princeton.edu/WebMedia/flash/lectures/20090330_conway_free_will.shtml]]

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]].

Links:
* [[WIKIPEDIA - Kolmogorov Complexity|http://en.wikipedia.org/wiki/Kolmogorov_complexity]]

Papers:
* [[A Path Integral Approach to the Kontsevich Quantization Formula - A. S. Cattaneo, G. Felder|http://arxiv.org/PS_cache/math/pdf/9902/9902090v3.pdf]]{{t100Cite{ [[pct. 455|http://scholar.google.de/scholar?cites=11329549912484812612&hl=de]]}}}
* [[Nonassociative Star Product Deformations for D-Brane World-Volumes in Curved Backgrounds - L. Cornalba, R. Schiappa|http://arxiv.org/PS_cache/hep-th/pdf/0101/0101219v3.pdf]] {{t100Cite{[[pct. 159|http://scholar.google.com/scholar?hl=de&lr=&cites=4175399176221946157&um=1&ie=UTF-8&ei=--StSrD9JtaM_AaIq525Bg&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]}}}

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 $\mathcal G$.

The $16$-dimensional ''$\Lambda_{16}$-Lattice'' (=''$BW_{16}$-Lattice'') was discovered 1959 by Barnes and Wall. It is an [[even 2-modular lattice |Lattice]] and part of the [[Barnes-Wall lattices|Barnes-Wall Lattice]] family.

It has $2\cdot240 + 16\cdot240  = 2\cdot240 + 15\cdot256 = 480 + 3.840 = 4.320$ units (vertices nearest the origin = [[kissing number|Kissing Number]]).
There are $0$, $4.320$, $61.440$, $5.222.720$ lattice points in shells $1$, $2$, $3$ and $4$, respectively.

Furthermore it is a sublattice of the [[Leech lattice|Leech Lattice]] and can be constructed from it.

$\Lambda_{16}$ contains the [[root lattice|Root Lattice]] of [[SO(32)]], $\mathbb D_{16}$, as a sub-lattice which consists of $480$ root vectors.

The [[automorphism group|Automorphism]] of $\Lambda_{16}$ is of order $89.181.388.800 = 2^{21} \cdot 3^5 \cdot  5^2 \cdot 7 = 2^7 \cdot \operatorname{ord}(Aut(\Lambda_8))$. (For the formula see [[Barnes-Wall lattices|Barnes-Wall Lattice]]).

!!!!Representation
The $\Lambda_{16}$-lattice can be represented in different ways, for example:
\begin{eqnarray}
\Lambda_{16} & = & 4\mathbb Z^{16} + 2(16, 15, 2)+ (16, 5, 8) \\
                      & = & 4\mathbb Z^{16} + 2\operatorname{RM}(3,4) + \operatorname{RM}(1,4) \\
                      & = & 2 \mathbb D_{16} + \operatorname{RM}(1,4) \\
                      & = & 2 \mathbb D_{16} + \operatorname{Had}(4)
\end{eqnarray}
with $\operatorname{RM}(1,4)$ the [[Reed-Muller code|Reed-Muller Code]] and $\operatorname{Had}(4)$ the [[Hadamard code|Hadamard Code]].

!!!!Construction
An example of a construction of the set of minimal vectors is as follows:
* $480$ vectors of the form $(\pm 2^2, 0^{14})$, where there are two nonzero components equal to $2$ or $?2$,
* $3.840$ vectors of the form $(\pm 1^8, 0^8)$, where the positions of the $\pm1$’s form one of the $30$ codewords of weight $8$ of the first order [[Reed-Muller code|Reed-Muller Code]] $\operatorname{RM}(1,4)$ (or equivalently the [[Hadamard code|Hadamard Code]] $\operatorname{Ham}(4)$) and there are an even number of minus signs.
A representation of the codewords is given by (see [1]):
\begin{eqnarray}
&& 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0, \;    0 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1, \\
&& 0 1 1 0 1 1 0 1 0 0 0 0 1 1 1 0, \;  0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 1, \\
&& 0 1 1 1 1 0 1 0 0 0 1 1 0 1 0 0, \; 1 0 0 1 1 1 0 0 1 0 0 1 0 1 1 0, \\
&& 1 0 1 0 1 1 0 1 0 0 1 1 0 0 0 1,  \; 1 0 1 1 0 1 0 0 0 1 1 0 1 1 0 0, \\
&& 1 0 1 1 1 0 1 0 0 0 0 0 1 0 1 1,  \; 1 1 0 0 1 1 1 0 0 1 0 1 1 0 0 0, \\
&& 1 1 0 1 0 1 1 1 0 0 0 0 0 1 0 1,  \; 1 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0,\\
&& 1 1 1 0 0 1 1 0 1 0 1 0 0 0 1 0,  \; 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1, \\
&& 1 1 1 1 0 0 0 1 1 0 0 1 1 0 0 0
\end{eqnarray}
with the other $15$ codewords complementary to the ones listed.
As the [[automorphism group|Automorphism]] of $\operatorname{RM}(1,4)$ is [[AGL(4,2)]], there exist $322.559$ other equivalent representations.

The [[weight enumerator|Weight Enumerator]] of $\operatorname{RM}(1,4)$ is given by $W_{\operatorname{RM}(1,4)}(x,y) = 1 + 30x^8 + x^{16}$.

An observation: The number of vertices of the second shell of the $E_8$-lattice is $2.160 = 9\cdot 240$ which is just half the number of vertices of the $\Lambda_{16}$-lattice or the number of shortest vectors of $\Lambda_{16}$. Does this mean anything ?

!!!![[MAGMA|http://magma.maths.usyd.edu.au/calc/]]^^[[Help|MAGMA]]^^ examples
{{{
L := Lattice("Lambda", 16);
IsEven(L);
IsIsomorphic(L, Dual(L));
IsIsomorphic(L, 2*Dual(L));
KissingNumber(L);
ShortestVectors(L);
G := AutomorphismGroup(L);
#G;
FactoredOrder(G);
CompositionFactors(G);
}}}

Papers:
* [[The Cell Structures of Certain Lattices - J. H. Conway|http://www.research.att.com/~njas/doc/structure.pdf]] [[pct. 35|http://scholar.google.de/scholar?cites=5213474273818532286&hl=de]]
* [[[1] Hypermetrics in Geometry of Numbers - M. Deza, V. P. Grishukhin, M. Lauren|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.32.3775&rep=rep1&type=pdf]] [[local|papers/10.1.1.32.3775.pdf]] [[pct. 23|http://scholar.google.com/scholar?cites=10261434390969144830&hl=de]]
* [[Octonions: E8 Lattice to Lambda16 - G. Dixon|http://arxiv.org/PS_cache/hep-th/pdf/9501/9501007v1.pdf]] [[local|papers/9501007v1.pdf]] [[pct. 8|http://scholar.google.de/scholar?cites=7729012755670111510&hl=de]]
* [[Lattice Quantisation for Image Sequences Coding at 16 Dimension - D. G. Sampson, M. Ghanbari|http://www.ask4research.info/Uploads/Files/Publications/En_Pubs/00184569.pdf]] pct. 0

Google books:
* [[Sphere Packings, Lattices, and Groups - J. H. Conway, N. J. A. Sloane, E. B.|http://books.google.com/books?id=upYwZ6cQumoC&pg=PA129&lpg=PA129&dq=%22Sphere+packings,+lattices,+and+groups%22+%22barnes+wall+lattice%22&source=bl&ots=_K0Q1ZjezW&sig=ewM3ycMixbZnq5IauvQHsgSsU1A&hl=de&ei=lGibSvr9H4LgnAPv2tTYBA&sa=X&oi=book_result&ct=result&resnum=1#v=onepage&q=%22Sphere%20packings%2C%20lattices%2C%20and%20groups%22%20%22barnes%20wall%20lattice%22&f=false]] [[local|google_books/SpherePackingsLatticesAndGroups.pdf]] [[chapt. 4.10|documents/BarnesWallLattice.pdf]] {{t1000Cite{[[bct. 2902|http://scholar.google.de/scholar?cites=9885698152233477682&hl=de]]}}} brl. 10 - Review by Gian\-Carlo Rota: "This is the best survey of the best work in one of the best fields of combinatorics, written by the best people. It will make the best reading by the best students interested in the best mathematics that is now going on."

The equation
\[
\boldsymbol \nabla^2 \boldsymbol \Phi = \mathbf  0
\]
is called the ''Laplace Equation''. The solutions to the equation are designated as ''Harmonic Functions''.
If $\boldsymbol \Phi: \mathbb C \rightarrow \mathbb C$ the solutions are determined by the [[Cauchy Riemann equations|Cauchy Riemann Equations]].
If $\boldsymbol \Phi: \mathbb A_n \rightarrow \mathbb A_n $ with $\mathbb A_n$ a [[Cayley-Dickson algebra|Cayley-Dickson Algebra]] and $n > 1$, the only solutions are of a linear form, i.e. $\boldsymbol\Phi(\mathbf X) = \mathbf A + \mathbf{BX}$ with $\boldsymbol A, \boldsymbol B\in \mathbb A_n$.
Fueter has therefore developed a generalisation of complex function theory, in that he regarded the equation ${\boldsymbol\nabla^2}^n \boldsymbol \Phi = \mathbf 0$ instead of the Laplace equation  which turns out to have less "trivial solutions".

The ''Large Number Hypothesis'' alludes to the fact that several (dimensionless) ratios of physical constants are of the order of $10^{40}$, a multiple or a simple power of it.
<html><center><img src="images/large_numbers.jpg" style="width: 540px; "/></center></html>
Links:
* [[Wikipedia, Dirac Large Number Hypothesis|http://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis]]

A ''Lattice'' is an algebra $\mathcal A$ with two operations ''$\wedge$'' (called ''Meet'' or ''And'') and ''$\vee$'' (called ''Join'' or ''Or'') for which, $\forall \mathbf A, \mathbf B, \mathbf C \in \mathcal A$, the following relations hold:
|!Relations|!Laws|
|$\mathbf A \wedge \mathbf A = \mathbf A$; $\;\mathbf A \vee \mathbf A = \mathbf A\quad$|[[Idempotency]]|
|$\mathbf A  \wedge \mathbf B = \mathbf B  \wedge \mathbf A$; $\;\mathbf A \vee \mathbf B = \mathbf B \vee \mathbf A\quad$ |''Commutativity''|
|$(\mathbf A  \wedge \mathbf B)  \wedge \mathbf C = \mathbf A  \wedge (\mathbf B  \wedge \mathbf C)$; $\;(\mathbf A \vee \mathbf B) \vee \mathbf C = \mathbf A \vee (\mathbf B \vee \mathbf C)$|''Associativity''|
|$\mathbf A \vee (\mathbf A \wedge \mathbf B) = \mathbf A$; $\;\mathbf A \wedge (\mathbf A \vee \mathbf B) = \mathbf A\quad$|''Absorption''|

''Lattice Gas Automata'' (''LGA'') or ''Lattice Gas Cellular Automata'' (''LGCA'') are [[cellular automata|Cellular Automaton]] that allow for the simulation of fluid flows. From the LGCA, it is possible to derive the macroscopic Navier\-Stokes equations.

Any numerical evolution of a discretized partial differential equation can be interpreted as the evolution of some LGA. In the continuous time and space limit such a [[cellular automaton|Cellular Automaton]] mimics the behavior of the partial differential equation.

In 2 dimensions, models based on square lattices lack rotational invariance. This problem can be cured by models based on hexagonal lattices.

In 3 dimensions, the only space filling regular polytope, the cube, lacks rotational invariance. The only other polytopes that come into consideration, the dodecahedron and the icosahedron on the other hand are not space filling. In order to come up with suitable models for three dimensions one considers emdeddings in higher dimensional spaces.

For a quantum version of LGA see: [[quantum cellular automata|Quantum Cellular Automaton]].

Links:
* [[WIKIPEDIA - Lattice Gas Automaton|http://en.wikipedia.org/wiki/Lattice_gas_automaton]]

/***
|''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);
}

//}}}

Given a loop, $\mathcal L$ a ''Left Translation'' $\mathbf L_{\mathbf A}: \mathcal L \rightarrow \mathcal L$  is defined as
\[
\mathbf L_{\mathbf A} (\mathbf X) = \mathbf {AX}
\]
Similarly a ''Right Translation'' $\mathbf R_{\mathbf A}: \mathcal L \rightarrow \mathcal L$ is defined as
\[
\mathbf R_{\mathbf B} (\mathbf X) = \mathbf {XB}
\]

The set of left- and right-translations $\{L_{\mathbf A}, R_{\mathbf A} : \mathbf A \in \mathcal L\}$ 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 $M(\mathcal L)$).

The 6 elementary particles electron, electron-neutrino, muon, muon-neutrino, tauon and tauon-neutrino are called ''Leptons''. Leptons are subject to the electro-weak and gravitational force.

The Fundamental Theorem of Riemannian Geometry states: On a [[Riemannian manifold|Riemann Space]] 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 expressed by means of the [[Christoffel symbols|Christoffel Symbols]].

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

A ''Lie Algebra'' or ''Binary Algebra'' is the [[tangent algebra|Tangent Algebra]] of a [[Lie Group]] at the identity.

A [[semi-simple Lie algebra|Simple Algebra]] is the sum of simple [[ideals|Ideal]]. Therefore most of its properties can be obtained by first considering the latter ones. I .e. simple Lie algebras are the "building blocks" of the semi-simple ones.

A Lie algebra is specified by its [[generators|Generator]] $\mathbf T_i$ ($i = 1, \ldots, n$) and their commutation relations
\[
[\mathbf T_i, \mathbf T_j] = C_{ij}^k \mathbf T_k
\]
with $C_{ij}^k$ the [[structure constants|Structure Constants]] of the algebra.
The ''Dimension'' of a Lie Algebra is defined by the number of its generators $n$, spanning a $n$-dimensional vector space.

A ''Representation'' of a Lie algebra is understood as a set of matrices $M_i$ which obey the same commutation relations as the generators $\mathbf T_i$. These define maps $R_i: \mathbf T_i \rightarrow M_i$.
A special set of such maps, given by $R_i : \mathbf X \rightarrow  [\mathbf T_i, \mathbf X]$ leads to what is called the [[adjoint representation|Adjoint]] $ad_{\mathbf T_i} \equiv [\mathbf T_i, \mathbf X] $.

<html><center><img src="images/LieTriangle.jpg" style="width: 440px; "/></center></html>

Theses:
* [[Particle Dynamics of Branes A. R. Ploegh|http://dissertations.ub.rug.nl/FILES/faculties/science/2008/a.r.ploegh/13-thesis.pdf]]

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.

A ''Lie Group'' is a [[group|Group]] $\mathcal G$ which is also a differentiable manifold, i.e. that the group operations of multiplication and inversion
\begin{eqnarray}
&&\mathcal G \times \mathcal G \rightarrow \mathcal G : (g_1, g_2) \mapsto g_1\cdot g_2  \\
&&\mathcal G \mathcal \rightarrow \mathcal G  : g \mapsto g^{-1} \\
\end{eqnarray}
are differentiable.

Lie groups represent the best developed theory of continuous symmetry of mathematical objects and structures. A generalisation are [[quasigroups|Quasigroup]].

The complete list of simple, real, compact, connected Lie groups was completed around 1890. They fall into four infinite families (the classical Lie groups) and five exceptional groups:
| ![[Rank]] | ![[Group]] | !A.k.a. | !Dim | !Name |!Isomorphisms|
| $r$ | $A_r$ | $SU(r+1)$ | $r(r+2)$ | Special unitary group | |
| $r$ | $B_r$ | $SO(2r+1) \equiv SO(N)$, $N$ odd | $r(2r+1) = \frac{1}{2} N(N-1)$ | Odd special [[orthogonal group|Orthogonal Group]] | |
| $r$ | $C_r$ | $Sp(2r) \equiv Sp(N)$, $N$ even| $r(2r+1) = \frac{1}{2} N(N+1)$ | Symplectic group | |
| $r>2$ | $D_r$ | $SO(2r)  \equiv SO(N)$, $N$ even | $r(2r-1) = \frac{1}{2} N(N-1)$ | Even special [[orthogonal group|Orthogonal Group]] |  |
| $2$ | $G_2$ |  | $14$ | [[G2]] |$Aut(\mathbb O)$|
| $4$ | $F_4$ |  | $52$ | [[F4]] |$Isom(\mathbb {OP}^2)$|
| $6$ | $E_6$ |  | $78$ | [[E6]] |$Isom((\mathbb C \otimes \mathbb O) \mathbb P^2)$|
| $7$ | $E_7$ |  | $133$ | [[E7]] |$Isom((\mathbb H \otimes \mathbb O) \mathbb P^2)$|
| $8$ | $E_8$ |  | $248$ | [[E8]] |$Isom((\mathbb O \otimes \mathbb O) \mathbb P^2)$|

The following coincidences hold:
\begin{eqnarray}
A_1 &=& B_1 = C_1  \Leftrightarrow SU(2) = SO(3) = Sp(2) \\
B_2 &=& C_2  \Leftrightarrow  SO(5) = Sp(4) \\
A_3 &=& D_3 \Leftrightarrow SU(4) = SO(6) \\
D_2 &=& A_1 × A_1 \Leftrightarrow SO(4) = SU(2) \times SU(2) \\
Spin(3) &=& SU(2) = Sp(1) \\
Spin(4)& =& SU(2) \times SU(2) \\
Spin(5) &= &Sp(2) \\
SU(4) &= &Spin(6) \\
\end{eqnarray}

See also:
* [[Lie group manifold|Lie Group Manifold]]
* [[Ado's theorem|Ado's Theorem]]

Lectures:
* [[Lie Groups in Physics - M.J.G. Veltman, B. Q. P. J. de Wit, G. 't Hooft|http://www.phys.uu.nl/~sahlmann/teaching/lecture%20notes/lie%20group%20lecture%20notes.pdf]] [[local|lectures/lie group lecture notes.pdf]]

Papers:
* [[Lie Groups in the Foundations of Geometry - H. Freudenthal|http://igitur-archive.library.uu.nl/math/2006-1218-201406/freudenthal_65_liegroups.pdf]] [[local|papers/freudenthal_65_liegroups.pdf]] {{t100Cite{[[bct. 115|http://scholar.google.de/scholar?cites=5752053240701341257&hl=de]]}}}
* [[Exceptional Groups and Elementary Particle Structures - L. C. Biedenharn, P. Truini|http://www.slac.stanford.edu/pubs/slacpubs/2750/slac-pub-2802.pdf]] [[local|papers/slac-pub-2802.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=1009577756658151379&hl=de]]

>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)]].

A ''Linear Blockcode'' is a linear subspace with dimension $k$ of a vector space $\mathbb{F}_q^n$, where $\mathbb{F}_q$ is the [[finite field|Galois Field]] with $q$ elements. A linear blockcode  is a special case of a [[blockcode|Blockcode]] which also comprises non-linear blockcodes.

A linear blockcode is entirely determined by $4$ parameters, its ''Code Length'' $n$, its ''Rank'' $k$ (i.e., having $k$ code words in its basis and $k$ rows in its [[generator matrix|Generator Matrix]]), its minimum [[Hamming weight|Hamming Distance]] $d$ and $q$, which equals the number of letters of the alphabet of the code.

A linear blockcode can be constructed from a generating matrix, by taking all linear combinations of its rows or, alternatively, form a [[parity check matrix|Parity Check Matrix]] $H$, requiring that a codeword satisfies $Hx =0$.

A linear block code can correct $\frac12(d-1)$ errors.

Alluding to the parameter $q$ one speaks of a $q$-ary code. Important examples are $q=2$ and $q=3$ which are denoted ''Binary Code'' and ''Ternary Code'' respectively.

The number of codewords (also referred to as vectors) of a $q$-ary linear code is given by $q^k$.

Linear blockcodes are advantageous in that methods of linear algebra apply to them. This makes coding and decoding them simpler. Most of the relevant codes are in fact linear, e.g. the [[Golay-|Golay Code]], [[Hamming-|Hamming Code]], [[Reed-Muller-|Reed-Muller Code]] or [[Hadamard-|Hadamard Code]] codes, all [[cyclic codes|Cyclic Code]] ([[BCH-codes|BCH Code]] and Reed\-Solomon\-codes) and the Goppa codes. One therefore often only speaks of a blockcode or a code, yet meaning a linear blockcode.

In the context of [[lattices|Lattice]] only linear codes play a role. Non-linear codes are important in the context with nonlattices (i.e. structures that are not lattices but still have periodic [[sphere packings|Kissing Number]]).

In technical applications of data transfer and error correction, linear codes are usually used. This is in parts due to the fact that algorithms for their coding and decoding can be made more efficient for them.

In general, for fewer than $32$ dimensions, the best block codes are linear and the best packings are [[lattice packings|Kissing Number]]. An exception occurs in $16$ dimensions, where the best known binary block code with $256$ codewords is the nonlinear [[Nordstrom-Robinson code|Nordstrom-Robinson Code]], which has a [[minimum Hamming distance|Hamming Distance]] $6$.

!!!!Notation
Since the number of codewords of a linear code is determined by the dimension of the subspace $k$, the $(n, M, d)$ notation for [[general codes|Blockcode]] is commonly replaced by ''$[n, k, d]_q \equiv (n, M, d)_q = (n, q^k, d)_q $'' in this case. If it is a binary code one also writes ''$[n, k, d]$'' for short (or sometimes ''$[n,k]$'', suppressing the Hamming weight).
Instead of $[n, k, d]_q$ the designation $(n, k, d)_q$ is also used for linear blockcodes which may be confused with the round brackets notation used for non-linear codes.

!!!!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]]).

{{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.
\[
(\mathbf{AB})\mathbf C \ne \mathbf A(\mathbf{BC})
\]
it is associative up to [[homotopy|Homotopy]], i.e.
\[
(\mathbf{AB})\mathbf C \sim \mathbf A(\mathbf{BC})
\]

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]]

The conventional ''Lorentz Group O(3,1)'' is the invariance group of Minkowski space $\mathbb{R}^{3+1}$, that is to say, the set of all linear [[automorphisms|Automorphism]] of Minkowski space that leave the (pseudo-)[[scalar product|Scalar Product]] invariant.

The Lorentz group is a $6$-dimensional non-compact [[Lie group|Lie Group]]. Its [[Lie algebra|Lie Algebra]] is isomorphic to $\mathfrak sl$$(2,\mathbb C)$ with $3$ generators $J_i$ of spacial rotations, representing the subalgebra $\mathfrak so$$(3)$ and $3$ generators $K_i$ of "spacetime-rotations" called ''Boosts'', satisfying the commutation relations:
\begin{eqnarray}
[J_i,J_j] & = &\varepsilon_{ijk} J_k \\
[K_i,K_j] & = -&\varepsilon_{ijk} J_k \\
\end{eqnarray}
Hence boosts do not form a group.

The full Lorentz group consists of $4$ disconnected spaces. Elements of a given connected space can be transformed into one another by smooth (infinitesimal) transformations, i. e. [[conjugations|Conjugation]]. Furthermore one has discrete transformations between the $4$ topologically separated pieces (conjugacy classes), namely space inversions $P$ and a time reversal $T$. (See also [[CPT-transformations|CPT-Transformations]]). The set of discrete transformations $\{1, P, T, PT\}$ forms a group which is isomorphic to the [[Klein four-group|Klein Four-group]].

The subgroup of the Lorentz group with determinant equal to $1$, ''SO(3,1)'', is called the ''Proper Lorentz Group'', also designated as ''$L_+$''. (The "+" indicates the positive sign of the determinant). It consists of the orientation preserving transformations. Its universal covering group is the group [[SL(2,C)]]. One has the isomorphism: $SO(3,1) \cong SL(2,\mathbb C)/\mathbb Z_2$.
$SO(3,1)$ is built up of of $2$ subgroups. One of them contains the identity transformation and is called ''Proper Orthochronous Lorentz Group'' or ''Restricted Lorentz Group'', also designated as ''$SO(3,1)^+$'' or ''$L_+^\uparrow$'' (where the up arrow stands for "orthochronous"). The [[quotient group|Quotient Group]] $O(1,3)/SO(1,3)^+$ is isomorphic to the Klein four-group mentioned above.

!!!!Generalisations
<html><center><img src="images/LorentzGroups.jpg" style="width: 500px; "/></center></html>

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]]

The ''Lucas' (Square Pyramid) Problem'', which was first raised in 1875 by Éduouard Lucas, states, that the only positive integer solutions $(s, t)$ to the Diophantine equation
\[
1^2 + 2^2 + \ldots + s^2 =  \frac{s(s+1)(2s+1)}{6} = t^2
\]
are given by $(s,t) = (1,1)$ and $(24,70)$.
It was not until 1918 that Watson was able to completely solve the equation. His proof depends upon properties of elliptic functions of modulus $1/\sqrt 2$ and arguably lacks simplicity. A second, more algebraic proof was found in 1952 by Ljunggren and also is somewhat on the complicated side.

Papers:
* [[Lucas' Square Pyramid Problem Revisited - M. A. Bennett|http://www.math.ubc.ca/~bennett/paper21.pdf]] [[pct. 6|http://scholar.google.de/scholar?cites=5568944220303255672&hl=de]]

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]]

''MOND'' is a modification of Newtonian dynamics, designed to reproduce the observed ‘flat’ galaxy rotation curves using only observed distributions of visible matter and reasonable assumptions about mass/light ratios as input data. It was proposed in 1983 by Moti Milgrom.
MOND applies at the very low accelerations which occur in the outer regions of spiral galaxies and in galaxy groups.

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
\[
W(C^\perp;x,y) = \frac{1}{\operatorname{ord}(C)} W(C;y-x,y+x).
\]

The ''Magnus Expansion'' can be regarded as the continuous analogue of the [[BCH formula|Baker Campbell Hausdorff Formula (BCH)]].

Papers:
* [[The Magnus Expansion and some of its Applications - S. Blanes, F. Casas, J. A. Oteo, J. Ros|http://arxiv.org/PS_cache/arxiv/pdf/0810/0810.5488v1.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=16415161858380004400&hl=de]]

An algebra (with a distributive multiplication) is called a ''Malcev Algebra'' if the following conditions are satisfied:
* Its elements [[anticommute|Anti-Commutator]], i.e. $ \{\mathbf A, \mathbf B\} = 0 $.
* The [[Malcev identity|Malcev Identity]] holds.
!!!!Examples
The following algebras are Malcev algebras:
* Every [[Lie algebra|Lie Algebra]].
* Any [[alternative algebra|Alternative Algebra]] equipped with the [[commutator product|Commutator]].
* The [[imaginary octonions|Octonion]] $\mathbb O^-$ equipped with the commutator product (which is in fact the only simple compact Malcev algebra over $\mathbb R$ that is not a Lie algebra).
For a given Malcev algebra there existsts (up to isomorphisms) a uniquely determined connected and simply connected analytical [[Moufang loop|Moufang Loop]]. The Malcev algebra is the [[tangent algebra|Tangent Algebra]] of this loop.

Papers:
* [[Speciality of Malcev Algebras - M. R. Bremner,  I. R. Hentzel, L. A. Peresi |http://www.birs.ca/workshops/2005/05rit020/report05rit020.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=16630602045831034957&hl=de]]

The ''Malcev Identity'' is given by:
\[
((\mathbf {AB})\mathbf C) \mathbf A + ((\mathbf{BC})\mathbf A)\mathbf A + ((\mathbf{CA})\mathbf A)\mathbf B - (\mathbf{AB})(\mathbf A \mathbf C) = 0
\]
The linearized form of the Malcev identity is also known as [[Sagle identity|Sagle Identity]].

The Malcev identity (or its linearized version) is regarded as a natural generalisation of the [[Jacobi identity|Jacobian]].

In any [[alternative algebra|Alternative Algebra]] the [[Lie bracket|Commutator]] satisfies anticommutativity and the Malcev identity.

The Malcev identity can be expressed in terms of the [[Jacobian|Jacobian]] (if characteristic $\ne 2,3$) as follows
\[
\mathbf J(\mathbf A, \mathbf B, \mathbf{AC}) = \mathbf J(\mathbf A, \mathbf B, \mathbf C)\mathbf A
\]
or equivalently
\[
\mathbf J (\mathbf{CA}, \mathbf B, \mathbf A) = \mathbf A \mathbf J( \mathbf C, \mathbf B, \mathbf A)
\]
__Proof:__
We proof the first identity. Written out this one reads:
\[
(\mathbf {AB})(\mathbf A\mathbf C) + (\mathbf {B (\mathbf A\mathbf C)})\mathbf A + ((\mathbf A\mathbf C)\mathbf A)\mathbf B = ((\mathbf {AB}) \mathbf C) \mathbf A + ((\mathbf {BC})\mathbf A) \mathbf A + ((\mathbf{CA})\mathbf B) \mathbf A
\]
To have equivalence with the Malcev identity the following condition has to be satisfied:
\[
((\mathbf{CA})\mathbf B) \mathbf A - (\mathbf {B (\mathbf A\mathbf C)})\mathbf A - ((\mathbf A\mathbf C)\mathbf A)\mathbf B =  ((\mathbf{CA})\mathbf A)\mathbf B
\]
which is given, taking into consideration the anticommutativity of the product of a [[Malcev algebra|Malcev Algebra]].

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The author of this WIKI ...

<html><center><img src="images/author.jpg" style="width: 315px; "/></center></html>

My philosophy: Strive for immortality. If you cannot make it there, you will not make it anywhere.

My passion: [["Hacking"|JHyperComplex]] the universe.

My education: Physicist with minors in relativistic quantum field theory, mathematics and a thesis in astronomy.

>Based on real experiments and computer simulations, quantum gauge theory in four dimensions is believed to have a mass gap. This is one of the most fundamental facts that makes the universe the way it is.
> - Edward Witten [1]

The ''Mass Gap'' is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle.

Theories with massless particles, like the photon or a Goldstone boson, have no mass gap.

Yang\-Mills theory is supposed to have one but a profound theoretical explanation for its existence is lacking. This is one of the seven Millennium Prize problems put forward by the Clay Mathematics Institute.

Papers:
* [[[1] The Problem of Gauge Theory - E. Witten|http://arxiv.org/PS_cache/arxiv/pdf/0812/0812.4512v3.pdf]] pct. 0

A ''Mathieu Group'' is a finite simple, [[sporadic group|Sporadic Group]]. There are 5 Mathieu groups, designated as ''M11'', ''M12'', ''M22'', ''M23'', ''M24''. They are the [[automorphism groups|Automorphism]] of the [[Steiner systems|Steiner Triple System]] S(4,5,11), S(5,6,12), S(3,6,22), S(4,7,23) and S(5,8,24) (= Witt Design) respectively. M24 is also the automorphism group of the extended binary Golay code.

Papers:
* [[Geometry of the Mathieu Groups and Golay Codes - E. A. Lord|http://materials.iisc.ernet.in/~lord/webfiles/eric/pdfs/42.pdf]] pct. 0

The ''Mathisson\-Papapetrou Equations'' or (''MP Equations'') are two coupled equations describing the motion of a particle with spin in a fixed gravitational background, given by:
\begin{eqnarray}
\frac{D\tilde p^\mu}{D\tau}  &=& -\frac{1}{2} {R^\mu}_{\nu\lambda\sigma} S^{\nu\lambda} u^\sigma \equiv f^\mu \\
\frac{DS^{\mu\nu}}{D\tau}   &=& \tilde{p}^{\mu}u^\nu - \tilde{p}^{\nu}u^\mu \equiv f^{\mu\nu}
\end{eqnarray}
with $\frac{D}{D\tau}$ the covariant derivative in respect to the proper time $\tau$, $u^\alpha$ the $4$-velocity, $S^{\mu\nu}$ the totally antisymmetric [[spin tensor|Spin Tensor]], $f_\mu$ the four-force acting on the particle and ${R^\mu}_{\nu\lambda\sigma}$ the [[Riemann tensor|Riemann Tensor]].

$\tilde{p}_\mu$ is called generalized momentum of the particle which in general is not aligned with the $4$-velocity and is given by
\[
\tilde{p}^{\mu} \equiv m u^\mu - \frac{DS^{\mu\nu}}{D\tau}u_\nu
\]
The first of the two equations is called ''Equation of Motion of the Particle'', the second one ''Equation Of Motion Of Spin''.
The equations can be derived by making a multipole expansion around the worldline of the particle.

Inserting the generalized momentum into the two equations one gets more explicitly
\begin{eqnarray}
\frac{D}{D\tau} \left (mu^\mu - \frac{DS^{\mu\nu}}{D\tau}u_\nu \right ) &=& -\frac{1}{2} {R^\mu}_{\nu\lambda\sigma} S^{\nu\lambda} u^\sigma  \\
\frac{DS^{\mu\nu}}{D\tau} &=& \left (m u^\mu - \frac{DS^{\mu\rho}}{D\tau}u_\rho \right ) u^\nu - \left (m u^\nu - \frac{DS^{\nu\rho}}{D\tau}u_\rho \right )u^\mu
\end{eqnarray}
hence
\begin{eqnarray}
\frac{D (m u^\mu)}{D\tau} - \frac{DS^{\mu\nu}}{D\tau} \frac{Du^\mu}{D\tau} +  \frac{D^2S^{\mu\nu}}{D\tau^2} u_\nu + \frac{1}{2} {R^\mu}_{\nu\lambda\sigma} S^{\nu\lambda} u^\sigma &=& 0\\
\frac{DS^{\mu\nu}}{D\tau} + \left (\frac{DS^{\mu\rho}}{D\tau} u^\nu - \frac{DS^{\nu\rho}}{D\tau} u^\mu \right) u_\rho    &=& 0
\end{eqnarray}
!!!!Special cases
!!!!! I.
If there is no spin precession ("spin acceleration"), i.e. $\frac{DS^{\mu\nu}}{D\tau}  = 0$, the first equation simplifies to
\[
\frac{D (m u^\mu)}{D\tau}  = -\frac{1}{2} {R^\mu}_{\nu\lambda\sigma} S^{\nu\lambda} u^\sigma \equiv f^\mu
\]
and the second one to 
\[
p^{\mu}u^\nu - p^{\nu}u^\mu  =  f^{\mu\nu} = 0
\]

!!!!! II.
If in addition spin is zero, no Lorentz force acts upon it and one is left with the classical [[geodesic equation|Geodesic Equation]] of a point particle in a gravitational field:
\[
\frac{D(m u^\mu)}{D\tau}  = 0
\]
!!!! Supplementary Conditions
If the the two equations are regarded as a closed system for determining $p^\mu (\tau)$ and $S^{\mu\nu}(\tau)$, the number of equations is less then the number of unknown functions. For their full determination $3$ additional scalar supplementary conditions are required. 
E.g. one introduces 
the ''Corinaldesi\-Papapetrou\-Condition''
\[
S^{\mu0} u_\nu = 0
\]
or the ''Pirani Condition''
\[
S^{\mu\nu} u_\nu = 0
\]
or the ''Tulczyjew Condition''
\[
S^{\mu\nu} \tilde p_\nu = 0
\]
There seems to be no fundamental reason as to why to prefer the one or the other condition.
Therefore the most widely accepted description of spinning test particles in relativity is incomplete, at least for what concerns the arbitrary choice of supplementary conditions required to make the model compatible.

It was demonstrated by Barker and O'Connell (1974) that when keeping only second-order terms in the spin or velocity of the particle or in the gravitational radius of the centre of a gravitational source, the first two supplementary conditions lead to different non-geodesic equations of motion.

!!!!Historical
The derivation in the original papers of Mathisson and Papapetrou was done half-phenomenologically by integrating the multiple $3$-momentum of distributed matter in space and subsequently reducing it to a point and making the result covariant. A small body can be studied by a multipole expansion method: the body is equivalently described by a set of multipole (energy) moments defined along a central line. The cutoff at successive multipole orders defines a hierarchy of elementary multipole particles. The first step is the point particle (or monopole), governed by the geodesic equation of motion. The second one is the dipole ("spinning") particle with which we are concerned here.

The first general covariant derivations were given by Tulczyjew (1959), Taub (1964) and Dixon (1964; here also higher multipoles were included).

Papers:
* [[Physical Applications of a Generalized Clifford Calculus (Papapetrou Equations and Metamorphic Curvature) - W. M. Pezzaglia Jr.|http://arxiv.org/PS_cache/gr-qc/pdf/9710/9710027v1.pdf]] [[pct. 20|http://scholar.google.de/scholar?cites=17097194775030158597&hl=de&as_sdt=2000]]
* [[Particles as Wilson Lines of the Gravitational Field - L. Freidel, J. Kowalski–Glikman, A. Starodubtsev|http://arxiv.org/PS_cache/gr-qc/pdf/0607/0607014v2.pdf]] [[pct. 13|http://scholar.google.de/scholar?cites=13441903653276481488&hl=de&as_sdt=2000]]
* [[Spinning Test Particles in a Kerr Field - I - O. Semerák|http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1999MNRAS.308..863S&defaultprint=YES&filetype=.pdf]] [[local|papers/spinningTestParticleI.pdf]]  [[pct. 10|http://scholar.google.de/scholar?cites=13776176730641717923&hl=de&as_sdt=2000]] prl. 8
* [[Relativistic Motion of Spinning Particles in a Gravitational Field - C. Chicone, B. Mashhoon, B. Punsly|http://arxiv.org/PS_cache/gr-qc/pdf/0504/0504146v2.pdf]] [[pct. 7|http://scholar.google.de/scholar?cites=13830973532948109739&hl=de&as_sdt=2000]]
* [[Mathisson-Papapetrou Equations in Metric and Gauge Theories of Gravity in a Lagrangian Formulation - M. Leclerc|http://arxiv.org/PS_cache/gr-qc/pdf/0505/0505021v2.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=4973884521331268498&hl=de&as_sdt=2000]]
* [[Polydimensional Supersymmetric Principles - W. M. Pezzaglia Jr.|http://arxiv.org/PS_cache/gr-qc/pdf/9909/9909071v1.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=9266058051844227761&hl=de&as_sdt=2000]]
* [[The Plane Trajectories of Spin Particles in the Schwarzschild Field - K. Svirskas, K. Pyragas, A. Lozdiene|http://adsabs.harvard.edu/full/1988Ap&SS.149...39S  ]] [[local|papers/nph-iarticle_query.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=10111546496191620081&hl=de&as_sdt=2000]]
* [[Dirac Equations in Curved Space-Time versus Papapetrou Spinning Particles - F. Cianfrani, G. Montani|http://arxiv.org/PS_cache/arxiv/pdf/0810/0810.0447v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=2840262814515515661&hl=de]]
* [[Nongeodesic Motion of Spinless Particles in the Teleparallel Gravitational Wave Background - L. C. Garcia de Andrade|http://arxiv.org/PS_cache/gr-qc/pdf/0205/0205120v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=15631791624518052181&hl=de&as_sdt=2000]]
* [[On the Coupling Between Spinning Particles and Cosmological Gravitational Waves - I. Millillo, M. Lattanzi, G. Montani|http://arxiv.org/PS_cache/arxiv/pdf/0804/0804.0572v1.pdf]] pct. 0
* [[Deriving Mathisson - Papapetrou Equations from Relativistic Pseudomechanics - R. R. Lompay|http://arxiv.org/PS_cache/gr-qc/pdf/0503/0503054v1.pdf]] pct. 0
* [[Classical and Quantum Spins in Curved Spacetimes - A. J. Silenko|http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.4443v1.pdf]] pct. 0
* [[Quasi-Maxwell Interpretation of the Spin-Curvature Coupling - Jose Natario|http://arxiv.org/PS_cache/gr-qc/pdf/0701/0701067v4.pdf]] pct. 0
* [[Mathisson’ Spinning Electron : Noncommutative Mechanics & Exotic Galilean Symmetry, 66 Years Ago - P. A. Horvathy|http://arxiv.org/PS_cache/hep-th/pdf/0303/0303099v4.pdf]] pct. 0
* [[The Papapetrou Equations and Supplementary Conditions - O. B. Karpov|http://arxiv.org/PS_cache/gr-qc/pdf/0406/0406002v2.pdf]] pct. 0
* [[Rotation and Spin in Physics - R. F. O’Connell|http://www.phys.lsu.edu/faculty/oconnell/PDFfiles/308.%20Rotation%20and%20Spin%20in%20Physics.pdf]] pct. 0
* [[Mathisson's New Mechanics: Its Aims and Realisation - W. G. Dixon|http://th-www.if.uj.edu.pl/acta/sup1/pdf/s1p0027.pdf]] pct. 0

Presentations:
* [[The Papapetrou Equation Derived as a Geodesic in a Non-holonomic Clifford Manifold - W. M. Pezzaglia Jr.|http://www.clifford.org/wpezzag/talk/1998oregon/1998oregon.pdf]] [[local|presentations/1998oregon.pdf]]
* [[Motion in Brane World Models: The Bazanski Approach - M.E.Kahil|http://www.pascos07.org/programme/talks/Kahil.pdf]] [[local|presentations/Kahil.pdf]] - With a nice compilation of the relevant formula. trl. 7

Links:
* [[The Matrix Reference Manual|http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html#Intro]]

A ''Maximal Torus'' of a [[group|Group]] is a [[torus|Torus]] that is not strictly contained in another one.
In any compact [[Lie group|Lie Group]] one can find a maximal torus

A ''Menon Design'' is a $2-(4u^2, 2u^2 \pm u, u^2 \pm u)$ [[design|Balanced Incomplete Block Design]].

A ''Mersenne Prime'' $M_n$ is a prime number of the form
\[
 M_n = 2^n - 1
\]
with $n \in \mathbb N\,$. It can be shown that $M_n$ can only be prime if $n$ is prime.
Currently only $47$ Mersenne primes are known. The sequence goes like this: $7,31, 127, 8.191, 131.071, 524.287, 2.147.483.647, \ldots$.

Mersenne primes are related to [[perfect numbers|Perfect Number]]. In both cases it is still unknown if there exist infinitely many such numbers.

Links:
* [[WIKIPEDIA - Mersenne Primes|http://en.wikipedia.org/wiki/Mersenne_prime]]

The ''Miracle Ocatad Generator'' (''MOG'') is an arrangement of the $24$ points of the [[Steiner system|Steiner System]] [[S(5,8,24)]] into a $4\times6$-array in which the octads assume a particularly recognizable form so it is easy to read them off. The pairing of the columns $12$, $34$ and $56$ are known as the ''Bricks'' of the MOG.

There are two different realisations of \MOGs commonly found in literature, one due to Curtis and one due to Conway. To obtain the one from the other one need only swap the rightmost two columns.

The MOG facilitates calculations in the Mathieu group $M_{24}$ and is an indispensable tool for working in many of the other [[sporadic groups|Sporadic Group]].
Historically it was an essential ingredient in the constructions of the $J_4$ and [[Monster groups|Monster Group]].

Links:
* [[Finitegeometry.org - The Miracle Octad Generator (MOG) of R T. Curtis|http://finitegeometry.org/sc/24/MOG.html]]

The concept of a ''Module'' is a generalization of the notion of a vector space. The difference is that a module is defined over a ring, whereas a vector space is defined over a field. 

A module, like a vector space, is an additive [[abelian group|Group]]; a product is defined between elements of the ring and elements of the module, and this multiplication is associative and distributive.

In 1978/79, J. \McKay, J. Thompson, [[J. Conway|John Conway]] and S. Norton had discovered astounding "numerology" culminating in the ''Monstrous Moonshine'' conjectures relating the not-yet-proved-to-exist [[Monster group|Monster Group]] to modular functions in number theory, namely:
There should exist a (natural) infinite-dimensional $\mathbb{Z}$-graded module for $M$ (i.e., representation of $M$)
\[
V=\bigoplus_{n=-1,0,1,2,3,\dots}V_n
\]
such that
\[
\sum_{n=-1,0,1,2,3,\dots}({\rm dim}V_n)q^n=J(q)
\]
where
\[
J(q)=q^{-1}+0+196884q+{\mbox {higher-order terms}}
\]
is the classical modular function with its constant term set to $0$.

Videos:
* [[What is Moonshine? - R. Borcherds|http://www.newton.ac.uk/webseminars/vault/borcherds/]]

The following table shows the four known ''Moore Graphs'' (and fifth that might exist):
<html>
<center>
<table width="600"  border="0" align="center">
  <tr>
    <th scope="col"><div align="center">Name</div></th>
    <th scope="col"><div align="center">Degree (edges per vertex) </div></th>
    <th scope="col"><div align="center">Diameter </div></th>
    <th scope="col"><div align="center">Vertices </div></th>

  </tr>
  <tr>
    <td><div align="center">Pentagon</div></td>
    <td><div align="center">2</div></td>
    <td><div align="center">2</div></td>
    <td><div align="center">5</div></td>
  </tr>

  <tr>
    <td><div align="center">Heptagon</div></td>
    <td><div align="center">2</div></td>
    <td><div align="center">3</div></td>
    <td><div align="center">7</div></td>
  </tr>
  <tr>

    <td><div align="center">Petersen Graph </div></td>
    <td><div align="center">3</div></td>
    <td><div align="center">2</div></td>
    <td><div align="center">10</div></td>
  </tr>
  <tr>
    <td><div align="center">Hoffman-Singleton Graph </div></td>

    <td><div align="center">7</div></td>
    <td><div align="center">2</div></td>
    <td><div align="center">50</div></td>
  </tr>
  <tr>
    <td><div align="center">???</div></td>
    <td><div align="center">57</div></td>

    <td><div align="center">2</div></td>
    <td><div align="center">3250</div></td>
  </tr>
</table> </center>
</html>

Papers:
* [[Morita Equivalence in Geometry and Algebra - R. Mayer|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.35.3449&rep=rep1&type=pdf]]

Reformulate physics and mathematics and recast it in a (coherent and consistent) form, that at least oneself can understand.

Why ?
* A lot of redundancies. Things are rediscovered over and over again and often it is not realised that they are the same. This corresponds to the problem of different representations for the same thing and different "brain wirings" of different authors, prefering different representations.
* One cannot learn mathematics and physics, one has to discover it.
* A lot is copied and pasted by authors, who do not really understand what the true meaning of the content is. This is even worse when things are copied over and over again. Errors accumulate and propagate. (Therefore it is no wonder, that original works are often way more joyful to read).
* A lot is too abstract, too far from an application. A priorisation of the relevance of mathematical structures in respect to their applicability in physics is needed. Classification of structures in mathematics is essential but one deals a lot with practically uninteresting structures (one can easily get lost).
* Reading and writing (or WIKIing these days) is better what concerns understanding than just reading.

A ''Moufang loop'' is an [[algebra|Algebra]] which satisfies the following identities, called ''Moufang Identities'':
!!!!''Left Moufang Identity'':
\[
((\mathbf{AB})\mathbf A)\mathbf C = \mathbf A(\mathbf B(\mathbf{AC}))
\]

!!!!''Right Moufang Identity'':
\[
\mathbf A(\mathbf B(\mathbf{CB})) = ((\mathbf{AB})\mathbf C)\mathbf B
\]

!!!!''Middle Moufang Identities'':
\[
(\mathbf A(\mathbf{BC}))\mathbf A = (\mathbf{AB})(\mathbf{CA})
\]
\[
\mathbf A((\mathbf{BC})\mathbf A) = (\mathbf{AB})(\mathbf{CA})
\]
therefore
\[
(\mathbf A(\mathbf{BC}))\mathbf A = \mathbf A((\mathbf{BC})\mathbf A)
\]


If one sets $\mathbf C = 1$ in the right Moufang identity one gets $ [\mathbf A,\mathbf B,\mathbf B] = 0$, i.e. a Moufang loop is [[right alternative|Alternative Algebra]]
Equivalently if one sets $\mathbf B = 1$ in the left Moufang identity one gets $ [\mathbf A,\mathbf A,\mathbf C] = 0$, i.e. a Moufang loop is [[left alternative|Alternative Algebra]].
Setting  $\mathbf B = 1$  in the first middle Moufang identity leads to $ [\mathbf A,\mathbf C,\mathbf A] = 0$, i.e. a Moufang loop is [[flexible|Flexible Algebra]].

In fact for any [[loop|Loop]] the identities are equivalent, i.e. if one is satisfied the loop is a [[flexible|Flexible Algebra]] and [[alternative|Alternative Algebra]] algebra.

As a Moufang loop is a left and a right [[Bol loop|Bol Loop]] it also satisfies the left and right Bol loop identities.
?
!!!!Linearization
Linearizing the four Moufang identities one gets
\begin{eqnarray}
\mathbf M_l(\mathbf A, \mathbf  B, \mathbf  C,\mathbf  D) \equiv ((\mathbf{AB})\mathbf C)\mathbf D  - \mathbf A(\mathbf B(\mathbf{CD})) + ((\mathbf{CB})\mathbf A)\mathbf D - \mathbf C(\mathbf B(\mathbf{AD})) = 0 \\
\mathbf M_r(\mathbf A, \mathbf  B, \mathbf  C,\mathbf  D) \equiv \mathbf A(\mathbf B(\mathbf{CD})) - ((\mathbf{AB}) \mathbf C)\mathbf D + \mathbf A(\mathbf D(\mathbf{CB})) - ((\mathbf{AD})\mathbf C)\mathbf B = 0 \\
\mathbf M_{m_1}(\mathbf A, \mathbf  B, \mathbf  C,\mathbf  D) \equiv \mathbf A(\mathbf{BC}))\mathbf D - (\mathbf{AB})(\mathbf{CD}) + \mathbf D(\mathbf{BC}))\mathbf A - (\mathbf{DB})(\mathbf{CA}) = 0\\
\mathbf M_{m_2}(\mathbf A, \mathbf  B, \mathbf  C,\mathbf  D) \equiv  \mathbf A((\mathbf{BC})\mathbf D - (\mathbf{AB})(\mathbf{CD}) + \mathbf D((\mathbf{BC})\mathbf A - (\mathbf{DB})(\mathbf{CA}) = 0
\end{eqnarray}
and hence
\[
\mathbf M_{m_1}(\mathbf A, \mathbf  B, \mathbf  C,\mathbf  D) - \mathbf M_{m_2}(\mathbf A, \mathbf  B, \mathbf  C,\mathbf  D)  =  \mathbf A(\mathbf{BC}))\mathbf D -  \mathbf A((\mathbf{BC})\mathbf D + \mathbf D(\mathbf{BC}))\mathbf A - \mathbf D((\mathbf{BC})\mathbf A = 0
\]
Introducing $\mathbf Q(\mathbf A,\mathbf B,\mathbf C,\mathbf D) \equiv ((\mathbf{AB})\mathbf C)\mathbf D - \mathbf A(\mathbf B(\mathbf{CD}))$,
\begin{eqnarray}
\mathbf M_l(\mathbf A, \mathbf  B, \mathbf  C,\mathbf  D) = \mathbf Q (\mathbf A,\mathbf B,\mathbf C,\mathbf D) + \mathbf Q(\mathbf C,\mathbf B,\mathbf A,\mathbf D) = 0 \\
\mathbf M_r(\mathbf A, \mathbf  B, \mathbf  C,\mathbf  D) = - \mathbf Q (\mathbf A,\mathbf B,\mathbf C,\mathbf D) - \mathbf Q (\mathbf A,\mathbf D,\mathbf C,\mathbf B) = 0 \\
\end{eqnarray}
i.e. $ \mathbf Q (\mathbf A,\mathbf B,\mathbf C,\mathbf D)$ exhibits the symmetries:
\begin{eqnarray}
\mathbf Q (\mathbf A,\mathbf B,\mathbf C,\mathbf D) = -  \mathbf Q (\mathbf C,\mathbf B,\mathbf A,\mathbf D) = - \mathbf Q (\mathbf A,\mathbf D,\mathbf C,\mathbf B)
\end{eqnarray}

!!!!People
Kuzmin, Kerdman, Sagle (and many others).

Papers:
* [[The Varieties of Loops of Bol-Moufang Type - J. D. Phillips and P. Vojtechovsky | http://arxiv.org/PS_cache/math/pdf/0701/0701714v1.pdf]] [[pct. 27|http://scholar.google.de/scholar?cites=4648346781753324864&hl=de]]
* [[An Introduction to Moufang Symmetry - E.N. Paal|http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?198805191]] [[pct. 8|http://scholar.google.de/scholar?cites=16503153514823441353&hl=de&as_sdt=2000]] - (in Russian)
* [[Extension of Local Loop Isomorphisms - P. T. Nagy|http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN362162050_0112&DMDID=dmdlog17]] [[local|papers/LocalLoopIsomorphisms.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=9463945647390757835&hl=de]]
* [[Moufang Loops and Malcev Algebras - Peter T. Nagy|http://www.heldermann-verlag.de/jlt/jlt03/NAGYLAT.PDF]] [[pct. 2|http://scholar.google.de/scholar?cites=16975866641755798613&hl=de]] [[local|papers/NAGYLAT.PDF]]

<html><center><img src="images/math_clock.jpg" style="width: 280px; "/></center></html>

Papers:
* [[My Favorite Integer Sequences N. J. A. Sloane|http://www.research.att.com/~njas/doc/sg.pdf]] [[pct. 9|http://scholar.google.com/scholar?hl=de&lr=&cites=15076201712377063024&um=1&ie=UTF-8&ei=2A2lSpX8B5m4sgaHtoXTBA&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]

Links:
* [[Cognitive Theoretic Model of the Universe (CMTU) - C. Langan|http://www.megafoundation.org/CTMU/]] - How a man with an alledged I.Q. of around 200 "sees" the universe.
* [[Ben Goertzel Essays|http://www.goertzel.org/essays.htm]]
* [[University of Toronto Mathematics Network - Question Corner and Discussion Area|http://www.math.toronto.edu/mathnet/questionCorner/qc.ps]]
* [[Articles by S. M. Phillips|http://www.smphillips.8m.com/html/articles.html]] - Interesting stuff, but to be taken with a gran of salt. ("Octonion Algebra is isomorphic to E8 Lie Algebra").
* [[Bitmaps for a Digital Theory of Everything - R. Aschheim|http://www.cs.indiana.edu/~dgerman/2008midwestNKSconference/rasch.pdf]]
* [[Strings and Loops in Event Symmetric Space-Time - P. Gibbs|http://arxiv.org/PS_cache/hep-th/pdf/9407/9407136v1.pdf]]
* [[Rafiki Inc.|http://www.codefun.com/]]
* [[The Cellular Universe website - C. Ranzan|http://www.cellularuniverse.org/]]
* [[Tony Smith's Homepage, 240 Thoughts|http://www.valdostamuseum.org/hamsmith/SWTxt.html]]
* [[Verman University Mathematical Quotations Server|http://math.furman.edu/~mwoodard/mqs/mquot.shtml]]

Given a group $\mathcal G$ and an Abelian group $\mathcal G_A$, a ''(inhomogeneous) n\-Cochain'' is defined as a $n$-linear map $\alpha_n: \mathcal G \times \mathcal G \times \ldots \times \mathcal G \rightarrow \mathcal G_A$,
\[
\alpha_n: (g_1,\ldots, g_i, \ldots, g_n) \rightarrow \alpha_n (g_1,\ldots, g_i, \ldots, g_n) \in \mathcal G_A
\]
with $g_i \in \mathcal G$, i.e. $n$-cochains form an abelian group.

Remark: Sometimes one finds the definition with $\mathbb R$ instead of $\mathcal G_A$, which is a special case of the definition given here.

A $n$-cochain can be associated with a differential $n$-form $\boldsymbol\omega_n$ as follows:
\begin{equation}
    \alpha_n({x};g_1,g_2,\dots, g_{n})= \int_{\Sigma_n}\boldsymbol \omega_n,
\end{equation}
where $\Sigma_n$ is a $n$-simplex with $n+1$ vertices $(x, x g_1, \dots, x g_1\dots g_n)$.

!!!!Special cases
Let $\mathcal G$ be a group with neutral element $e$, then:

A ''1-Cochain'' $s$ in $\mathcal G$ is a map $s: \mathcal G \rightarrow \mathbb K$ (a nowhere-zero function with $s(e) = 1$), obeying $s^2 = 1$.

A ''2-Cochain'' $F$ in $\mathcal G$ is a map $F: \mathcal G \times \mathcal G \rightarrow \mathbb K\,$ satisfying $F(e,x)= F(x,e) = 1,\, \forall x \in G$.
Given a finite group $\mathcal G$  with elements by $\{\mathbf e_1 \equiv \mathbf e, \mathbf e_2, \ldots, \mathbf e_n\}$, the cochain $F$ can be represented by a $n \times n$ matrix with entries $F_{ij} = F(\mathbf e_i, \mathbf e_j)$. It has $1$ in the first row and column and all entries are non-zero. Conversely, any such matrix will do for a cochain. Therefore it is a wide-open question what other groups and cochains might be of interest. The $F_{ij}$ can be identified with the [[structure constant|Structure Constants]] $C_{ijk}$ of the respective algebra if one fixes $k$. Or, if furthermore one only allows for $c_{ijk} = \pm1$, with its [[sign matrix|Sign Tables]].

!!!!Cochain twist
We restrict ourselves here to the example of a ''${{\mathbb Z}_2}^n$'' ''$2$-Cochain Twist''.

In this case we can define the following maps:
$\mathbb Z_2 = \{0,1\} \rightarrow \{\mathbf e, \mathbf e_1\}\$
$\mathbb Z_2 \times \mathbb Z_2 = \{00,01,10,11\} \rightarrow  \{\mathbf e, \mathbf e_1 ,\mathbf e_2, \mathbf e_1 \mathbf e_2 \}$
$\mathbb Z_2 \times \mathbb Z_2 \times \mathbb Z_2 = \{000,001,010,100, 011,101, 110, 111\}\ \rightarrow \{\mathbf e, \mathbf e_1 ,\mathbf e_2, \mathbf e_3, \mathbf e_1 \mathbf e_2,  \mathbf e_1 \mathbf e_3,  \mathbf e_2 \mathbf e_3,  \mathbf e_1 \mathbf e_2  \mathbf e_3 \}   $
$\ldots$
which demonstrates the $n$-grading of the basis.
We will also use the notation $\mathbf e_{\vec x}$ with $\vec x \in \mathbb Z^n_2$. E.g. for $n = 2$ the base elements would be coded as follows:
$\mathbf e_{\vec x} = \mathbf e_{(x_1,x_2)} = \{\mathbf e_{00}, \mathbf e_{10}, \mathbf e_{01}, \mathbf e_{11} \} \equiv \{\mathbf e, \mathbf e_1 ,\mathbf e_2, \mathbf e_1 \mathbf e_2 \}$.

$\mathbb Z_2^n$ naturally forms an (Abelian) group $\mathcal G$ with a product denoted $*$.
For $\mathbb Z^2 \times \mathbb Z^2$ for example one has the multiplication table
|$*$|$\mathbf e$|$\mathbf e_1$|$\mathbf e_2$|$\mathbf e_1\mathbf e_2$|
|$\mathbf e$|$\mathbf e$|$\mathbf e_1$|$\mathbf e_2$|$\mathbf e_1\mathbf e_2$|
|$\mathbf e_1$|$\mathbf e_1$|$\mathbf e$|$\mathbf e_1\mathbf e_2$|$\mathbf e_2$|
|$\mathbf e_2$ |$\mathbf e_2$|$\mathbf e_1\mathbf e_2$|$\mathbf e$|$\mathbf e_1$|
|$\mathbf e_1 \mathbf e_2$|$\mathbf e_1\mathbf e_2$|$\mathbf e_2$|$\mathbf e_1$|$\mathbf e$|
which is isomorphic to the [[Klein four-group|Klein Four-group]].

A $\mathbb Z_2^n$-twist of $\mathcal G$ now means, that one modifies (or deforms) the group product such that a new product ("twisted product") $*_F$ results which is given by
\[
\mathbf e_i  *_F \mathbf e_j \equiv F(\mathbf e_i, \mathbf e_j) \mathbf e_i * \mathbf e_j
\]
with $\mathbf e_i, \mathbf e_j \in \mathcal G$ and $F(\mathbf e_i, \mathbf e_j)$ a $2$-cochain.

There are many examples of algebras letting $\mathcal G$ be any group and $F$ any cochain.
Even if one requires $F^2 = 1$ for the cochain, one can still obtain very different types of algebras. Examples are [[Clifford algebras|Clifford Algebra]] and [[Cayley-Dickson algebras|Cayley-Dickson Algebra]], but also other types of algebras, like ones based on symmetric [[Hadamard matrices|Hadamard Matrix]].

In the following we assume $F^2(\mathbf e_i,\mathbf e_j)  = 1 = \left [ (-1)^{f(\mathbf e_i,\mathbf e_j)} \right ]^2$ with a scalar function $f$. This condition still captures many interesting types of algebras.
Notice, that $F$ defines an [[orthogonal transformation|Orthogonal Transformation]] and as the matrix elements $F(\mathbf e_i,\mathbf e_j)$ are either $+1$ or $-1$ in this case, $F$ is a [[Hadamard matrix|Hadamard Matrix]].

For specific choices of $f$ that are associated with [[normalized Hadamard matrices|Hadamard Matrix]], one gets the classical (non-split) [[Cayley-Dickson algebras|Cayley-Dickson Algebra]].
These are given in the following:

i) [[Complex numbers|Complex Number]]:
\[
f(\mathbf e_x,\mathbf e_y)=xy,\quad x,y\in \mathcal G = \mathbb Z_2
\]

ii) [[Quaternions|Quaternion]]:
\[
f(\mathbf e_{\vec{x}},\mathbf e_{\vec{y}})=  \sum_{i\le j} x_iy_j = x_1y_1+ x_1y_2 + x_2y_2,\quad \vec{x},\vec{y}\in \mathcal G = \mathbb Z_2^2
\]

iii) [[Octonion|Octonion]]:
\begin{eqnarray}
f(\mathbf e_{\vec{x}},\mathbf e_{\vec{y}}) & = & \sum_{i\le j}x_iy_j+y_1x_2x_3+x_1y_2x_3+x_2x_2y_3,\quad \vec{x},\vec{y}\in \mathcal G = \mathbb Z_2^3 \\
& = & (x_1y_1 + x_1y_2 + x_1y_3 + x_2y_2 + x_2y_3 +  x_3y_3) + \\
&& (x_2x_3y_1 + x_1x_3y_2 +x_2x_2y_3)
\end{eqnarray}

iv) [[Sedenions|Sedenion]]:
\begin{eqnarray}
f (\mathbf e_{\vec{x}},\mathbf e_{\vec{y}})&=&\sum_{i\le j}x_iy_j+ \sum_{i\ne j\ne k\ne i}x_ix_jy_k+\sum_{{\rm distinct}\ i,j,k,l} x_ix_jy_ky_l+\sum_{i\ne
j\ne k\ne i}x_ix_4y_jy_k,\quad \vec{x},\vec{y}\in \mathcal G = \mathbb Z_2^4 \\
& = & (x_1y_1 + x_1y_2 + x_1y_3 + x_1y_4 +  x_2y_2 + x_2y_3 +  x_2y_4 + x_3y_3  + x_3y_4 + x_4y_4 ) + \\
& & (x_1x_2y_3 + x_1x_2y_4 + x_1x_3 y_2 + x_1x_3 y_4 + x_1x_4 y_2  + x_1x_4 y_3 + x_2x_3 y_1 + x_2x_3 y_4 + x_2x_4 y_1 + x_2x_4 y_3 +  x_3x_4 y_1 + x_3x_4 y_2) +\\
& & x_1 x_2 y_3 y_4 +  x_1 x_3 y_2 y_4 +  x_2 x_3 y_1 y_4 +  2x_1 x_4 y_2 y_3  + 2x_2 x_4 y_1 y_3 + 2x_3 x_4 y_1 y_2 +  x_4 x_4 y_1 y_2 \\
\end{eqnarray}
Note, that the cochain and the group are not independent. E.g. it can be demonstrated that if one exchanges the sign tables of certain of the [[480 different octonion algebras|480 Octonion Multiplication Tables]], this leads to algebras that are no longer octonion algebras, as they do not satisfy the [[Moufang identity|Moufang Loop]] any more.

In case of the complex numbers and the quaternions, the Cayley\-Dickson algebra construction and the Clifford algebra construction coincide.
For the higher order algebras however, the resulting algebras differ, in that Clifford algebras do not contain cubic or higher order terms in $f$ contrary to the Cayley\-Dickson algebras. The latter therefore have a richer structure than do have Clifford algebras and hence "code" more information.
Also, Clifford algebra only allow for $2^{nd}$-order [[tangent algebras|Tangent Algebra]] ([[Lie algebras|Lie Algebra]]), whereas Cayley\-Dickson algebras allow for higher order tangent algebras and therefore for a generalisation of the concept of  Lie algebras. I.e. in comparison, Clifford algebras are the more "trivial" algebras.

Papers:
* [[Quasialgebra Structure of the Octonions - H. Albuquerque, S. Majid|http://arxiv.org/PS_cache/math/pdf/9802/9802116v1.pdf]] [[pct. 49|http://scholar.google.de/scholar?cites=15697425696231458277&hl=de&as_sdt=2000]]
* [[From Clifford Algebras to Cayley Algebras - H. Albuquerque|http://www.cim.pt/files/FromLieAlgebrasToQuantumGroups.pdf#page=7]]  [[local|papers/FromCliffordToCayley.pdf]] pct. 0

>Cocycles are everywhere !
>K. J. Horadam - Hadamard Matrices and their Applications

A ''n\-Cocycle'' is a [[n-cochain|N-Cochain]] $\alpha_n(x; g_1, g_2,\ldots, g_n)$ for which the following condition (a.k.a ''N\-Cocycle Condition'') holds:
\begin{eqnarray}
\delta \alpha_n & \equiv& \alpha_n(x g_1;g_2, \ldots, g_{n+1}) -  \alpha_n(x; g_1 g_2, \ldots , g_{n+1}) + \ldots\\
&& (−1)^{i} \alpha_n(x; g_1, \ldots , g_i g_{i+1}, \ldots , g_{n+1})
+ (−1)^{n+1} \alpha_n(x; g_1, \ldots , g_n) \\
&\equiv& 0
\end{eqnarray}
with $g_i$ elements of a group.

$\delta$ is called ''Coboundary Operator'' which maps a $n$-cochain to a $(n+1)$-cochain, i.e. $\delta: \alpha_n \rightarrow \alpha_{n+1}$. 
It satisfies $\delta \delta \alpha_n = 0$.

If $\alpha_n = \delta \alpha_{n-1}$ the cochain $\alpha_n$ is called a ''Coboundary''. A coboundary is always a cocycle. 

A cocycle is said to be nontrivial (or to have nontrivial cohomology) if it is not a coboundary, i.e. $\delta \alpha_n = 0$ but $\alpha_n \ne \delta \alpha_{n-1}$.

A $n$-cocycle condition can be interpreted as being equivalent to a [[coherence law|Coherence Law]] of order $n+1$.

!!!!Examples
\begin{eqnarray}
\delta \alpha_0 &=& \alpha_0(x g_1) \\
\delta \alpha_1 &=&  \alpha_1(x g_1;g_2) −  \alpha_1(x; g_1 g_2) +  \alpha_1(x; g_1) \\ \\
\delta \alpha_2 &=&  \alpha_2(x g_1;g_2,g_{3}) −  \alpha_2(x; g_1 g_2, g_{3}) + \alpha_2(x; g_1, g_2 g_{3}) −  \alpha_2(x; g_1,g_2) \\
\delta \alpha_3 &= &\alpha_3(x g_1;g_2,g_3,g_4) − \alpha_3(x; g_1 g_2, g_3,  g_4) + \\
&&  \alpha_3(x; g_1, g_2 g_3, g_4) - \alpha_3(x; g_1, g_2, g_3 g_4) + \alpha_3(x; g_1, g_2, g_3) \\
\delta \alpha_4 &= &\alpha_4(x g_1;g_2,g_3,g_4,g_5) −  \alpha_4(x; g_1 g_2, g_3, g_4) + \\
&& \alpha_4(x; g_1, g_2 g_3, g_4,g_5) - \alpha_4(x; g_1, g_2, g_3 g_4,g_5) +  \alpha_4(x; g_1, g_2, g_3, g_4 g_5) - \\
&&  \alpha_4(x; g_1, g_2, g_3, g_4)
\end{eqnarray}

$\alpha_n$ can be represented by means of an integral along the edges of a $n$-simplex $\Sigma_n$ over a differential $n$-form $\omega_n$:
\[
\alpha_n(x; g_1, g_2,..., g_{n+1}) = \int_{\Sigma_n} \omega_n  = \sum^{n-1}_{i=0} \int \limits_{x_i}^{x_{i+1}} w_n
\]
with the $n+1$ vertices $(x, xg_1 \ldots , xg_1, \ldots, g_n) \equiv  (x_0, \ldots, x_n)$.

e.g. for $n = 3$:
\[
\alpha_3(x;g_1,g_2,g_3) = \int \limits_{x}^{xg_1} w_3 + \int \limits_{x g_1}^{x g_1 g_2} w_3 + \int \limits_{x g_1 g2}^{x g1 g2 g3} w_3
\]
!!!!Examples
* [[2-cocycle|2-Cocycle]]
* [[3-cocycle|3-Cocycle]]

!!!!Applications
In quantum mechanics a cocycle can be interpreted as a phase.
!!!!!1-cocycle
Given a complex wavefunction $\Psi(\mathbf x)$, we let a translation operator $g(\Delta \mathbf x)$ act on it.
$g(\Delta \mathbf x) \Psi(\mathbf x) = \Psi(\mathbf x + \Delta \mathbf x) = \exp (i \alpha_1 (\mathbf x; \Delta \mathbf x)) \Psi(\mathbf x)$.
The $1$-cocycle can be interpreted as representing a measure of "ordering" (e.g. $|g(\Delta \mathbf x) \Psi(\mathbf x)|$ < $|g(\Delta \mathbf x')\Psi(\mathbf x)|$). This distinguishes it from the $0$-cocycle which is independent of the choice of a group element. 

As $g$ "lives" in the space of $\Psi$ we require that it is an Abelian group. Therefore two translations $\Delta \mathbf x$ and $\Delta \mathbf x'$ have to commute, i. e.
\[
g(\Delta \mathbf x) g(\Delta \mathbf x') \Psi (\mathbf x)  =g(\Delta \mathbf x') g(\Delta \mathbf x) \Psi (\mathbf x) 
\]
This is equivalent to the $1$-cocycle condition, as  
\begin{equation}
\exp (i \alpha_1 (\mathbf x; \Delta \mathbf x + \Delta \mathbf x')) \Psi(\mathbf x) = \exp (i \alpha_1 (\mathbf x + \Delta \mathbf x; \Delta \mathbf x')  \exp ( i \alpha_1 (\mathbf x; \Delta \mathbf x)) \Psi(\mathbf x) 
\end{equation}
or
\begin{equation}
\exp (i \alpha_1 (\mathbf x; \Delta \mathbf x + \Delta \mathbf x') - i \alpha_1 (\mathbf x + \Delta \mathbf x; \Delta \mathbf x') - i \alpha_1 (\mathbf x; \Delta \mathbf x)) \Psi(\mathbf x) = 1 
\end{equation}
and finally 
\[
\alpha_1 (\mathbf x; \Delta \mathbf x + \Delta \mathbf x') - i \alpha_1 (\mathbf x + \Delta \mathbf x; \Delta \mathbf x') - \alpha_1 (\mathbf x; \Delta \mathbf x) = 0 \,(\operatorname{mod} 2 \pi \mathbb Z)
\]
i.e.  
\[
\delta \alpha_1 = 0 \,(\operatorname{mod} 2 \pi \mathbb Z)
\]
!!!!!2-cocycle
The $2$-cocycle generalizes the $1$-cocycle to non-Abelian groups. This situation applies for example to a [[quaternion|Quaternion]]-valued function $\Psi$ (and hence also to a [[Dirac spinor|Dirac Spinor]]).
One now has:
\[
g(\Delta \mathbf x) g(\Delta \mathbf x') \Psi (\mathbf x) \ne g (\Delta \mathbf x') g(\Delta \mathbf x) \Psi (\mathbf x)
\]
or equivalently
\[
[g(\Delta \mathbf x), g(\Delta \mathbf x')] \Psi (\mathbf x) \ne 0
\]
One says that the [[commutator|Commutator]] is anomalous. 

The anomaly can be "absorbed" in a $2$-cochain as follows: 
\[
g(\Delta \mathbf x) g(\Delta \mathbf x') \Psi (\mathbf x) = \exp (i \alpha_2 (\Delta \mathbf x, \Delta \mathbf x')) g(\Delta \mathbf x') g(\Delta \mathbf x) \Psi (\mathbf x)
\]

As the $g$'s are elements of an Abelian group, one has to satisfy the condition of associativity for $3$ consecutive translations $\Delta \mathbf x$, $\Delta \mathbf x'$ and $\Delta \mathbf x''$, i.e.
\[
[g(\Delta \mathbf x), g(\Delta \mathbf x'), g(\Delta \mathbf x'')]  \Psi (\mathbf x) = 0
\]
This is equivalent to the $2$-cocycle-condition, as 
\begin{eqnarray}
&&\exp (i \alpha_2 (\mathbf x; \Delta \mathbf x \Delta \mathbf x', \Delta \mathbf x'')) g(\Delta \mathbf x'') (g(\Delta \mathbf x)g (\Delta \mathbf x')) \Psi (\mathbf x)  = \exp (i \alpha_2 (\mathbf x; \Delta \mathbf x', \Delta \mathbf x''))  (g (\Delta \mathbf x) g (\Delta \mathbf x'')) g (\Delta \mathbf x')  \Psi (\mathbf x) \Leftrightarrow \\
&&\exp (i \alpha_2 (\mathbf x; \Delta \mathbf x \Delta \mathbf x', \Delta \mathbf x'')) \exp (i \alpha_2 (\mathbf x; \Delta \mathbf x, \Delta \mathbf x')) g (\Delta \mathbf x'') g (\Delta \mathbf x') g(\Delta \mathbf x) \Psi (\mathbf x)  =  \\
&& \exp (i \alpha_2 (\mathbf x; \Delta \mathbf x, \Delta \mathbf x' \Delta \mathbf x'')) \exp (i \alpha_2 (\mathbf x + \Delta \mathbf x', \Delta \mathbf x', \Delta \mathbf x''))  g (\Delta \mathbf x'') g (\Delta \mathbf x') g(\Delta \mathbf x) \Psi (\mathbf x)
\end{eqnarray}
and so
\[
\alpha_2 (\mathbf x; \Delta \mathbf x \Delta \mathbf x', \Delta \mathbf x'') + \alpha_2 (\mathbf x; \Delta \mathbf x, \Delta \mathbf x')  -
\alpha_2 (\mathbf x; \Delta \mathbf x, \Delta \mathbf x' \Delta \mathbf x'') - \alpha_2 (\mathbf x + \Delta \mathbf x', \Delta \mathbf x', \Delta \mathbf x'')
 = 0 \,(\operatorname{mod} 2 \pi \mathbb Z)
\]
or 
\[
\delta \alpha_2 = 0 \,(\operatorname{mod} 2 \pi \mathbb Z)
\]

Some remarks:
The $2$-cocyle captures the essence of (conventional) quantum mechanics. The $2$-cocycle condition is actually the [[quatisation condition|Quantization]] of quantum mechanics. In other words, quantisation of a classical system is equal to introducing a commutator anomaly. This suggests a straightforward generalisation of quantum mechanics by taking into consideration non-trivial higher order cocycles. 
As the $2$-cocycle condition is equal to an associativity condition (as seen above), conventional quantum mechanics relies on the fact that physics is associative. That is to say, algebraically it can be based on (non-Abelian) groups. 
One either sticks to this assumption, i.e. one must