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These [[InterfaceOptions]] for customising [[TiddlyWiki]] are saved in your browser

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<<tiddler [[include_tiddlers/(4,2) - Association Type Identities.html#"(4,2) - Association Type Identities"]]>>
<<tiddler [[include_tiddlers/(4,3) - Association Type Identities.html#"(4,3) - Association Type Identities"]]>>
<<tiddler [[include_tiddlers/(Co)homology.html#"(Co)homology"]]>>
A ''*-Algebra'' is an algebra equipped with an [[involution|Involution]] ''*''.

Links:
* [[WIKIPEDIA - *-Algebra|http://en.wikipedia.org/wiki/*-algebra]]
<<tiddler [[include_tiddlers/15-Sphere.html#"15-Sphere"]]>>
<<tiddler [[include_tiddlers/16-Vector.html#"16-Vector"]]>>
<<tiddler [[include_tiddlers/2-(15,7,3) Design.html#"2-(15,7,3) Design"]]>>
<<tiddler [[include_tiddlers/2-(31,15,7) Design.html#"2-(31,15,7) Design"]]>>
<<tiddler [[include_tiddlers/2-Qubit State.html#"2-Qubit State"]]>>
<<tiddler [[include_tiddlers/2-d Gravity.html#"2-d Gravity"]]>>
<<tiddler [[include_tiddlers/23andMe.html#"23andMe"]]>>
<<tiddler [[include_tiddlers/24-cell.html#"24-cell"]]>>
<<tiddler [[include_tiddlers/3-Curvature.html#"3-Curvature"]]>>
<<tiddler [[include_tiddlers/3-Manifold.html#"3-Manifold"]]>>
<<tiddler [[include_tiddlers/3-Qubit State.html#"3-Qubit State"]]>>
<<tiddler [[include_tiddlers/3-Web.html#"3-Web"]]>>
<<tiddler [[include_tiddlers/3-d Gravity.html#"3-d Gravity"]]>>
<<tiddler [[include_tiddlers/4-Cube.html#"4-Cube"]]>>
<<tiddler [[include_tiddlers/4-Curvature.html#"4-Curvature"]]>>
<<tiddler [[include_tiddlers/4-Manifold.html#"4-Manifold"]]>>
<<tiddler [[include_tiddlers/4-Qubit State.html#"4-Qubit State"]]>>
<<tiddler [[include_tiddlers/4-Web.html#"4-Web"]]>>
<<tiddler [[include_tiddlers/4-fermion Interactions.html#"4-fermion Interactions"]]>>
<<tiddler [[include_tiddlers/5-Cube.html#"5-Cube"]]>>
<<tiddler [[include_tiddlers/5-dimensional Spacetime.html#"5-dimensional Spacetime"]]>>
<<tiddler [[include_tiddlers/7-Sphere.html#"7-Sphere"]]>>
<<tiddler [[include_tiddlers/8-Form.html#"8-Form"]]>>
<<tiddler [[include_tiddlers/A 'Brief History of Time' Revisited.html#"A 'Brief History of Time' Revisited"]]>>
<<tiddler [[include_tiddlers/A Moment of Full Awareness of Death.html#"A Moment of Full Awareness of Death"]]>>
<<tiddler [[include_tiddlers/A-Theorem.html#"A-Theorem"]]>> 
<<tiddler [[include_tiddlers/AGER1.html#"AGER1"]]>>
''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]]
<<tiddler [[include_tiddlers/AJAX.html#"AJAX"]]>>
<<tiddler [[include_tiddlers/AMPK.html#"AMPK"]]>>
<<tiddler [[include_tiddlers/ASCOM.html#"ASCOM"]]>>
<<tiddler [[include_tiddlers/Aberrant Protein.html#"Aberrant Protein"]]>>
<<tiddler [[include_tiddlers/About Spacetime and the Moon.html#"About Spacetime and the Moon"]]>>
<<tiddler [[include_tiddlers/About the Scientific Method.html#"About the Scientific Method"]]>>
<<tiddler [[include_tiddlers/Absolute Horizon.html#"Absolute Horizon"]]>>
<<tiddler [[include_tiddlers/Absolute Treshold.html#"Absolute Treshold"]]>>
<<tiddler [[include_tiddlers/Abstract Algebra.html#"Abstract Algebra"]]>>
<<tiddler [[include_tiddlers/Acoustic Black Hole.html#"Acoustic Black Hole"]]>>
<<tiddler [[include_tiddlers/Action.html#"Action"]]>>
<<tiddler [[include_tiddlers/Activated Charcoal.html#"Activated Charcoal"]]>>
An ''Active Transformation'' transforms the basis elements of an algebra. See also [[passive transformations|Passive Transformation]].
> Polchinski admits that the condensed-matter sceptics have a point. "I don't think that string theorists have yet come up with anything that condensed-matter theorists don't already know," he says. The quantitative results tend to be re-derivations of answers that condensed-matter theorists had already calculated using more mundane methods.
> - [1] -

> Unfortunately, so far that promise of a truly background independent formulation of string theory has not been achieved. For several reasons this cannot be the \AdS/CFT correspondence. One reason is that the cosmological constant has the wrong sign, another is that a cosmological theory cannot have boundaries or asymptotic regions ...
> - Lee Smolin -


The ''\AdS/CFT Correspondence'' is a realization of the [[holographic principle|Holographic Principle]].

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

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


Papers:
* [[Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories (1998) - E. Witten|http://arxiv.org/abs/hep-th/9803131]] [[local|papers/9803131.pdf]] {{t1000Cite{[[pct. 2297|http://scholar.google.de/scholar?cites=6457584981350275502&as_sdt=2005&sciodt=0,5&hl=en]]}}}
* [[Josephson Junctions and AdS/CFT Networks (2011) - E. Kiritsis, V. Niarchos|http://arxiv.org/abs/1105.6100]] [[local|papers/1105.6100.pdf]] [[pct. 14|http://scholar.google.de/scholar?cites=10143861866020688593&as_sdt=2005&sciodt=0,5&hl=en]]

Magazines:
* [[A Black Hole Full of Answers (2007) - J. Zaanen|http://hep.physics.uoc.gr/ads-cond-mat/zaanen-nature.pdf]] [[local|magazines/zaanen-nature.pdf]] [[mct. 15|http://scholar.google.de/scholar?cites=9573698935225144409&as_sdt=2005&sciodt=0,5&hl=en]]
* [[Strange and Stringy (2013) - S. Sachdev|http://qpt.physics.harvard.edu/c63.pdf]] [[local|magazines/c63.pdf]] [[mct. 2|http://scholar.google.de/scholar?cites=14538583843972156913&as_sdt=2005&sciodt=0,5&hl=en]]
* [[[1] String Theory Finds a Bench Mate (2011) - Z. Merali|http://www.nature.com/news/2011/111019/pdf/478302a.pdf]] [[local|magazines/478302a.pdf]] [[mct. 1|http://scholar.google.de/scholar?cites=17665837402351397811&as_sdt=2005&sciodt=0,5&hl=en]]

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

Videos:
* [[KITP Program: Holographic Duality and Condensed Matter Physics 2011|http://online.kitp.ucsb.edu/online/adscmt11/]]
''Adams' Theorem'' states:
If there exists a Hopf map $f: S^n \rightarrow S^{(n + 1)/2}$ with integer valued Hopf invariant $\gamma (f)$, then $n$ must equal to $1$, $3$, $7$ or $15$.
<<tiddler [[include_tiddlers/Adenosine Triphosphate.html#"Adenosine Triphosphate"]]>>
<<tiddler [[include_tiddlers/Adiabatic Quantum Computation.html#"Adiabatic Quantum Computation"]]>>
<<tiddler [[include_tiddlers/Adjoint.html#"Adjoint"]]>>
<<tiddler [[include_tiddlers/Adjoint Properties.html#"Adjoint Properties"]]>>
<<tiddler [[include_tiddlers/Adjoint Representation.html#"Adjoint Representation"]]>>
<<tiddler [[include_tiddlers/Advanced Lipoxidation End-product.html#"Advanced Lipoxidation End-product"]]>>
<<tiddler [[include_tiddlers/Affine General Linear Group.html#"Affine General Linear Group"]]>>
<<tiddler [[include_tiddlers/Affinely Connected Space.html#"Affinely Connected Space"]]>>
<<tiddler [[include_tiddlers/Afterlife.html#"Afterlife"]]>>
<<tiddler [[include_tiddlers/Akbulut Cork.html#"Akbulut Cork"]]>>
<<tiddler [[include_tiddlers/Akivis Algebra.html#"Akivis Algebra"]]>>
<<tiddler [[include_tiddlers/Akivis Element.html#"Akivis Element"]]>>
<<tiddler [[include_tiddlers/Alain Connes.html#"Alain Connes"]]>>
''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]]
<<tiddler [[include_tiddlers/Albert Algebra.html#"Albert Algebra"]]>>
<<tiddler [[include_tiddlers/Albert Einstein.html#"Albert Einstein"]]>>
<<tiddler [[include_tiddlers/Alcubierre Metric.html#"Alcubierre Metric"]]>>
<<tiddler [[include_tiddlers/Alexander Grothendieck.html#"Alexander Grothendieck"]]>>
<<tiddler [[include_tiddlers/Alexander Horned Sphere.html#"Alexander Horned Sphere"]]>>
<<tiddler [[include_tiddlers/Alexander Vilenkin.html#"Alexander Vilenkin"]]>>
<<tiddler [[include_tiddlers/Alexander's Trick.html#"Alexander's Trick"]]>>
<<tiddler [[include_tiddlers/Algebra.html#"Algebra"]]>>
<<tiddler [[include_tiddlers/Algebraic Quantum Field Theory.html#"Algebraic Quantum Field Theory"]]>>
<<tiddler [[include_tiddlers/Algebraic Reachability and Reality.html#"Algebraic Reachability and Reality"]]>>
<<tiddler [[include_tiddlers/Algebraic Structure.html#"Algebraic Structure"]]>>
<<tiddler [[include_tiddlers/Algebraic Topology.html#"Algebraic Topology"]]>>
<<tiddler [[include_tiddlers/Algebraic Variety.html#"Algebraic Variety"]]>>
<<tiddler [[include_tiddlers/Algebras and Signature.html#"Algebras and Signature"]]>>
Minimum encoding length principles, rooted in (algorithmic) information theory, quantify Ockhamââ¬â¢s razor principle, and lead to a solid pragmatic foundation of inductive reasoning. Essentially, one can show that the more one can compress, the better one can predict, and vice versa.

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

Links:
* [[WIKIPEDIA - Algorithmic Probability|http://en.wikipedia.org/wiki/Algorithmic_probability]]
* [[WIKIPEDIA - Ray Solomonoff|http://en.wikipedia.org/wiki/Ray_Solomonoff]]
* [[SCHOLARPEDIA - Algorithmic probability|http://www.scholarpedia.org/article/Algorithmic_probability]]
* [[WIKIPEDIA - Inductive Inference|http://en.wikipedia.org/wiki/Inductive_inference]]
* [[WIKIPEDIA - Prior Probability|http://en.wikipedia.org/wiki/Diffuse_prior#Uninformative_priors]]
@@display:block;text-align:right;[img[My comments ...|images/Solomonoff.jpg][Comments]]&nbsp;@@
/***
name: AllTagsExceptPlugin
author: Garrett
version: 0.1.0
This is a revision of Clint Checketts' allTagsExcept plugin, which lists all tags except those listed.

<<option chkDisableExcept>> show hidden system tags

!!Usage
{{{
<<AllTagsExcept tag1 tag2 ...>>
}}}
!!!Code
***/
/*{{{*/
version.extensions.AllTagsExcept = {major: 0, minor: 1, revision: 0};

if (!config.options.chkDisableExcept) config.options.chkDisableExcept=false; // default to standard action

config.macros.AllTagsExcept = {tooltip: "Show notes tagged with '%0'",noTags: "There are no tags to display"};

config.macros.AllTagsExcept.handler = function(place,macroName,params)
{
	var tags = store.getTags();
	var theDateList = createTiddlyElement(place,"ul");
	if(tags.length == 0)
		createTiddlyElement(theDateList,"li",null,"listTitle",this.noTags);
	for(var t=0; t<tags.length; t++)
		{
		var includeTag = true;
            	for (var p=0;p<params.length; p++) if ((tags[t][0] == params[p])&&(!config.options.chkDisableExcept)) includeTag = false;
            	if (includeTag)
			{
			var theListItem =createTiddlyElement(theDateList,"li");
			var theTag = createTiddlyButton(theListItem,tags[t][0] + " (" + tags[t][1] + ")",this.tooltip.format([tags[t][0]]),onClickTag);
			theTag.setAttribute("tag",tags[t][0]);
			}
		}
}
/*}}}*/
<<tiddler [[include_tiddlers/Allometric Scaling Law.html#"Allometric Scaling Law"]]>>
<<tiddler [[include_tiddlers/Alternating Group.html#"Alternating Group"]]>>
<<tiddler [[include_tiddlers/Alternative Algebra.html#"Alternative Algebra"]]>>
<<tiddler [[include_tiddlers/Alternative Quantum Mechanics.html#"Alternative Quantum Mechanics"]]>>
<<tiddler [[include_tiddlers/Amitsur-Levitzki Theorem.html#"Amitsur-Levitzki Theorem"]]>>
<<tiddler [[include_tiddlers/Analogue Gravity.html#"Analogue Gravity"]]>>
<<tiddler [[include_tiddlers/Analytic Function.html#"Analytic Function"]]>>
<<tiddler [[include_tiddlers/Analytic Loop.html#"Analytic Loop"]]>>
<<tiddler [[include_tiddlers/Analytic Manifold.html#"Analytic Manifold"]]>>
<<tiddler [[include_tiddlers/Anatomical Terms of Location and Direction.html#"Anatomical Terms of Location and Direction"]]>>
<<tiddler [[include_tiddlers/Andrei Linde.html#"Andrei Linde"]]>>
<<tiddler [[include_tiddlers/Andromeda-Milky Way Collision.html#"Andromeda-Milky Way Collision"]]>>
<<tiddler [[include_tiddlers/Angular Momentum.html#"Angular Momentum"]]>>
<<tiddler [[include_tiddlers/AngularJS.html#"AngularJS"]]>>
<<tiddler [[include_tiddlers/Anharmonic Oscillator.html#"Anharmonic Oscillator"]]>>
<<tiddler [[include_tiddlers/Anholonomy.html#"Anholonomy"]]>>
<<tiddler [[include_tiddlers/Animal Consciousness.html#"Animal Consciousness"]]>>
<<tiddler [[include_tiddlers/Anomalous Magnetic Dipole Moment.html#"Anomalous Magnetic Dipole Moment"]]>>
<<tiddler [[include_tiddlers/Anomaly.html#"Anomaly"]]>>
<<tiddler [[include_tiddlers/Anthocyanidins.html#"Anthocyanidins"]]>>
<<tiddler [[include_tiddlers/Anthocyanins.html#"Anthocyanins"]]>>
<<tiddler [[include_tiddlers/Anthropic Principle.html#"Anthropic Principle"]]>>
<<tiddler [[include_tiddlers/Anti-Commutator.html#"Anti-Commutator"]]>>
<<tiddler [[include_tiddlers/Anti-De Sitter Space.html#"Anti-De Sitter Space"]]>>
<<tiddler [[include_tiddlers/Anti-Hermiticity.html#"Anti-Hermiticity"]]>>
<<tiddler [[include_tiddlers/Anti-Unitarity.html#"Anti-Unitarity"]]>>
<<tiddler [[include_tiddlers/Anti-aging Compound.html#"Anti-aging Compound"]]>>
<<tiddler [[include_tiddlers/Antibracket Formalism.html#"Antibracket Formalism"]]>>
<<tiddler [[include_tiddlers/Antoine's Necklace.html#"Antoine's Necklace"]]>>
<<tiddler [[include_tiddlers/Anyon.html#"Anyon"]]>>
<<tiddler [[include_tiddlers/Apache Maven.html#"Apache Maven"]]>>
Links:
* [[WIKIPEDIA - Apophis|http://en.wikipedia.org/wiki/99942_Apophis]]
<<tiddler [[include_tiddlers/Apoptosis.html#"Apoptosis"]]>>
<<tiddler [[include_tiddlers/Apparent Horizon.html#"Apparent Horizon"]]>>
<<tiddler [[include_tiddlers/Appelquist-Carazzone Decoupling Theorem.html#"Appelquist-Carazzone Decoupling Theorem"]]>>
Common lore is that [[Planck Units|Planck Units]] are fundamental in the sense that the expectations are that around these values new physics "kicks in".
For instance it is expected that around the [[Planck length|Planck Length]] [[general relativity|General Relativity]] and/or [[quantum mechanics|Quantum Mechanics]] breaks down and some more fundamental theory ([[quantum gravity|Quantum Gravity]]) takes over.

But there is a catch to this argument - the [[Planck mass|Planck Mass]], which is about $10^{-8} \rm kg$. Nothing unusual seems to happen for objects of this mass. Rather, it's way smaller masses like the quark masses or the electron mass that appear to be fundamental.
Maybe we have overlooked something, but if Planck unit-related effects are generally subtle and difficult to observe, what are the chances that we will ever see such an effect at the Planck scale, the Planck temperature, etc. ?

<html><center><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_60.html" width=51% height=86></iframe></center></html>
Contrary to Planck units, [[large numbers|Large Numbers Hypothesis]] seem to be regarded as more esoteric, maybe just an accident. But aren't they in the same manner the result of some ad hoc algebraic manipulation ? Yet, contrary to Planck units, these are at least within experimental reach and therefore may deserve more attention.

There is no doubt that the Planck units are universal numbers that one can form by combining (supposedly) fundamental constants of nature, but does this really imply that they are fundamental ? I have begun daring to doubt.

© 2014 Markus Maute
Papers:
* [[Radial Motion into an Einstein-Rosen Bridge - N. J. Pop?awski|http://www.physics.indiana.edu/~nipoplaw/PLB_687_110.pdf]] [[local|papers/PLB_687_110.pdf]] pct. 0

Links:
* [[NATIONAL GEOGRAPHIC: Every Black Hole Contains Another Universe? - K. Than|http://news.nationalgeographic.com/news/2010/04/100409-black-holes-alternate-universe-multiverse-einstein-wormholes/]]
* [[Is the Big Bang a Black Hole? - P. Gibbs|http://www.desy.de/user/projects/Physics/Relativity/BlackHoles/universe.html]]
* [[The Universe is Not a Black Hole - S. Carroll|http://blogs.discovermagazine.com/cosmicvariance/2010/04/28/the-universe-is-not-a-black-hole/]]
@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/Wheeler.jpg][Comments]]}}}&nbsp;@@
<<tiddler [[include_tiddlers/Area Derivative.html#"Area Derivative"]]>>
<<tiddler [[include_tiddlers/Arrow of Time.html#"Arrow of Time"]]>>
<<tiddler [[include_tiddlers/Arthur Stanley Eddington.html#"Arthur Stanley Eddington"]]>>
<<tiddler [[include_tiddlers/Artificial Intelligence.html#"Artificial Intelligence"]]>>
<<tiddler [[include_tiddlers/Artificial Life.html#"Artificial Life"]]>>
<<tiddler [[include_tiddlers/Ashtekar Formalism.html#"Ashtekar Formalism"]]>>
An ''Associahedron $\mathcal K(n)$'' is an $(n?2)$-dimensional solid [[polytope|Polytope]] (or polyhedron). There is exactly one associahedron of each dimension.

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

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

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

Links:
* [[Strange Associations|http://www.ams.org/featurecolumn/archive/associahedra.html]]
<<tiddler [[include_tiddlers/Association Type.html#"Association Type"]]>>
<<tiddler [[include_tiddlers/Association Type Expansions.html#"Association Type Expansions"]]>>
<<tiddler [[include_tiddlers/Association Type Identities.html#"Association Type Identities"]]>>
The ''Associator'' is defined as:
\[
[\mb A,\mb B, \mb C] \equiv (\mb{AB})\mb C - \mb A(\mb{BC}) \equiv \mb{AB} \cdot \mb C -\mb A \cdot \mb{BC}
\]
The latter notation is found frequently in literature.

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

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

!!!!Properties
1. ''Linearity''
\[
[\sum_i \lambda_i \mb A_i,\sum_j  \mu_j \mb B_j,\sum_k  \nu_k \mb C_k ] = \sum_{i,j,k} \lambda_i  \mu_j  \nu_k \mb [ \mb A_i,\mb B_j, \mb C_k]
\]
@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/Albert.jpg][Comments]]}}}@@
<<tiddler [[include_tiddlers/Associator Expansion.html#"Associator Expansion"]]>>
<<tiddler [[include_tiddlers/Associator Quantization.html#"Associator Quantization"]]>>
<<tiddler [[include_tiddlers/Astronomical Objects.html#"Astronomical Objects"]]>>
<<tiddler [[include_tiddlers/Astronomy Software.html#"Astronomy Software"]]>>
<<tiddler [[include_tiddlers/Astrophotography.html#"Astrophotography"]]>>
<<tiddler [[include_tiddlers/Asymptotic Conditions.html#"Asymptotic Conditions"]]>>
<<tiddler [[include_tiddlers/Asymptotic Expansion.html#"Asymptotic Expansion"]]>>
<<tiddler [[include_tiddlers/Asymptotic Freedom.html#"Asymptotic Freedom"]]>>
<<tiddler [[include_tiddlers/Asymptotic Safety.html#"Asymptotic Safety"]]>>
<<tiddler [[include_tiddlers/Asymptotically Flat Spacetime.html#"Asymptotically Flat Spacetime"]]>>
<<tiddler [[include_tiddlers/Asymptotically Safe Gravity.html#"Asymptotically Safe Gravity"]]>>
<<tiddler [[include_tiddlers/Atheism.html#"Atheism"]]>>
<<tiddler [[include_tiddlers/Atiyah-Singer Index Theorem.html#"Atiyah-Singer Index Theorem"]]>>
<<tiddler [[include_tiddlers/Atom Laser.html#"Atom Laser"]]>>
<<tiddler [[include_tiddlers/Autocorrelation.html#"Autocorrelation"]]>>
<<tiddler [[include_tiddlers/Automated Reasoning.html#"Automated Reasoning"]]>>
<<tiddler [[include_tiddlers/Automorphism.html#"Automorphism"]]>>
<<tiddler [[include_tiddlers/Autoparallelity.html#"Autoparallelity"]]>>
<<tiddler [[include_tiddlers/Autophagy.html#"Autophagy"]]>>
<<tiddler [[include_tiddlers/Autotopy.html#"Autotopy"]]>>
<<tiddler [[include_tiddlers/Axial Age.html#"Axial Age"]]>>
<<tiddler [[include_tiddlers/Axial Torsion.html#"Axial Torsion"]]>>
<<tiddler [[include_tiddlers/Axiom Computer Algebra System.html#"Axiom Computer Algebra System"]]>>
<<tiddler [[include_tiddlers/Axion.html#"Axion"]]>>
<<tiddler [[include_tiddlers/Axion-like Particle.html#"Axion-like Particle"]]>>
<<tiddler [[include_tiddlers/B-L.html#"B-L"]]>>
''BCH Code'' = ''Bose\-Chaudhuri\-Hocquenghem Code'' belong to a large class of cyclic [[blockcodes|Blockcode]].

Lectures:
* [[Notes on Coding Theory, The Definition of BCH and RS Codes - J. Beachy|http://www.math.niu.edu/~beachy/courses/523/08coding.pdf]]
<<tiddler [[include_tiddlers/BICEP2.html#"BICEP2"]]>>
<<tiddler [[include_tiddlers/BKL Conjecture.html#"BKL Conjecture"]]>>
<<tiddler [[include_tiddlers/BPS State.html#"BPS State"]]>>
<<tiddler [[include_tiddlers/BRST Quantization.html#"BRST Quantization"]]>>
<<tiddler [[include_tiddlers/BTZ Black Hole.html#"BTZ Black Hole"]]>>
<<tiddler [[include_tiddlers/Baby Monster Group.html#"Baby Monster Group"]]>>
<<tiddler [[include_tiddlers/Bach Brackets.html#"Bach Brackets"]]>>
<<tiddler [[include_tiddlers/Background Independence.html#"Background Independence"]]>>
<<tiddler [[include_tiddlers/Baker-Campbell-Hausdorff Formula.html#"Baker-Campbell-Hausdorff Formula"]]>>
<<tiddler [[include_tiddlers/Banach Algebra.html#"Banach Algebra"]]>>
<<tiddler [[include_tiddlers/Bargmann-Hall-Wightman Theorem.html#"Bargmann-Hall-Wightman Theorem"]]>>
<<tiddler [[include_tiddlers/Barnes-Wall Lattice.html#"Barnes-Wall Lattice"]]>>
<<tiddler [[include_tiddlers/Barnum Effect.html#"Barnum Effect"]]>>
<<tiddler [[include_tiddlers/Baryogenesis.html#"Baryogenesis"]]>>
<<tiddler [[include_tiddlers/Baryon Asymmetry.html#"Baryon Asymmetry"]]>>
<<tiddler [[include_tiddlers/Baryonic Tully-Fisher Relation.html#"Baryonic Tully-Fisher Relation"]]>>
!Summary
This tiddler contains some basic $\LaTeX$ macros.  Documentation can be found [[here|BasicMacros Documentation]].

!Version
version: 1.0.4

1.0: (2011-04-15) Major release.
1.0.1: (2011-04-16) Bugfix. {{{#1}}} was added to the definition of {{{\closure}}}.
1.0.2: (2011-04-16) Command name change.  The command {{{\blackboard}}} was added (to the supplemental list).  {{{\blackBoard}}} is kept for backwards compatability.
1.0.3: (2011-04-17) Command name added.  The abbreviation {{{\fns}}} for {{{\functions}}} was added (to the supplemental list).
1.0.4: (2011-04-17) Comments changed.  Consistent capitalisation was applied to the section names.

!Code
$
\def\BasicMacros{

% Miscellaneous basics

\def\parentheses#1{{\left( {#1} \right)}}
\def\brackets#1{{\left[ {#1} \right]}}
\def\solidus#1/#2{{\left. {#1} \left/ \vphantom{#1} {#2} \right. \right.}}
\def\of{\parentheses}
\def\operator{\operatorname}

\def\script{\mathcal}
\def\gothic{\mathfrak}
\def\blackBoard{\mathbb}
\def\mb{\mathbf}
\def\bs{\boldsymbol}
\def\cunit{\mathbf i_{\small \mathbb C}}
\def\slash#1{{#1}\!\!\!/}
\def\Chr#1#2{\left\{ {\begin{array}{*{20}c}  #1\\ {#2} \\ \end{array} } \right\}}


% Symbols

\def\epsilon{\varepsilon}
\def\infinity{\infty}
\def\continuum{\gothic{c}}

\def\naturals{\blackBoard{N}}
\def\integers{\blackBoard{Z}}
\def\rationals{\blackBoard{Q}}
\def\reals{\blackBoard{R}}
\def\complexNumbers{\blackBoard{C}}

\def\half{\frac{1}{2}}
\def\third{\frac{1}{3}}
\def\quarter{\frac{1}{4}}


% Sets and set operations

\def\Set#1{{\left\lbrace {#1} \right\rbrace}}
\def\set#1:#2{\Set{{#1} \colon {#2}}}
\def\singleton{\Set}
\def\emptySet{\emptyset}

\def\Sequence#1{{\left\langle {#1} \right\rangle}}
\def\sequence#1:#2{\Sequence{{#1} \colon {#2}}}
\def\tuple{\parentheses}
\def\emptyTuple{\tuple{}}

\def\family#1:#2{{\parentheses{#1}_{#2}}}

\def\subset{\subseteq}
\def\superset{\supseteq}
\def\union{\cup}
\def\Union{\bigcup}
\def\intersect{\cap}
\def\Intersection{\bigcap}
\def\setMinus{\setminus}
\def\cross{\times}
\def\product{\prod}

\def\powerSet#1{{\script{P}\of{#1}}}
\def\functionSpace#1#2{{{#2}^{#1}}}
\def\cardinality#1{{\left\lvert {#1} \right\rvert}}
\def\isEmpty{= \emptySet}
\def\isNonempty{\not= \emptySet}


% Functions and function operations

\def\function#1:#2->#3{{{#1} \colon {#2} \to {#3}}}
\def\mapsTo{\mapsto}
\def\map#1->#2{\parentheses{{#1} \mapsTo {#2}}}

\def\compose{\circ}
\def\inverse#1{{{#1}^{-1}}}
\def\restriction#1#2{{{\left. {#1} \right\lvert}_{#2}}}

\def\image#1#2{{{#2}\of{#1}}}
\def\preimage#1#2{{{#2}^{-1}\of{#1}}}
\def\fiber#1#2{\preimage{\singleton{#1}}{#2}}

\def\identity#1{{\operator{id}_{#1}}}


% Topology related notation

\def\homeomorphic{\cong}
\def\convergesTo{\to}

\def\closure#1{\overline{#1}}
\def\closureIn#1#2{{\closure{#2}^{#1}}}
\def\interior#1{{\operator{int}\of{#1}}}
\def\interiorIn#1#2{{\operator{int}_{#1}\of{#2}}}

\def\collection{\script}
\def\collections{\gothic}
\def\topology{\collection}
\def\cover{\collection}
\def\filter{\collection}
\def\ideal{\collection}

\def\interval#1#2,#3#4{{\left#1 {#2}, {#3} \right#4}}
\def\unitInterval{\interval[0, 1]}
\def\plane{{\functionSpace{2}{\reals}}}
\def\stoneCechCompactification#1{{\beta{#1}}}


% General notation

\def\modulus#1{{\left\lvert {#1} \right\rvert}}
\def\norm#1{{\left\Vert {#1} \right\Vert}}
\def\equivalent{\sim}
\def\quotient{\solidus}


% Aliases

\def\functions{\function}

}

\def\BasicMacroAbbreviations{

\def\p{\parentheses}
\def\b{\brackets}

\def\N{\naturals}
\def\Z{\integers}
\def\Q{\rationals}
\def\R{\reals}
\def\C{\complexNumbers}

\def\sng{\singleton}
\def\Seq{\Sequence}
\def\seq{\sequence}
\def\pSet{\powerSet}
\def\fnSpace{\functionSpace}
\def\card{\cardinality}

\def\fn{\function}
\def\id{\identity}

\def\homeo{\homeomorphic}
\def\sCC{\stoneCechCompactification}

}

% The following is a supplement to \BasicMacros  and \BasicMacroAbbreviations.  This macroset can be thought of as the list of changes that will be made for the next version.
\def\BasicMacrosSupplement
{

% Because 'blackboard' is one word the second 'b' should not be capitalised in the command name.
\def\blackboard{\blackBoard}

% Given the commands \function, \fn, and \functions, it is only natural that we have a command \fns.
\def\fns{\functions}

}
$
<<tiddler [[include_tiddlers/Beam Splitter.html#"Beam Splitter"]]>>
<<tiddler [[include_tiddlers/Bekenstein-Hawking Entropy.html#"Bekenstein-Hawking Entropy"]]>>
Links:
* [[Bel and Bel-Robinson Tensors|http://www.phy.olemiss.edu/~luca/Topics/b/bel.html]] [[local|html/bel.html]]
[[General Relativity|General Relativity]] describes macroscopic (spinless) matter. Therefore a symmetric momentum current appears in the [[Einstein equation|Einstein Field Equations]]. In order to deal with spinor matter, which in general has an asymmetric canonical momentum tensor, one has to execute a symmetrisation procedure. This procedure has to transform the canonical momentum and spin currents into new ones, which also fulfil the conservation laws of momentum and total angular momentum. Furthermore the new momentum current has to be symmetric.
 In general there are many possibilities for such a transformation. In GR, however, this transformation is further restricted by the fact, that there is no equation for the spin current. Therefore the new spin current must vanish. The ''Procedure'' which accomplishes this, is the one of ''Belinfante and Rosenfeld''. No such complicated operation is needed in the [[Einstein-Cartan theory|Einstein-Cartan Theory]]. From this point of view the EC theory is the natural extension of GR into microphysics. 
<<tiddler [[include_tiddlers/Bell Inequality.html#"Bell Inequality"]]>>
<<tiddler [[include_tiddlers/Bell's Theorem.html#"Bell's Theorem"]]>>
<<tiddler [[include_tiddlers/Belousov-Zhabotinsky Reaction.html#"Belousov-Zhabotinsky Reaction"]]>>
<<tiddler [[include_tiddlers/Berberine.html#"Berberine"]]>>
<<tiddler [[include_tiddlers/Berezin Calculus.html#"Berezin Calculus"]]>>
<<tiddler [[include_tiddlers/Berry Phase.html#"Berry Phase"]]>>
Links:
* [[WIKIPEDIA - Bertrand Russell|http://de.wikipedia.org/wiki/Bertrand_Russell]]

Videos:
* [[Bertrand Russell - To our Descendants|http://www.youtube.com/watch?v=g3jnEqXhDNI&feature=related]]
* [[Bertrand Russell on God (1959)|http://www.youtube.com/watch?v=2aPOMUTr1qw&feature=related]]
* [[Bertrand Russell on Clarity and Exact Thinking|http://www.youtube.com/watch?v=mpJcn0Otk7I&feature=related]]
<<tiddler [[include_tiddlers/Bertrand's Theorem.html#"Bertrand's Theorem"]]>>
Links:
* [[WIKIPEDIA - Bessel Function|http://en.wikipedia.org/wiki/Bessel_function]]
<<tiddler [[include_tiddlers/Betti Number.html#"Betti Number"]]>>
A ''Bialgebra'' over a [[field|Field]] $\mathbb K$ is a vector space over $\mathbb K$ which is both a unital associative algebra and a [[coalgebra|Coalgebra]], such that the algebraic- and coalgebraic structure satisfy certain compatiblility relations.


Links:
* [[WIKIPEDIA - Bialgebra|http://en.wikipedia.org/wiki/Bialgebra]]
<<tiddler [[include_tiddlers/Bianchi Classification.html#"Bianchi Classification"]]>>
<<tiddler [[include_tiddlers/Big Bang.html#"Big Bang"]]>>
<<tiddler [[include_tiddlers/Big Bang Nucleosynthesis.html#"Big Bang Nucleosynthesis"]]>>
<<tiddler [[include_tiddlers/Big Bounce.html#"Big Bounce"]]>>
<<tiddler [[include_tiddlers/Big Desert.html#"Big Desert"]]>>
* Is there a principle - supposedly mathematical in nature - which allows one to derive all of physics and if so, what is it ?
* What is [[consciousness|Consciousness]] ?
* Can we be [[immortal|Immortality]] ? If we are not by nature, can we engineer immortality or at least (considerably) increase human lifespan ?
<<tiddler [[include_tiddlers/Bilinear Covariants.html#"Bilinear Covariants"]]>>
A ''Binary Code'' of length $n$ and dimension $k$ is a $k$?dimensional vector subspace of $\mathbb F^n_2$. The ''(Hamming) Weight'' of a vector of  $\mathbb F^n_2$ is the number of non-zero coordinates it contains. 
<<tiddler [[include_tiddlers/Binary Icosahedral Group.html#"Binary Icosahedral Group"]]>>
<<tiddler [[include_tiddlers/Binary Octahedral Group.html#"Binary Octahedral Group"]]>>
<<tiddler [[include_tiddlers/Binary Polyhedral Group.html#"Binary Polyhedral Group"]]>>
<<tiddler [[include_tiddlers/Binary Tetrahedral Group.html#"Binary Tetrahedral Group"]]>>
<<tiddler [[include_tiddlers/Binding Problem.html#"Binding Problem"]]>>
<<tiddler [[include_tiddlers/Binomial Coefficient.html#"Binomial Coefficient"]]>>
<<tiddler [[include_tiddlers/Bioctonion.html#"Bioctonion"]]>>
<<tiddler [[include_tiddlers/Bioctonionic Projective Plane.html#"Bioctonionic Projective Plane"]]>>
<<tiddler [[include_tiddlers/Biological Aging.html#"Biological Aging"]]>>
<<tiddler [[include_tiddlers/Biological Cell.html#"Biological Cell"]]>>
<<tiddler [[include_tiddlers/Biological Cells from Fundamental Physics.html#"Biological Cells from Fundamental Physics"]]>>
<<tiddler [[include_tiddlers/Biological Complementarity.html#"Biological Complementarity"]]>>
<<tiddler [[include_tiddlers/Biological Evolution.html#"Biological Evolution"]]>>
<<tiddler [[include_tiddlers/Biological Immortality.html#"Biological Immortality"]]>>
<<tiddler [[include_tiddlers/Biological Life.html#"Biological Life"]]>>
<<tiddler [[include_tiddlers/Biological Life and Quantum Gravity.html#"Biological Life and Quantum Gravity"]]>>
<<tiddler [[include_tiddlers/Biological Network.html#"Biological Network"]]>>
<<tiddler [[include_tiddlers/Biological Thermodynamics.html#"Biological Thermodynamics"]]>>
<<tiddler [[include_tiddlers/Biology and Criticality.html#"Biology and Criticality"]]>>
<<tiddler [[include_tiddlers/Biomarker of Aging.html#"Biomarker of Aging"]]>>
<<tiddler [[include_tiddlers/Biophoton.html#"Biophoton"]]>>
<<tiddler [[include_tiddlers/Biquaternion.html#"Biquaternion"]]>>
''Birefringence'' or ''Double Refraction'' is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through an anisotropic material.
For a medium having no birefringence means that it has a single lightcone. 
<<tiddler [[include_tiddlers/Birkhoff's Theorem.html#"Birkhoff's Theorem"]]>>
<<tiddler [[include_tiddlers/Bisedenion.html#"Bisedenion"]]>>
<<tiddler [[include_tiddlers/Bisognano-Wichmann Theorem.html#"Bisognano-Wichmann Theorem"]]>>
<<tiddler [[include_tiddlers/Bit String Physics.html#"Bit String Physics"]]>>
<<tiddler [[include_tiddlers/Black Body Radiation.html#"Black Body Radiation"]]>>
<<tiddler [[include_tiddlers/Black Hole.html#"Black Hole"]]>>
<<tiddler [[include_tiddlers/Black Hole Complementarity.html#"Black Hole Complementarity"]]>>
<<tiddler [[include_tiddlers/Black Hole Condensate.html#"Black Hole Condensate"]]>>
<<tiddler [[include_tiddlers/Black Hole Entanglement Entropy.html#"Black Hole Entanglement Entropy"]]>>
<<tiddler [[include_tiddlers/Black Hole Evaporation.html#"Black Hole Evaporation"]]>>
<<tiddler [[include_tiddlers/Black Hole Hair.html#"Black Hole Hair"]]>>
<<tiddler [[include_tiddlers/Black Hole Remnant.html#"Black Hole Remnant"]]>>
<<tiddler [[include_tiddlers/Black Hole Thermodynamics.html#"Black Hole Thermodynamics"]]>>
<<tiddler [[include_tiddlers/Black Holes at the LHC.html#"Black Holes at the LHC"]]>>
<<tiddler [[include_tiddlers/Blaschke Conjecture.html#"Blaschke Conjecture"]]>>
<<tiddler [[include_tiddlers/Block Universe.html#"Block Universe"]]>>
<<tiddler [[include_tiddlers/Blockcode.html#"Blockcode"]]>>
<<tiddler [[include_tiddlers/Body Mass Index.html#"Body Mass Index"]]>>
<<tiddler [[include_tiddlers/Bogoliubov Transformation.html#"Bogoliubov Transformation"]]>>
<<tiddler [[include_tiddlers/Bol Algebra.html#"Bol Algebra"]]>>
<<tiddler [[include_tiddlers/Bol Loop.html#"Bol Loop"]]>>
<<tiddler [[include_tiddlers/Boltzmann Brain Paradox.html#"Boltzmann Brain Paradox"]]>>
<<tiddler [[include_tiddlers/Boltzmann Constant.html#"Boltzmann Constant"]]>>
<<tiddler [[include_tiddlers/Bondi-Metzner-Sachs Group.html#"Bondi-Metzner-Sachs Group"]]>>
<<tiddler [[include_tiddlers/Borde-Vilenkin-Guth Theorem.html#"Borde-Vilenkin-Guth Theorem"]]>>
<<tiddler [[include_tiddlers/Bore Hole Anomaly.html#"Bore Hole Anomaly"]]>>
<<tiddler [[include_tiddlers/Borel Normality.html#"Borel Normality"]]>>
<<tiddler [[include_tiddlers/Born Rule.html#"Born Rule"]]>>
<<tiddler [[include_tiddlers/Bose-Einstein Condensate.html#"Bose-Einstein Condensate"]]>>
<<tiddler [[include_tiddlers/Bose-Einstein Statistics.html#"Bose-Einstein Statistics"]]>>
<<tiddler [[include_tiddlers/Bosenova.html#"Bosenova"]]>>
<<tiddler [[include_tiddlers/Bosonic Quantum Harmonic Oscillator.html#"Bosonic Quantum Harmonic Oscillator"]]>>
<<tiddler [[include_tiddlers/Bosonic Spectral Action.html#"Bosonic Spectral Action"]]>>
<<tiddler [[include_tiddlers/Bosonic String.html#"Bosonic String"]]>>
<<tiddler [[include_tiddlers/Bound State.html#"Bound State"]]>>
<<tiddler [[include_tiddlers/Boundary Condition.html#"Boundary Condition"]]>>
<br><<tiddler [[include_tiddlers/Bra-Ket Notation.html#"Bra-Ket Notation"]]>>
<<tiddler [[include_tiddlers/Braided Monoidal Category.html#"Braided Monoidal Category"]]>>
<<tiddler [[include_tiddlers/Brain.html#"Brain"]]>>
<<tiddler [[include_tiddlers/Brain Aging.html#"Brain Aging"]]>>
<<tiddler [[include_tiddlers/Brain Microtubule.html#"Brain Microtubule"]]>>
<<tiddler [[include_tiddlers/Brain Repair.html#"Brain Repair"]]>>
<<tiddler [[include_tiddlers/Brain-Computer-Interface.html#"Brain-Computer-Interface"]]>>
<<tiddler [[include_tiddlers/Brainstem.html#"Brainstem"]]>>
<<tiddler [[include_tiddlers/Branched-chain Amino Acid.html#"Branched-chain Amino Acid"]]>>
<<tiddler [[include_tiddlers/Brane Scan.html#"Brane Scan"]]>>
<<tiddler [[include_tiddlers/Brane World Scenario.html#"Brane World Scenario"]]>>
<<tiddler [[include_tiddlers/Brans Conjecture.html#"Brans Conjecture"]]>>
<<tiddler [[include_tiddlers/Brans-Dicke Theory.html#"Brans-Dicke Theory"]]>>
/***
|Name|BreadcrumbsPlugin|
|Author|Eric Shulman|
|Source|http://www.TiddlyTools.com/#BreadcrumbsPlugin|
|Documentation|http://www.TiddlyTools.com/#BreadcrumbsPluginInfo|
|Version|2.1.5|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|list/jump to tiddlers viewed during this session plus "back" button/macro|
This plugin provides a list of links to all tiddlers opened during the session, creating a "trail of breadcrumbs" from one tiddler to the next, allowing you to quickly navigate to any previously viewed tiddler, or select 'home' to reset the display to the initial set of tiddlers that were open at the start of the session (i.e., when the document was loaded into the browser).
!!!!!Documentation
<<<
see [[BreadcrumbsPluginInfo]]
<<<
!!!!!Configuration
<<<
<<option chkCreateDefaultBreadcrumbs>> automatically create breadcrumbs display (if needed)
<<option chkShowBreadcrumbs>> show/hide breadcrumbs display
<<option chkReorderBreadcrumbs>> re-order breadcrumbs when visiting a previously viewed tiddler
<<option chkBreadcrumbsHideHomeLink>> omit 'Home' link from breadcrumbs display
<<option chkBreadcrumbsSave>> prompt to save breadcrumbs when 'Home' link is pressed
<<option chkShowStartupBreadcrumbs>> show breadcrumbs for 'startup' tiddlers
<<option chkBreadcrumbsReverse>> show breadcrumbs in reverse order (most recent first)
<<option chkBreadcrumbsLimit>> limit breadcrumbs display to {{twochar{<<option txtBreadcrumbsLimit>>}}} items
<<option chkBreadcrumbsLimitOpenTiddlers>> limit open tiddlers to {{twochar{<<option txtBreadcrumbsLimitOpenTiddlers>>}}} items

<<<
!!!!!Revisions
<<<
2012.06.10 2.1.5 refactored default options to eliminate global variable and use init() handling
| Please see [[BreadcrumbsPluginInfo]] for previous revision details |
2006.02.01 1.0.0 initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.BreadcrumbsPlugin = { major: 2, minor: 1, revision: 5, date: new Date(2012,6,10) };
config.macros.breadcrumbs = {
	crumbs: [], // the list of current breadcrumbs
	askMsg: "Save current breadcrumbs before clearing?\n"
		+"Press OK to save, or CANCEL to continue without saving.",
	saveMsg: 'Enter the name of a tiddler in which to save the current breadcrumbs',
	saveTitle: 'SavedBreadcrumbs',
	options: {
		chkShowBreadcrumbs:		true,
		chkReorderBreadcrumbs:		true,
		chkCreateDefaultBreadcrumbs:	true,
		chkShowStartupBreadcrumbs:	false,
		chkBreadcrumbsReverse:		false,
		chkBreadcrumbsLimit:		true,
		txtBreadcrumbsLimit:		5,
		chkBreadcrumbsLimitOpenTiddlers:false,
		txtBreadcrumbsLimitOpenTiddlers:5,
		chkBreadcrumbsHideHomeLink:	false,
		chkBreadcrumbsSave:		false,
		txtBreadcrumbsHomeSeparator:	'|',
		txtBreadcrumbsCrumbSeparator:	''
	},
	init: function() {
		merge(config.options,this.options,true);
	},
	handler: function(place,macroName,params,wikifier,paramString,tiddler) {
		var area=createTiddlyElement(place,"span",null,"breadCrumbs",null);
		area.setAttribute("homeSep",params[0]||config.options.txtBreadcrumbsHomeSeparator);
		area.setAttribute("crumbSep",params[1]||config.options.txtBreadcrumbsCrumbSeparator);
		this.render(area);
	},
	add: function (title) {
		var thisCrumb = title;
		var ind = this.crumbs.indexOf(thisCrumb);
		if(ind === -1)
			this.crumbs.push(thisCrumb);
		else if (config.options.chkReorderBreadcrumbs)
			this.crumbs.push(this.crumbs.splice(ind,1)[0]); // reorder crumbs
		else
			this.crumbs=this.crumbs.slice(0,ind+1); // trim crumbs
		// if (config.options.chkBreadcrumbsLimitOpenTiddlers)
		//	this.limitOpenTiddlers();
		this.refresh();
		return false;
	},
	getAreas: function() {
		var crumbAreas=[];
		// find all DIVs with classname=="breadCrumbs"
		var all=document.getElementsByTagName("*");
		for (var i=0; i<all.length; i++)
			try{ if (hasClass(all[i],"breadCrumbs")) crumbAreas.push(all[i]); } catch(e) {;}
		// or, find single DIV w/fixed ID (backward compatibility)
		var byID=document.getElementById("breadCrumbs")
		if (byID && !hasClass(byID,"breadCrumbs")) crumbAreas.push(byID);
		if (!crumbAreas.length && config.options.chkCreateDefaultBreadcrumbs) {
			// no crumbs display... create one
			var defaultArea = createTiddlyElement(null,"span",null,"breadCrumbs",null);
		 	defaultArea.style.display= "none";
			var targetArea= document.getElementById("tiddlerDisplay");
		 	targetArea.parentNode.insertBefore(defaultArea,targetArea);
			crumbAreas.push(defaultArea);
		}
		return crumbAreas;
	},
	refresh: function() {
		var crumbAreas=this.getAreas();
		for (var i=0; i<crumbAreas.length; i++) {
			crumbAreas[i].style.display = config.options.chkShowBreadcrumbs?"inline":"none";
			removeChildren(crumbAreas[i]);
			this.render(crumbAreas[i]);
		}
	},
	render: function(here) {
		var co=config.options; var out=""
		if (!co.chkBreadcrumbsHideHomeLink) {
			createTiddlyButton(here,"Home",null,this.home,"tiddlyLink tiddlyLinkExisting");
			out+=here.getAttribute("homeSep")||config.options.txtBreadcrumbsHomeSeparator;
		}
		for (c=0; c<this.crumbs.length; c++) // remove non-existing tiddlers from crumbs
			if (!store.tiddlerExists(this.crumbs[c]) && !store.isShadowTiddler(this.crumbs[c]))
				this.crumbs.splice(c,1);
		var count=this.crumbs.length;
		if (co.chkBreadcrumbsLimit && co.txtBreadcrumbsLimit<count) count=co.txtBreadcrumbsLimit;
		var list=[];
		for (c=this.crumbs.length-count; c<this.crumbs.length; c++) list.push('[['+this.crumbs[c]+']]');
		if (co.chkBreadcrumbsReverse) list.reverse();
		out+=list.join(here.getAttribute("crumbSep")||config.options.txtBreadcrumbsCrumbSeparator);
		wikify(out,here);
	},
	home: function() {
		var cmb=config.macros.breadcrumbs;
		if (config.options.chkBreadcrumbsSave && confirm(cmb.askMsg)) cmb.saveCrumbs();
		story.closeAllTiddlers(); restart();
		cmb.crumbs = []; var crumbAreas=cmb.getAreas();
		for (var i=0; i<crumbAreas.length; i++) crumbAreas[i].style.display = "none";
		return false;
	},
	saveCrumbs: function() {
		var tid=prompt(this.saveMsg,this.saveTitle); if (!tid||!tid.length) return; // cancelled by user
		var t=store.getTiddler(tid);
		if(t && !confirm(config.messages.overwriteWarning.format([tid]))) return;
		var who=config.options.txtUserName;
		var when=new Date();
		var text='[['+this.crumbs.join(']]\n[[')+']]';
		var tags=t?t.tags:[]; tags.pushUnique('story');
		var fields=t?t.fields:{};
		store.saveTiddler(tid,tid,text,who,when,tags,fields);
		story.displayTiddler(null,tid);
		story.refreshTiddler(tid,null,true);
		displayMessage(tid+' has been '+(t?'updated':'created'));
	},
	limitOpenTiddlers: function() {
		var limit=config.options.txtBreadcrumbsLimitOpenTiddlers; if (limit<1) limit=1;
		for (c=this.crumbs.length-1; c>=0; c--) {
			var tid=this.crumbs[c];
			var elem=story.getTiddler(tid);
			if (elem) { // tiddler is displayed
				if (limit <=0) { // display limit has been reached
					if (elem.getAttribute("dirty")=="true") { // tiddler is being edited
						var msg= "'"+tid+"' is currently being edited.\n\n"
							+"Press OK to save and close this tiddler\n"
							+"or press Cancel to leave it opened";
						if (confirm(msg)) {
							story.saveTiddler(tid);
							story.closeTiddler(tid);
						}
					}
					else story.closeTiddler(this.crumbs[c]);
				}
				limit--;
			}
		}
	}
};
//}}}
// // PreviousTiddler ('back') command and macro
//{{{
config.commands.previousTiddler = {
	text: 'back',
	tooltip: 'view the previous tiddler',
	handler: function(event,src,title) {
		var crumbs=config.macros.breadcrumbs.crumbs;
		if (crumbs.length<2) config.macros.breadcrumbs.home();
		else story.displayTiddler(story.findContainingTiddler(src),crumbs[crumbs.length-2]);
		return false;
	}
};
config.macros.previousTiddler= {
	label: 'back',
	prompt: 'view the previous tiddler',
	handler: function(place,macroName,params,wikifier,paramString,tiddler) {
		var label=params.shift(); if (!label) label=this.label;
		var prompt=params.shift(); if (!prompt) prompt=this.prompt;
		createTiddlyButton(place,label,prompt,function(ev){
			return config.commands.previousTiddler.handler(ev,this)
		});
	}
}
//}}}
// // HIJACKS
//{{{
// update crumbs when a tiddler is displayed
if (Story.prototype.breadCrumbs_coreDisplayTiddler==undefined)
	Story.prototype.breadCrumbs_coreDisplayTiddler=Story.prototype.displayTiddler;
Story.prototype.displayTiddler = function(srcElement,tiddler) {
	var title=(tiddler instanceof Tiddler)?tiddler.title:tiddler;
	this.breadCrumbs_coreDisplayTiddler.apply(this,arguments);
	if (!startingUp || config.options.chkShowStartupBreadcrumbs)
		config.macros.breadcrumbs.add(title);
}

// update crumbs when a tiddler is deleted
if (TiddlyWiki.prototype.breadCrumbs_coreRemoveTiddler==undefined)
	TiddlyWiki.prototype.breadCrumbs_coreRemoveTiddler=TiddlyWiki.prototype.removeTiddler;
TiddlyWiki.prototype.removeTiddler= function() {
	this.breadCrumbs_coreRemoveTiddler.apply(this,arguments);
	config.macros.breadcrumbs.refresh();
}
//}}}
<<tiddler [[include_tiddlers/Breakthrough Initiatives.html#"Breakthrough Initiatives"]]>>
<<tiddler [[include_tiddlers/Breakthrough Starshot Initiative.html#"Breakthrough Starshot Initiative"]]>>
<<tiddler [[include_tiddlers/Broca's Area.html#"Broca's Area"]]>>
<<tiddler [[include_tiddlers/Bronstein Cube.html#"Bronstein Cube"]]>>
A ''(Left) Bruck Loop'' or ''K\-Loop'' is a [[(left) Bol loop|Bol Loop]], satisfying the ''Left Bruck Identity''
\[
(\mb A \mb B) (\mb A  \mb B) = \mb A (\mb B (\mb B \mb A))
\]
or equivalently the ''Automorphic Inverse Identity''
\[
(\mb A \mb B)^{-1}  = \mb A^{-1} \mb B^{-1}
\]
Left Bruck loops are equivalent to Ungar's [[gyrocommutative gyrogroups|Gyrogroup]].
<<tiddler [[include_tiddlers/Buddhism.html#"Buddhism"]]>>
<<tiddler [[include_tiddlers/Bunch-Davies Vacuum.html#"Bunch-Davies Vacuum"]]>>
A ''Burgers Vector'' characterizes a [[dislocation|Dislocation]].

{{center{[img(407px+, )[images/BurgersVector.jpg]]}}}
Links:
* [[Dislocations - Institut für Angewandte Physik der Technischen Universität Wien|http://www.iap.tuwien.ac.at/www/surface/stm_gallery/dislocations]]
<<tiddler [[include_tiddlers/Buy a Website !!!!!.html#"Buy a Website !!!!!"]]>>
A ''C''${}^*$''-Algebra'' is a [[Banach*-Algebra|Banach Algebra]] $\mathcal A$ over the field of complex numbers, satisfying the so called ''C${}^*$-Identity''
\[
\|\mb A^* \mb A\| =  \|\mb A\| \|\mb A\| = \|\mb A\|^2
\]
or equivalently
\[
\|\mb A \mb A^* \| = \|\mb A\|^2
\]
$\forall \mb A \in \mathcal A$.

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

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

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

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

!!!!Generalizations
See [1].


Papers:
* [[Jordan C*-Algebras (1977) - J. D. Maitland Wright|http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.mmj/1029001946]] [[local|papers/JordanCStar.pdf]] [[pct. 93|http://scholar.google.de/scholar?cites=9473281614977063834&as_sdt=2005&sciodt=0,5&hl=de]]
* [[State Spaces of C*-Algebras (1980) - E. M. Alfsen, H. Hanche-Olsen, F. W. Shultz|http://www.kryakin.com/files/Acta_Mat_%282_55%29/acta150_107/144/144_8.pdf]] [[local|papers/144_8.pdf]] [[pct. 84|http://scholar.google.de/scholar?cites=15606370926646522510&as_sdt=2005&sciodt=0,5&hl=de]]
* [[Ring C*-Algebras (2009) - X. Li|http://arxiv.org/pdf/0905.4861]] [[local|papers/0905.4861v1.pdf]] [[pct. 8|http://scholar.google.de/scholar?hl=de&lr=&cites=14927717313074666055&um=1&ie=UTF-8&ei=2kZJTt72DYmWswajz4W1Bw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCcQzgIwAA]]
* [[[1] C*-algebras in Tensor Categories (2007) - P. Bouwknegt, K. Hannabuss, V. Mathai|http://arxiv.org/abs/math/0702802]] [[local|papers/0702802v2.pdf]] [[pct. 5|http://scholar.google.com/scholar?hl=de&lr=&cites=878409271660475324&um=1&ie=UTF-8&ei=0S4MT4Vdx8-zBvDkrMAE&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCcQzgIwAA]]
* [[Algebraic Quantum Mechanics (2009) - N. P. Landsman|http://www.math.ru.nl/~landsman/algebraicQM.pdf]] [[local|papers/algebraicQM.pdf]] [[pct. 2|http://scholar.google.com/scholar?hl=de&lr=&cites=18015674315809583263&um=1&ie=UTF-8&ei=3O6XT46GPIP3sgbRgbn0AQ&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCkQzgIwAA]]
* [[On the Definition of C*-algebra (2007) - J.M. Egger|http://cheng.staff.shef.ac.uk/pssl85/egger.pdf]] [[local|papers/egger.pdf]] pct. 0

Links:
* [[WIKIPEDIA - C*-Algebra|http://en.wikipedia.org/wiki/C*-algebra#Some_history:_B.2A-algebras_and_C.2A-algebras]]
* [[WIKIPEDIA - Von Neumann Algebra|http://en.wikipedia.org/wiki/Von_Neumann_algebra]]
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Links:
* [[Cages - A. E. Brouwer| http://www.win.tue.nl/~aeb/graphs/cages/cages.html]]
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<<tiddler [[include_tiddlers/Canonical Polyvector Klein-Gordon Field Quantization.html#"Canonical Polyvector Klein-Gordon Field Quantization"]]>>
The idea of ''Canonical Polyvector Quantization'' is to lift a non-linear field theory to [[polyvector space|Polyvector Space]], casting it to a quasi-linear formulation. This should allow for applying the classical tools of canonical field quantization.
Therefore on the level of polyvector geometry a quantized polyvector field can be seen as represented by states of a collection of [[harmonic polyvector oscillators|Harmonic Oscillator]] which in fact can be (highly) [[anharmonic oscillators|Anharmonic Oscillator]] on the level of conventional field theory.

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

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

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

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

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

The largest $k$ for which a $k$-cap in $PG(n, 2)$ exists is equal to $2^n$ (complement of a hyperplane).
The ''Cardinality'' of a finite set $S$, denoted $\operatorname{card}(S)$ is equal to the number of elements of the set. E.g. $\operatorname {card}(\{2, 4, 6\}) = 3$.
<<tiddler [[include_tiddlers/Cardy Formula.html#"Cardy Formula"]]>>
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<<tiddler [[include_tiddlers/Carl Gustav Jung.html#"Carl Gustav Jung"]]>>
<<tiddler [[include_tiddlers/Carlos Castro.html#"Carlos Castro"]]>>
<<tiddler [[include_tiddlers/Carnosine.html#"Carnosine"]]>>
The ''Cartan Matrix'' $g_{ij}$ of a rank $r$ [[root system|Root Lattice]] is a $r \times r$ matrix given by
\[
g_{ij} = 2\frac{\langle\mb e_i|\mb e_j\rangle}{\langle \mb e_i|\mb e_i \rangle}
\]
where $\mb e_i$ are the [[simple roots|Simple Root]]. The entries are independent of the choice of simple roots (up to ordering).

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

Papers:
* [[Strings on Orbifolds: An Introduction - H.-P. Nilles|http://cdsweb.cern.ch/record/184239/files/198802208.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=16984624762816364962&hl=de&as_sdt=2000]]
<<tiddler [[include_tiddlers/Cartan Subalgebra.html#"Cartan Subalgebra"]]>>
The ''Cartan Tensor'' $C^\rho_{\mu\nu}$ (a.k.a. ''Modified Torsion Tensor'') is defined by
\begin{equation}
C^\rho_{\mu\nu} = T^\rho_{\mu\nu} + T_\mu \delta_\nu^\rho + T_\nu \delta_\mu^\rho
\end{equation}
with $T^\rho_{\mu\nu}$ the uncontracted and $T_\mu = T^\nu_{\mu\nu}$ the contracted [[Cartan torsion tensor|Torsion]] (a.k.a ''Torsion Vector''). For the latter different normalisations are found in literature.

Papers:
* [[On a Completely Antisymmetric Cartan Torsion Tensor - L. Fabbri|http://arxiv.org/PS_cache/gr-qc/pdf/0608/0608090v2.pdf]] [[pct. 2|http://scholar.google.de/scholar?hl=de&lr=&cites=1653535039168438446]] - There exists yet another different version.
The ''Cartan\-Laptev Method'' is a reinterpretation/generalization of the methods of the mobile frame and of equivalence of Ã⬰lie Cartan by the Russian geometer German Fedorovich Laptev. The method is used in [[web-theory|Web]].

Papers:
* [[The Cartan-Laptev Method in the Study of G-structures on Manifolds - N. M. Ostianu|http://83.149.209.141/php/getFT.phtml?jrnid=into&paperid=22&volume=30&year=2002&issue=&fpage=5&what=fullt&option_lang=eng]] pct. 0 - Russian
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<<tiddler [[include_tiddlers/Character Table.html#"Character Table"]]>>
Given a $n \times n$-matrix $\mb M$ the ''Characteristic Polynomial'' $p_{\mb M} (\lambda)$ is defined by
\begin{equation}
p_{\mb M} (\lambda) \equiv \det(\lambda \mb I_n- \mb M)
\end{equation}
It is the solution to the [[eigenvalue problem|Eigenvalue Theory]] $ \mb M \mb A = \lambda \mb A$.
The equation
\begin{equation}
p_{\mb M} (\lambda)=0
\end{equation}
is called the ''Characteristic Equation''.

!!!!Examples
$1 \times 1$''-matrix''
\begin{equation}
p_{m} (\lambda)= \lambda-m
\end{equation}
$2 \times 2$''-matrix''
\begin{eqnarray}
p_{\mb M} (\lambda) &=&  \lambda^2 ? \lambda (m_{11} + m_{22}) + (m_{11}m_{22} - m_{12}m_{21}) \\
& =& \lambda^2 ? \operatorname{Tr} (\mb M) \lambda + \det (\mb M)
\end{eqnarray}
<<tiddler [[include_tiddlers/Charles Musès.html#"Charles Musès"]]>>
<<tiddler [[include_tiddlers/Chebyshev Polynomial.html#"Chebyshev Polynomial"]]>>
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The ''Chevalley Groups'' are the [[automorphism groups|Automorphism]] of the [[Lie algebras|Lie Algebra]] defined over the [[finite fields|Galois Field]].
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The ''Christoffel Symbols'' describe the symmetric ([[Levi-Civita-part|Levi-Civita Connection]]) of a general [[gravitational connection|Connection]].

One distinguishes:
''Christoffel Symbols of first kind''
\[
\{\lambda,\mu,\nu\}
\]
''Christoffel Symbols of second kind''
\[
\Chr{\lambda}{\mu\nu} \equiv g^{\lambda\rho} \{\rho,\mu,\nu\}
\]
A definition of the latter is required as the Christoffel connection is not tensorial which would imply the possibility of raising and lowering indices.
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<<tiddler [[include_tiddlers/Coleman-Mandula Theorem.html#"Coleman-Mandula Theorem"]]>>
<<tiddler [[include_tiddlers/Coleman-Weinberg Mechanism.html#"Coleman-Weinberg Mechanism"]]>>
<<tiddler [[include_tiddlers/Collapse of the Wavefunction.html#"Collapse of the Wavefunction"]]>>
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A ''Collineation'' (or ''Projective Transformation'', ''Projectivity'') is a bijection between [[projective planes or spaces|Projective Space]], that maps straight lines to straight lines. Hence rectangles are mapped to rectangles (in particular squares are mapped to rectangles). Collineations do not preserve sizes or angles but do preserve coincidences and cross-ratios: two properties which are important in projective geometry. Collineations form a [[group|Group]].

<html><center><img src="images/colllineation.jpg" style="width: 380px; "/></center></html>
<<tiddler [[include_tiddlers/Color Algebra.html#"Color Algebra"]]>>
<<tiddler [[include_tiddlers/Color Charge.html#"Color Charge"]]>>
Papers:
* [[Combinatorics Entering the Third Millennium - P. J. Cameron|http://www.maths.qmul.ac.uk/~pjc/preprints/pfhist.pdf]] pct. 0
Your comments are very welcome. Please refer to a [[tiddler|What is a Tiddler]] or topic if possible. Thanx a lot !

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<<tiddler [[include_tiddlers/Commission E Monographs.html#"Commission E Monographs"]]>>
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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]]}}}
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<<tiddler [[include_tiddlers/Conformal Transformation.html#"Conformal Transformation"]]>>
The ''Conformal Weyl Group'' is the $10$-parameter [[Poincaré group|Poincaré Transformation]] supplemented with a $1$-parameter group of scale transformations.
\[
x'_{\mu}  = e^{\theta} x_\mu\text{,} \quad \bs \psi' (\mb x') = e^{?k\theta} \bs \psi (\mb x)\text{;} \quad k,\theta= const.
\]
<<tiddler [[include_tiddlers/Conic Sedenion.html#"Conic Sedenion"]]>>
Links:
* [[Website Kevin Carmody - Hypernumbers|http://web.archive.org/web/20041204062721/kevincarmody.com/math/hypernumbers.html]]
* [[Tony Smith's Homepage - Zero Divisor Algebras|http://www.valdostamuseum.org/hamsmith/NDalg.html#rulebim]]

Papers:
* [[Circular and Hyperbolic Quaternions, Octonions, and Sedenions - K. Carmody|http://web.archive.org/web/20050130075442/kevincarmody.com/math/sedenions1.pdf]] [[local|papers/sedenions1.pdf]] [[pct. 18|http://scholar.google.de/scholar?cites=12950820281512531271&hl=de]]
* [[Circular and Hyperbolic Quaternions, Octonions, and Sedenions - Further Results - K. Carmody|http://web.archive.org/web/20050130102121/kevincarmody.com/math/sedenions2.pdf]] [[local|papers/sedenions2.pdf]] [[pct. 18|http://scholar.google.de/scholar?cites=10942444327132985935&hl=de]]
<<tiddler [[include_tiddlers/Conjugation.html#"Conjugation"]]>>
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<<tiddler [[include_tiddlers/Continuum Hypothesis.html#"Continuum Hypothesis"]]>>
<<tiddler [[include_tiddlers/Contorsion.html#"Contorsion"]]>>
<<tiddler [[include_tiddlers/Conway Groups.html#"Conway Groups"]]>>
/***
<<tiddler CookieManager>>
***/
/***
!!![[Portable cookies:|CookieSaverPlugin]] {{fine{<<option chkPortableCookies>>enable <<option chkMonitorCookieJar>>monitor}}}
^^This section is ''//__automatically maintained__//'' by [[CookieSaverPlugin]].  To block specific cookies, see [[CookieSaverPluginConfig]].^^
***/
// // /% end portable cookies %/
/***
!!![[Baked cookies:|CookieManagerPlugin]]
^^Press {{smallform{<<cookieManager button>>}}} to save the current browser cookies... then hand-edit this section to customize the results.^^
***/

// 5 cookies saved on Tuesday, April 8th 2008 at 17:30:05 by ELSDesignStudios//
//^^(edit/remove username check and/or individual option settings as desired)^^//
//{{{
if (["ELSDesignStudios","ELS","Eric"].contains(config.options.txtUserName)) {
	config.options["chkInsertTabs"]=true;
	config.options["chkSearchByDate"]=true;
	config.options["chkTOCIncludeHidden"]=true;
	config.options["txtUploadUserName"]="ELSDesignStudios";
	config.options["txtUploadBackupDir"]="backup";
	config.options["txtUploadStoreUrl"]="http://www.tiddlytools.com/store.php";
	config.options["txtPrivWizardDefaultStep"]="2";
	config.options["txtSliderunsaved"]=true;
}
//}}}
// The following settings are applied as defaults for all users:           //
//{{{
	config.options["txtMaxEditRows"]=25;
	config.options["txtTiddlerLinkTootip"]="%0 - %2 (%3 bytes) - %4";

	if (!config.options.txtTheme.length) config.options.txtTheme='StyleSheet';

	// Make sure FAQ panels are *always* closed on startup (so load-on-demand can be used later on):
	config.options.chkSliderWelcomeShowFAQ=false;
	saveOptionCookie('chkSliderWelcomeShowFAQ');
	config.options.chkSliderMainMenu_faq=false;
	saveOptionCookie('chkSliderMainMenu_faq');
	config.options.chkSliderSiteMenu_faq=false;
	saveOptionCookie('chkSliderSiteMenu_faq');
//}}}
// 0 cookies saved on Sunday, March 25th 2012 at 11:12:42 by Markus Maute//
//^^(edit/remove username check and/or individual option settings as desired)^^//
//{{{
if (config.options.txtUserName=="Markus Maute") {
}
//}}}
/***
|Name|CookieManagerPlugin|
|Source|http://www.TiddlyTools.com/#CookieManagerPlugin|
|Version|2.4.1|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|view/add/delete browser-based cookies and "bake" cookies to CookieJar tiddler for 'sticky' settings|
!!!!!Usage
<<<
This plugin provides an interactive control panel that lets you select, view, modify, or delete any of the current values for TiddlyWiki options that have been stored as local, private, //browser cookies//.  You can also use the control panel to "bake cookies", which generates a set of javascript statements that assign hard-coded option values to the TiddlyWiki internal variables that correspond to the current browser cookie settings.  These hard-coded values are then stored in the [[CookieJar]] tiddler, which is tagged with<<tag systemConfig>> so that each time the document is loaded, the baked cookie settings will be automatically applied.

When a set of baked cookies is added to the [[CookieJar]], it is automatically surrounded by a conditional test so that the hard-coded settings will only be applied for the username that was in effect when they were initially generated.  In this way, if you publish or share your document with others, //your// particular baked cookie settings are not automatically applied to others, so that their own browser-based cookie settings (if defined) will be applied as usual.

Whenever you "bake cookies", new hard-coded javascript assignment statements are *appended* to the end of the [[CookieJar]].  However, any baked cookies that were previously generated and stored in the [[CookieJar]] are not automatically removed from the tiddler.  As a result, because the most recently baked cookie settings in the [[CookieJar]] are always the last to be processed, the values assigned by older baked cookies are immediately overridden by the values from the newest baked cookies, so that the newest values will be in effect when the CookieJar startup processing is completed.

Each time you bake a new batch of cookies, it is recommended that you should review and hand-edit the [[CookieJar]] to remove any "stale cookies" or merge the old and new sets of baked cookies into a single block to simplify readability (as well as saving a little tiddler storage space).  Of course, you can also hand-edit the [[CookieJar]] tiddler at any time simply to remove a few individual //baked cookies// if they are no longer needed, and you can even delete the entire [[CookieJar]] tiddler and start fresh, if that is appropriate.  Please note that changing or deleting a baked cookie does not alter the current value of the corresponding option setting, and any changes you make to the [[CookieJar]] will only be applied after you have saved and reloaded the document in your browser.
<<<
!!!!!Examples
<<<
{{{<<cookieManager>>}}}
{{smallform small center{
@@display:block;width:35em;<<cookieManager>>@@}}}
<<<
!!!!!Configuration
<<<
<<option chkAllowBrowserCookies>> store ~TiddlyWiki option settings using private browser cookies
<<option chkMonitorBrowserCookies>> monitor browser cookie activity (show a message whenever a cookie is set or deleted)
<<option chkCookieManagerAddToAdvancedOptions>> display [[CookieManager]] in [[AdvancedOptions]]
//note: this setting does not take effect until you reload the document//
<<<
!!!!!Revisions
<<<
2011.01.16 2.4.1 in init(), corrected double addition of CookieManager to backstage
2009.08.05 2.4.0 changed CookieJar output format to support odd symbols in option names (e.g. '@')
2008.09.14 2.3.2 fixed handling for blocked cookies (was still allowing some blocked cookies to be set)
2008.09.12 2.3.1 added blocked[] array and allowBrowserCookie() test function for selective blocking of changes to browser cookies based on cookie name
2008.09.08 2.3.0 extensive code cleanup: defined removeCookie(), renamed cookies, added 'button' param for stand-alone "bake cookies" button, improved init of shadow [[CookieManager]] and [[CookieJar]] tiddlers for compatibility with new [[CookieSaverPlugin]].
2008.07.11 2.2.1 fixed recursion error in hijack for saveOptionCookie()
2008.06.26 2.2.0 added chkCookieManagerNoNewCookies option
2008.06.05 2.1.3 replaced hard-coded definition for "CookieJar" title with option variable
2008.05.12 2.1.2 add "cookies" task to backstage (moved from BackstageTasks)
2008.04.09 2.1.0 added options: chkCookieManagerAddToAdvancedOptions
2008.04.08 2.0.1 automatically include CookieManager control panel in AdvancedOptions shadow tiddler
2007.08.02 2.0.0 converted from inline script
2007.04.29 1.0.0 initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.CookieManagerPlugin= {major: 2, minor: 4, revision: 1, date: new Date(2011,1,16)};
//}}}
//{{{
config.macros.cookieManager = {
	target:
		config.options.txtCookieJar||"CookieJar",
	blockedCookies:
		[],
	allowBrowserCookie: function(name) {
		return true;
	},
	displayStatus: function(msg) {
		if (config.options.chkMonitorBrowserCookies && !startingUp)
			displayMessage("CookieManager: "+msg);
	},
	init: function() {
		if (config.options.txtCookieJar===undefined)
			config.options.txtCookieJar=this.target;
		if (config.options.chkAllowBrowserCookies===undefined)
			config.options.chkAllowBrowserCookies=false;
		if (config.options.chkMonitorBrowserCookies===undefined)
			config.options.chkMonitorBrowserCookies=false;

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

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

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

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

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

// ... and then hijack it to add cookie blocking and monitoring messages
config.macros.cookieManager.removeCookie=removeCookie;
window.removeCookie=function(name,force)
{
	var cmc=config.macros.cookieManager; // abbrev
	if (!cmc.getCookieList().contains(name))
		return; // not a current cookie!
	if (force || ((config.options.chkAllowBrowserCookies || name=="chkAllowBrowserCookies")
		&& !cmc.blockedCookies.contains(name) && cmc.allowBrowserCookie(name))) {
		cmc.removeCookie.apply(this,arguments);
		cmc.displayStatus("deleted "+name);
	} else cmc.displayStatus("deletion of '"+name+"' is blocked");
}
//}}}
/***
|Name|CookieManagerPluginConfig|
|Source|http://www.TiddlyTools.com/#CookieManagerPluginConfig|
|Requires|CookieManagerPlugin|
|Description|custom settings for [[CookieManagerPlugin]]|
!!!!!Browser Cookie Configuration:
***/
// // <<option chkAllowBrowserCookies>> store ~TiddlyWiki option settings using private browser cookies
// // <<option chkMonitorBrowserCookies>> monitor browser cookie activity (shows a message whenever a cookie is updated)
//{{{
// default settings:
config.options.chkAllowBrowserCookies=false;	// if FALSE, this blocks *all* cookies
config.options.chkMonitorBrowserCookies=false;
//}}}

// // Individual cookie names can be prevented from being created, modified, or deleted in your browser's stored cookies by adding them to the {{{config.macros.cookieManager.blockedCookies}}} array:
//{{{
var bc=config.macros.cookieManager.blockedCookies; // abbreviation
bc.push("txtMainTab");			// TiddlyWiki:  SideBarTabs
bc.push("txtTOCSortBy");		// TiddlyTools: TableOfContentsPlugin
bc.push("txtCatalogTab");		// TiddlyTools: CatalogTabs
//}}}
// // You can also define a javascript test function that determines whether or not any particular cookie name should be blocked or allowed.  The following function should return FALSE if the browser cookie should be blocked, or TRUE if changes to the cookie should be allowed:
//{{{
config.macros.cookieManager.allowBrowserCookie=function(name) {
	// add tests based on specific cookie names and runtime conditions
	return true;
}
//}}}
<<tiddler [[include_tiddlers/Cooper Pair.html#"Cooper Pair"]]>>
<<tiddler [[include_tiddlers/Copernican Principle.html#"Copernican Principle"]]>>
<<tiddler [[include_tiddlers/Coquaternion.html#"Coquaternion"]]>>
<<tiddler [[include_tiddlers/Core-cusp Problem.html#"Core-cusp Problem"]]>>
/***
|Name|CoreTweaks|
|Source|http://www.TiddlyTools.com/#CoreTweaks|
|Version|M.M. - 831 removed due to collision with MathJax.|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.2.0|
|Type|plugin|
|Description|a small collection of overrides to TW core functions|
This tiddler contains small changes to TW core functions that correct or enhance standard features or behaviors.
***/
//{{{
// calculate TW version number - used to determine which tweaks should be applied
var ver=version.major+version.minor/10+version.revision/100;
//}}}
/***
----

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

window.coreTweaks_missingMacro_invokeMacro = window.invokeMacro;
window.invokeMacro = function(place,macro,params,wikifier,tiddler) {
	if (!config.macros[macro] || !config.macros[macro].handler)
		if (config.options.chkHideMissingMacros) return;
	window.coreTweaks_missingMacro_invokeMacro.apply(this,arguments);
}
//}}}
// // }}}}}}
// // <<foldHeadings>>
<<tiddler [[include_tiddlers/Cornelius Lanczos.html#"Cornelius Lanczos"]]>>
<<tiddler [[include_tiddlers/Correspondence Principle.html#"Correspondence Principle"]]>>
<<tiddler [[include_tiddlers/Cosmic Coincidences.html#"Cosmic Coincidences"]]>>
<<tiddler [[include_tiddlers/Cosmic Creation and God.html#"Cosmic Creation and God"]]>>
<<tiddler [[include_tiddlers/Cosmic Horizon.html#"Cosmic Horizon"]]>>
<<tiddler [[include_tiddlers/Cosmic Microwave Background.html#"Cosmic Microwave Background"]]>>
<<tiddler [[include_tiddlers/Cosmic String.html#"Cosmic String"]]>>
<<tiddler [[include_tiddlers/Cosmic Topology.html#"Cosmic Topology"]]>>
<<tiddler [[include_tiddlers/Cosmological Constant.html#"Cosmological Constant"]]>>
<<tiddler [[include_tiddlers/Cosmological Equation of State.html#"Cosmological Equation of State"]]>>
<br><<tiddler [[include_tiddlers/Cosmological Natural Selection.html#"Cosmological Natural Selection"]]>>
<<tiddler [[include_tiddlers/Cosmological Principle.html#"Cosmological Principle"]]>>
<<tiddler [[include_tiddlers/Cosmology.html#"Cosmology"]]>>
<<tiddler [[include_tiddlers/Cosmology Large Angular Scale Surveyor.html#"Cosmology Large Angular Scale Surveyor"]]>>
<<tiddler [[include_tiddlers/Counterfactual Conditional.html#"Counterfactual Conditional"]]>>
<<tiddler [[include_tiddlers/Coupled Map Lattice.html#"Coupled Map Lattice"]]>>
<<tiddler [[include_tiddlers/Coupled Oscillator.html#"Coupled Oscillator"]]>>
<<tiddler [[include_tiddlers/Covariant Derivative.html#"Covariant Derivative"]]>>
<<tiddler [[include_tiddlers/Covariant Entropy Bound.html#"Covariant Entropy Bound"]]>>
<<tiddler [[include_tiddlers/Coxeter Group.html#"Coxeter Group"]]>>
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]]
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<<tiddler [[include_tiddlers/D'Alembert Equation.html#"D'Alembert Equation"]]>>
The ''D'Alembert Operator'' (a.k.a. ''d'Alembertian'' or ''Wave Operator'') generalizes the [[Laplace operator|Laplace Equation]] $\Delta$ to Minkowski-space and is given by

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

Links:
* [[WIKIPEDIA - D'Alembert Operator|http://en.wikipedia.org/wiki/D'Alembert_operator]]
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[[Welcome]]
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<<tiddler [[include_tiddlers/Dimensionality of the World.html#"Dimensionality of the World"]]>>
<<tiddler [[include_tiddlers/Dirac Algebra.html#"Dirac Algebra"]]>>
<<tiddler [[include_tiddlers/Dirac Delta Function.html#"Dirac Delta Function"]]>>
<<tiddler [[include_tiddlers/Dirac Equation.html#"Dirac Equation"]]>>
Papers:
* [[Mechanism for Vanishing Zero Point Energy (2003) - R. D. Klauber|http://arxiv.org/abs/astro-ph/0309679]] [[local|papers/0309679v3.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=3350334982651691078&as_sdt=2005&sciodt=0,5&hl=de]]

Videos:
* [[Field Theory I - Lecture 6 (2011) - K. Zarembo|http://pirsa.org/displayFlash.php?id=11100014]]
* [[Lecture Notes and Videos (2011) - P. Siegel|http://www.csupomona.edu/~pbsiegel/phy404/classnotes.html]]
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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!}$
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<<tiddler [[include_tiddlers/E6 Unification.html#"E6 Unification"]]>>
''$E_7$'' 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]]
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The ''ECE (Einstein\-Cartan\-Evans) Theory'' was developed by Myron Evans.

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

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

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

Links:
* [[WIKIPEDIA - Electronic Voice Phenomenon|http://en.wikipedia.org/wiki/Electronic_voice_phenomena]]
* [[Stimmen aus einer anderen Welt - Chronik und Technik der Tonbandstimmenforschung - von Hildegard Schäfer|http://www.rodiehr.de/a_27_s_stimmen_inhalt.htm#I N H A L T]]
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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]]
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<<tiddler [[include_tiddlers/Einstein Metric.html#"Einstein Metric"]]>>
<<tiddler [[include_tiddlers/Einstein Space.html#"Einstein Space"]]>>
<<tiddler [[include_tiddlers/Einstein Tensor.html#"Einstein Tensor"]]>>
<<tiddler [[include_tiddlers/Einstein's Constant.html#"Einstein's Constant"]]>>
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{{center{[img(513px+, )[images/CyclicModel.jpg]]}}}

Papers:
* [[The Ekpyrotic Universe: Colliding Branes and the Origin of the Hot Big Bang (2001) - J. Khoury, B. A. Ovrut, P. J. Steinhardt, N. Turok|http://arxiv.org/abs/hep-th/0103239]] [[local|papers/0103239v3.pdf]] {{t500Cite{[[pct. 961|http://scholar.google.de/scholar?cites=4222507034535707019&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[From Big Crunch to Big Bang (2002) - J. Khoury., B. A. Ovrut, N. Seiberg, P. J. Steinhardt, N. Turok|http://arxiv.org/pdf/hep-th/0108187]] [[local|papers/0108187v4.pdf]] {{t500Cite{[[pct. 473|http://scholar.google.de/scholar?cites=6718511135230793892&as_sdt=2005&sciodt=0,5&hl=de]]}}}

Links:
* [[WIKIPEDIA - Ekpyrotic Universe|http://en.wikipedia.org/wiki/Ekpyrotic_universe]]

Videos:
* [[Inflationary Cosmology on Trial (2011) - P. J. Steinhardt|http://www.youtube.com/watch?v=IcxptIJS7kQ&feature=endscreen&NR=1]]
* [[Interview with Neil Turok (2008)|http://www.youtube.com/watch?v=l7X1AU9x0ZY&feature=related]]
* [[Authors@Google (2007) - Paul Steinhardt|http://www.youtube.com/watch?v=kb2USbR6_DQ]]
* [[PERIMETER INSTITUTE RECORDED SEMINAR ARCHIVE - J.-L. Lehners|http://pirsa.org/index.php?p=speaker&name=Jean-Luc_Lehners]]
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''Equations of Motion'' are [[differential equations|Differential Equation]] which remain valid if transformed differentially to new coordinates, even if the transformation is not integrable in the Schwarz sense.
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<<tiddler [[include_tiddlers/Exotic 4-Geometry and Gravity.html#"Exotic 4-Geometry and Gravity"]]>>
<<tiddler [[include_tiddlers/Exotic 4-Geometry and Physics.html#"Exotic 4-Geometry and Physics"]]>>
<<tiddler [[include_tiddlers/Exotic 4-Geometry and Quantum Gravity.html#"Exotic 4-Geometry and Quantum Gravity"]]>>
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<<tiddler [[include_tiddlers/Exotic Geometry.html#"Exotic Geometry"]]>>
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<<tiddler [[include_tiddlers/Exotic Sphere.html#"Exotic Sphere"]]>>
<<tiddler [[include_tiddlers/Explicit Symmetry Breaking.html#"Explicit Symmetry Breaking"]]>>
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]]
<<tiddler [[include_tiddlers/Exponential Integral.html#"Exponential Integral"]]>>
<<tiddler [[include_tiddlers/Exterior Derivative.html#"Exterior Derivative"]]>>
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|Source|http://www.TiddlyTools.com/#ExternalTiddlersPlugin|
|Documentation|http://www.TiddlyTools.com/#ExternalTiddlersPluginInfo|
|Version|1.3.3|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Requires|TemporaryTiddlersPlugin, SectionLinksPlugin (optional, recommended)|
|Description|retrieve and wikify content from external files or remote URLs|
This plugin extends the {{{<<tiddler>>}}} macro syntax so you can retrieve and wikify content directly from external files or remote URLs.  You can also define alternative "fallback" sources to provide basic "import on demand" handling by automatically creating/importing tiddler content from external sources when the specified ~TiddlerName does not already exist in your document.
!!!!!Documentation
>see [[ExternalTiddlersPluginInfo]]
!!!!!Configuration
<<<
<<option chkExternalTiddlersImport>> automatically create/import tiddlers when using external fallback references
{{{usage: <<option chkExternalTiddlersImport>>}}}
<<option chkExternalTiddlersQuiet>> don't display messages when adding tiddlers ("quiet mode")
{{{usage: <<option chkExternalTiddlersQuiet>>}}}
<<option chkExternalTiddlersTemporary>> tag retrieved tiddlers as 'temporary'(requires [[TemporaryTiddlersPlugin]])
{{{usage: <<option chkExternalTiddlersTemporary>>}}}
tag retrieved tiddlers with: <<option txtExternalTiddlersTags>>
{{{usage: <<option txtExternalTiddlersTags>>}}}

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

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

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

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

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

		// tiddler doesn't exist, and not an external file/URL reference... skip it
		if (!config.formatterHelpers.isExternalLink(src)) {
			if (!fallback.length) fallback=src; // title to use when importing external tiddler
			continue;
		}
		// separate 'permaview' list of tiddlers (if any) from file/URL (i.e., '#name name name..." suffix)
		var p=src.split("#"); src=p.shift(); var tids=p.join('#').readBracketedList(false);
		// if reference is to a remotely hosted document or the current document is remotely hosted...
		if (src.substr(0,4)=="http" || document.location.protocol.substr(0,4)=="http") {
			if (src.substr(0,4)!="http") // fixup URL for relative remote references
				{ var h=document.location.href; src=h.substr(0,h.lastIndexOf("/")+1)+src; }
			var wrapper = createTiddlyElement(place,"span",getGUID(),className); // create placeholder for async rendering
			var callback=function(success,params,text,src,xhr) { // ASYNC CALLBACK
				if (!success) { displayMessage(xhr.status); return; } // couldn't read remote file... report the error
				if (params.fallback.length)
					addTiddler(params.url,text,params.tids.length?params.tids:[params.fallback]); // import tiddler
				var wrapper=document.getElementById(params.id); if (!wrapper) return;
				wikify(substitute(extract(text,params.tids),params.args),wrapper); // ASYNC RENDER
			};
			var callbackparams={ url:src, id:wrapper.id, args:args, tids:tids, fallback:fallback }  // ASYNC PARAMS
			var name=config.options.txtRemoteUsername; // optional value
			var pass=config.options.txtRemotePassword; // optional value
			var x=doHttp("GET",src,null,null,name,pass,callback,callbackparams,null)
			if (typeof(x)=="string") // couldn't start XMLHttpRequest... report error
				{ displayMessage("error: cannot access "+src); displayMessage(x); }
			break; // can't tell if async read will succeed.... stop processing alternatives anyway.
		}
		else { // read file from local filesystem
			var text=loadFile(getLocalPath(src));
			if (!text) { // couldn't load file... fixup path for relative reference and retry...
				var h=document.location.href;
				var text=loadFile(getLocalPath(decodeURIComponent(h.substr(0,h.lastIndexOf("/")+1)))+src);
			}
			if (text) { // test it again... if file was loaded OK, render it in a class wrapper
				if (fallback.length) // create new tiddler using primary source name (if any)
					addTiddler(src,text,tids.length?tids:[fallback]);
				var wrapper=createTiddlyElement(place,"span",null,className);
				wikify(substitute(extract(text,tids),args),wrapper); // render
				break; // stop processing alternatives
			}
		}
	}
};
//}}}
<<tiddler [[include_tiddlers/Extra Loop.html#"Extra Loop"]]>>
<<tiddler [[include_tiddlers/Extrasensory Perception.html#"Extrasensory Perception"]]>>
An ''Extraspecial Group $2^{1+2d}$'' is a subgroup of $GL(2d, \mathbb F)$, for a field $\mathbb F$ of characteristic $0$.
<<tiddler [[include_tiddlers/Extraterestrial Life.html#"Extraterestrial Life"]]>>
<<tiddler [[include_tiddlers/Extremal Black Hole.html#"Extremal Black Hole"]]>>
Papers:
* [[Extremal Lattices - R. Scharlau, R. Schulze-Pillot|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.46.1119&rep=rep1&type=ps]] [[pct. 30|http://scholar.google.de/scholar?cites=16121390943809304545&hl=de&as_sdt=2000]]
<<tiddler [[include_tiddlers/F-theory.html#"F-theory"]]>>
<<tiddler [[include_tiddlers/F4.html#"F4"]]>>
<<tiddler [[include_tiddlers/FOCAL (Spacecraft).html#"FOCAL (Spacecraft)"]]>>
<<tiddler [[include_tiddlers/Faber-Jackson Relation.html#"Faber-Jackson Relation"]]>>
<<tiddler [[include_tiddlers/Factor.html#"Factor"]]>>
<<tiddler [[include_tiddlers/Failure Rate.html#"Failure Rate"]]>>
<<tiddler [[include_tiddlers/False Vacuum.html#"False Vacuum"]]>>
<<tiddler [[include_tiddlers/Fano Plane.html#"Fano Plane"]]>>
<<tiddler [[include_tiddlers/Fano Planes - Classification.html#"Fano Planes - Classification"]]>>
<<tiddler [[include_tiddlers/Fano Spaces.html#"Fano Spaces"]]>>
<<tiddler [[include_tiddlers/Fermi Gamma-ray Space Telescope.html#"Fermi Gamma-ray Space Telescope"]]>>
<<tiddler [[include_tiddlers/Fermi Paradox.html#"Fermi Paradox"]]>>
<<tiddler [[include_tiddlers/Fermi's Golden Rule.html#"Fermi's Golden Rule"]]>>
<<tiddler [[include_tiddlers/Fermi-Dirac Statistics.html#"Fermi-Dirac Statistics"]]>>
<<tiddler [[include_tiddlers/Fermion Doubling.html#"Fermion Doubling"]]>>
<<tiddler [[include_tiddlers/Fermionic Path Integral.html#"Fermionic Path Integral"]]>>
<<tiddler [[include_tiddlers/Ferroelectricity.html#"Ferroelectricity"]]>>
* [[A Possible Mechanism for Evading Temperature Quantum Decoherence in Living Matter by Feshbach Resonance (2009) - N. Poccia, A. Ricci, D. Innocenti, A. Bianconi|http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2695269/pdf/ijms-10-02084.pdf]] [[local|papers/ijms-10-02084.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=1650174702112680860&hl=de&as_sdt=2000]]

Lectures:
* [[Bose-Einstein Condensation in a Dilute Gas; the First 70 Years and some Recent Experiments - E. A. Cornell, E. Wieman|http://nobelprize.org/nobel_prizes/physics/laureates/2001/cornellwieman-lecture.pdf]] [[local|lectures/cornellwieman-lecture.pdf]]
The ''Feshbach\-Villars Representation'' casts the [[Klein-Gordon equation|Klein-Gordon Equation]] into two equations, both of which are first order in time.
<br><<tiddler [[include_tiddlers/Feynman Checkerboard.html#"Feynman Checkerboard"]]>>
<<tiddler [[include_tiddlers/Fiber Bundle.html#"Fiber Bundle"]]>>
<<tiddler [[include_tiddlers/Fibration.html#"Fibration"]]>>
<<tiddler [[include_tiddlers/Field.html#"Field"]]>>
<<tiddler [[include_tiddlers/Field Strength Tensor.html#"Field Strength Tensor"]]>>
<<tiddler [[include_tiddlers/Field Theory.html#"Field Theory"]]>>
Links:
* [[Les Médailles Fields|http://serge.mehl.free.fr/anx/med_fields.html]]
<<tiddler [[include_tiddlers/Fifth Force.html#"Fifth Force"]]>>
<<tiddler [[include_tiddlers/Fine Structure Constant.html#"Fine Structure Constant"]]>>
<<tiddler [[include_tiddlers/Fine Tuning.html#"Fine Tuning"]]>>
<<tiddler [[include_tiddlers/Finite Geometry.html#"Finite Geometry"]]>>
<<tiddler [[include_tiddlers/Finite Spectral Triple.html#"Finite Spectral Triple"]]>>
<<tiddler [[include_tiddlers/Finite Subgroups of Exceptional Lie Groups.html#"Finite Subgroups of Exceptional Lie Groups"]]>>
<<tiddler [[include_tiddlers/Finslerian Geometry.html#"Finslerian Geometry"]]>>
<<tiddler [[include_tiddlers/Firewall Paradox.html#"Firewall Paradox"]]>>
<<tiddler [[include_tiddlers/First Bianchi Identity.html#"First Bianchi Identity"]]>>
<<tiddler [[include_tiddlers/First Fundamental Form.html#"First Fundamental Form"]]>>
<<tiddler [[include_tiddlers/First Law of Thermodynamics.html#"First Law of Thermodynamics"]]>>
<<tiddler [[include_tiddlers/Fisetin.html#"Fisetin"]]>>
Given a [[projective geometry|Projective Geometry]] $PG(n,q)$, a ''Flat'' is a subspace of dimension $k?1$.

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

The complement of a $(n-1)$-flat in $PG(n, q)$ is isomorphic to $AG(n, q)$.
<<tiddler [[include_tiddlers/Flatness Problem.html#"Flatness Problem"]]>>
<<tiddler [[include_tiddlers/Flavans.html#"Flavans"]]>>
<<tiddler [[include_tiddlers/Flavour.html#"Flavour"]]>>
<<tiddler [[include_tiddlers/Flavour SU(4).html#"Flavour SU(4)"]]>>
<<tiddler [[include_tiddlers/Flexible Algebra.html#"Flexible Algebra"]]>>
<<tiddler [[include_tiddlers/Floating Point Operations Per Second.html#"Floating Point Operations Per Second"]]>>
<<tiddler [[include_tiddlers/Floating Water Bridge.html#"Floating Water Bridge"]]>>
The ''Floor Function'' maps a real number $x$ to the next smallest integer, i.e.
\[
\operatorname {floor} (x) \equiv \lfloor x \rfloor \equiv \max\, \{n\in\mathbb{Z}\mid n\le x\}
\]
Links:
* [[WIKIPEDIA - Floor Function|http://en.wikipedia.org/wiki/Floor_and_ceiling_functions]]
<<tiddler [[include_tiddlers/Flyby Anomaly.html#"Flyby Anomaly"]]>>
<<tiddler [[include_tiddlers/Fock Space.html#"Fock Space"]]>>
<<tiddler [[include_tiddlers/Fokker-Planck Equation.html#"Fokker-Planck Equation"]]>>
<<tiddler [[include_tiddlers/Foliation.html#"Foliation"]]>>
/***
|Name|FontSizePlugin|
|Created by|SaqImtiaz|
|Location|http://tw.lewcid.org/#FontSizePlugin|
|Version|1.0|
|Requires|~TW2.x|
!Description:
Resize tiddler text on the fly. The text size is remembered between sessions by use of a cookie.
You can customize the maximum and minimum allowed sizes.
(only affects tiddler content text, not any other text)

Also, you can load a TW file with a font-size specified in the url.
Eg: http://tw.lewcid.org/#font:110

!Demo:
Try using the font-size buttons in the sidebar, or in the MainMenu above.

!Installation:
Copy the contents of this tiddler to your TW, tag with systemConfig, save and reload your TW.
Then put {{{<<fontSize "font-size:">>}}} in your SideBarOptions tiddler, or anywhere else that you might like.

!Usage
{{{<<fontSize>>}}} results in <<fontSize>>
{{{<<fontSize font-size: >>}}} results in <<fontSize font-size:>>

!Customizing:
The buttons and prefix text are wrapped in a span with class fontResizer, for easy css styling.
To change the default font-size, and the maximum and minimum font-size allowed, edit the config.fontSize.settings section of the code below.

!Notes:
This plugin assumes that the initial font-size is 100% and then increases or decreases the size by 10%. This stepsize of 10% can also be customized.

!History:
*27-07-06, version 1.0 : prevented double clicks from triggering editing of containing tiddler.
*25-07-06,  version 0.9

!Code
***/

//{{{
config.fontSize={};

//configuration settings
config.fontSize.settings =
{
            defaultSize : 135,  // all sizes in %
            maxSize : 500,
            minSize : 40,
            stepSize : 10
};

//startup code
var fontSettings = config.fontSize.settings;

if (!config.options.txtFontSize)
            {config.options.txtFontSize = fontSettings.defaultSize;
            saveOptionCookie("txtFontSize");}
setStylesheet(".tiddler .viewer {font-size:"+config.options.txtFontSize+"%;}\n","fontResizerStyles");
setStylesheet("#contentWrapper .fontResizer .button {display:inline;font-size:110%; font-weight:bold; margin:0 1px; padding: 0 3px; text-align:center !important;}\n .fontResizer {margin:0 0.5em;}","fontResizerButtonStyles");

//macro
config.macros.fontSize={};
config.macros.fontSize.handler = function (place,macroName,params,wikifier,paramString,tiddler)
{

               var sp = createTiddlyElement(place,"span",null,"fontResizer");
               sp.ondblclick=this.onDblClick;
               if (params[0])
                           createTiddlyText(sp,params[0]);
               createTiddlyButton(sp,"-","decrease font-size",this.decFont);
               createTiddlyButton(sp,"=","reset font-size",this.resetFont);
               createTiddlyButton(sp,"+","increase font-size",this.incFont);
}

config.macros.fontSize.onDblClick = function (e)
{
             if (!e) var e = window.event;
             e.cancelBubble = true;
             if (e.stopPropagation) e.stopPropagation();
             return false;
}

config.macros.fontSize.setFont = function ()
{
               saveOptionCookie("txtFontSize");
               setStylesheet(".tiddler .viewer {font-size:"+config.options.txtFontSize+"%;}\n","fontResizerStyles");
}

config.macros.fontSize.incFont=function()
{
               if (config.options.txtFontSize < fontSettings.maxSize)
                  config.options.txtFontSize = (config.options.txtFontSize*1)+fontSettings.stepSize;
               config.macros.fontSize.setFont();
}

config.macros.fontSize.decFont=function()
{

               if (config.options.txtFontSize > fontSettings.minSize)
                  config.options.txtFontSize = (config.options.txtFontSize*1) - fontSettings.stepSize;
               config.macros.fontSize.setFont();
}

config.macros.fontSize.resetFont=function()
{

               config.options.txtFontSize=fontSettings.defaultSize;
               config.macros.fontSize.setFont();
}

config.paramifiers.font =
{
               onstart: function(v)
                  {
                   config.options.txtFontSize = v;
                   config.macros.fontSize.setFont();
                  }
};
//}}}
<<tiddler [[include_tiddlers/Form.html#"Form"]]>>
<<tiddler [[include_tiddlers/Four Colour Theorem.html#"Four Colour Theorem"]]>>
<<tiddler [[include_tiddlers/Fourier Transform.html#"Fourier Transform"]]>>
<<tiddler [[include_tiddlers/Fourth Order Bol Identities Expansions.html#"Fourth Order Bol Identities Expansions"]]>>
<<tiddler [[include_tiddlers/Fourth Order Theory.html#"Fourth Order Theory"]]>>
<<tiddler [[include_tiddlers/Fractal.html#"Fractal"]]>>
<<tiddler [[include_tiddlers/Fractal Spacetime.html#"Fractal Spacetime"]]>>
<<tiddler [[include_tiddlers/Frame Bundle.html#"Frame Bundle"]]>>
<<tiddler [[include_tiddlers/Fred Hoyle.html#"Fred Hoyle"]]>>
<<tiddler [[include_tiddlers/Free Algebra.html#"Free Algebra"]]>>
<<tiddler [[include_tiddlers/Free Parameters of the Standard Model.html#"Free Parameters of the Standard Model"]]>>
<<tiddler [[include_tiddlers/Free Radical Theory of Aging.html#"Free Radical Theory of Aging"]]>>
<<tiddler [[include_tiddlers/Free Will.html#"Free Will"]]>>
<<tiddler [[include_tiddlers/Free Will Theorem.html#"Free Will Theorem"]]>>
<br><<tiddler [[include_tiddlers/Freeman Dyson.html#"Freeman Dyson"]]>>
<<tiddler [[include_tiddlers/Freudental's Magic Square.html#"Freudental's Magic Square"]]>>
<<tiddler [[include_tiddlers/Friedmann Equations.html#"Friedmann Equations"]]>>
<<tiddler [[include_tiddlers/Friedmann-Lemaître-Robertson-Walker Metric.html#"Friedmann-Lemaître-Robertson-Walker Metric"]]>>
<<tiddler [[include_tiddlers/Fringe Science.html#"Fringe Science"]]>>
<<tiddler [[include_tiddlers/Frobenius Inner Product.html#"Frobenius Inner Product"]]>>
According to the ''Frobenius Theorem'' the only finite dimensional associative division algebras over the real numbers are the real numbers, the complex numbers and the [[quaternions|Quaternion]].
An extended version states that every alternative [[division algebra|Division Algebra]] is isomorphic to one of the following: the algebra of real numbers, the algebra of complex numbers, the [[quaternions|Quaternion]] and the [[Cayley numbers|Octonion]].
<<tiddler [[include_tiddlers/Fructation.html#"Fructation"]]>>
<<tiddler [[include_tiddlers/Fröhlich Condensation.html#"Fröhlich Condensation"]]>>
<<tiddler [[include_tiddlers/Fubini-Study Metric.html#"Fubini-Study Metric"]]>>
<<tiddler [[include_tiddlers/Fun Stuff.html#"Fun Stuff"]]>>
<<tiddler [[include_tiddlers/Functional.html#"Functional"]]>>
<<tiddler [[include_tiddlers/Functional Determinant.html#"Functional Determinant"]]>>
<<tiddler [[include_tiddlers/Fundamental Group.html#"Fundamental Group"]]>>
<<tiddler [[include_tiddlers/Fundamental Physics Prize.html#"Fundamental Physics Prize"]]>>
<<tiddler [[include_tiddlers/Fundamental Representation.html#"Fundamental Representation"]]>>
<<tiddler [[include_tiddlers/Fuzzy Dark Matter.html#"Fuzzy Dark Matter"]]>>
<<tiddler [[include_tiddlers/Fuzzy Sphere.html#"Fuzzy Sphere"]]>>
<<tiddler [[include_tiddlers/G-Structure.html#"G-Structure"]]>>
<<tiddler [[include_tiddlers/G2.html#"G2"]]>>
<<tiddler [[include_tiddlers/G2 Gauge Theory.html#"G2 Gauge Theory"]]>>
<<tiddler [[include_tiddlers/G2(2).html#"G2(2)"]]>>
<<tiddler [[include_tiddlers/G2(C).html#"G2(C)"]]>>
<<tiddler [[include_tiddlers/G2-Manifold.html#"G2-Manifold"]]>>
<<tiddler [[include_tiddlers/GAP.html#"GAP"]]>>
<<tiddler [[include_tiddlers/GL(4,2).html#"GL(4,2)"]]>>
<<tiddler [[include_tiddlers/GNS Construction.html#"GNS Construction"]]>>
<<tiddler [[include_tiddlers/GTG.html#"GTG"]]>>
<<tiddler [[include_tiddlers/GUT.html#"GUT"]]>>
<<tiddler [[include_tiddlers/Gaia Hypothesis.html#"Gaia Hypothesis"]]>>
<<tiddler [[include_tiddlers/Galaxy.html#"Galaxy"]]>>
<<tiddler [[include_tiddlers/Galaxy Rotation Curve.html#"Galaxy Rotation Curve"]]>>
<<tiddler [[include_tiddlers/Galois Field.html#"Galois Field"]]>>
<<tiddler [[include_tiddlers/Galois Theory.html#"Galois Theory"]]>>
<<tiddler [[include_tiddlers/Gamma 16 Lattice.html#"Gamma 16 Lattice"]]>>
<<tiddler [[include_tiddlers/Gamma Function.html#"Gamma Function"]]>>
<<tiddler [[include_tiddlers/Gamma Matrices.html#"Gamma Matrices"]]>>
<<tiddler [[include_tiddlers/Gamma Ray Burst.html#"Gamma Ray Burst"]]>>
<<tiddler [[include_tiddlers/Gamow State.html#"Gamow State"]]>>
<<tiddler [[include_tiddlers/Gauge Anomaly.html#"Gauge Anomaly"]]>>
<<tiddler [[include_tiddlers/Gauge Fields Self-couplings Lagrangian.html#"Gauge Fields Self-couplings Lagrangian"]]>>
<<tiddler [[include_tiddlers/Gauge Fixing.html#"Gauge Fixing"]]>> 
<<tiddler [[include_tiddlers/Gauge Theory.html#"Gauge Theory"]]>>
<<tiddler [[include_tiddlers/Gauge Theory of Gravity.html#"Gauge Theory of Gravity"]]>>
The ''(q-ary) Gaussian Binomial Coefficient'' (a.k.a. ''q-binomial Coefficient'', ''Gaussian Number'' or ''Gaussian Polynomial'') is defined by:
\[
\begin{bmatrix} m \\ r \end{bmatrix}_q \equiv \frac{(1-q^m)(1-q^{m-1})\dots(1-q^{m-r+1})}{(1-q)(1-q^2)\dots(1-q^r)} = \prod_{i=0}^{r-1} \frac {1-q^{m-i}} {1-q^{i+1}}
\]
For $q = 1$ it coincides with the classical [[binomial coefficient|Binomial Coefficient]], hence it is a generalization thereof.

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

Links:
* [[WIKIPEDIA - Gaussian Binomial|http://en.wikipedia.org/wiki/Gaussian_binomial]]
* [[WolframMathWorld - q-Binomial Coefficient|http://mathworld.wolfram.com/q-BinomialCoefficient.html]]
<<tiddler [[include_tiddlers/Gaussian Integer.html#"Gaussian Integer"]]>>
<<tiddler [[include_tiddlers/Gaussian Integral.html#"Gaussian Integral"]]>>
<<tiddler [[include_tiddlers/Gelfand-Naimark Theorem.html#"Gelfand-Naimark Theorem"]]>>
Links:
* [[WIKIPEDIA - Geomerical Frustration|http://en.wikipedia.org/wiki/Geometrical_frustration]]
* [[Geomerical Frustration - Physics Today 02/2006|http://www.physics.rutgers.edu/grad/681/GFrustration_physics.today.pdf]]
<<tiddler [[include_tiddlers/Gene Drive.html#"Gene Drive"]]>>
<<tiddler [[include_tiddlers/General Covariance.html#"General Covariance"]]>>
<<tiddler [[include_tiddlers/General Relativity.html#"General Relativity"]]>>
<<tiddler [[include_tiddlers/Generalized Poincaré Conjecture.html#"Generalized Poincaré Conjecture"]]>> 
<<tiddler [[include_tiddlers/Generalized Second Law of Thermodynamics.html#"Generalized Second Law of Thermodynamics"]]>>
<<tiddler [[include_tiddlers/Generalized Uncertainty Principle.html#"Generalized Uncertainty Principle"]]>>
!!!!Rotation group:
The number $n$ of generators of $SO(p,q)$ with $p+q = N$ is given by:
\[
n = \frac{N(N-1)}{2}
\]
This number is equal to the number of (maximally) different off-diagonal elements of a symmetric $N\times N$-matrix.

!!!!!Examples
* $SO(3)$: The classical $3$ Euler angles.
* [[SO(4)]]: $n = 6$, i.e. the classical $3$ Euler angles + $3$ additional angles
* $SO(3,1)$: $n = 6$, i.e. the classical $3$ Euler angles and due to the Minkowski metric $3$ "imaginary angles" which correspond to Lorentz boosts.
* [[SO(7)]]: $n = 21$
* [[SO(8)]]: $n = 28$
* $SO(15)$: $n = 105$
* [[SO(16)]]: $n = 120$
!!!![[Codes|Blockcode]]
The ''Generator Matrix $G$'' of a code is a matrix with code words in its rows such that all linear combinations of the rows generate the whole of a linear code $[n,k,d]$ (i.e. all of its $2^k$ words).
$G$ is therefore a $k \times n$-matrix.
!!!!!Example
$[8,4]$-code:
\[
\mb{G} := \begin{pmatrix} 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1\\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1\\ 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \end{pmatrix}
\]
!!!![[Lattices|Lattice]]
Given a lattice $L$ with basis vectors $\{\mb e_1, \ldots, \mb e_n$\}, its ''Generator Matrix'' (or ''Basis Matrix'') ''$B$'' is defined by $B_{ij} \equiv (\mb e_i)_j$. I.e. it is a matrix with the rows holding the components of the basis vectors of the lattice.

By means of $B$, $L$ can be represented as follows:
\[
L = \{\vec x' : \vec x' = B \vec x = \sum_{i=1}^n x_i\mb e_i, \, x_i \in \mathbb Z\}
\]
<<tiddler [[include_tiddlers/Genetic Engineering.html#"Genetic Engineering"]]>>
<<tiddler [[include_tiddlers/Genome.html#"Genome"]]>>
<<tiddler [[include_tiddlers/Geodesic Equation.html#"Geodesic Equation"]]>>
<<tiddler [[include_tiddlers/Geodesic Loop.html#"Geodesic Loop"]]>>
<<tiddler [[include_tiddlers/Geometric Algebra.html#"Geometric Algebra"]]>>
<<tiddler [[include_tiddlers/Geometric Phase.html#"Geometric Phase"]]>>
<<tiddler [[include_tiddlers/Geometric Product.html#"Geometric Product"]]>>
<<tiddler [[include_tiddlers/Geometric Quantization.html#"Geometric Quantization"]]>>
<<tiddler [[include_tiddlers/Geometrodynamics.html#"Geometrodynamics"]]>>
The ''\Georgi-Glashow Model'' is a [[grand unified theory|GUT]] (GUT) based on the [[Lie group|Lie Group]] [[SU(5)]].

$SU(5)$ is the smallest Lie group containing the [[gauge group|Gauge Theory]] $SU(3)_c\times SU(2)_w\times U(1)_y$ of the$\;\;\,$[[Standard Model]].

The \Georgi-Glashow model was one of the first attempts to construct a GUT. It combines [[leptons|Lepton]] and [[quarks|Quark]] into single [[irreducible representations|Representation Theory]], therefore they might have interactions that do not conserve [[baryon number|Baryon Number]], still conserving the difference between baryon and [[lepton number|Lepton Number]] ([[B-L]]).

The simplest version of the non-[[supersymmetric|Supersymmetry]] \Georgi-Glashow model, based on $SU(5)$, has been ruled out by the accurate experimental results on the [[strong coupling constant|Strong Interactions]], $sin^2 \theta_W$ at the $M_Z$ scale, and the lower limit on the [[proton lifetime|Proton Decay]].

See also:
* [[Pati-Salam model|Pati-Salam Model]]
* [[SO(10)]]
* [[E₆|E6]]


Papers:
* [[Unity of all Elementary-particle Forces (1974) - H. Georgi, S. L. Glashow|http://www.physics.princeton.edu/~mcdonald/examples/EP/georgi_prl_32_438_74.pdf]] [[local|papers/georgi_prl_32_438_74.pdf]] {{t1000Cite{[[pct. 4394|http://scholar.google.com/scholar?cites=17118915033857368619&as_sdt=2005&sciodt=0,5&hl=de]]}}}

Documents:
* [[Introduction to Group Theory for Physicists (2011) - M. von Steinkirch|http://mysbfiles.stonybrook.edu/~mvonsteinkir/books/group.pdf]] [[local|documents/group.pdf]]

Links:
* [[WIKIPEDIA - Georgi-Glashow Model|http://en.wikipedia.org/wiki/Georgi%E2%80%93Glashow_model]]
<<tiddler [[include_tiddlers/Gerard 't Hooft.html#"Gerard 't Hooft"]]>>
<<tiddler [[include_tiddlers/Geroch's Theorem.html#"Geroch's Theorem"]]>>
<<tiddler [[include_tiddlers/Gerson Therapy.html#"Gerson Therapy"]]>>
<<tiddler [[include_tiddlers/Ghost Condensation.html#"Ghost Condensation"]]>>
<<tiddler [[include_tiddlers/Ghost Field.html#"Ghost Field"]]>>
<<tiddler [[include_tiddlers/Gibbs Free Energy.html#"Gibbs Free Energy"]]>>q
<<tiddler [[include_tiddlers/Gibbs State.html#"Gibbs State"]]>>
<<tiddler [[include_tiddlers/Ginkgo Biloba.html#"Ginkgo Biloba"]]>>
<<tiddler [[include_tiddlers/Git.html#"Git"]]>>
''Gleason's Theorem'' which might be regarded as the most fundamental theorem of algebraic [[coding theory|Coding Theory]] states, that every even, [[self-dual error correcting code|Dual Code]] can be generated by the [[Hamming code|Hamming Code]] and the [[Golay code|Golay Code]].


Papers:
* [[Weight Polynomials of Self-dual Codes and the MacWilliams Identities (1970) - A. M. Gleason|http://www.mathunion.org/ICM/ICM1970.3/Main/icm1970.3.0211.0216.ocr.pdf]] [[local|papers/icm1970.3.0211.0216.ocr.pdf]] {{t100Cite{[[pct. 108|http://scholar.google.de/scholar?cites=2273257787172737719&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Gleason's Theorem on Self-Dual Codes and its Generalizations (2006) - N. J. A. Sloane|http://arxiv.org/abs/math/0612535]] [[local|papers/0612535v1.pdf]] [[pct. 2|http://scholar.google.com/scholar?hl=de&lr=&cites=17406637980407761989&um=1&ie=UTF-8&sa=X&ei=xJQsUIunE8ndsgadvYHIAQ&ved=0CGYQzgIwAA]]

Presentations:
* [[The Type of a Code (2008) - G. Nebe|http://www.math.rwth-aachen.de/~Gabriele.Nebe/talks/nato1.pdf]] [[local|presentations/nato1.pdf]]
<<tiddler [[include_tiddlers/Gleason's Theorem (Quantum Mechanics).html#"Gleason's Theorem (Quantum Mechanics)"]]>>
<<tiddler [[include_tiddlers/Global Anomaly.html#"Global Anomaly"]]>>
<<tiddler [[include_tiddlers/Global Symmetry.html#"Global Symmetry"]]>>
<<tiddler [[include_tiddlers/Globally Hyperbolic Manifold.html#"Globally Hyperbolic Manifold"]]>>
<<tiddler [[include_tiddlers/Glueball.html#"Glueball"]]>>
<<tiddler [[include_tiddlers/Gluon.html#"Gluon"]]>>
<<tiddler [[include_tiddlers/Glutathione-S-transferase.html#"Glutathione-S-transferase"]]>>
<<tiddler [[include_tiddlers/Glycation.html#"Glycation"]]>>
<<tiddler [[include_tiddlers/Glycerophospholipid.html#"Glycerophospholipid"]]>>
''Gnuplot'' is a venerable open source plotting software, also part of UNIX distributions. (It already ran quite well on my $50$ Mhz laptop under Windows in the 90s and I did most of the plots of my thesis with it).
 
Since 2014 a cool [[online version|http://gnuplot.respawned.com]] is available. 

Links:
* [[Guplot Homepage|http://www.gnuplot.info/]]
* [[Gnuplot Manual (Version 4.6)|http://www.gnuplot.info/docs_4.6/gnuplot.pdf]] [[local|documents/gnuplot.pdf]]
* [[WIKIPEDIA - Gnuplot|http://en.wikipedia.org/wiki/Gnuplot]]
<<tiddler [[include_tiddlers/God and Physics.html#"God and Physics"]]>>
<<tiddler [[include_tiddlers/Godbillon-Vey Invariant.html#"Godbillon-Vey Invariant"]]>>
<<tiddler [[include_tiddlers/Golay Code.html#"Golay Code"]]>>
<<tiddler [[include_tiddlers/Golden Ratio.html#"Golden Ratio"]]>>
<<tiddler [[include_tiddlers/Goldstone Boson.html#"Goldstone Boson"]]>>
<<tiddler [[include_tiddlers/Golgi Apparatus.html#"Golgi Apparatus"]]>>
<<tiddler [[include_tiddlers/Gompertz-Makeham Law of Mortality.html#"Gompertz-Makeham Law of Mortality"]]>>
<<tiddler [[include_tiddlers/Google App Engine.html#"Google App Engine"]]>>
<<tiddler [[include_tiddlers/Google Web Toolkit.html#"Google Web Toolkit"]]>>
/***
|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]]
<<tiddler [[include_tiddlers/Grading.html#"Grading"]]>>
Given a vector $\vec {\mb X} = \{\mb A_1, \ldots, \mb A_N\}$ of elements of an algebra, having a [[inner product|Scalar Product]] $<.|.>$, the ''Gram Matrix'' is defined as
\[
\mb M(\mb A_1, \ldots,\mb A_N) \equiv \langle \vec {\mb X}| \vec {\mb X}^t \rangle = \left ( \begin{matrix}  \langle \mb  A_1,\mb  A_1\rangle & \langle \mb  A_1, \mb  A_2\rangle & \ldots & \langle \mb  A_1,\mb  A_N\rangle \\  \langle \mb  A_2, \mb  A_1\rangle & \langle \mb  A_2, \mb  A_2\rangle & \ldots & \langle \mb  A_2, \mb  A_N\rangle \\  \ldots & \ldots & \ldots & \ldots \\  \langle \mb  A_N,\mb  A_1\rangle & \langle \mb  A_N,\mb  A_2\rangle & \ldots & \langle \mb  A_N, \mb A_N\rangle \\  \end{matrix} \right )
\]
or in component form
\[
M_{ij} = \langle \mb A_i|\mb A_j \rangle
\]
!!!!Properties
* Any Gram matrix is symmetric, since inner products are symmetric.
* Given a Gram matrix the vectors $\vec {\mb X}$ are determined up to [[isometry|Isometry]].
* Given a real symmetric positive semidefinite $N \times N$-matrix $A$, then $A$ is a Gram matrix. I.e. Gram matrices provide a concrete realization of all positive semidefinite matrices.
<<tiddler [[include_tiddlers/Grand Canonical Ensemble.html#"Grand Canonical Ensemble"]]>>
Links:
* [[WolframMathWorld - Distance-Transitive Graph|http://mathworld.wolfram.com/Distance-TransitiveGraph.html]]
<<tiddler [[include_tiddlers/Graphene.html#"Graphene"]]>>
<<tiddler [[include_tiddlers/Grassmann Algebra.html#"Grassmann Algebra"]]>>
<<tiddler [[include_tiddlers/Gravastar.html#"Gravastar"]]>>
<<tiddler [[include_tiddlers/Graviphoton.html#"Graviphoton"]]>>
<<tiddler [[include_tiddlers/Gravitation.html#"Gravitation"]]>>
<<tiddler [[include_tiddlers/Gravitational Constant.html#"Gravitational Constant"]]>>
<<tiddler [[include_tiddlers/Gravitational Hydrogen Atom Analogue.html#"Gravitational Hydrogen Atom Analogue"]]>>
<<tiddler [[include_tiddlers/Gravitational Instanton.html#"Gravitational Instanton"]]>>
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Links:
* [[Prefixes & Words Based On Greek Number Names|http://home.comcast.net/~igpl/NWG.html]]
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<<tiddler [[include_tiddlers/Gregory Chaitin.html#"Gregory Chaitin"]]>>
The ''Griess Algebra'' is the weight-$2$ subspace of the [[Moonshine VOA|Monstrous Moonshine]] $V^\natural$. It is a non-associative but commutative algebra of dimension $196.884 =196.883+1$ with a positive definite invariant bilinear form.

It has $48$-dimensional associative subalgebras.

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

[[Conway|John Conway]] constructed a slightly modified version of it, called the ''Conway\-Griess Algebra''.
> But who are we to tell the genius, what he should or should not do ?
> - C. Lanczos (in William Rowan Hamilton: An Appreciation) -

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

Videos:
* [[SWR2 Wissen 31.03.2008: Die Perelman-Vermutung|http://www.ardmediathek.de/ard/servlet/content/3517136?documentId=3230166]]
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An algebra $\mathcal A$ is called a ''H*-Algebra'' (or ''Hilbert Algebra'') if it is a [[Banach algebra|Banach Algebra]] with an involution as well as a [[Hilbert space|Hilbert Space]]. Furthermore it has to satisfy conditions, relating the involution with the Hilbert space structure.

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

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

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

Links:
* [[WIKIPEDIA - H*-Algebra|http://de.wikipedia.org/wiki/H*-Algebra]]
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In 1975, Rudolf Haag, Jan T. AopuszaDski and Martin Sohnius published a proof (''Haag-AopuszaDski-Sohnius Theorem'') which shows, that by weakening the assumptions of the [[Coleman-Mandula theorem|Coleman-Mandula Theorem]] allowing both commuting and anticommuting symmetry generators, there is a nontrivial extension of the [[Poincaré algebra|Poincaré Transformation]], namely the [[supersymmetry algebra|Supersymmetry]].


Journals:
* All Possible Generators of Supersymmetries of the S Matrix (1975) - R. Haag, J. T. AopuszaDski, M. Sohnius {{t500Cite{[[jct. 959|http://scholar.google.de/scholar?hl=de&lr=&cites=7651531531072320161&um=1&ie=UTF-8&ei=JGB3TpbeE-Hm4QTapeC8DQ&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB8QzgIwAA]]}}}
A ''Hadamard Code $\operatorname{Had}(m)$'' is a (binary) $[2^m, m + 1, 2^{m?1}]$ - [[linear error-correcting code|Linear Blockcode]] which is equivalent to a [[first order Reed-Muller code|Reed-Muller Code]] $\operatorname{RM}(1,m)$. (It is a special Reed\-Muller code, having an equal number of "1"'s and "0"'s).

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

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

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

Papers:
* [[Z4-linear Hadamard and Extended Perfect Codes -D. S.Krotov|http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.0199v1.pdf]] [[pct. 51|http://scholar.google.de/scholar?cites=10926960367645449155&hl=de&as_sdt=2000]]
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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]]. 
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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]]: $w(x) \in 4\mathbb Z$. 
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>For many years whenever I got into a different topic I found out who was behind the scene, and sure enough, it was Hermann Weyl.
> - Michael Atiyah [1] -

Links:
* [[[1] An Interview with Michael Atiyah|http://kryakin.com/files/Atiyah.pdf]]
* [[WIKIPEDIA - Hermann Weyl|http://en.wikipedia.org/wiki/Hermann_Weyl]]
* [[Weylmann.com|http://www.weylmann.com/]]
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Given the real-valued function $f(x_1, x_2, \dots, x_n)$ for which it is assumed that all second partial derivatives exist, the ''Hessian Matrix'' of $f$ is defined as:
\[
(\mb H_f)_{ij}(\mb{x}) = \frac{\partial^2 f(\mb{x})}{\partial x_i\partial x_j} = \partial_i \partial_j f(\mb{x})\,\!
\]
Written out explicitly it is:
\[
\mb H_f(\mb{x}) = \begin{pmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1\,\partial x_n} \\  \\ \frac{\partial^2 f}{\partial x_2\,\partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2\,\partial x_n} \\  \\ \vdots & \vdots & \ddots & \vdots \\  \\ \frac{\partial^2 f}{\partial x_n\,\partial x_1} & \frac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{pmatrix}
\]
The ''Hessian Matrix'' describes the second order change of the function $f$. It appears as the term in the Taylor series expansion of $f$ which corresponds to this change:
\[
\Delta f(\mb{x}) =f(\mb{x}+\Delta\mb{x})\approx f(\mb{x}) + \mb J_f(\mb{x})\Delta \mb{x} +\frac{1}{2} \Delta\mb{x}^\mathrm{T} \mb H_f(\mb{x}) \Delta\mb{x}
\]
The first order change of $f$ is described by the [[Jacobian matrix|Jacobi Matrix]] $\mb J_f$.

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

Links:
* [[The Curvature of a Hessian Metric - B. Totaro|http://arxiv.org/PS_cache/math/pdf/0401/0401381v2.pdf]] [[pct. 7|http://scholar.google.de/scholar?cites=3803753193891904130&hl=de&as_sdt=2000]]
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This graph made its official appearance in the context of the construction of the [[sporadic simple group|Sporadic Group]] $HS$ which is a subgroup in the [[automorphism group|Automorphism]] of the graph.

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

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

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

Videos:
* [[Projections of the Higman-Sims Graph from the Leech Lattice|http://www.youtube.com/watch?v=neUd794Gbg0]] [[local|videos/Projections of the Higman Sims graph from the Leech lattice.wmv]]
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Links:
* [[Stanford Encyclopedia of Philosophy|http://plato.stanford.edu/entries/spacetime-holearg/]]
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A ''Hopf Algebra'' is a [[bialgebra|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:

{{center{[img(233px+, )[images/HopfAlgebra.png]]}}}
!!!! Examples
* [[Groups|Group]]
* Polynomial [[rings|Ring]]
* [[Graphs|Graph]]
* Permutations
* [[Cohomology|(Co)homology]] ring of a [[Lie Goup|Lie Group]]
* [[Universal enveloping algebra|Universal Enveloping Algebra]] of a [[Lie algebra|Lie Algebra]]
* [[Quantum groups|Quantum Group]]
and many more, demonstrating the relevance of the concept.

!!!! Generalisations
* [[Quasi-Hopf Algebras]]
* Weak Hopf algebras
* Hopf algebroids
* Hopf monads
* Hopfish algebras
* [[Braided Hopf algebras|Braided Hopf Algebra]]
For details, e.g. see [1].


Papers:
* [[Hopf Algebras, Renormalization and Noncommutative Geometry (1998) - A. Connes, D. Kreimer|http://arxiv.org/abs/hep-th/9808042]] [[local|papers/9808042v1.pdf]] {{t100Cite{[[pct. 482|http://scholar.google.de/scholar?cites=5769109935345409798&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[A Primer of Hopf Algebras (2006) - P. Cartier|http://www.math.osu.edu/~kerler.2/VIGRE/InvResPres-Sp07/Cartier-IHES.pdf]] [[local|papers/M-06-40.pdf]] [[pct. 44|http://scholar.google.de/scholar?cites=14988245623391773438&as_sdt=2005&sciodt=0,5&hl=de]]
* [[Quantum Groups and Noncommutative Geometry (2000) - S. Majid|http://arxiv.org/abs/hep-th/0006167]] [[local|papers/0006167v1.pdf]] [[pct. 39|http://scholar.google.de/scholar?cites=3737741067961111195&as_sdt=2005&sciodt=0,5&hl=de]]
* [[Hopf Algebras in Noncommutative Geometry (2001) - J. C. Várilly|http://arxiv.org/abs/hep-th/0109077]] [[local|papers/0109077v2.pdf]] [[pct. 18|http://scholar.google.de/scholar?cites=13040727115811140653&as_sdt=2005&sciodt=0,5&hl=de]]
* [[[1] On Hopf Algebras and their Generalizations (2008) - G. Karaali|http://arxiv.org/abs/math/0703441]] [[local|papers/0703441.pdf]] [[pct. 10|http://scholar.google.de/scholar?cites=2789398302986250753&as_sdt=2005&sciodt=0,5&hl=de]]
* [[The Hopf Algebra Structure of GL(1,Hq) and the Isomorphism between SPq(1) and SUq(2) - S. Celik|http://arxiv.org/abs/math/0112118]] [[local|papers/0112118v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=16877251551747315291&as_sdt=2005&sciodt=0,5&hl=de]]

Videos:
* [[Lecture 1 - Hopf Algebras and Combinatorics (Federico Ardila)|http://www.youtube.com/watch?v=FzVhjCRuXus]] - Excellent lectures (Math olymia... )
* [[Compact Quantum Groups & their Representation Categories|http://www.dailymotion.com/relevance/search/+Compact+quantum+groups+%26+their+representation+categories]] - 10 lectures by Sergey Neshveyev.

Links:
* [[WIKIPEDIA - Hopf Algebra|http://en.wikipedia.org/wiki/Hopf_algebra]]@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/HeinzHopf.jpg][Comments]]}}}&nbsp;@@
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The ''Howeââ¬â¬STucker 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 energyââ¬â¬Smomentum tensor has vanishing trace.
''Hořava-Lifshitz [[Gravity|Gravitation]]''  (short ''HL Gravity'') is a proposal for a [[power-counting renormalizable|Renormalization]] model which is [[not Lorentz invariant|Lorentz Violation]]. It is inspired by [[critical|Critical Phenomenon]] and [[condensed-matter|Condensed Matter]] systems.

An essential idea is to [[give up Lorentz invariance|Lorentz Violation]] in order to keep a [[quantum field theory|Quantum Field Theory]] finite.

A property of HL gravity, stemming from the [[running of the couplings|Renormalization Group]] effective dimension, is that the [[spectral dimension|Spectral Dimension]] $d_S$ at short scales is $d_S \approx 2$.


Papers:
* [[Quantum Gravity at a Lifshitz Point (2009) - P. Hořava|http://arxiv.org/abs/0901.3775]] [[local|papers/0901.3775v2.pdf]] {{t1000Cite{[[pct. 1030|http://scholar.google.de/scholar?hl=de&lr=&cites=3289619117987180248&um=1&ie=UTF-8&ei=1zmMTrfNN4744QTnouisCQ&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCIQzgIwAA]]}}}
* [[Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point (2009) - P. Hořava|http://arxiv.org/abs/0902.3657]] [[local|papers/0902.3657.pdf]] {{t100Cite{[[pct. 339|http://scholar.google.de/scholar?cites=14868975567692566615&as_sdt=2005&sciodt=0,5&hl=de]]}}}
* [[Quantum Gravity without Lorentz Invariance (2009) - T. P. Sotiriou, M. Visser, S. Weinfurtner|http://arxiv.org/abs/0905.2798]] [[local|papers/0905.2798v3.pdf]] {{t100Cite{[[pct. 182|http://scholar.google.de/scholar?hl=de&lr=&cites=4756775468544563841&um=1&ie=UTF-8&ei=nXaMTtntNYKG4gSW3OWsCQ&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCUQzgIwAA]]}}}

Presentations:
* [[Quantum Gravity with Anisotropic Scaling (2010) - P. Hořava|http://www.theorie.physik.uni-muenchen.de/activities/workshops/archive_2010/fundaments_gravity/horava.pdf]] [[local|presentations/horava.pdf]]

Videos:
* [[Quantum Gravity with Anisotropic Scaling and the Multicritical Universe (2011) - P. Hořava|http://vimeo.com/76445152]]
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|Name|ImageSizePlugin|
|Source|http://www.TiddlyTools.com/#ImageSizePlugin|
|Version|1.2.3|
|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
<<<
2011.09.03 [1.2.3] bypass addStretchHandlers() if no '+' suffix is used (i.e., not resizable)
2010.07.24 [1.2.2] moved tip/dragtip text to config.formatterHelpers.imageSize object to enable customization
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: 3, date: new Date(2011,9,3)};
//}}}
//{{{
var f=config.formatters[config.formatters.findByField("name","image")];
f.match="\\[[<>]?[Ii][Mm][Gg](?:\\([^,]*,[^\\)]*\\))?\\[";
f.lookaheadRegExp=/\[([<]?)(>?)[Ii][Mm][Gg](?:\(([^,]*),([^\)]*)\))?\[(?:([^\|\]]+)\|)?([^\[\]\|]+)\](?:\[([^\]]*)\])?\]/mg;
f.handler=function(w) {
	this.lookaheadRegExp.lastIndex = w.matchStart;
	var lookaheadMatch = this.lookaheadRegExp.exec(w.source)
	if(lookaheadMatch && lookaheadMatch.index == w.matchStart) {
		var floatLeft=lookaheadMatch[1];
		var floatRight=lookaheadMatch[2];
		var width=lookaheadMatch[3];
		var height=lookaheadMatch[4];
		var tooltip=lookaheadMatch[5];
		var src=lookaheadMatch[6];
		var link=lookaheadMatch[7];

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

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

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

config.formatterHelpers.imageSize={
	tip: 'SHIFT-CLICK=show full size, CTRL-CLICK=restore initial size',
	dragtip: 'DRAG=stretch/shrink, '
}

config.formatterHelpers.addStretchHandlers=function(e,stretchW,stretchH) {
	e.title=((stretchW||stretchH)?this.imageSize.dragtip:'')+this.imageSize.tip;
	e.statusMsg='width=%0, height=%1';
	e.style.cursor='move';
	e.originalW=e.style.width;
	e.originalH=e.style.height;
	e.minW=Math.max(e.offsetWidth/20,10);
	e.minH=Math.max(e.offsetHeight/20,10);
	e.stretchW=stretchW;
	e.stretchH=stretchH;
	e.onmousedown=function(ev) { var ev=ev||window.event;
		this.sizing=true;
		this.startX=!config.browser.isIE?ev.pageX:(ev.clientX+findScrollX());
		this.startY=!config.browser.isIE?ev.pageY:(ev.clientY+findScrollY());
		this.startW=this.offsetWidth;
		this.startH=this.offsetHeight;
		return false;
	};
	e.onmousemove=function(ev) { var ev=ev||window.event;
		if (this.sizing) {
			var s=this.style;
			var currX=!config.browser.isIE?ev.pageX:(ev.clientX+findScrollX());
			var currY=!config.browser.isIE?ev.pageY:(ev.clientY+findScrollY());
			var newW=(currX-this.offsetLeft)/(this.startX-this.offsetLeft)*this.startW;
			var newH=(currY-this.offsetTop )/(this.startY-this.offsetTop )*this.startH;
			if (this.stretchW) s.width =Math.floor(Math.max(newW,this.minW))+'px';
			if (this.stretchH) s.height=Math.floor(Math.max(newH,this.minH))+'px';
			clearMessage(); displayMessage(this.statusMsg.format([s.width,s.height]));
		}
		return false;
	};
	e.onmouseup=function(ev) { var ev=ev||window.event;
		if (ev.shiftKey) { this.style.width=this.style.height=''; }
		if (ev.ctrlKey)  { this.style.width=this.originalW; this.style.height=this.originalH; }
		this.sizing=false;
		clearMessage();
		return false;
	};
	e.onmouseout=function(ev) { var ev=ev||window.event;
		this.sizing=false;
		clearMessage();
		return false;
	};
}
//}}}
<<tiddler [[include_tiddlers/Immortality.html#"Immortality"]]>>
<<tiddler [[include_tiddlers/Immune System.html#"Immune System"]]>>
<<tiddler [[include_tiddlers/Incidence Matrix.html#"Incidence Matrix"]]>>
<<tiddler [[include_tiddlers/Incidence Structure.html#"Incidence Structure"]]>>
''Index of a Subgroup''
The index of a subgroup $H$ of a group $G$ (usually denoted $|G:H|$ or $[G:H]$) is the "relative size" of $H$ in respect to $G$.
If $G$ and $H$ are finite, the index is simply the quotient of the [[orders|Order]] of $G$ and $H$. By Lagrange's theorem, this number is always a positive integer.
If $G$ and $H$ are infinite, the index is defined as the number of cosets of $H$ in $G$. If $H$ is a [[normal subgroup|Normal Subgroup]] of $G$, then the index is equal to the order of the [[quotient group|Quotient Group]] $G/H$.
!!!!Example
The special [[orthogonal group|Orthogonal Group]] $SO(n)$ has index 2 in respect to the orthogonal group $O(n)$.
<<tiddler [[include_tiddlers/Index Theorem.html#"Index Theorem"]]>>
<<tiddler [[include_tiddlers/Indistinguishable Particles.html#"Indistinguishable Particles"]]>>
The ''Induced Metric'' (or [[first fundamental form|First Fundamental Form]]) of a manifold $\mathcal M$ is the assignment of an [[inner product|Scalar Product]] to each point in the manifold:
\begin{equation}
\langle \; , \; \rangle: T\mathcal M \times T\mathcal M \rightarrow \mathbb R
\end{equation}
I.e. the induced metric is the scalar product restricted to the tangent spaces of $\mathcal M$.
<<tiddler [[include_tiddlers/Induced Pluripotent Stem Cell.html#"Induced Pluripotent Stem Cell"]]>>
<<tiddler [[include_tiddlers/Inertial Induction.html#"Inertial Induction"]]>>
<<tiddler [[include_tiddlers/Infinite Number of Degrees of Freedom.html#"Infinite Number of Degrees of Freedom"]]>>
<<tiddler [[include_tiddlers/Infinity.html#"Infinity"]]>>
<<tiddler [[include_tiddlers/Inflation.html#"Inflation"]]>>
<<tiddler [[include_tiddlers/Inflationary Models.html#"Inflationary Models"]]>>
<<tiddler [[include_tiddlers/Information Loss Paradox.html#"Information Loss Paradox"]]>>
<<tiddler [[include_tiddlers/Infraparticle.html#"Infraparticle"]]>>
/***
|Name|InlineJavascriptPlugin|
|Source|http://www.TiddlyTools.com/#InlineJavascriptPlugin|
|Documentation|http://www.TiddlyTools.com/#InlineJavascriptPluginInfo|
|Version|1.9.5|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|Insert Javascript executable code directly into your tiddler content.|
''Call directly into TW core utility routines, define new functions, calculate values, add dynamically-generated TiddlyWiki-formatted output'' into tiddler content, or perform any other programmatic actions each time the tiddler is rendered.
!!!!!Documentation
>see [[InlineJavascriptPluginInfo]]
!!!!!Revisions
<<<
2009.04.11 [1.9.5] pass current tiddler object into wrapper code so it can be referenced from within 'onclick' scripts
2009.02.26 [1.9.4] in $(), handle leading '#' on ID for compatibility with JQuery syntax
|please see [[InlineJavascriptPluginInfo]] for additional revision details|
2005.11.08 [1.0.0] initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.InlineJavascriptPlugin= {major: 1, minor: 9, revision: 5, date: new Date(2009,4,11)};

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

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

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

// // GLOBAL FUNCTION: $(...) -- 'shorthand' convenience syntax for document.getElementById()
//{{{
if (typeof($)=='undefined') { function $(id) { return document.getElementById(id.replace(/^#/,'')); } }
//}}}
<<tiddler [[include_tiddlers/Instanton.html#"Instanton"]]>>
An ''$n$-dimensional Integer Lattice $\mathbb Z^n$'' (not to be confused with an [[integral lattice|Lattice]]), a.k.a ''$n$-dimensional Cubic Lattice'', is defined as
\[
\mathbb Z^n \equiv  \{(x_1,x_2,\ldots,x_i, \ldots, x_n) : x_i \in \mathbb Z\}
\]
$\mathbb Z^n$ is [[self-dual|Lattice]] and its [[kissing number|Kissing Number]] is $2n$.

The [[automorphism group|Automorphism]] $Aut(\mathbb Z^n)$ consists of all sign changes of the $n$ coordinates ($= 2^n$) and all permutations ($= n!$). Hence $N(n) \equiv \operatorname{ord} (Aut (\mathbb Z^n)) = 2^n n! = (2n)!!$. (The latter is the [[Double factorial|Double Factorial]]).
Examples:
* $N(2) = 8$
* $N(4) = 384$
* $N(8) = 10.321.920$
* $N(16) = 1.371.195.958.099.968.000$
See also: [[Sloane's A000165|http://www.research.att.com/~njas/sequences/A000165]].
<<tiddler [[include_tiddlers/Integrability.html#"Integrability"]]>>
<<tiddler [[include_tiddlers/Integral Bioctonion.html#"Integral Bioctonion"]]>>
<<tiddler [[include_tiddlers/Integral Elements.html#"Integral Elements"]]>>
<<tiddler [[include_tiddlers/Integral Octonion.html#"Integral Octonion"]]>>
<<tiddler [[include_tiddlers/Intelligent Design.html#"Intelligent Design"]]>>
<<tiddler [[include_tiddlers/Interaction Picture.html#"Interaction Picture"]]>>
<<tiddler [[include_tiddlers/Interface Theory of Perception.html#"Interface Theory of Perception"]]>>
<<tiddler [[include_tiddlers/Intermittent Fasting.html#"Intermittent Fasting"]]>>
<<tiddler [[include_tiddlers/Internet of Things.html#"Internet of Things"]]>>
<<tiddler [[include_tiddlers/Interpretation of Quantum Mechanics.html#"Interpretation of Quantum Mechanics"]]>>
<<tiddler [[include_tiddlers/Intersection Form.html#"Intersection Form"]]>>
<<tiddler [[include_tiddlers/Interstellar Medium.html#"Interstellar Medium"]]>>
Links:
* [[WIKIPEDIA - Indra's Net|http://en.wikipedia.org/wiki/Indra's_net]]
<<tiddler [[include_tiddlers/Invariant Mass.html#"Invariant Mass"]]>>
<<tiddler [[include_tiddlers/Involution.html#"Involution"]]>>
> I regard consciousness as fundamental. I regard matter as derivative from consciousness. We cannot get behind consciousness. Everything that we talk about, everything that we regard as existing, postulates consciousness.
> - [[Max Planck]] -

>The mind-stuff of the world is, of course, something more general than our individual conscious minds.... The mind-stuff is not spread in space and time; these are part of the cyclic scheme ultimately derived out of it.... It is necessary to keep reminding ourselves that all knowledge of our environment from which the world of physics is constructed, has entered in the form of messages transmitted along the nerves to the seat of consciousness.... Consciousness is not sharply defined, but fades into subconsciousness; and beyond that we must postulate something indefinite but yet continuous with our mental nature.... It is difficult for the matter-of-fact physicist to accept the view that the substratum of everything is of mental character. But no one can deny that mind is the first and most direct thing in our experience, and all else is remote inference.
> - [[Arthur Stanley Eddington]] -

In philosophy the idea that [[consciousness|Consciousness]] is fundamental has a long tradition. It comes in many guises like idealism, [[panpsychism|Panpsychism]] or [[dualism|Dualism]].
Fundamental in this context means that consciousness cannot be explained as an [[emergent phenomenon|Emergence]], based on known laws. In other words, reductionism fails.

Several variants are conceivable:
* Consciousness has to be adjoined to the known physical laws, being not derivable from them. <br><br>
* Consciousness is inherent in everything, i.e. also the known laws are fundamentally related to consciousness. In this case one can try to make postulates as to what the (mathematical) properties of consciousness are and attempt to derive the known physical laws. This approach has been pursued by D. D. Hoffman in his theory of conscious agents.

See also:
* [[Hard problem of consciousness|Hard Problem of Consciousness]]

Papers:
* [[Conscious Realism and the Mind-Body Problem (2008) - D. D. Hoffman|http://www.cogsci.uci.edu/~ddhoff/ConsciousRealism2.pdf]] [[local|papers/ConsciousRealism2.pdf]] [[pct. 14|http://scholar.google.de/scholar?cites=640471342280185625&as_sdt=2005&sciodt=0,5&hl=en]]
* [[A Theory of Consciousness (2009) - C. Koch|http://www.klab.caltech.edu/koch/CR/CR-Complexity-09.pdf]] [[local|papers/CR-Complexity-09.pdf]] [[pct. 7|http://scholar.google.de/scholar?cites=1840188834486456137&as_sdt=2005&sciodt=0,5&hl=en]]
* [[Objects of Consciousness (2014) - D. D.Hoffman, C. Prakash|http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4060643/]] [[local|papers/fpsyg-05-00577.pdf]] pct. 0

Videos:
* [[Consciousness and the Interface Theory of Perception - D. Hoffman|https://www.youtube.com/watch?v=dqDP34a-epI]]
* [[Physics From Consciousness - D. Hoffman|https://www.youtube.com/watch?v=vQUpU4W6SDs]]
* [[Donald Hoffman- Sages and Scientists 2013|https://www.youtube.com/watch?v=kZ5QeiCxOh0]]
<html><br><center><a href="http://radicaldreaming-gfx.deviantart.com/art/Infinite-Regression-193282497"><img src="images/infinite_regression.jpg" style="width: 355px; "/></a></center></html>
Can we decide if physical reality is
# finite,
# countably infinite or
# uncountably infinite ?
The answer seems to depend crucially on the nature of [[consciousness|Consciousness]] of the observer. We'll assume that consciousness is part of physical reality (no [[Cartesian dualism|Dualism]]). If so and the mind is finite, then we'll never know if 2. or 3. hold or not, for all we can extract from nature and "map" to our brain is a finite amount of information.
If the states of the brain of the observer are countably infinite then we'll never know if 3. is true or not. But if the brain's capacity is that of the [[Cantor continuum|Cardinal Number]], then we may able be find out that this is so.

So what is the conclusion ? The only question about the [[cardinality|Cardinality]] of the number of states of nature that is potentially decidable is the question if nature is uncountably infinite ? However we'll never be able to say that this is not so, for if we could this would imply that our brain's capacity is countably infinite or less and we're back at 1. or 2. from above. But we can hope for finding a "clean" example system that requires its state space to be uncountably infinite.

Where to look for that ? [[Quantum field theory|Quantum Field Theory]] appears to be based on 3., but this could be just an idealization and there is a way to describe any physical system without invoking Cantor's continuum. (E.g. infinities that require [[renormalization|Renormalization]] are mere artifacts). Is there a solid and acceptable proof that this cannot be so ? I have some doubts, because a conventional proof is based on a finite sequence of arguments and to proof something this way a finite brain will do (i.e we're back to 1.). But if we then come up with the result that reality must be infinite, our brain must have "cheated" us. But what is the alternative to a conventional proof ? Intuition, introspection ? I don't know. Maybe we are just doomed to live with uncertainty. I.e. philosophy will thrive forever :-)

See also:
* [[Infinity]]
* [[Non-computability|Non-Computability]]

© 2012 Markus Maute


Videos:
* [[BBC Horizon - Infinity|http://www.youtube.com/watch?v=FiMigmLwwTM&feature=related]]
@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/Cantor.jpg][Comments]]}}}&nbsp;@@
What is meant by ''"Is Quantum Gravity Trivial ?"'' is that [[quantum gravity|Quantum Gravity]] is "trivial" in the UV (e.g. the [[Planck scale|Planck Length]]) in the sense that there is not much phenomenology to be discovered there, but not so in the IR.

Some ideas:
* At the [[GUT]] scale physics is already quite "simple". <br><br>
* The early universe is quite "simple".<br><br>
* The [[vacuum energy|Zero Point Energy]] calculated by means of [[QFT|Quantum Field Theory]] is way to high ([[cosmological constant problem|Cosmological Constant]]). A recipe for a resolution: "Unpopulate" the vacuum, make it "simple".<br><br>
* The [[renormalization group|Renormalization Group]] flow is from a simple [[effective action|Effective Action]] at the GUT scale to a complex and messy IR action (possibly nonperturbative). Compare this also to confinement in [[QCD]].<br><br>
* There may be a [[UV fixed point|Fixed Point]]. Nothing new happens there when further "zooming in" due to self-similarity. [[Universality]] reigns, i.e. spacetime is in a universality class and the microscopic structure is unimportant. (Be there a [[superstring|Superstring Theory]] description or any other description which leads to this universality class, physically it would make no difference). <br><br>
* If the [[Bekenstein-Hawking formula|Bekenstein-Hawking Entropy]] of [[entropy|Entropy]] is correct and the bulk of entropy stems from [[dark energy|Dark Energy]] modes in the IR, there is nothing left for UV entropy. (See also [[dark energy for dummies|Dark Energy for Dummies]]).
So maybe the interesting physics is the one around us, which is already mostly discovered, because this is the realm of complexity, and a theory of quantum gravity, if we can ever come up with such thing, will just add some "boring" new physics, which on the other hand may have important philosophical implications.
''Is [[Quantum Mechanics|Quantum Mechanics]] Fundamental?'' It is not clear if this question makes sense at all, because for answering it one had to know a proper definition of the notion of "fundamental", which is probably not available. But one can weaken the question somewhat and ask if something is more fundamental than something else in the sense that the "something else" can be derived from the "something". This seems to be a well defined question. A very prominent and still controversial question is if quantum mechanics is more fundamental than [[classical physics|Classicality]] or vice versa. Most experts will probably say that the former is true. On the physics side there are many good arguments for believing that this is so (e.g. [[Bell inequalites|Bell Inequality]]), but recently experimental findings in psychology (namely in the field of [[quantum cognition|Quantum Cognition]]) have lead to a dilemma facing those who believe in the more fundamental nature of quantum mechanics, which it seems has hardly ever been clearly formulated, if at all, and therefore is the concern of what follows.

{{center{[img(451px+, )[images/digital_brain.jpg]]}}}
To get started, we make one crucial assumption, namely that the [[brain|Brain]] operates classically. This is current state of the arts of thinking. Although there have been many attempts to attest a generic quantum behaviour to the brain and even a whole discipline, known as [[quantum consciousness|Quantum Consciousness]], has emerged, it is probably save to say that there is no convincing experimental evidence for such thing yet. As the argumentation is slightly subtle, one more comment in this context: By quantum consciousness we mean a macroscopic effect which requires [[Planck's constant|Planck's Constant]] $\hbar$ for its description indicating that its origins are in the quantum mechanics as we know it. Of course if we zoom in to the molecular level of the brain the description is the usual quantum one that applies to biochemical processes. 
Now come the finding: Certain types of decision processes show a quantum-like behaviour, meaning that nearly the full fledged mathematical formalism of quantum mechanics can be applied for describing such phenomena. This is a very robust finding that has been independently replicated over and over again. It is tempting to assume that there is some conventional quantum mechanics at work, but most researchers in field of quantum cognition insist on it being just a quantum-like phenomenon, which as far as we know has nothing to do with the quantum mechanics at the atomic level. The crucial argument seems to be that no $\hbar$ shows up in such experiments.

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If we accept this point of view that there are two quantum mechanical (more or less mathematically equivalent) descriptions applying to different phenomena in nature, the one having the alleged status of being the more fundamental description than the classical one of the macroscopic world, but the other one being "on top" of the classical description of the brain, suggesting that it is derived from a classical description of the brain and hence is less fundamental, we have the dilemma alluded to. Formally the contradiction is this:
 
Conventional quantum mechanics of atoms and molecules (QM) ⇒ classical physics (CP)
Classical physics of the brain (CP) ⇒ quantum-like processes (QML)

If on the mathematical (descriptive) level QM = QML, we face a contradiction.

There have been suggestions how noncommutativity of observables and hence quantum behaviour could emerge in classical systems (e.g. [1]). If this could be backed  experimentally the fundamental nature of conventional quantum mechanics would be severely at stake.

Interestingly the quantum mechanical concept of [[complementarity|Complementarity]], which [[Niels Bohr]] introduced into quantum mechanics, was not his original idea, rather it was taken from earlier works of  William James in psychology. 

See also:
* [[Deterministic quantum mechanics|Deterministic Quantum Mechanics]]
* [[Emergent quantum mechanics|Emergent Quantum Mechanics]]

Papers:
* [[[1] Complementarity in Classical Dynamical Systems (2004) - P. beim Graben, H. Atmanspacher|http://arxiv.org/abs/nlin/0407046]] [[local|papers/0407046v5.pdf]] [[pct. 26|http://scholar.google.de/scholar?cites=16951494988481892183&as_sdt=2005&sciodt=0,5&hl=en]]
> It's like figuring out what is in a room behind a locked door, and we don't even know if there's a room there.
> - Roberto Casadio -

The big problem in [[quantum gravity|Quantum Gravity]] seems to be that we do not know what to [[quantize|Quantization]]. Usually one takes a classical geometry and quantizes it. In case of [[General Relativity]] this would be a [[Riemannian geometry|Riemann Space]], but this doesn't work. What can we make out of this ? Should we quantize [[gravity|Gravitation]] at all or replace Riemannian geometry by something else, e.g. [[metric affine spaces|Metric Affine Space]], [[noncommutative spaces|Noncommutative Geometry]] ? How can we know what the classical geometry should be ? One thing that could help us out is the observation of a violations of General Relativity. But at the moment there seem to be no hints thereof, whatsoever.

Could it be that there is just no theory of quantum gravity (at least for alle practical purposes) ?
Here are some arguments/lines of thought in this respect:
* The spectral lattice of spacetime is [[aperiodic|Tiling]] and the spectrum is [[non-computable|Non-Computability]]. <br><br>
* The spectral lattice could be a superposition of many lattices, making a resolution impossible.<br><br>
* Not everything in physics needs to be quantized. For example, does it make sense to quantize temperature ?<br><br>
* As spacetime has a Lorentzian metric, the spectrum could be continuous, making it indistinguishable from a smooth classical background.<br><br>
* Spacetime is secondary, all degrees of freedom are already encoded by matter.


Papers:
* [[Is Quantum Gravity Necessary? (2008) - S. Carlip|http://arxiv.org/abs/0803.3456]] [[local|papers/0803.3456.pdf]] [[pct. 46|http://scholar.google.de/scholar?cites=1346475142959222848&as_sdt=2005&sciodt=0,5&hl=en]]
* [[Are the Spectra of Geometrical Operators in Loop Quantum Gravity Really Discrete? (2007) - B. Dittrich, T. Thiemann|http://arxiv.org/abs/0708.1721]] [[local|papers/0708.1721v2.pdf]] [[pct. 39|http://scholar.google.com/scholar?cites=6904359822207054749&as_sdt=2005&sciodt=0,5&hl=de]]
* [[Is Quantum Gravity Necessary? - J. Mattingly|http://www.jmattingly.org/papers/IsGravityNecessarilyQuantized.proofs.pdf]] [[local|papers/IsGravityNecessarilyQuantized.proofs.pdf]] [[pct. 16|http://scholar.google.de/scholar?cites=9891950535489077454&as_sdt=2005&sciodt=0,5&hl=en]]

Links:
* [[The World on a String (2004) - F. Dyson|http://www.nybooks.com/articles/archives/2004/may/13/the-world-on-a-string/?pagination=false]]
<<tiddler [[include_tiddlers/Isometry.html#"Isometry"]]>>
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!!!![[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. $q(v) < 0, \; \forall \,v$.
<<tiddler [[include_tiddlers/JAlgebra.html#"JAlgebra"]]>>
<<tiddler [[include_tiddlers/JHyperComplex.html#"JHyperComplex"]]>>
<<tiddler [[include_tiddlers/JSON.html#"JSON"]]>>
<<tiddler [[include_tiddlers/JUnit.html#"JUnit"]]>>
{{center{[img(485px+, )[images/JacobianDeterminant.gif]]}}}
The ''Jacobi Determinant'' is the determinant of a square [[Jacobi matrix|Jacobi Matrix]]:
\[
\det (\mb J_{\mb f}(\mb{x})) = \begin{vmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\  \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_2} \\  \\ \vdots & \vdots & \ddots & \vdots \\  \\ \frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_n} \end{vmatrix}
\]
Its value gives the following information about the behavior of the function $\mb f$ in the point $\mb x$:
*  $> 0$  orientation preserving
* $< 0$ orientation reversing
* $= 0$ not invertible
* $= 1$ volume preserving

''General Relativity''
For a coordinate transformation $x_\mu \mapsto x'_\nu(\mb x) $ in general relativity, the Jacobi determinant $\det (\mb J_{\mb{x'}}(\mb{x}))$ can be written as
\[
\det (J_{\mb{\mb x'}}(\mb{x})) = \epsilon_{\mu\nu\rho\sigma} \frac{\partial x'^\mu}{\partial x^0} \frac{\partial x'^\nu}{\partial x^1} \frac{\partial x'^\rho}{\partial x^2} \frac{\partial x'^\sigma}{\partial x^3}
\]
Links:
* [[The Jacobian Determinant - Jeff Knisley|http://math.etsu.edu/MultiCalc/Chap3/Chap3-5/index.htm]] - Doesn't work well with firefox, better use other browser.
<<tiddler [[include_tiddlers/Jacobi Matrix.html#"Jacobi Matrix"]]>>
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<<tiddler [[include_tiddlers/JavaScript Library.html#"JavaScript Library"]]>>
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<<tiddler [[include_tiddlers/John Conway.html#"John Conway"]]>>
<<tiddler [[include_tiddlers/John Von Neumann.html#"John Von Neumann"]]>>
<<tiddler [[include_tiddlers/Jordan Algebra.html#"Jordan Algebra"]]>>
<<tiddler [[include_tiddlers/Jordan Identity.html#"Jordan Identity"]]>>
<<tiddler [[include_tiddlers/Jordan Structure.html#"Jordan Structure"]]>>
<<tiddler [[include_tiddlers/Jordan Triple Product.html#"Jordan Triple Product"]]>>
<<tiddler [[include_tiddlers/Jordan Triple System.html#"Jordan Triple System"]]>>
<<tiddler [[include_tiddlers/Josephson Junction.html#"Josephson Junction"]]>>
<<tiddler [[include_tiddlers/Julian Schwinger.html#"Julian Schwinger"]]>>
<<tiddler [[include_tiddlers/K-Theory.html#"K-Theory"]]>>
<<tiddler [[include_tiddlers/KO-Theory.html#"KO-Theory"]]>>
<<tiddler [[include_tiddlers/KP Equation.html#"KP Equation"]]>>
<<tiddler [[include_tiddlers/Kalb-Ramond Field.html#"Kalb-Ramond Field"]]>>
<<tiddler [[include_tiddlers/Kardashev Scale.html#"Kardashev Scale"]]>>
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 Diracââ¬â¢s electron theory; practically all the definitions of physical quantities like spin, magnetic moment etc. have their exact counterpart.
<<tiddler [[include_tiddlers/Kerr Effect.html#"Kerr Effect"]]>>
<<tiddler [[include_tiddlers/Ketogenic Diet.html#"Ketogenic Diet"]]>>
<<tiddler [[include_tiddlers/Khovanov Homology.html#"Khovanov Homology"]]>>
A ''Kikkawa Space'' is a manifolds with affine connection such that all [[geodesic loops|Geodesic Loop]] of some neighborhood (at some point) are [[right-monoalternative|Alternative Algebra]].

Papers:
* [[On Kikkawa Spaces - L. Sabinina|http://www.iop.org/EJ/article/0036-0279/58/4/L13/RMS_58_4_L13.pdf?request-id=79c53b95-8005-40d6-b9d1-08c30b8dab36]]
A ''Kirkman Triple System'' of order $v$ (shortly denoted by $KTS(v)$) is a [[resolvable|Resolvable Design]] [[Steiner triple system|Steiner Triple System]] $STS(v)$. The case $v = 15$ became known as [[Kirkman's schoolgirl problem|Kirkman's Schoolgirl Problem]].

In 1971 D. K. Ray\-Choudhury and R. M. Wilson proved that at least one Kirkman triple system for every (non-negative) order exists, provided a STS exists for that order.

The smallest possibility has $v = 3$ with exactly one block and one parallel class, hence it is trivial.
For $v=9$ (which is related to the $3 \times 3$ magic square [1]) there is a single unique (up to an isomorphism) solution, while there are $7$ different systems for $v=15$.

Links:
* [[[1] WIKIPEDIA - Magic Square|http://en.wikipedia.org/wiki/Magic_square]]
<<tiddler [[include_tiddlers/Kirkman's Schoolgirl Problem.html#"Kirkman's Schoolgirl Problem"]]>>
<<tiddler [[include_tiddlers/Kirlian Photography.html#"Kirlian Photography"]]>>
<<tiddler [[include_tiddlers/Kissing Number.html#"Kissing Number"]]>>
<<tiddler [[include_tiddlers/Klaus Volkamer.html#"Klaus Volkamer"]]>>
<<tiddler [[include_tiddlers/Kleiber's Law.html#"Kleiber's Law"]]>>
<<tiddler [[include_tiddlers/Klein Four-group.html#"Klein Four-group"]]>>
Papers:
* [[History and Physics of the Klein Paradox (1999) - A. Calogeracos, N. Dombey|http://arxiv.org/PS_cache/quant-ph/pdf/9905/9905076v1.pdf]] [[local|papers/9905076v1.pdf]] [[pct. 43|http://scholar.google.de/scholar?cites=5733264893586937786&as_sdt=2005&sciodt=2000&hl=de]]
<<tiddler [[include_tiddlers/Klein-Gordon Equation.html#"Klein-Gordon Equation"]]>>
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<<tiddler [[include_tiddlers/Kleinfeld Identities.html#"Kleinfeld Identities"]]>>
<<tiddler [[include_tiddlers/Knot Complement.html#"Knot Complement"]]>>
<<tiddler [[include_tiddlers/Knot Theory.html#"Knot Theory"]]>>
<<tiddler [[include_tiddlers/Kochen-Specker Theorem.html#"Kochen-Specker Theorem"]]>>
<<tiddler [[include_tiddlers/Kolmogorov Complexity.html#"Kolmogorov Complexity"]]>>
<<tiddler [[include_tiddlers/Kolmogorov-Sinai Entropy.html#"Kolmogorov-Sinai Entropy"]]>>
<<tiddler [[include_tiddlers/Korteweg-De Vries Equation.html#"Korteweg-De Vries Equation"]]>>
<<tiddler [[include_tiddlers/Krein Space.html#"Krein Space"]]>>
The ''Kretschmann Scalar K'' for an n-dimensional Riemann manifold is given by
\begin{equation}
K = R_{\mu\nu\rho\sigma} \, R^{\mu\nu\rho\sigma} = C_{\mu\nu\rho\sigma} \, C^{\mu\nu\rho\sigma} +\frac{4}{d-2} R_{\mu\nu}\, R^{\mu\nu} - \frac{2}{(n-1)(n-2)}R^2
\end{equation}
with $ C_{\mu\nu\rho\sigma}$ the [[Weyl tensor|Weyl Tensor]], $R_{\mu\nu}$ the [[Ricci tensor|Ricci Tensor]] and $R$ the [[Ricci scalar|Ricci Scalar]].
In 4 dimensions one has:
\begin{equation}
K = C_{\mu\nu\rho\sigma} \, C^{\mu\nu\rho\sigma} +\frac{1}{2} R_{\mu\nu}\, R^{\mu\nu} - \frac{1}{3}R^2
\end{equation}

<<tiddler [[include_tiddlers/Kähler Manifold.html#"Kähler Manifold"]]>>
<<tiddler [[include_tiddlers/Kähler-Dirac Equation.html#"Kähler-Dirac Equation"]]>>
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<<tiddler [[include_tiddlers/LOFAR.html#"LOFAR"]]>>
<<tiddler [[include_tiddlers/LSZ Reduction Formula.html#"LSZ Reduction Formula"]]>>
<<tiddler [[include_tiddlers/Lagrange's Theorem.html#"Lagrange's Theorem"]]>>
<<tiddler [[include_tiddlers/Lambda 16 Lattice.html#"Lambda 16 Lattice"]]>>
<<tiddler [[include_tiddlers/Lambda Calculus.html#"Lambda Calculus"]]>>
<<tiddler [[include_tiddlers/Lambda-CDM Model.html#"Lambda-CDM Model"]]>>
<<tiddler [[include_tiddlers/Landau Ghost.html#"Landau Ghost"]]>>
<<tiddler [[include_tiddlers/Landau Pole.html#"Landau Pole"]]>>
<<tiddler [[include_tiddlers/Landauer's Principle.html#"Landauer's Principle"]]>>
Links:
* [[WIKIPEDIA - Langlands Program|http://en.wikipedia.org/wiki/Langlands_program]]
<<tiddler [[include_tiddlers/Laplace Equation.html#"Laplace Equation"]]>>
<<tiddler [[include_tiddlers/Laplace Transform.html#"Laplace Transform"]]>>
The ''Laplace\-Beltrami Operator'' $\square$ is a generalisation of the [[Laplace operator|Laplace Equation]] for [[Riemannian|Riemann Space]] and [[pseudo-Riemannian manifolds|Pseudo-Riemannian Space]].
It is given by:
\[
 \square_g \equiv \Delta_g \equiv \ \frac{1} {\sqrt{g}} \ \partial_\mu \left ( \sqrt{g} g^{\mu\nu} \partial_{\nu}  \right )
\]
<<tiddler [[include_tiddlers/Large Hadron Collider.html#"Large Hadron Collider"]]>>
<<tiddler [[include_tiddlers/Large Numbers Hypothesis.html#"Large Numbers Hypothesis"]]>>
<<tiddler [[include_tiddlers/Large Numbers and the Wavefunction Collapse.html#"Large Numbers and the Wavefunction Collapse"]]>>
<<tiddler [[include_tiddlers/Large Telescopes.html#"Large Telescopes"]]>>
<<tiddler [[include_tiddlers/Laser.html#"Laser"]]>>
<<tiddler [[include_tiddlers/Late-life Mortality Convergence.html#"Late-life Mortality Convergence"]]>>
!Summary
Here we initialise the ~MathJax environment.

The Init section of this tiddler is rendered automatically when the site is loaded.  This effect was achieved by adding the line
{{{
<div style='display:none' macro='tiddler LatexNucleus##Init'></div>
}}}
to [[PageTemplate]].  It is important to add this line before the <div id='displayArea'> tag so that mathematics which appears in a default tiddle is rendered correctly.  The goal of Init is to initialise the ~MathJax environment and then call \SaveGlobals so that this freshly initialised state can be returned to at any time by use of the command \Clean.

!Init
$
%We hook the usual macro-creating Latex commands up to Richard's javascript routines.
\def\newcommand{\MetaNewCom}
\def\renewcommand{\MetaNewCom}
\def\def{\MetaDef}

%Here we make copies of the standard Latex commands which we may displace.
\let\LaTeXepsilon\epsilon
\let\LaTeXsubset\subset
\let\LaTeXrestriction\restriction

%We transclude all of the macroset tiddlers so that the macroset macros are always available.  If you create a new macroset then you'll need to add the appropriate line here so that it is detected when the site loads.
$<<tiddler BasicMacros##Code>>$

%We also include a dummy macroset, \LocalMacros, which can be used for the main extra macros of a particular tiddler.  This can be useful, for example, if you want to comment on someones work using some of your personal macros but don't want to replace any of the local definitions.
\def\LocalMacros{}

%We preload the macrosets which are considered global.
\BasicMacros
\BasicMacroAbbreviations
\BasicMacrosSupplement

%And finally we save the current state of the ~MathJax definition tables.  This is the state we return to whenever \Clean is called.  We also call \Clean here to keep the ~MathJax environment in good condition when this tiddler is viewed.  This is put in a separate mathematics block because \SaveGlobals will throw an error on every call other than the first and this error will prevent \Clean from being called if \Clean is later on in the same block.
\SaveGlobals
$$
\Clean
$
<<tiddler [[include_tiddlers/Latin Hypercube.html#"Latin Hypercube"]]>>
A ''Lattice'' is an algebra $\mathcal A$ with two operations ''$\wedge$'' (called ''Meet'' or ''And'') and ''$\vee$'' (called ''Join'' or ''Or'') for which, $\forall \mb A, \mb B, \mb C \in \mathcal A$, the following relations hold:
|!Relations|!Laws|
|$\mb A \wedge \mb A = \mb A$; $\;\mb A \vee \mb A = \mb A\quad$|[[Idempotency]]|
|$\mb A  \wedge \mb B = \mb B  \wedge \mb A$; $\;\mb A \vee \mb B = \mb B \vee \mb A\quad$ |''Commutativity''|
|$(\mb A  \wedge \mb B)  \wedge \mb C = \mb A  \wedge (\mb B  \wedge \mb C)$; $\;(\mb A \vee \mb B) \vee \mb C = \mb A \vee (\mb B \vee \mb C)$|''Associativity''|
|$\mb A \vee (\mb A \wedge \mb B) = \mb A$; $\;\mb A \wedge (\mb A \vee \mb B) = \mb A\quad$|''Absorption''|
<<tiddler [[include_tiddlers/Lattice Gas Cellular Automaton.html#"Lattice Gas Cellular Automaton"]]>>
<<tiddler [[include_tiddlers/Lattice QCD.html#"Lattice QCD"]]>>
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<<tiddler [[include_tiddlers/Left- and Right Translation.html#"Left- and Right Translation"]]>>
<<tiddler [[include_tiddlers/Left-right Symmetry.html#"Left-right Symmetry"]]>>
<<tiddler [[include_tiddlers/Leggett-Garg Inequality.html#"Leggett-Garg Inequality"]]>>
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<<tiddler [[include_tiddlers/Lepton Number.html#"Lepton Number"]]>>
<<tiddler [[include_tiddlers/Leptoquark.html#"Leptoquark"]]>>
The fundamental theorem of [[Riemannian geometry|Riemann Space]] states: On a Riemannian manifold there is a unique [[connection|Connection]] which is [[torsion-free|Torsion]] and [[compatible with the metric|Metric Compatibility]].

This connection is called the ''Levi\-Civita Connection'' (a.k.a. ''Riemannian\-'' or ''Christoffel Connection''). The connection coefficients are at times expressed by means of the [[Christoffel symbols|Christoffel Symbols]].

The relation with the [[metric tensor|Metric Tensor]] is given by:
\[
\Chr{\lambda}{\mu\nu} = \frac{1}{2} g^{\lambda \rho} (\partial_\mu g_{\rho \nu}  + \partial_\nu g_{\rho\mu}  - \partial_\rho g_{\mu \nu}  )
\]
which means that the metric tensor completely determines the Christoffel coefficients and vice versa which is characteristic of a [[Riemann manifold|Riemann Space]].

The Levi\-Civita connection is a symmetric connection, therefore
\[
\Chr{\lambda}{\mu\nu} = \Chr{\lambda}{\nu\mu}
\]
Due to its symmetry, it consists of $40 = 10\cdot 4$ independent components.


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<<tiddler [[include_tiddlers/Levinthal Paradox.html#"Levinthal Paradox"]]>>
The ''Lewis\-Tolman Lever Paradox'' (or ''Right\-Angle Lever Paradox'') is one of the first paradoxes of special relativity proposed in 1909.

Papers:
* [[Right Angle Lever Paradox - J. C. Nickerson, R. T. McAdory|http://polaris.deas.harvard.edu/galileo/images/material/1469/351/reltorque.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=14416921559306520612&hl=de]]
* [[The Lewis-Tolman Lever Paradox - J. W. Butler|http://www.physics.princeton.edu/~mcdonald/examples/mechanics/butler_ajp_38_360_70.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=7742882603248677411&hl=de]]
* [[Covariant Formulation of Hooke's Law - O. Gron|http://www.physics.princeton.edu/~mcdonald/examples/mechanics/gron_ajp_49_28_81.pdf]] pct. 0
* [[The Lack of Rotation in a Moving Right Angle Lever - J. Franklin|http://arxiv.org/PS_cache/arxiv/pdf/0805/0805.1196v2.pdf]] pct. 0
* [[Relativistic Angular Momentum - N. Menicucci|http://panda.unm.edu/Courses/Finley/P495/TermPapers/relangmom.pdf]] pct. 0
<<tiddler [[include_tiddlers/Lie Algebra.html#"Lie Algebra"]]>>
The ''Lie Derivative'' $\mathcal L_V$ in respect to two vectors $W$ and $V$ is defined by:
\[
\mathcal{L}_V(W)_\mu = V^\nu D_\nu W_\mu ? W^\nu D_\nu V_\mu
\]
with $D$ the [[covariant derivative|Covariant Derivative]].
The Lie derivative can be generalized involving tensors.
<<tiddler [[include_tiddlers/Lie Group.html#"Lie Group"]]>>
<<tiddler [[include_tiddlers/Lie Triple System.html#"Lie Triple System"]]>>
>It is Lie's most remarkable insight that the bracket is determined by the degree two terms in the Taylor expansion of the product, and that is suffices as a basis for the entire local theory.
> - K. H. Hofmann, K. Strambach -

''I. Theorem''
Each local real analytic [[Lie group|Lie Group]] determines a [[Lie algebra|Lie Algebra]] in its [[tangent space|Tangent Algebra]] at the identity element.

''III. Theorem (also called: "Converse Lie Theorem")''
Any finite-dimensional Lie algebra over the real numbers is the Lie algebra associated to some (unique) Lie group.

''Generalisation'' of Lie's 3-rd theorem for [[quasigroups|Quasigroup]]:
In general an [[Akivis algebra|Akivis Algebra]] does not determine a local differentiable [[quasigroup|Quasigroup]] uniquely.
However for [[Malcev-|Malcev Algebra]] and [[Bol- algebras|Bol Algebra]] (which are particular cases of Akivis algebras) local Moufang and Bol quasigroups are determined in a unique way respectively.
As for [[monoassociative|Monoassociativity]] quasigroups, their local Akivis algebras do not determine them uniquely. However, a prolonged Akivis algebra defined in a fourth-order differential neighborhood determines a monoassociative quasigroup uniquely. Note that besides the operations of commutation and association, a prolonged Akivis algebra has two quaternary operations called quaternators. It is therefore a so called binary-ternary-quaternary algebra.
A key ingedient in the proof of Lie's Third Theorem is the [[(generalized) Baker Campbell Hausdorff formula|Baker Campbell Hausdorff Formula (BCH)]].
<<tiddler [[include_tiddlers/Life Extension.html#"Life Extension"]]>>
<<tiddler [[include_tiddlers/Life Extension Companies.html#"Life Extension Companies"]]>>
<<tiddler [[include_tiddlers/Light Cone.html#"Light Cone"]]>>
<<tiddler [[include_tiddlers/Light Cone Coordinates.html#"Light Cone Coordinates"]]>>
<<tiddler [[include_tiddlers/Light Cone Quantization.html#"Light Cone Quantization"]]>>
<<tiddler [[include_tiddlers/Light Scattering.html#"Light Scattering"]]>>
<<tiddler [[include_tiddlers/Lindblad Equation.html#"Lindblad Equation"]]>>
<<tiddler [[include_tiddlers/Linear Blockcode.html#"Linear Blockcode"]]>>
<<tiddler [[include_tiddlers/Linearizability.html#"Linearizability"]]>>
<<tiddler [[include_tiddlers/Lipschitz Units.html#"Lipschitz Units"]]>>
<<tiddler [[include_tiddlers/Liquid Biopsy.html#"Liquid Biopsy"]]>>
<<tiddler [[include_tiddlers/Liquid Crystal.html#"Liquid Crystal"]]>>
<<tiddler [[include_tiddlers/Lisi's E8 Model.html#"Lisi's E8 Model"]]>>
<<tiddler [[include_tiddlers/Lithium.html#"Lithium"]]>>
<<tiddler [[include_tiddlers/Lithium Problem.html#"Lithium Problem"]]>>
<<tiddler [[include_tiddlers/Logic.html#"Logic"]]>>
<<tiddler [[include_tiddlers/Logic Gate.html#"Logic Gate"]]>>
<<tiddler [[include_tiddlers/Longevity Gene.html#"Longevity Gene"]]>>
<<tiddler [[include_tiddlers/Longevity Pathway.html#"Longevity Pathway"]]>>
<<tiddler [[include_tiddlers/Loop.html#"Loop"]]>>
<<tiddler [[include_tiddlers/Loop Quantum Gravity.html#"Loop Quantum Gravity"]]>>
<<tiddler [[include_tiddlers/Lorentz Group.html#"Lorentz Group"]]>>
<<tiddler [[include_tiddlers/Lorentz-Dirac Equation.html#"Lorentz-Dirac Equation"]]>>
<<tiddler [[include_tiddlers/Lorentzian Lattice.html#"Lorentzian Lattice"]]>>
<<tiddler [[include_tiddlers/Lorentzian Spectral Triple.html#"Lorentzian Spectral Triple"]]>>
''Lovelock Theory of Gravity'' (short ''Lovelock Gravity'') represents a unique class of higher curvature gravity theories with field equations that do not involve derivatives of the [[Riemann curvature tensor|Riemann Tensor]].

It was introduced by D. Lovelock in 1971 and can be regarded as a generalization of [[Einstein's theory of general relativity|General Relativity]] to higher dimensions. In dimension $D= 3$ and $ D= 4$ Lovelock's theory coincides with Einstein's theory, but in higher dimension both theories are different.
For $D > 4$ Einstein gravity can be thought of as a particular case of Lovelock gravity since the [[Einstein-Hilbert action|Einstein-Hilbert Action]] is one of several terms that constitute the Lovelock action.

Links:
* [[WIKIPEDIA - Lovelock Theory of Gravity|http://en.wikipedia.org/wiki/Lovelock_theory_of_gravity]]
<<tiddler [[include_tiddlers/Low Energy Quantum Gravity.html#"Low Energy Quantum Gravity"]]>>
<<tiddler [[include_tiddlers/Low Protein Diet.html#"Low Protein Diet"]]>>
A ''Low-density Parity-check Code'' (''LDPC Code'' or ''Gallagar Code'') is a [[linear error-correcting code|Linear Blockcode]] which was introduced in 1960 by Robert G. Gallager.

LDPC codes have parity-check matrices with a low density of "1's" (i.e. they are "sparse"), which renders low complexity decoding and leads to simple implementations.

It has been shown that these codes can achieve a good error performance that is very close to Shannon limit.

A special class are [[finite geometry|Finite Geometry]] LDPC codes, based on [[euclidean|Affine Geometry]] and [[projective geometries|Projective Space]].
One distinguishes four classes of such codes:
# Type\-I Euclidean geometry (EG)\-LDPC codes,
# type\-II EG\-LDPC codes,
# type\-I projective geometry (PG)\-LDPC codes,
# type\-II PG\-LDPC codes.

Papers:
* [[Low Density Parity Check Codes Based on Finite Geometries: A Rediscovery and New Results - Y. Kou, S. Lin, M. P.C. Fossorier|http://www.stanford.edu/class/ee379b/class_reader/ucd1.pdf]] [[local|papers/ucd1.pdf]] {{t500Cite{[[pct. 598|http://scholar.google.de/scholar?cites=10652403661149396541&hl=de]]}}}
* [[Structured Low-Density Parity-Check Codes - J. M. F. Moura, J. Lu, H. Zhang|http://www.ece.cmu.edu/~moura/papers/spm-jan04-moura-lu-zhang-ieeeexplore.pdf]] [[local|papers/spm-jan04-moura-lu-zhang-ieeeexplore.pdf]] [[pct. 35|http://scholar.google.de/scholar?cites=12666190617458772897&hl=de]]
<<tiddler [[include_tiddlers/Lucas' Problem.html#"Lucas' Problem"]]>>
<<tiddler [[include_tiddlers/Ludwig Wittgenstein.html#"Ludwig Wittgenstein"]]>>
<<tiddler [[include_tiddlers/Lyman-alpha Forest.html#"Lyman-alpha Forest"]]>>
<<tiddler [[include_tiddlers/Lyth Bound.html#"Lyth Bound"]]>>
<<tiddler [[include_tiddlers/M-Algebra.html#"M-Algebra"]]>>
<<tiddler [[include_tiddlers/M-Theory.html#"M-Theory"]]>>
!!!![[MAGMA|http://magma.maths.usyd.edu.au/calc/]]^^[[Help|MAGMA]]^^
One if the outstanding features of MAGMA is that it allows for the generation of [[lattices|Lattice]], a feature that is often missing in other computer algebra systems.

Links:
* [[MAGMA Computational Algebra System Home Page|http://magma.maths.usyd.edu.au/]]
* [[MAGMA Online Calculator|http://magma.maths.usyd.edu.au/calc/]]
* [[WIKIPEDIA - MAGMA Computer Algebra System|http://en.wikipedia.org/wiki/Magma_computer_algebra_system]]
* [[Solving Problems with MAGMA - W. Bosma, J. Cannon, C. Playoust, A. Steel|http://www.dms.auburn.edu/research/manuals/magma2.6/examples.pdf]]  [[local|lectures/SolvingProblemsWithMAGMA.pdf]] [[lct. 10|http://scholar.google.com/scholar?hl=de&lr=&cites=9486123372688473527&um=1&ie=UTF-8&ei=YuE2S46dNKfesAbnzbHSCA&sa=X&oi=science_links&resnum=10&ct=sl-citedby&ved=0CDkQzgIwCTgK]]
* [[Handbook of MAGMA Functions|http://www.msri.org/about/computing/docs/magma/]] [[local|documents/MAGMA]]
** [[Lattices|http://www.msri.org/about/computing/docs/magma/html/text826.htm]] [[local|documents/MAGMA/html/text826.htm]]
** [[Coding Theory|http://www.msri.org/about/computing/docs/magma/html/part16.htm]] [[local|documents/MAGMA/html/part16.htm]]
** [[Hadamard Matrices|http://www.msri.org/about/computing/docs/magma/html/text1517.htm]] [[local|file:///E:/Trajectory/documents/MAGMA/html/text1517.htm]]
** [[Incidence Structures and Designs|http://www.msri.org/about/computing/docs/magma/html/text1502.htm]] [[local|file:///E:/Trajectory/documents/MAGMA/html/text1502.htm]]

Examples:
* [[Applied Abstract Algebra - D. Joyner, R. Kreminski, J. Turisco|http://www.usna.edu/Users/math/wdj/book/book.html]]
Links:
* [[Maxima website|http://maxima.sourceforge.net]]
<<tiddler [[include_tiddlers/MOND Derivations.html#"MOND Derivations"]]>>
<<tiddler [[include_tiddlers/MOND and Cosmology.html#"MOND and Cosmology"]]>>
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
\[
W(C^\perp;x,y) = \frac{1}{\operatorname{ord}(C)} W(C;y-x,y+x).
\]
<<tiddler [[include_tiddlers/Mach's Principle.html#"Mach's Principle"]]>>
<<tiddler [[include_tiddlers/Magma.html#"Magma"]]>>
<<tiddler [[include_tiddlers/Magnet Ball.html#"Magnet Ball"]]>>
<<tiddler [[include_tiddlers/Magnetoception.html#"Magnetoception"]]>>
<<tiddler [[include_tiddlers/Magnetoreception.html#"Magnetoreception"]]>>
<<tiddler [[include_tiddlers/Magnus Expansion.html#"Magnus Expansion"]]>>
<<tiddler [[include_tiddlers/Majorana Mass Matrix.html#"Majorana Mass Matrix"]]>>
<<tiddler [[include_tiddlers/Majorana Spinor.html#"Majorana Spinor"]]>>
<<tiddler [[include_tiddlers/Malament-Hogarth Spacetime.html#"Malament-Hogarth Spacetime"]]>>
<<tiddler [[include_tiddlers/Malcev Algebra.html#"Malcev Algebra"]]>>
<<tiddler [[include_tiddlers/Malcev Identity.html#"Malcev Identity"]]>>
<<tiddler [[include_tiddlers/Manfred Eigen.html#"Manfred Eigen"]]>>
<<tiddler [[include_tiddlers/Many Worlds Interpretation.html#"Many Worlds Interpretation"]]>>
<<tiddler [[include_tiddlers/Mappings of the Dirac Equation.html#"Mappings of the Dirac Equation"]]>>
<<tiddler [[include_tiddlers/Margolus-Levitin Theorem.html#"Margolus-Levitin Theorem"]]>>
<<tiddler [[include_tiddlers/Markov Process.html#"Markov Process"]]>>
<!--{{{-->
<link rel='alternate' type='application/rss+xml' title='RSS' href='index.xml' />
<!--}}}-->

<style type="text/css">#contentWrapper {display:none;}</style><div id="SplashScreen" style="border: 3px solid #ccc; display: block; text-align: center; width: 320px; margin: 100px auto; padding: 50px; color:#000; font-size: 28px; font-family:Tahoma; background-color:#eee;"><b>Trajectory of the Universe</b><br><small> - Notes on Mathematics, Physics, Philosophy and all that ...</small><br> &nbsp; &nbsp; <small> <small> 1.5 MB </small></small> loading<blink> <FONT SIZE="4"><br> (... this may take a moment). </FONT> </blink>
<center><br><img src="images/trajectory.jpg" style="width: 320px;"/></center>
</blink><span style="font-size: 20px; color:red;"><small> Requires Javascript.</small></span></div>
<<tiddler [[include_tiddlers/Markus' Wisdom.html#"Markus' Wisdom"]]>>
<<tiddler [[include_tiddlers/Mars.html#"Mars"]]>>
<<tiddler [[include_tiddlers/Martinus J. G. Veltman.html#"Martinus J. G. Veltman"]]>>
<<tiddler [[include_tiddlers/Mass Dimension.html#"Mass Dimension"]]>>
<<tiddler [[include_tiddlers/Mass Gap.html#"Mass Gap"]]>>
<<tiddler [[include_tiddlers/Mass Inflation.html#"Mass Inflation"]]>>
<<tiddler [[include_tiddlers/Mass Shell Condition.html#"Mass Shell Condition"]]>>
<<tiddler [[include_tiddlers/Massive Graviton.html#"Massive Graviton"]]>>
<<tiddler [[include_tiddlers/Massive Photon.html#"Massive Photon"]]>>
<<tiddler [[include_tiddlers/Master Equation.html#"Master Equation"]]>>
/***
|Name|MatchTagsPlugin|
|Source|http://www.TiddlyTools.com/#MatchTagsPlugin|
|Documentation|http://www.TiddlyTools.com/#MatchTagsPluginInfo|
|Version|2.0.3|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|'tag matching' with full boolean expressions (AND, OR, NOT, and nested parentheses)|
!!!!!Documentation
> see [[MatchTagsPluginInfo]]
!!!!!Revisions
<<<
2010.03.02 2.0.3 added %6 format (tags)
| please see [[MatchTagsPluginInfo]] for additional revision details |
2008.02.28 1.0.0 initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.MatchTagsPlugin= {major: 2, minor: 0, revision: 3, date: new Date(2010,3,2)};

// store.getMatchingTiddlers() processes boolean expressions for tag matching
//    sortfield (optional) sets sort order for tiddlers - default=title
//    tiddlers (optional) use alternative set of tiddlers (instead of current store)
TiddlyWiki.prototype.getMatchingTiddlers = function(tagexpr,sortfield,tiddlers) {

	var debug=config.options.chkDebug; // abbreviation
	var cmm=config.macros.matchTags; // abbreviation
	var r=[]; // results are an array of tiddlers
	var tids=tiddlers||store.getTiddlers(sortfield||"title");
	if (tiddlers && sortfield) store.sortTiddlers(tids,sortfield);
	if (debug) displayMessage(cmm.msg1.format([tids.length]));

	// try simple lookup to quickly find single tags or tags that
	// contain boolean operators as literals, e.g. "foo and bar"
	for (var t=0; t<tids.length; t++)
		if (tids[t].isTagged(tagexpr)) r.pushUnique(tids[t]);
	if (r.length) {
		if (debug) displayMessage(cmm.msg4.format([r.length,tagexpr]));
		return r;
	}

	// convert expression into javascript code with regexp tests,
	// so that "tag1 AND ( tag2 OR NOT tag3 )" becomes
	// "/\~tag1\~/.test(...) && ( /\~tag2\~/.test(...) || ! /\~tag3\~/.test(...) )"

	// normalize whitespace, tokenize operators, delimit with "~"
	var c=tagexpr.trim(); // remove leading/trailing spaces
	c = c.replace(/\s+/ig," "); // reduce multiple spaces to single spaces
	c = c.replace(/\(\s?/ig,"~(~"); // open parens
	c = c.replace(/\s?\)/ig,"~)~"); // close parens
	c = c.replace(/(\s|~)?&&(\s|~)?/ig,"~&&~"); // &&
	c = c.replace(/(\s|~)AND(\s|~)/ig,"~&&~"); // AND
	c = c.replace(/(\s|~)?\|\|(\s|~)?/ig,"~||~"); // ||
	c = c.replace(/(\s|~)OR(\s|~)/ig,"~||~"); // OR
	c = c.replace(/(\s|~)?!(\s|~)?/ig,"~!~"); // !
	c = c.replace(/(^|~|\s)NOT(\s|~)/ig,"~!~"); // NOT
	c = c.replace(/(^|~|\s)NOT~\(/ig,"~!~("); // NOT(
	// change tag terms to regexp tests
	var terms=c.split("~"); for (var i=0; i<terms.length; i++) { var t=terms[i];
		if (/(&&)|(\|\|)|[!\(\)]/.test(t) || t=="") continue; // skip operators/parens/spaces
		if (t==config.macros.matchTags.untaggedKeyword)
			terms[i]="tiddlertags=='~~'"; // 'untagged' tiddlers
		else
			terms[i]="/\\~"+t+"\\~/.test(tiddlertags)";
	}
	c=terms.join(" ");
	if (debug) { displayMessage(cmm.msg2.format([tagexpr])); displayMessage(cmm.msg3.format([c])); }

	// scan tiddlers for matches
	for (var t=0; t<tids.length; t++) {
	 	// assemble tags from tiddler into string "~tag1~tag2~tag3~"
		var tiddlertags = "~"+tids[t].tags.join("~")+"~";
		try { if(eval(c)) r.push(tids[t]); } // test tags
		catch(e) { // error in test
			displayMessage(cmm.msg2.format([tagexpr]));
			displayMessage(cmm.msg3.format([c]));
			displayMessage(e.toString());
			break; // skip remaining tiddlers
		}
	}
	if (debug) displayMessage(cmm.msg4.format([r.length,tagexpr]));
	return r;
}
//}}}
//{{{
config.macros.matchTags = {
	msg1: "scanning %0 input tiddlers",
	msg2: "looking for '%0'",
	msg3: "using expression: '%0'",
	msg4: "found %0 tiddlers matching '%1'",
	noMatch: "no matching tiddlers",
	untaggedKeyword: "-",
	untaggedLabel: "no tags",
	untaggedPrompt: "show tiddlers with no tags",
	defTiddler: "MatchingTiddlers",
	defTags: "",
	defFormat: "[[%0]]",
	defSeparator: "\n",
	reportHeading: "Found %0 tiddlers tagged with: '{{{%1}}}'\n----\n",
	handler: function(place,macroName,params,wikifier,paramString,tiddler) {
		var mode=params[0]?params[0].toLowerCase():'';
		if (mode=="inline")
			params.shift();
		if (mode=="report" || mode=="panel") {
			params.shift();
			var target=params.shift()||this.defTiddler;
		}
		if (mode=="popup") {
			params.shift();
			if (params[0]&&params[0].substr(0,6)=="label:") var label=params.shift().substr(6);
			if (params[0]&&params[0].substr(0,7)=="prompt:") var prompt=params.shift().substr(7);
		} else {
			var fmt=(params.shift()||this.defFormat).unescapeLineBreaks();
			var sep=(params.shift()||this.defSeparator).unescapeLineBreaks();
		}
		var sortBy="+title";
		if (params[0]&&params[0].substr(0,5)=="sort:") sortBy=params.shift().substr(5);
		var expr = params.join(" ");
		if (mode!="panel" && (!expr||!expr.trim().length)) return;
		if (expr==this.untaggedKeyword)
			{ var label=this.untaggedLabel; var prompt=this.untaggedPrompt };
		switch (mode) {
			case "popup": this.createPopup(place,label,expr,prompt,sortBy); break;
			case "panel": this.createPanel(place,expr,fmt,sep,sortBy,target); break;
			case "report": this.createReport(target,this.defTags,expr,fmt,sep,sortBy); break;
			case "inline": default: this.createInline(place,expr,fmt,sep,sortBy); break;
		}
	},
	formatList: function(tids,fmt,sep) {
		var out=[];
		for (var i=0; i<tids.length; i++) { var t=tids[i];
			var title=t.title;
			var who=t.modifier;
			var when=t.modified.toLocaleString();
			var text=t.text;
			var first=t.text.split("\n")[0];
			var desc=store.getTiddlerSlice(t.title,"description");
			desc=desc||store.getTiddlerSlice(t.title,"Description");
			desc=desc||store.getTiddlerText(t.title+"##description");
			desc=desc||store.getTiddlerText(t.title+"##Description");
			var tags=t.tags.length?'[['+t.tags.join(']] [[')+']]':'';
			out.push(fmt.format([title,who,when,text,first,desc,tags]));
		}
		return out.join(sep);
	},
	createInline: function(place,expr,fmt,sep,sortBy) {
		wikify(this.formatList(store.sortTiddlers(store.getMatchingTiddlers(expr),sortBy),fmt,sep),place);
	},
	createPopup: function(place,label,expr,prompt,sortBy) {
		var btn=createTiddlyButton(place,
			(label||expr).format([expr]),
			(prompt||config.views.wikified.tag.tooltip).format([expr]),
			function(ev){ return config.macros.matchTags.showPopup(this,ev||window.event); });
		btn.setAttribute("sortBy",sortBy);
		btn.setAttribute("expr",expr);
	},
	showPopup: function(here,ev) {
		var p=Popup.create(here); if (!p) return false;
		var tids=store.getMatchingTiddlers(here.getAttribute("expr"));
		store.sortTiddlers(tids,here.getAttribute("sortBy"));
		var list=[]; for (var t=0; t<tids.length; t++) list.push(tids[t].title);
		if (!list.length) createTiddlyText(p,this.noMatch);
		else {
			var b=createTiddlyButton(createTiddlyElement(p,"li"),
				config.views.wikified.tag.openAllText,
				config.views.wikified.tag.openAllTooltip,
				function() {
					var list=this.getAttribute("list").readBracketedList();
					story.displayTiddlers(null,tids);
				});
			b.setAttribute("list","[["+list.join("]] [[")+"]]");
			createTiddlyElement(p,"hr");
		}
		var out=this.formatList(tids," &nbsp;[[%0]]&nbsp; ","\n"); wikify(out,p);
		Popup.show();
		ev.cancelBubble=true;
		if(ev.stopPropagation) ev.stopPropagation();
		return false;
	},
	createReport: function(target,tags,expr,fmt,sep,sortBy) {
		var tids=store.sortTiddlers(store.getMatchingTiddlers(expr),sortBy);
		if (!tids.length) { displayMessage('no matches for: '+expr); return false; }
		var msg=config.messages.overwriteWarning.format([target]);
		if (store.tiddlerExists(target) && !confirm(msg)) return false;
		var out=this.reportHeading.format([tids.length,expr])
		out+=this.formatList(tids,fmt,sep);
		store.saveTiddler(target,target,out,config.options.txtUserName,new Date(),tags,{});
		story.closeTiddler(target); story.displayTiddler(null,target);
	},
	createPanel: function(place,expr,fmt,sep,sortBy,tid) {
		var s=createTiddlyElement(place,"span"); s.innerHTML=store.getTiddlerText("MatchTagsPlugin##html");
		var f=s.getElementsByTagName("form")[0];
		f.expr.value=expr; f.fmt.value=fmt; f.sep.value=sep.escapeLineBreaks();
		f.tid.value=tid; f.tags.value=this.defTags;
	}
};
//}}}
/***
//{{{
!html
<form style='display:inline;white-space:nowrap'>
<input type='text'    name='expr' style='width:50%' title='tag expression'><!--
--><input type='text'    name='fmt'  style='width:10%' title='list item format'><!--
--><input type='text'    name='sep'  style='width:5%'  title='list item separator'><!--
--><input type='text'    name='tid'  style='width:12%' title='target tiddler title'><!--
--><input type='text'    name='tags' style='width:10%' title='target tiddler tags'><!--
--><input type='button'  name='go'   style='width:8%'  value='go' onclick="
	var expr=this.form.expr.value;
	if (!expr.length) { alert('Enter a boolean tag expression'); return false; }
	var fmt=this.form.fmt.value;
	if (!fmt.length) { alert('Enter the list item output format'); return false; }
	var sep=this.form.sep.value.unescapeLineBreaks();
	var tid=this.form.tid.value;
	if (!tid.length) { alert('Enter a target tiddler title'); return false; }
	var tags=this.form.tags.value;
	config.macros.matchTags.createReport(tid,tags,expr,fmt,sep,'title');
	return false;">
</form>
!end
//}}}
***/
//{{{
// SHADOW TIDDLER for displaying default panel input form
config.shadowTiddlers.MatchTags="<<matchTags panel>>";
//}}}
//{{{
// TWEAK core filterTiddlers() for enhanced boolean matching in [tag[...]] syntax:
// use getMatchingTiddlers instead getTaggedTiddlers
var fn=TiddlyWiki.prototype.filterTiddlers;
fn=fn.toString().replace(/getTaggedTiddlers/g,"getMatchingTiddlers");
eval("TiddlyWiki.prototype.filterTiddlers="+fn);
//}}}
//{{{
// REDEFINE core handler for enhanced boolean matching in tag:"..." paramifier
// use filterTiddlers() instead of getTaggedTiddlers() to get list of tiddlers.
config.paramifiers.tag = {
	onstart: function(v) {
		var tagged = store.filterTiddlers("[tag["+v+"]]");
		story.displayTiddlers(null,tagged,null,false,null);
	}
};
//}}}
<<tiddler [[include_tiddlers/MathJax.html#"MathJax"]]>>
$A_B$ now renders correctly, so does $A_{B}$.
Similarly, $ABc$, $AB c$, $A Bc$, $aBCd$, $ABC$ work now.
!The Lorenz Equations
\[\begin{eqnarray}
\dot{x} & = & \sigma(y-x) \\
\dot{y} & = & \rho x - y - xz \\
\dot{z} & = & -\beta z + xy
\end{eqnarray}\]
!The ~Cauchy-Schwarz Inequality
\[ \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2
\right) \left( \sum_{k=1}^n b_k^2 \right) \]
!A Cross Product Formula
\[\mathbf{V}_1 \times \mathbf{V}_2 =  \left|\begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial X}{\partial u} &  \frac{\partial Y}{\partial u} & 0 \\
            \frac{\partial X}{\partial v} &  \frac{\partial Y}{\partial v}
& 0
\end{array}\right|  \]
!The probability of getting \(k\) heads when flipping \(n\) coins is:
\[P(E)   = {n \choose k} p^k (1-p)^{ n-k} \]
!An Identity of Ramanujan
\[ \frac{1}{\left(\sqrt{\phi \sqrt{5}}-\phi\right) e^{\frac25 \pi}} =
1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}
{1+\frac{e^{-8\pi}} {1+\ldots} } } } \]
!A ~Rogers-Ramanujan Identity
\[  1 +  \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =
\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},
\quad\quad \textrm{for}\quad |q|<1. \]
!Maxwell's Equations
\[\begin{eqnarray}
\nabla \times \vec{\mathbf{B}} -\, \frac1c\,\frac{\partial\vec{\mathbf{E}}}{\partial t} & = & \frac{4\pi}{c}\vec{\mathbf{j}} \\
\nabla \cdot \vec{\mathbf{E}} & = & 4 \pi \rho \\
\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\,\frac{\partial\vec{\mathbf{B}}}{\partial t} & = &\vec{\mathbf{0}} \\
\nabla \cdot \vec{\mathbf{B}} & = & 0
\end{eqnarray}\]
Finally, while display equations look good for a page of samples, the ability to mix math and text
in a paragraph is also important. This expression \(\sqrt{3x-1}+(1+x)^2\) is an example of an
inline equation.  As you see, ~MathJax equations can be used this way as well, without unduly
disturbing the spacing between lines.
<<tiddler [[include_tiddlers/Mathematica.html#"Mathematica"]]>>
<<tiddler [[include_tiddlers/Mathematical Structure.html#"Mathematical Structure"]]>>
Links:
* [[INTERNATIONAL CONFERENCE OF MATHEMATICIANS MADRID 2006|http://www.dmmm.uniroma1.it/~prastaro/POSTER_ICM2006_APDEG.html]] - Tags of mathematicians.
* [[FamousWhy|http://www.famouswhy.com/Tags/mathematician]] - Dto.
<<tiddler [[include_tiddlers/Mathieu Group.html#"Mathieu Group"]]>>
<<tiddler [[include_tiddlers/Mathisson-Papapetrou Equations.html#"Mathisson-Papapetrou Equations"]]>>
Links:
* [[The Matrix Reference Manual|http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html#Intro]]
<<tiddler [[include_tiddlers/Max Planck.html#"Max Planck"]]>>
<<tiddler [[include_tiddlers/Maximal Acceleration.html#"Maximal Acceleration"]]>>
<<tiddler [[include_tiddlers/Maximal Torus.html#"Maximal Torus"]]>>
<<tiddler [[include_tiddlers/Maximum Entropy Production Principle.html#"Maximum Entropy Production Principle"]]>>
<<tiddler [[include_tiddlers/Maximum Lifespan.html#"Maximum Lifespan"]]>>
<<tiddler [[include_tiddlers/Measure Problem.html#"Measure Problem"]]>>
<<tiddler [[include_tiddlers/Measurement Algebra.html#"Measurement Algebra"]]>>
<<tiddler [[include_tiddlers/Measurement Theory.html#"Measurement Theory"]]>>
<<tiddler [[include_tiddlers/Median.html#"Median"]]>>
<<tiddler [[include_tiddlers/Membrane Theory of Aging.html#"Membrane Theory of Aging"]]>>
<<tiddler [[include_tiddlers/Memory.html#"Memory"]]>>
A ''Menon Design'' is a $2-(4u^2, 2u^2 \pm u, u^2 \pm u)$ [[design|Balanced Incomplete Block Design]].
<<tiddler [[include_tiddlers/Meromorphic Function.html#"Meromorphic Function"]]>>
A ''Mersenne Prime'' $M_n$ is a prime number of the form
\[
 M_n = 2^n - 1
\]
with $n \in \mathbb N\,$. It can be shown that $M_n$ can only be prime if $n$ is prime.
Currently only $47$ Mersenne primes are known. The sequence goes like this: $7,31, 127, 8.191, 131.071, 524.287, 2.147.483.647, \ldots$.

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

Links:
* [[WIKIPEDIA - Mersenne Primes|http://en.wikipedia.org/wiki/Mersenne_prime]]
<<tiddler [[include_tiddlers/Metabolic Network.html#"Metabolic Network"]]>>
<<tiddler [[include_tiddlers/Metabolic Theory of Cancer.html#"Metabolic Theory of Cancer"]]>>
<<tiddler [[include_tiddlers/Metamaterial.html#"Metamaterial"]]>>
<<tiddler [[include_tiddlers/Methuselah Fly.html#"Methuselah Fly"]]>>
<<tiddler [[include_tiddlers/Methylglyoxal.html#"Methylglyoxal"]]>>
<<tiddler [[include_tiddlers/Metric Affine Gravity.html#"Metric Affine Gravity"]]>>
<<tiddler [[include_tiddlers/Metric Affine Space.html#"Metric Affine Space"]]>>
<<tiddler [[include_tiddlers/Metric Compatibility.html#"Metric Compatibility"]]>>
<<tiddler [[include_tiddlers/Metric Space.html#"Metric Space"]]>>
<<tiddler [[include_tiddlers/Metric Tensor.html#"Metric Tensor"]]>>
<<tiddler [[include_tiddlers/Michael Atiyah.html#"Michael Atiyah"]]>>
<<tiddler [[include_tiddlers/Michael Freedman.html#"Michael Freedman"]]>>
<<tiddler [[include_tiddlers/Microcausality.html#"Microcausality"]]>>
<<tiddler [[include_tiddlers/Microtubule.html#"Microtubule"]]>>
<<tiddler [[include_tiddlers/Mie Scattering.html#"Mie Scattering"]]>>
<<tiddler [[include_tiddlers/Minimal Coupling.html#"Minimal Coupling"]]>>
<<tiddler [[include_tiddlers/Minimal Mass.html#"Minimal Mass"]]>>
<<tiddler [[include_tiddlers/Minisuperspace Model.html#"Minisuperspace Model"]]>>
<<tiddler [[include_tiddlers/Miracle Ocatad Generator (MOG).html#"Miracle Ocatad Generator (MOG)"]]>>
<<tiddler [[include_tiddlers/Mitochondrial Biogenesis.html#"Mitochondrial Biogenesis"]]>>
<<tiddler [[include_tiddlers/Mitochondrial DNA.html#"Mitochondrial DNA"]]>>
<<tiddler [[include_tiddlers/Mitochondrial Eve.html#"Mitochondrial Eve"]]>>
<<tiddler [[include_tiddlers/Mitochondrial Theory of Aging.html#"Mitochondrial Theory of Aging"]]>>
<<tiddler [[include_tiddlers/Mitochondrion.html#"Mitochondrion"]]>>
<<tiddler [[include_tiddlers/Mitophagy.html#"Mitophagy"]]>>
<<tiddler [[include_tiddlers/Mixed Anomaly.html#"Mixed Anomaly"]]>>
<<tiddler [[include_tiddlers/Model Organism.html#"Model Organism"]]>>
<<tiddler [[include_tiddlers/Model-view-controller.html#"Model-view-controller"]]>>
<<tiddler [[include_tiddlers/Modified Gravity.html#"Modified Gravity"]]>>
<<tiddler [[include_tiddlers/Modified Newtonian Dynamics.html#"Modified Newtonian Dynamics"]]>>
<<tiddler [[include_tiddlers/Modular Group.html#"Modular Group"]]>>
<<tiddler [[include_tiddlers/Modular Lattice.html#"Modular Lattice"]]>>
<<tiddler [[include_tiddlers/Module.html#"Module"]]>>
<<tiddler [[include_tiddlers/Monism.html#"Monism"]]>>
A [[loop|Loop]] is called ''monoassociative'' (or ''3-''[[power-associative|Power Associative Algebra]], short ''3PA'')  if each of its elements generates an associative subloop.
This implies that the [[associators|Associator]] of its local algebras satisfy the condition
\[
[\mb A, \mb A, \mb A] = 0
\]

A [[quasigroup|Quasigroup]] is said to be local if all its local loops are monoassociative.
It follows that any tangent vector $\bs \zeta$ of such a local loop satisfies
\begin{eqnarray}
[\bs \zeta, \bs \zeta, \bs \zeta] = \zeta^\mu \zeta^\nu \zeta^\rho [\bs e_\mu, \bs e_\nu, \bs e_\rho] = \zeta^\mu \zeta^\nu \zeta^\rho A_{\mu\nu\rho}^\sigma \mb e_\sigma= 0
\end{eqnarray}
As
\[
\zeta^\mu \zeta^\nu \zeta^\rho A_{\mu\nu\rho}^\sigma \mb e_\sigma = 0 = \zeta^\mu \zeta^\nu \zeta^\rho A_{(\mu\nu\rho)}^\sigma \mb e_\sigma
\]
the monoassociativity condition in tensorial form reads
\[
A_{(\mu\nu\rho)}^\sigma = 0
\]

Monoassociativity is a weaker condition than is [[power-associativity|Power Associative Algebra]].

Monoassociativity is akin to [[alternativity|Alternative Algebra]] in that in the former case the associator vanishes if all three arguments are identical, whereas in the latter case the associator vanishes if two arguments are identical.
Thus monoassociativity is also weaker than is alternativity.

!!!!Example
Any [[flexible algebra|Flexible Algebra]] is monoassociative, as it satisfies $[\mb A,\mb B,\mb A] = 0$.
<<tiddler [[include_tiddlers/Monogamy of Entanglement.html#"Monogamy of Entanglement"]]>>
<<tiddler [[include_tiddlers/Monster Group.html#"Monster Group"]]>>
<<tiddler [[include_tiddlers/Monstrous Moonshine.html#"Monstrous Moonshine"]]>>
<<tiddler [[include_tiddlers/Monty Hall Problem.html#"Monty Hall Problem"]]>>
<<tiddler [[include_tiddlers/Moon Landing Hoax.html#"Moon Landing Hoax"]]>>
<<tiddler [[include_tiddlers/Moore Graph.html#"Moore Graph"]]>>
<<tiddler [[include_tiddlers/Moore's Law.html#"Moore's Law"]]>>
Papers:
* [[Morita Equivalence in Geometry and Algebra - R. Mayer|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.35.3449&rep=rep1&type=pdf]]
Reformulate physics and mathematics and recast it in a (coherent and consistent) form, that at least oneself can understand.

Why ?
* A lot of redundancies. Things are rediscovered over and over again and often it is not realised that they are the same. This corresponds to the problem of different representations for the same thing and different "brain wirings" of different authors, prefering different representations.
* One cannot learn mathematics and physics, one has to discover it.
* A lot is copied and pasted by authors, who do not really understand what the true meaning of the content is. This is even worse when things are copied over and over again. Errors accumulate and propagate. (Therefore it is no wonder, that original works are often way more joyful to read).
* A lot is too abstract, too far from an application. A priorisation of the relevance of mathematical structures in respect to their applicability in physics is needed. Classification of structures in mathematics is essential but one deals a lot with practically uninteresting structures (one can easily get lost).
* Reading and writing (or WIKIing these days) is better what concerns understanding than just reading.
<<tiddler [[include_tiddlers/Moufang Loop.html#"Moufang Loop"]]>>
<<tiddler [[include_tiddlers/Multipartite Entanglement.html#"Multipartite Entanglement"]]>>
<<tiddler [[include_tiddlers/Multiplet.html#"Multiplet"]]>>
<<tiddler [[include_tiddlers/Multitier Architecture.html#"Multitier Architecture"]]>>
<<tiddler [[include_tiddlers/Multivalued Function.html#"Multivalued Function"]]>>
<<tiddler [[include_tiddlers/Multivector Derivative.html#"Multivector Derivative"]]>>
<<tiddler [[include_tiddlers/Multiverse.html#"Multiverse"]]>>
<<tiddler [[include_tiddlers/Multiverse and Consciousness.html#"Multiverse and Consciousness"]]>>
<<tiddler [[include_tiddlers/Murray Gell-Mann.html#"Murray Gell-Mann"]]>>
<<tiddler [[include_tiddlers/Musean Hypernumber.html#"Musean Hypernumber"]]>>
<html><center><img src="images/math_clock.jpg" style="width: 280px; "/></center></html>

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

Links:
* [[Cognitive Theoretic Model of the Universe (CMTU) - C. Langan|http://www.megafoundation.org/CTMU/]] - How a man with an alledged I.Q. of around 200 "sees" the universe.
* [[Ben Goertzel Essays|http://www.goertzel.org/essays.htm]]
* [[University of Toronto Mathematics Network - Question Corner and Discussion Area|http://www.math.toronto.edu/mathnet/questionCorner/qc.ps]]
* [[Articles by S. M. Phillips|http://www.smphillips.8m.com/html/articles.html]] - Interesting stuff, but to be taken with a grain of salt. ("Octonion Algebra is isomorphic to E8 Lie Algebra").
* [[Bitmaps for a Digital Theory of Everything - R. Aschheim|http://www.cs.indiana.edu/~dgerman/2008midwestNKSconference/rasch.pdf]]
* [[Strings and Loops in Event Symmetric Space-Time - P. Gibbs|http://arxiv.org/PS_cache/hep-th/pdf/9407/9407136v1.pdf]]
* [[Rafiki Inc.|http://www.codefun.com/]]
* [[The Cellular Universe website - C. Ranzan|http://www.cellularuniverse.org/]]
* [[Tony Smith's Homepage, 240 Thoughts|http://www.valdostamuseum.org/hamsmith/SWTxt.html]]
* [[Verman University Mathematical Quotations Server|http://math.furman.edu/~mwoodard/mqs/mquot.shtml]]
* [[God does not play Dice|http://www.god-does-not-play-dice.net/]]
* [[Gennady I. Shipov|http://www.shipov.com/]] - Torsion, warp drives and all that ...
* [[The Orientation Congruent Algebra and the Native Exterior Calculus of Twisted Differential Forms|http://felicity.freeshell.org/math/index.htm#vis-tw-objs]]
* [[viXra.org|http://www.vixra.org/]] - The alternative arXiv.
<<tiddler [[include_tiddlers/Mysterious Analogies.html#"Mysterious Analogies"]]>>
<<tiddler [[include_tiddlers/Möbius Function.html#"Möbius Function"]]>>
<<tiddler [[include_tiddlers/Möbius Transformation.html#"Möbius Transformation"]]>>
<<tiddler [[include_tiddlers/N-Cochain.html#"N-Cochain"]]>>
<<tiddler [[include_tiddlers/N-Cocycle.html#"N-Cocycle"]]>>
<<tiddler [[include_tiddlers/N-Curvature.html#"N-Curvature"]]>>
<<tiddler [[include_tiddlers/N-Point Function.html#"N-Point Function"]]>>
<<tiddler [[include_tiddlers/N-Quasigroup.html#"N-Quasigroup"]]>>
<<tiddler [[include_tiddlers/N-Sphere.html#"N-Sphere"]]>>
See also:
* [[N-Quasigroup]]
* [[Ternary Algebra]]


Papers:
* [[N-ary Algebras: A Review with Applications (2010) - J. A. de Azcárraga, J. M. Izquierdo|http://arxiv.org/abs/1005.1028]] [[local|papers/1005.1028v2.pdf]] [[pct. 17|http://scholar.google.de/scholar?cites=16756209416460701104&as_sdt=5&sciodt=0&hl=de]]
<<tiddler [[include_tiddlers/N-body Simulation.html#"N-body Simulation"]]>>
<<tiddler [[include_tiddlers/Naked Singularity.html#"Naked Singularity"]]>>
<<tiddler [[include_tiddlers/Nambu Mechanics.html#"Nambu Mechanics"]]>>
<<tiddler [[include_tiddlers/Nambu-Jona-Lasinio Model.html#"Nambu-Jona-Lasinio Model"]]>>
<<tiddler [[include_tiddlers/Nanobot.html#"Nanobot"]]>>
The (even, selfdual) ''Narain Lattice'' is given by $\Lambda_{6,22} = \Lambda_{6,6} \otimes \Lambda_{16}$ with $\Lambda_{16}$ the $Spin(32)/\mathbb{Z}_2$ lattice ([[Barnes-Wall Lattice|Barnes-Wall Lattice]]).
<<tiddler [[include_tiddlers/Near Death Experience.html#"Near Death Experience"]]>>
<<tiddler [[include_tiddlers/Necrosis.html#"Necrosis"]]>>
<<tiddler [[include_tiddlers/Negative Mass.html#"Negative Mass"]]>>
To get used to the very similar looking tensor expressions for nested [[commutators|Commutator]] and [[associators|Associator]] (i.e. [[torsion-|Torsion]] and [[nonassociativity tensors|Nonassociativity Tensor]]), in the following a list of some objects occuring:
!!!!Inner commutators
\begin{eqnarray}
[[\mb A, \mb B], \mb C, \mb D]  &= & A^i B^j C^k D^l [[\mb e_i, \mb e_j], \mb e_k, \mb e_l] \\
&= & A^i B^j C^k D^l  A_{mkl} T_{ij}^m
\end{eqnarray}
accordingly
\begin{eqnarray}
[\mb A, [\mb B, \mb C], \mb D]  &= & A^i B^j C^k D^l  A_{iml} T_{jk}^m
\end{eqnarray}
and
\begin{eqnarray}
[\mb A, \mb B, [\mb C, \mb D]]  &= & A^i B^j C^k D^l  A_{ijm} T_{kl}^m
\end{eqnarray}
Hence the position of the commutator equals the position of the summation index of the associator.

!!!!Inner associators
\begin{eqnarray}
[[\mb A, \mb B, \mb C], \mb D]  &= & A^i B^j C^k D^l [[\mb e_i, \mb e_j, \mb e_k], \mb e_l] \\
&= & A^i B^j C^k D^l  A_{ijk}^m T_{ml}
\end{eqnarray}
and accordingly
\begin{eqnarray}
[\mb A, [\mb B, \mb C, \mb D]] &= & A^i B^j C^k D^l [\mb e_i, [\mb e_j, \mb e_k, \mb e_l]] \\
&= & A^i B^j C^k D^l  A_{jkl}^m T_{im}
\end{eqnarray}
Hence the position of the summation index of the torsion tensor indicates, if the associator is to the left or to the right.

Furthermore as a mnemonic, the tensor with the additional index is the inner one.
<<tiddler [[include_tiddlers/Network.html#"Network"]]>>
<<tiddler [[include_tiddlers/Network Model.html#"Network Model"]]>>
<<tiddler [[include_tiddlers/Network Robustness.html#"Network Robustness"]]>>
<<tiddler [[include_tiddlers/Network Theory of Aging.html#"Network Theory of Aging"]]>>
<<tiddler [[include_tiddlers/Neuron.html#"Neuron"]]>>
<<tiddler [[include_tiddlers/Neurotheology.html#"Neurotheology"]]>>
<<tiddler [[include_tiddlers/Neutral Evolution.html#"Neutral Evolution"]]>>
<<tiddler [[include_tiddlers/Neutrino.html#"Neutrino"]]>>
<<tiddler [[include_tiddlers/Neutrino Minimal Standard Model.html#"Neutrino Minimal Standard Model"]]>>
<<tiddler [[include_tiddlers/Newman-Janis Algorithm.html#"Newman-Janis Algorithm"]]>>
<<tiddler [[include_tiddlers/News.html#"News"]]>>
<<tiddler [[include_tiddlers/Newton's Law of Gravitation.html#"Newton's Law of Gravitation"]]>>
<<tiddler [[include_tiddlers/Nicotinamid Ribose.html#"Nicotinamid Ribose"]]>>
<<tiddler [[include_tiddlers/Nicotinamide Adenine Dinucleotide.html#"Nicotinamide Adenine Dinucleotide"]]>>
<<tiddler [[include_tiddlers/Niels Bohr.html#"Niels Bohr"]]>>
<<tiddler [[include_tiddlers/Nielsen-Ninomiya Theorem.html#"Nielsen-Ninomiya Theorem"]]>>
<<tiddler [[include_tiddlers/Nilpotency.html#"Nilpotency"]]>>
Links:
* [[WIKIPEDIA - Nilradical of a Ring|http://en.wikipedia.org/wiki/Nilradical_of_a_ring]]
<<tiddler [[include_tiddlers/No-Ghost Theorem.html#"No-Ghost Theorem"]]>>
<<tiddler [[include_tiddlers/No-boundary Proposal.html#"No-boundary Proposal"]]>>
<<tiddler [[include_tiddlers/No-cloning Theorem.html#"No-cloning Theorem"]]>>
<<tiddler [[include_tiddlers/Noise.html#"Noise"]]>>
<<tiddler [[include_tiddlers/Non-Computability.html#"Non-Computability"]]>>
<<tiddler [[include_tiddlers/Non-Hermiticity.html#"Non-Hermiticity"]]>>
<<tiddler [[include_tiddlers/Non-Lagrangian System.html#"Non-Lagrangian System"]]>>
<<tiddler [[include_tiddlers/Non-Metricity Tensor.html#"Non-Metricity Tensor"]]>>
<<tiddler [[include_tiddlers/Non-Unitarity.html#"Non-Unitarity"]]>>
<<tiddler [[include_tiddlers/Non-equilibrium Thermodynamics.html#"Non-equilibrium Thermodynamics"]]>>
<<tiddler [[include_tiddlers/Nonassociative Algebra.html#"Nonassociative Algebra"]]>>
<<tiddler [[include_tiddlers/Nonassociative Geometry.html#"Nonassociative Geometry"]]>>
<<tiddler [[include_tiddlers/Nonassociative Physics.html#"Nonassociative Physics"]]>>
<<tiddler [[include_tiddlers/Nonassociative Quantum Field Theory.html#"Nonassociative Quantum Field Theory"]]>>
<<tiddler [[include_tiddlers/Nonassociative Quantum Mechanics.html#"Nonassociative Quantum Mechanics"]]>>
<<tiddler [[include_tiddlers/Nonassociative Quantum Model of Consciousness.html#"Nonassociative Quantum Model of Consciousness"]]>>
<<tiddler [[include_tiddlers/Nonassociative Standard Model.html#"Nonassociative Standard Model"]]>>
<<tiddler [[include_tiddlers/Nonassociativity Tensor.html#"Nonassociativity Tensor"]]>>
<<tiddler [[include_tiddlers/Noncommutative Cosmology.html#"Noncommutative Cosmology"]]>>
<<tiddler [[include_tiddlers/Noncommutative Geometry.html#"Noncommutative Geometry"]]>>
<<tiddler [[include_tiddlers/Noncommutative Geometry and Quantization.html#"Noncommutative Geometry and Quantization"]]>>
<<tiddler [[include_tiddlers/Noncommutative Jordan Algebra.html#"Noncommutative Jordan Algebra"]]>>
<<tiddler [[include_tiddlers/Noncommutative Pati-Salam Model.html#"Noncommutative Pati-Salam Model"]]>>
<<tiddler [[include_tiddlers/Noncommutative Quantum Field Theory.html#"Noncommutative Quantum Field Theory"]]>>
<<tiddler [[include_tiddlers/Noncommutative Spacetime.html#"Noncommutative Spacetime"]]>>
<<tiddler [[include_tiddlers/Noncommutative Standard Model Dirac Operator.html#"Noncommutative Standard Model Dirac Operator"]]>>
<<tiddler [[include_tiddlers/Noncommutative Standard Model Lagrangian.html#"Noncommutative Standard Model Lagrangian"]]>>
Papers:
* [[On the Notion of Lower Central Series for Loops - J. Mostovoy|http://arxiv.org/PS_cache/math/pdf/0410/0410515v1.pdf]]
<<tiddler [[include_tiddlers/Nongeometric String Background.html#"Nongeometric String Background"]]>>
<<tiddler [[include_tiddlers/Nonholonomic Mapping Principle.html#"Nonholonomic Mapping Principle"]]>>
<<tiddler [[include_tiddlers/Nonlinear Dirac Equation.html#"Nonlinear Dirac Equation"]]>>
<<tiddler [[include_tiddlers/Nonlinear Quantum Mechanics.html#"Nonlinear Quantum Mechanics"]]>>
<<tiddler [[include_tiddlers/Nonlinear Sigma Model.html#"Nonlinear Sigma Model"]]>>
''Nonsymmetric Gravity Theory (NGT)'' allows for antisymmetric metrics. Nonsymmetric metrics were already studied by Einstein & Straus (1946) in their search for a unified theory for gravity and electromagnetism, yet this unification was not successful.

Links:
* [[WIKIPEDIA - Nonsymmetric Gravitational Theory|http://en.wikipedia.org/wiki/Nonsymmetric_gravitational_theory]]
<<tiddler [[include_tiddlers/Nordstrom-Robinson Code.html#"Nordstrom-Robinson Code"]]>>
<<tiddler [[include_tiddlers/Norm.html#"Norm"]]>>
<<tiddler [[include_tiddlers/Normal Order.html#"Normal Order"]]>>
<<tiddler [[include_tiddlers/Normal Subgroup.html#"Normal Subgroup"]]>>
<<tiddler [[include_tiddlers/Normal Subloop.html#"Normal Subloop"]]>>
<<tiddler [[include_tiddlers/Nothing Comes from Nothing.html#"Nothing Comes from Nothing"]]>>
<<tiddler [[include_tiddlers/Nucleus.html#"Nucleus"]]>>
<<tiddler [[include_tiddlers/Null Geodesic.html#"Null Geodesic"]]>>
<<tiddler [[include_tiddlers/Nutrient Sensor.html#"Nutrient Sensor"]]>>
<<tiddler [[include_tiddlers/Nutrigenomics.html#"Nutrigenomics"]]>>
<<tiddler [[include_tiddlers/ORAC.html#"ORAC"]]>>
<<tiddler [[include_tiddlers/Observable.html#"Observable"]]>>
<<tiddler [[include_tiddlers/Observer Design Pattern.html#"Observer Design Pattern"]]>>
<<tiddler [[include_tiddlers/Occam's Razor.html#"Occam's Razor"]]>>
The ''Octacode'' is a [[linear|Linear Blockcode]] [[self-dual code|Dual Code]] over $\mathbb Z_4$ of length $8$ and minimal Lee distance $6$.
It is the $\mathbb Z_4$-analogue of a [[Hamming code|Hamming Code]].
<<tiddler [[include_tiddlers/Octonion.html#"Octonion"]]>>
<<tiddler [[include_tiddlers/Octonion Loop.html#"Octonion Loop"]]>>
<<tiddler [[include_tiddlers/Octonion Multiplication Tables.html#"Octonion Multiplication Tables"]]>>
<<tiddler [[include_tiddlers/Octonionic Analysis.html#"Octonionic Analysis"]]>>
<<tiddler [[include_tiddlers/Octonionic Geometric Algebra.html#"Octonionic Geometric Algebra"]]>>
<<tiddler [[include_tiddlers/Octonionic Hilbert Space.html#"Octonionic Hilbert Space"]]>>
<<tiddler [[include_tiddlers/Octonionic Physics.html#"Octonionic Physics"]]>>
The ''Octonionic Projective Plane'' $\mathbb{OP}^2$ is an example of a geometry not satisfying the full [[Desargue's theorem|Desargues' Theorem]]. This was shown by Ruth Moufang in 1933. However, as the [[octonions|Octonion]] are [[alternative|Alternative Algebra]], the "small Desargue's theorem" is still valid.


Papers:
* [[The Octonions - J. C. Baez|http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf]] [[local|papers/S0273-0979-01-00934-X.pdf]]  {{t100Cite{[[pct. 318|http://scholar.google.de/scholar?cites=12472142092836538554&hl=de]]}}}
* [[Submanifolds of the Cayley Projective Plane with bounded Second Fundamental Form - P. Coulton, J. Glazenbrook|http://www.springerlink.com/content/l370j13w60666g39/fulltext.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=17130347132186689525&hl=de]]
* [[OP2 Bundles in M-theory - H. Sati|http://www.hausdorff-research-institute.uni-bonn.de/files/preprints/OP2paper-Aug10.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=801582279879132875&hl=de]]
* [[Plurisubharmonic Functions on the Octonionic Plane and Spin(9)-Invariant Valuations on Convex Sets - S. Alesker|http://www.springerlink.com/content/50340775283p6712/fulltext.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=9030141090597490996&hl=de]]
* [[On the Geometry of the Supermultiplet in M-theory - H. Sati|http://arxiv.org/PS_cache/arxiv/pdf/0909/0909.4737v1.pdf]] pct. 0
* [[On the Cell Structure of the Octonion Projective Plane - I. Yokota|http://dlisv03.media.osaka-cu.ac.jp/infolib/user_contents/sugaku/DB-M-006-01-03.pdf]] pct. 0
<<tiddler [[include_tiddlers/Octooctonionic Projective Plane.html#"Octooctonionic Projective Plane"]]>>
<<tiddler [[include_tiddlers/Oil Pulling.html#"Oil Pulling"]]>>
Links:
* [[WIKIPEDIA - Olber's Paradox|http://en.wikipedia.org/wiki/Olbers'_paradox]]

Videos:
* [[Physics Lecture:- Olbers Paradox and the Distant Universe - R. Nemiroff|http://www.youtube.com/watch?v=sIDSf8-aHXI&feature=results_main&playnext=1&list=PLCAB7A268352F7836]]
<<tiddler [[include_tiddlers/Oligomeric Proanthocyanidins.html#"Oligomeric Proanthocyanidins"]]>>
<<tiddler [[include_tiddlers/Omega Point.html#"Omega Point"]]>>
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<<tiddler [[include_tiddlers/Operationalism.html#"Operationalism"]]>>
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<<tiddler [[include_tiddlers/Orbifold.html#"Orbifold"]]>>
Given a set $X$ and a [[group|Group]] $\mathcal G$, then for a $x \in X$ the set
\[
\operatorname{Orb}(x) = \left\{ g x \mid g \in \mathcal G \right\}
\]
is called the ''Orbit'' of $x$. (It is the set of elements of $X$ to which $x$ can be "moved to" by the elements of $\mathcal G$).

The number of elements of an orbit is called the ''length of the orbit''.

The orbits of this action are exactly the [[left cosets|Coset]] of $X$ in $\mathcal G$.

Orbits form equivalence classes. The equivalence relation $x \sim y$ for $x,y \in X$ is given by the group action, i. e. $y = gx$ with a $g \in \mathcal G$.

The set of all orbits of $X$ under the action of $\mathcal G$ forms a partition of $X$. It is written as $X/\mathcal G$ (or, less frequently: $\mathcal G$\\$X$) and is called the quotient of the action; in geometric situations it may be called the ''Orbit Space''.
<<tiddler [[include_tiddlers/Orch-OR Model.html#"Orch-OR Model"]]>>
In [[group theory|Group]] the term ''Order'' is used in two closely related senses:
#  the order of a group is the number of its elements.
#  the order of an element $\mb A$ of a group is the smallest number $n \in \mathbb N$ such that $\mb A^n = \mb e$. All elements of finite groups have finite order.
<<tiddler [[include_tiddlers/Order Parameter.html#"Order Parameter"]]>>
<<tiddler [[include_tiddlers/Organic Universe.html#"Organic Universe"]]>>
<<tiddler [[include_tiddlers/Origin of Life.html#"Origin of Life"]]>>
<<tiddler [[include_tiddlers/Orthogonal Array.html#"Orthogonal Array"]]>>
A ''(Real) Orthogonal Design (ROD)'' is a $m\times n$-matrix $G$ in $k$ real variables $x_1, x_2, \ldots , x_k$ such that each non-zero entry of the matrix is a real linear combinations of the variables $G G^T = (x_1^2+x_2^2+ \ldots +x_k^2)I_n$
with $I_n$ the $n\times n$ identity matrix.

Papers:
* [[Algebraic Methods in the Design of Space-time Codes - J- Hiltunen|http://users.utu.fi/jakahi/lisuri.pdf]] - pct. 0
* [[Delay-optimal Rectangular Real Orthogonal Designs from Cayley-Dickson Algebra - S. Das, B. S. Rajan|http://www.ncc.org.in/download.php?f=NCC2009/file1.pdf]] pct. 0
The ''r+s-dimensional Orthogonal Group $O(r, s)$'' is the [[automorphism group|Automorphism]] of the [[norm|Norm]], i.e. the norm is preserved under its action.

It is defined by:
\[
O(r, s) \equiv \{\mb O \in GL(n,\mathbb R) :  \langle \mb O \mb A | \mb O \mb B \rangle = \langle \mb A | \mb B \rangle \;\, \forall \mb A, \mb B \in \mathbb R^{r+s}\}
\]

The ''Special Orthogonal $SO(r, s)$'' is defined by:
\[
SO(r, s) \equiv \{\mb A \in O(r, s) : \det(\mb A) = 1\}
\]
Orthogonal transforms are [[isometries|Isometry]], i.e. they preserve distances and angles, hence they are [[conformal maps|Conformal Transformation]]. However not all conformal linear transforms are orthogonal.

!!!!Orthogonal group over finite fields
Orthogonal groups can also be defined over [[finite fields|Galois Field]] $\mathbb F_q$, where $q$ is a power of a prime $p$. In this case they come in two types in even dimension: $O^+ (2n,q)$ and $O^? (2n,q)$ and one type in odd dimension: $O(2n + 1,q)$.
Their [[order|Order]] is given by:
\[
\operatorname{ord}(O(2n,q))=2(q^n+(-1)^{n+1})\prod_{i=1}^{n-1}(q^{2n}-q^{2i}).
\]
\[
\operatorname{ord}(O(2n+1,q))=2q^n\prod_{i=0}^{n-1}(q^{2n}-q^{2i})
\]
<<tiddler [[include_tiddlers/Orthogonal Latin Square.html#"Orthogonal Latin Square"]]>>
An ''Orthogonal Tranfsormation'' $\mb O$ is a linear transformation that preserves the scalar product
\begin{equation}
\langle \mb O(\mb A)|\mb O (\mb B)\rangle = \langle \mb A| \mb B \rangle
\end{equation}
<<tiddler [[include_tiddlers/P-Brane.html#"P-Brane"]]>>
Links:
* [[PARI/GP Website|http://pari.math.u-bordeaux.fr/]]
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According to the ''Palatini Principle (or first order formalism)'', [[metric|Metric Tensor]] and [[connection|Connection]] are regarded as independent. For a Lagrangian this means that the variation is carried out for both of them independently.
For the [[Einstein Hilbert action|Einstein-Hilbert Action]] this leads - besides the [[Einstein equations|Einstein Field Equations]] when varying in respect to the [[metric|Metric Tensor]] - to a second set of equations, namely
\[
\partial_\lambda g_{\mu\nu} - \Gamma^\eta_{\lambda\mu} g_{\eta\nu} - \Gamma^\eta_{\lambda\nu} g_{\mu\eta} = 0
\]
which is the condition of [[metric compatibility|Metric Compatibility]]. These equations besides the Einstein equations can be regarded as additional field equations describing the gravitational field.

In general the Palatini principle allows the geometry to have an affine structure, that is that the associated space-time is a [[metric affine space|Metric Affine Space]]. In four dimensions, i.e. for a $(\mathbb L_4, g)$-space, there are $10$ equations for the metric tensor and $64$ for the connection.

The advantage of deriving the field equations using the Palatini method is that the geometry of space-time is less restrictive. In general the Palatini variation allows for the existence of [[torsion|Torsion]] and [[non-metricity|Non-Metricity Tensor]].

Although the principle carries the name of Palatini its introduction is usually ascribed to Einstein.

Papers:
* [[The Universality of Einstein Equations (1993) - M. Ferraris, M. Francaviglia, I. Volovich|http://arxiv.org/PS_cache/gr-qc/pdf/9303/9303007v2.pdf]] [[pct. 72|http://scholar.google.de/scholar?cites=9818932703691021574&hl=de&as_sdt=2000]]
*[[On the so-called "Palatini Method" of Variation in Covariant Gravitational Theories (1973) - A. A. El-Kholy, R. U. Sexl, H. K. Urbantke|http://archive.numdam.org/ARCHIVE/AIHPA/AIHPA_1973__18_2/AIHPA_1973__18_2_121_0/AIHPA_1973__18_2_121_0.pdf]] pct. 0
*[[Palatini Variational Principle for an Extended Einstein-Hilbert Action (1997) - H. Burton, R. B. Mann|http://arxiv.org/PS_cache/gr-qc/pdf/9711/9711003v1.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=9052327792572994752&hl=de&as_sdt=2000]]

Theses:
* [[On the Palatini Variation and Connection Theories of Gravity - H. S. Burton (1998)|http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape9/PQDD_0009/NQ38225.pdf]] [[local|theses/NQ38225.pdf]] tct. 0
<<tiddler [[include_tiddlers/Paleo Diet.html#"Paleo Diet"]]>>
<<tiddler [[include_tiddlers/Panpsychism.html#"Panpsychism"]]>>
<<tiddler [[include_tiddlers/Panspermia.html#"Panspermia"]]>>
The conjecture is that the days of ''Paper and Pencil Physics and Mathematics'' are over.

Most of the reasonable work has been done, often long time ago. Everybody who is doing paper and pencil mathematics or physics these days is competing with those famous figures in history, that have already pushed the envelope as far as one can do by means of the (classical) method - at least in respect to reasonable mathematics and physics.
To get a paper published these days, in physics for example, often requires a very exotic topic to avoid duplicating old results, thus often having little to do with reality, i.e. nature in the first place, rather with the fact that as a scientist one is urged to publish on a regular basis.

On the other hand, due to [[Moore's law|Moore's Law]], computational mathematics and physics opens up new possibilities every day and therefore appears more promising.

Compare this to the game of chess. These days the best chess players are computers ...
<html><center><img src="images/paralleltransport.jpg" style="width: 320px; "/></center></html>
!!!!Historical
The concept of parallel transport of vectors was introduced independently by Levi\-Civita and Schouten in 1917 (two years after Einstein's general relativity theory was published).

!!!!Generalisations
Instead of vectors one can also transport oriented autoparallel segments on a manifold. For details see [[geodesic loop|Geodesic Loop]].

See also: [[autoparallelity|Autoparallelity]].

Papers:
* [[On the Parallel Transport of Tetrad in a Riemann-Cartan Spacetime - D.-C. Chern|http://psroc.phys.ntu.edu.tw/cjp/v19/45.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=13757775929773306320&as_sdt=2005&sciodt=2000&hl=de]]
A manifold $\mathcal M$ is said to be ''parallelizable'' if there exists a [[torsion|Torsion]] $T_{\nu\rho\sigma}$ which \"flattens\ââ¬Â it, i.e. makes the [[Riemann curvature tensor|Riemann Tensor]] $\tilde{R}_{\mu\nu\sigma\lambda}$ vanish. (This definition is due to Cartan and Schouten and is mostly used in physics literature. In mathematics it is the definition for absolute parallelism and is a stronger condition than parallelism. That is, that in general absolute parallelism implies parallelism, but not vice-versa.
One has the condition:
\begin{equation}
R_{\mu\nu\rho\sigma} =\tilde{R}_{\mu\nu\rho\sigma} + D_\rho T_{\mu\nu\sigma} - D_\sigma T_{\mu\nu\rho} + T_{\mu\sigma\lambda} {T^{\lambda}}_{\nu\rho} - T_{\mu\rho\lambda} {T^{\lambda}}_{\nu\sigma} = 0
\end{equation}
with $\tilde{R}_{\mu\nu\rho\sigma}$ stemming from the [[Christoffel part|Christoffel Symbols]] of the [[connection|Connection]] and the rest from the torsion part.

Analogously a manifold is said to be ''Ricci-parallizable'' if the [[Ricci tensor|Ricci Tensor]] $R_{\mu\nu}$ vanishes, i.e.
\begin{equation}
R_{\mu\nu} =\tilde{R}_{\mu\nu}  + D_\lambda {T^{\lambda}}_{\mu\nu} + T_{\mu\sigma\lambda} {T^{\sigma\lambda}}_{\nu} = 0
\end{equation}
This implies that the symmtric and the antisymmetric parts of the Ricci tensor vanish, i.e.
\begin{equation}
\tilde{R}_{\mu\nu}  + T_{\mu\sigma\lambda} {T^{\sigma\lambda}}_{\nu} = 0
\end{equation}
\begin{equation}
D_\lambda {T^{\lambda}}_{\mu\nu} = 0
\end{equation}
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<<tiddler [[include_tiddlers/Particle Number Operator.html#"Particle Number Operator"]]>>
<<tiddler [[include_tiddlers/Partition Function.html#"Partition Function"]]>>
<<tiddler [[include_tiddlers/Pascual Jordan.html#"Pascual Jordan"]]>>
A ''Passive Transformation'' only changes the representation of any of its elements in respect to coordinates and a basis (i.e. the way it is decomposed into a basis and coordinates). It leaves the physical state of the system unchanged. It implies merely a change in the method of describing it.

In concrete: Given an element $\mb A$ of an algebra and a basis $\{\mb e_\mu \}$ one has the coordinate representation $\mb A = A_\mu \mb e_\mu$. If a transformation to new coordinates  $A'_\mu$ and a new basis $\{\mb e'_\mu \}$ is carried out such that $\mb A = A'_\mu \mb e'_\mu$ it satisfies the requirements for a passive transformation.
Obviously an infinitesimal line element isn't changed either as it is just the scalar product of two elements an hence the physical action stays the same.

Examples:
[[Passive Lorentz transformations|Lorentz Transformation]] that only change the reference frame for the description of the same phenomena. However Lorentz transformations can also be used to describe [[active transformations|Active Transformation]], so called particle boost transformations.

!!!!Polyvectors
Given a [[polyvector|Polyvector Space]] $\mb A = A_\alpha \mb E_\alpha$, a passive transformation is required to not change it. Therefore
\begin{eqnarray}
d\mb A & = & 0\\
& = & dA_\alpha \mb E_\alpha + A_\alpha d  \mb E_\alpha \\
& = & dA_\alpha \mb E_\alpha + A_\alpha \Gamma_{\beta\alpha}  \mb E_\alpha \\
& = & ( dA_\alpha + A_\beta \Gamma_{\beta\alpha}) \mb E_\alpha \\
\end{eqnarray}
Hence the [[covariant polyvector derivative|Polyvector Covariant Derivative]] must be zero.
<<tiddler [[include_tiddlers/Past Life Regression.html#"Past Life Regression"]]>>
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<<tiddler [[include_tiddlers/Perfect Fluid.html#"Perfect Fluid"]]>>
A ''Perfect Number'' is a number which is the sum of all its positive divisors except for itself. (I.e. it cannot be a prime number).
The first four perfect numbers were already described by Euklid and obey the formula $2^{n?1}(2^n ? 1) = 2^{n?1} M_n$ with $M_n$ a [[Mersenne prime number|Mersenne Prime]].
Hence
\begin{eqnarray}
6 & = &2^1(2^2-1) = 1+2+3  \\
28 &=& 2^2(2^3-1) = 1+2+3+4+5+6+7 = 1^3+3^3 \\
496 & =& 2^4(2^5-1) = 1+2+3+\cdots+29+30+31 = 1^3+3^3+5^3+7^3  \\
8128 &= & 2^6(2^7-1) = 1+2+3+\cdots+125+126+127 = 1^3+3^3+5^3+7^3+9^3+11^3+13^3+15^3
\end{eqnarray}

Currently $47$ perfect numbers are known which equals the number of Mersenne primes.

Links:
* [[WIKIPEDIA - Perfect Number|http://en.wikipedia.org/wiki/Perfect_number]]
* [[The MacTutor History of Mathematics Archive - Perfect Numbers|http://www-history.mcs.st-andrews.ac.uk/HistTopics/Perfect_numbers.html]]
<<tiddler [[include_tiddlers/Permittivity.html#"Permittivity"]]>>
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<<tiddler [[include_tiddlers/Perron-Frobenius Theorem.html#"Perron-Frobenius Theorem"]]>>
<<tiddler [[include_tiddlers/Personalized Medicine.html#"Personalized Medicine"]]>>
<<tiddler [[include_tiddlers/Perturbation Theory.html#"Perturbation Theory"]]>>
<<tiddler [[include_tiddlers/Petrov Classification.html#"Petrov Classification"]]>>
<<tiddler [[include_tiddlers/Pfaffian.html#"Pfaffian"]]>>
A ''Pfaffian Equation'' is defined by relationship between [[forms|Form]]. The theory of Pfaffian equations is a rather difficult subject.
!!!!Example
$xdy ? ydx = 0$
''Pfister forms'' are a special class of [[quadratic forms|Quadratic Form]] and play a key role in the theory of quadratic forms.

A ''$n$-Pfister Form'' (a.k.a ''$n$-fold Pfister Form'' or '' Pfister $n$-Form'') is defined by
\[
\langle \langle a_1, a_2, ... , a_n \rangle \rangle  \equiv \otimes_{i = 1}^n  \langle 1, -a_i \rangle
\]
i.e. it is a the quadratic form with coefficients obtained by an $n$-fold tensor product of the coefficients $\{1, a_i\}$ of a quadratic forms given by $ 1\cdot x_1^2 - a_ix_2^2$.

$n$-Pfister forms for $n \le 3$ are [[norm forms|Norm]] of [[composition algebras|Composition Algebra]].

!!!!Examples
$n = 1$: $\langle\langle a_1\rangle\rangle = \langle 1, -a_1 \rangle = x_1^2 - a_1x_2^2$.
Depending on the sign of $a_1$ one has either [[signature|Signature]] $(2,0)$ or $(1,1)$.

$n = 2$: $\langle\langle a_1, a_2 \rangle\rangle = \langle 1, -a_1 \rangle \otimes \langle 1, -a_2 \rangle =  x_1^2 - a_1x_2^2 - a_2x_3^2 + a_1a_2 x_4^2$.
Depending on the signs of $a_1$ and $a_2$  one has either signature $(4,0)$ or $(2,2)$.

$n = 3$: $\langle\langle a_1, a_2, a_3 \rangle\rangle = \langle 1, -a_1 \rangle \otimes \langle 1, -a_2 \rangle  \otimes  \langle 1, -a_3 \rangle =  x_1^2 - a_1x_2^2 - a_2x_3^2 -  a_3x_4^2 + a_1 a_2 x_5^2 + a_1 a_3 x_6^2 + a_2 a_3 x_7^2 - a_1a_2a_3 x_8^2$.

The quadratic form of an [[octonion algebra|Octonion]] is always a Pfister $3$-form. There are two Pfister $3$-forms over $\mathbb R$ in dimension $8$, the one is [[hyperbolic|Isotropy]] (signature $(4,4)$) and the other one is anisotropic (signature $(8,0)$).

Lectures:
* [[Topics on Quadratic Forms - A. Vishik|http://users.ictp.it/~pub_off/lectures/lns023/Vishik/Vishik.pdf]]
Papers:
* [[Sums of Squares - O. Taussky|http://www.dm.unito.it/~cerruti/ntlab2007/taussky-squares.pdf]] [[pct. 16|http://scholar.google.de/scholar?cites=5052403810380338764&hl=de]]
* [[The Degen-Graves-Cayley Eight-Square Identity - T. Piezas III|http://www.geocities.com/titus_piezas/DegenGraves1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=16420083384206827699&hl=de]]
* [[Pfister's Theorem on Sums of Squares - K. Conrad|http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/pfister.pdf]] pct. 0
> By any measure, phase space is the natural arena for both classical and quantum mechanics.
> - John R. Klauder -

A ''Phase Space'' is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space.


Papers:
* [[Complex Coordinates and Quantum Mechanics (1996) - F. Strocchi|http://www.ing.unibs.it/~beretta/www.quantumthermodynamics.org/FStrocchi-RevModPhys-38-36-1966.pdf]] [[local|papers/FStrocchi-RevModPhys-38-36-1966.pdf]] {{t100Cite{[[pct. 105|http://scholar.google.de/scholar?cites=10560228305715457362&as_sdt=2005&sciodt=0,5&hl=de]]}}}

Links:
* [[WIKIPEDIA - Phase Space|http://en.wikipedia.org/wiki/Phase_space]]
<<tiddler [[include_tiddlers/Phase Transition.html#"Phase Transition"]]>>
<<tiddler [[include_tiddlers/Phenotypic Plasticity.html#"Phenotypic Plasticity"]]>>
<<tiddler [[include_tiddlers/Philip Warren Anderson.html#"Philip Warren Anderson"]]>>
<<tiddler [[include_tiddlers/Philosophical Aspects of Quantum Field Theory.html#"Philosophical Aspects of Quantum Field Theory"]]>>
<<tiddler [[include_tiddlers/Phosphatidylcholine.html#"Phosphatidylcholine"]]>>
<<tiddler [[include_tiddlers/Phospholipid.html#"Phospholipid"]]>>
<<tiddler [[include_tiddlers/Photon Bose-Einstein Condensate.html#"Photon Bose-Einstein Condensate"]]>>
<<tiddler [[include_tiddlers/Physical Constant.html#"Physical Constant"]]>>
<<tiddler [[include_tiddlers/Physical Exercise.html#"Physical Exercise"]]>>
<<tiddler [[include_tiddlers/Physical Interpretation of Nonassociativity.html#"Physical Interpretation of Nonassociativity"]]>>
<<tiddler [[include_tiddlers/Physics and Mathematics Blogs.html#"Physics and Mathematics Blogs"]]>>
<<tiddler [[include_tiddlers/Physics and the Soul.html#"Physics and the Soul"]]>>
<<tiddler [[include_tiddlers/Piecewise Linear Manifold.html#"Piecewise Linear Manifold"]]>>
Given an algebra $\mathcal A$, its ''Pierce Decomposition'' relative to an [[idempotent|Idempotency]] $\mb E$ is given by
\[
\mathcal A = \mathcal A_0 \oplus \mathcal A_{1/2} \oplus  \mathcal A_1
\]
with
\begin{eqnarray}
\mathcal A_k &= &\{\mb A \in \mathcal A: \mb{AE} = \mb{EA} = k \mb A\}, \quad k = 0,1 \\
\mathcal A_{1/2} &= &\{\mb A \in \mathcal A:  \mb{AE} + \mb{EA}  = \mb A\}
\end{eqnarray}
<<tiddler [[include_tiddlers/Pin Group.html#"Pin Group"]]>>
<<tiddler [[include_tiddlers/Pion.html#"Pion"]]>>
<<tiddler [[include_tiddlers/Pioneer Anomaly.html#"Pioneer Anomaly"]]>>
<<tiddler [[include_tiddlers/Planck Angular Momentum.html#"Planck Angular Momentum"]]>>
<<tiddler [[include_tiddlers/Planck Entropy.html#"Planck Entropy"]]>>
<<tiddler [[include_tiddlers/Planck Force.html#"Planck Force"]]>>
<<tiddler [[include_tiddlers/Planck Lattice.html#"Planck Lattice"]]>>
<<tiddler [[include_tiddlers/Planck Length.html#"Planck Length"]]>>
<<tiddler [[include_tiddlers/Planck Mass.html#"Planck Mass"]]>>
<<tiddler [[include_tiddlers/Planck Mass Plasma Vacuum Conjecture.html#"Planck Mass Plasma Vacuum Conjecture"]]>>
<<tiddler [[include_tiddlers/Planck Particle.html#"Planck Particle"]]>>
<<tiddler [[include_tiddlers/Planck Temperature.html#"Planck Temperature"]]>>
<<tiddler [[include_tiddlers/Planck Units.html#"Planck Units"]]>>
<<tiddler [[include_tiddlers/Planck's Constant.html#"Planck's Constant"]]>>
<<tiddler [[include_tiddlers/Plant Intelligence.html#"Plant Intelligence"]]>>
<<tiddler [[include_tiddlers/Plants and Indoor Climate.html#"Plants and Indoor Climate"]]>>
<<tiddler [[include_tiddlers/Platform as a Service.html#"Platform as a Service"]]>>
<<tiddler [[include_tiddlers/Platonic Solid.html#"Platonic Solid"]]>>
<<tiddler [[include_tiddlers/Platonism.html#"Platonism"]]>>
/***
|Name|PlayerPlugin|
|Source|http://www.TiddlyTools.com/#PlayerPlugin|
|Version|1.1.4|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|Embed a media player in a tiddler|
!!!!!Usage
<<<
{{{<<player [id=xxx] [type] [URL] [width] [height] [autoplay|true|false] [showcontrols|true|false] [extras]>>}}}

''id=xxx'' is optional, and specifies a unique identifier for each embedded player.  note: this is required if you intend to display more than one player at the same time.

''type'' is optional, and is one of the following: ''windows'', ''realone'', ''quicktime'', ''flash'', ''image'' or ''iframe''.  If the media type is not specified, the plugin automatically detects Windows, Real, QuickTime, Flash video or JPG/GIF images by matching known file extensions and/or specialized streaming-media transfer protocols (such as RTSP:).  For unrecognized media types, the plugin displays an error message.

''URL'' is the location of the media content

''width'' and ''height'' are the dimensions of the video display area (in pixels)

''autoplay'' or ''true'' or ''false'' is optional, and specifies whether the media content should begin playing as soon as it is loaded, or wait for the user to press the "play" button.  Default is //not// to autoplay.

''showcontrols'' or ''true'' or ''false'' is optional, and specifies whether the embedded media player should display its built-in control panel (e.g., play, pause, stop, rewind, etc), if any.  Default is to display the player controls.

''extras'' are optional //pairs// of parameters that can be passed to the embedded player, using the {{{<param name=xxx value=yyy>}}} HTML syntax.

''If you use [[AttachFilePlugin]] to encode and store a media file within your document, you can play embedded media content by using the title of the //attachment tiddler//'' as a parameter in place of the usual reference to an external URL.  When playing an attached media content, you should always explicitly specify the media type parameter, because the name used for the attachment tiddler may not contain a known file extension from which a default media type can be readily determined.
<<<
!!!!!Configuration
<<<
Default player size:
width: <<option txtPlayerDefaultWidth>> height: <<option txtPlayerDefaultHeight>>
<<<
!!!!!Revisions
<<<
2008.05.10 [1.1.4] in handlers(), immediately return if no params (prevents error in macro).  Also, refactored auto-detect code to make type mapping configurable.
2007.10.15 [1.1.3] in loadURL(), add recognition for .PNG (still image), fallback to iframe for unrecognized media types
2007.08.31 [1.1.2] added 'click-through' link for JPG/GIF images
2007.06.21 [1.1.1] changed "hidecontrols" param to "showcontrols" and recognize true/false values in addition to 'showcontrols', added "autoplay" param (also recognize true/false values), allow "auto" as value for type param
2007.05.22 [1.1.0] added support for type=="iframe" (displays src URL in an IFRAME)
2006.12.06 [1.0.1] in handler(), corrected check for config.macros.attach (instead of config.macros.attach.getAttachment) so that player plugin will work when AttachFilePlugin is NOT installed.  (Thanks to Phillip Ehses for bug report)
2006.11.30 [1.0.0] support embedded media content using getAttachment() API defined by AttachFilePlugin or AttachFilePluginFormatters.  Also added support for 'image' type to render JPG/GIF still images
2006.02.26 [0.7.0] major re-write.  handles default params better.  create/recreate player objects via loadURL() API for use with interactive forms and scripts.
2006.01.27 [0.6.0] added support for 'extra' macro params to pass through to object parameters
2006.01.19 [0.5.0] Initial ALPHA release
2005.12.23 [0.0.0] Started
<<<
!!!!!Code
***/
//{{{
version.extensions.PlayerPlugin= {major: 1, minor: 1, revision: 4, date: new Date(2008,5,10)};

config.macros.player = {};
config.macros.player.html = {};
config.macros.player.handler= function(place,macroName,params) {
	if (!params.length) return; // missing parameters - do nothing
	var id=null;
	if (params[0].substr(0,3)=="id=") id=params.shift().substr(3);
	var type="";
	if (!params.length) return; // missing parameters - do nothing
	var p=params[0].toLowerCase();
	if (p=="auto" || p=="windows" || p=="realone" || p=="quicktime" || p=="flash" || p=="image" || p=="iframe")
		type=params.shift().toLowerCase();
	var url=params.shift(); if (!url || !url.trim().length) url="";
	if (url.length && config.macros.attach!=undefined) // if AttachFilePlugin is installed
		if ((tid=store.getTiddler(url))!=null && tid.isTagged("attachment")) // if URL is attachment
			url=config.macros.attach.getAttachment(url); // replace TiddlerTitle with URL
	var width=params.shift();
	var height=params.shift();
	var autoplay=false;
	if (params[0]=='autoplay'||params[0]=='true'||params[0]=='false')
		autoplay=(params.shift()!='false');
	var show=true;
	if (params[0]=='showcontrols'||params[0]=='true'||params[0]=='false')
		show=(params.shift()!='false');
	var extras="";
	while (params[0]!=undefined)
		extras+="<param name='"+params.shift()+"' value='"+params.shift()+"'> ";
	this.loadURL(place,id,type,url,width,height,autoplay,show,extras);
}

if (config.options.txtPlayerDefaultWidth==undefined) config.options.txtPlayerDefaultWidth="100%";
if (config.options.txtPlayerDefaultHeight==undefined) config.options.txtPlayerDefaultHeight="480"; // can't use "100%"... player height doesn't stretch right :-(

config.macros.player.typeMap={
	windows: ['mms', '.asx', '.wvx', '.wmv', '.mp3'],
	realone: ['rtsp', '.ram', '.rpm', '.rm', '.ra'],
	quicktime: ['.mov', '.qt'],
	flash: ['.swf', '.flv'],
	image: ['.jpg', '.gif', '.png'],
	iframe: ['.htm', '.html', '.shtml', '.php']
};

config.macros.player.loadURL=function(place,id,type,url,width,height,autoplay,show,extras) {

	if (id==undefined) id="tiddlyPlayer";
	if (!width) var width=config.options.txtPlayerDefaultWidth;
	if (!height) var height=config.options.txtPlayerDefaultHeight;
	if (url && (!type || !type.length || type=="auto")) { // determine type from URL
		u=url.toLowerCase();
		var map=config.macros.player.typeMap;
		for (var t in map) for (var i=0; i<map[t].length; i++)
			if (u.indexOf(map[t][i])!=-1) var type=t;
	}
	if (!type || !config.macros.player.html[type]) var type="none";
	if (!url) var url="";
	if (show===undefined) var show=true;
	if (!extras) var extras="";
	if (type=="none" && url.trim().length) type="iframe"; // fallback to iframe for unrecognized media types

	// adjust parameter values for player-specific embedded HTML
	switch (type) {
		case "windows":
			autoplay=autoplay?"1":"0"; // player-specific param value
			show=show?"1":"0"; // player-specific param value
			break;
		case "realone":
			autoplay=autoplay?"true":"false";
			show=show?"block":"none";
			height-=show?60:0; // leave room for controls
			break;
		case "quicktime":
			autoplay=autoplay?"true":"false";
			show=show?"true":"false";
			break;
		case "image":
			show=show?"block":"none";
			break;
		case "iframe":
			show=show?"block":"none";
			break;
	}

	// create containing div for player HTML
	// and add or replace player in TW DOM structure
	var newplayer = document.createElement("div");
	newplayer.playerType=type;
	newplayer.setAttribute("id",id+"_div");
	var existing = document.getElementById(id+"_div");
	if (existing && !place) place=existing.parentNode;
	if (!existing)
		place.appendChild(newplayer);
	else {
		if (place==existing.parentNode) place.replaceChild(newplayer,existing)
		else { existing.parentNode.removeChild(existing); place.appendChild(newplayer); }
	}

	var html=config.macros.player.html[type];
	html=html.replace(/%i%/mg,id);
	html=html.replace(/%w%/mg,width);
	html=html.replace(/%h%/mg,height);
	html=html.replace(/%u%/mg,url);
	html=html.replace(/%a%/mg,autoplay);
	html=html.replace(/%s%/mg,show);
	html=html.replace(/%x%/mg,extras);
	newplayer.innerHTML=html;
}
//}}}

// // Player-specific API functions: isReady(id), isPlaying(id), toggleControls(id), showControls(id,flag)

//{{{
// status values:
// Windows: 0=Undefined, 1=Stopped, 2=Paused, 3=Playing, 4=ScanForward, 5=ScanReverse
//          6=Buffering, 7=Waiting, 8=MediaEnded, 9=Transitioning, 10=Ready, 11=Reconnecting
// RealOne: 0=Stopped, 1=Contacting, 2=Buffering, 3=Playing, 4=Paused, 5=Seeking
// QuickTime: 'Waiting', 'Loading', 'Playable', 'Complete', 'Error:###'
// Flash: 0=Loading, 1=Uninitialized, 2=Loaded, 3=Interactive, 4=Complete
config.macros.player.isReady=function(id)
{
	var d=document.getElementById(id+"_div"); if (!d) return false;
	var p=document.getElementById(id); if (!p) return false;
	if (d.playerType=='windows') return !((p.playState==0)||(p.playState==7)||(p.playState==9)||(p.playState==11));
	if (d.playerType=='realone') return (p.GetPlayState()>1);
	if (d.playerType=='quicktime') return !((p.getPluginStatus()=='Waiting')||(p.getPluginStatus()=='Loading'));
	if (d.playerType=='flash') return (p.ReadyState>2);
	return true;
}
config.macros.player.isPlaying=function(id)
{
	var d=document.getElementById(id+"_div"); if (!d) return false;
	var p=document.getElementById(id); if (!p) return false;
	if (d.playerType=='windows') return (p.playState==3);
	if (d.playerType=='realone') return (p.GetPlayState()==3);
	if (d.playerType=='quicktime') return (p.getPluginStatus()=='Complete');
	if (d.playerType=='flash') return (p.ReadyState<4);
	return false;
}
config.macros.player.showControls=function(id,flag) {
	var d=document.getElementById(id+"_div"); if (!d) return false;
	var p=document.getElementById(id); if (!p) return false;
	if (d.playerType=='windows') { p.ShowControls=flag; p.ShowStatusBar=flag; }
	if (d.playerType=='realone') { alert('show/hide controls not available'); }
	if (d.playerType=='quicktime')      // if player not ready, retry in one second
		{ if (this.isReady(id)) p.setControllerVisible(flag); else setTimeout('config.macros.player.showControls("'+id+'",'+flag+')',1000); }
	if (d.playerType=='flash') { alert('show/hide controls not available'); }
}
config.macros.player.toggleControls=function(id) {
	var d=document.getElementById(id+"_div"); if (!d) return false;
	var p=document.getElementById(id); if (!p) return false;
	if (d.playerType=='windows') var flag=!p.ShowControls;
	if (d.playerType=='realone') var flag=true; // TBD
	if (d.playerType=='quicktime') var flag=!p.getControllerVisible();
	if (d.playerType=='flash') var flag=true; // TBD
	this.showControls(id,flag);
}
config.macros.player.fullScreen=function(id) {
	var d=document.getElementById(id+"_div"); if (!d) return false;
	var p=document.getElementById(id); if (!p) return false;
	if (d.playerType=='windows') p.DisplaySize=3;
	if (d.playerType=='realone') p.SetFullScreen();
	if (d.playerType=='quicktime') { alert('full screen not available'); }
	if (d.playerType=='flash') { alert('full screen not available'); }
}
//}}}

// // Player HTML

//{{{
// placeholder (no player)
config.macros.player.html.none=' \
	<table id="%i%" width="%w%" height="%h%" style="background-color:#111;border:0;margin:0;padding:0;"> \
	<tr style="background-color:#111;border:0;margin:0;padding:0;"> \
	<td width="%w%" height="%h%" style="background-color:#111;color:#ccc;border:0;margin:0;padding:0;text-align:center;"> \
	&nbsp; \
	%u% \
	&nbsp; \
	</td></tr></table>';
//}}}

//{{{
// JPG/GIF/PNG still images
config.macros.player.html.image='\
	<a href="%u%" target="_blank"><img width="%w%" height="%h%" style="display:%s%;" src="%u%"></a>';
//}}}

//{{{
// IFRAME web page viewer
config.macros.player.html.iframe='\
	<iframe id="%i%" width="%w%" height="%h%" style="display:%s%;background:#fff;" src="%u%"></iframe>';
//}}}

//{{{
// Windows Media Player
// v7.1 ID: classid=CLSID:6BF52A52-394A-11d3-B153-00C04F79FAA6
// v9	ID: classid=CLSID:22d6f312-b0f6-11d0-94ab-0080c74c7e95
config.macros.player.html.windows=' \
	<object id="%i%" width="%w%" height="%h%" style="margin:0;padding:0;width:%w%;height:%h%px;" \
		classid="CLSID:22d6f312-b0f6-11d0-94ab-0080c74c7e95" \
		codebase="http://activex.microsoft.com/activex/controls/mplayer/en/nsmp2inf.cab#Version=6,4,5,715" \
		align="baseline" border="0" \
		standby="Loading Microsoft Windows Media Player components..." \
		type="application/x-oleobject"> \
		<param name="FileName" value="%u%"> <param name="ShowControls" value="%s%"> \
		<param name="ShowPositionControls" value="1"> <param name="ShowAudioControls" value="1"> \
		<param name="ShowTracker" value="1"> <param name="ShowDisplay" value="0"> \
		<param name="ShowStatusBar" value="1"> <param name="AutoSize" value="1"> \
		<param name="ShowGotoBar" value="0"> <param name="ShowCaptioning" value="0"> \
		<param name="AutoStart" value="%a%"> <param name="AnimationAtStart" value="1"> \
		<param name="TransparentAtStart" value="0"> <param name="AllowScan" value="1"> \
		<param name="EnableContextMenu" value="1"> <param name="ClickToPlay" value="1"> \
		<param name="InvokeURLs" value="1"> <param name="DefaultFrame" value="datawindow"> \
		%x% \
		<embed src="%u%" style="margin:0;padding:0;width:%w%;height:%h%px;" \
			align="baseline" border="0" width="%w%" height="%h%" \
			type="application/x-mplayer2" \
			pluginspage="http://www.microsoft.com/windows/windowsmedia/download/default.asp" \
			name="%i%" showcontrols="%s%" showpositioncontrols="1" \
			showaudiocontrols="1" showtracker="1" showdisplay="0" \
			showstatusbar="%s%" autosize="1" showgotobar="0" showcaptioning="0" \
			autostart="%a%" autorewind="0" animationatstart="1" transparentatstart="0" \
			allowscan="1" enablecontextmenu="1" clicktoplay="0" invokeurls="1" \
			defaultframe="datawindow"> \
		</embed> \
	</object>';
//}}}

//{{{
// RealNetworks' RealOne Player
config.macros.player.html.realone=' \
	<table width="%w%" style="border:0;margin:0;padding:0;"><tr style="border:0;margin:0;padding:0;"><td style="border:0;margin:0;padding:0;"> \
	<object id="%i%" width="%w%" height="%h%" style="margin:0;padding:0;" \
		CLASSID="clsid:CFCDAA03-8BE4-11cf-B84B-0020AFBBCCFA"> \
		<PARAM NAME="CONSOLE" VALUE="player"> \
		<PARAM NAME="CONTROLS" VALUE="ImageWindow"> \
		<PARAM NAME="AUTOSTART" Value="%a%"> \
		<PARAM NAME="MAINTAINASPECT" Value="true"> \
		<PARAM NAME="NOLOGO" Value="true"> \
		<PARAM name="BACKGROUNDCOLOR" VALUE="#333333"> \
		<PARAM NAME="SRC" VALUE="%u%"> \
		%x% \
		<EMBED width="%w%" height="%h%" controls="ImageWindow" type="audio/x-pn-realaudio-plugin" style="margin:0;padding:0;" \
			name="%i%" \
			src="%u%" \
			console=player \
			maintainaspect=true \
			nologo=true \
			backgroundcolor=#333333 \
			autostart=%a%> \
		</OBJECT> \
	</td></tr><tr style="border:0;margin:0;padding:0;"><td style="border:0;margin:0;padding:0;"> \
	<object id="%i%_controls" width="%w%" height="60" style="margin:0;padding:0;display:%s%" \
		CLASSID="clsid:CFCDAA03-8BE4-11cf-B84B-0020AFBBCCFA"> \
		<PARAM NAME="CONSOLE" VALUE="player"> \
		<PARAM NAME="CONTROLS" VALUE="All"> \
		<PARAM NAME="NOJAVA" Value="true"> \
		<PARAM NAME="MAINTAINASPECT" Value="true"> \
		<PARAM NAME="NOLOGO" Value="true"> \
		<PARAM name="BACKGROUNDCOLOR" VALUE="#333333"> \
		<PARAM NAME="SRC" VALUE="%u%"> \
		%x% \
		<EMBED WIDTH="%w%" HEIGHT="60" NOJAVA="true" type="audio/x-pn-realaudio-plugin" style="margin:0;padding:0;display:%s%" \
			controls="All" \
			name="%i%_controls" \
			src="%u%" \
			console=player \
			maintainaspect=true \
			nologo=true \
			backgroundcolor=#333333> \
		</OBJECT> \
	</td></tr></table>';
//}}}

//{{{
// QuickTime Player
config.macros.player.html.quicktime=' \
	<OBJECT ID="%i%" WIDTH="%w%" HEIGHT="%h%" style="margin:0;padding:0;" \
		CLASSID="clsid:02BF25D5-8C17-4B23-BC80-D3488ABDDC6B" \
		CODEBASE="http://www.apple.com/qtactivex/qtplugin.cab"> \
		<PARAM name="SRC" VALUE="%u%"> \
		<PARAM name="AUTOPLAY" VALUE="%a%"> \
		<PARAM name="CONTROLLER" VALUE="%s%"> \
		<PARAM name="BGCOLOR" VALUE="#333333"> \
		<PARAM name="SCALE" VALUE="aspect"> \
		<PARAM name="SAVEEMBEDTAGS" VALUE="true"> \
		%x% \
		<EMBED name="%i%" WIDTH="%w%" HEIGHT="%h%" style="margin:0;padding:0;" \
			SRC="%u%" \
			AUTOPLAY="%a%" \
			SCALE="aspect" \
			CONTROLLER="%s%" \
			BGCOLOR="#333333" \
			EnableJavaSript="true" \
			PLUGINSPAGE="http://www.apple.com/quicktime/download/"> \
		</EMBED> \
	</OBJECT>';
//}}}

//{{{
// Flash Player
config.macros.player.html.flash='\
	<object id="%i%" width="%w%" height="%h%" style="margin:0;padding:0;" \
		classid="clsid:D27CDB6E-AE6D-11cf-96B8-444553540000" \
		codebase="http://download.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=6,0,29,0"> \
		<param name="movie" value="%u%"> \
		<param name="quality" value="high"> \
		<param name="SCALE" value="exactfit"> \
		<param name="bgcolor" value="333333"> \
		%x% \
		<embed name="%i%" src="%u%" style="margin:0;padding:0;" \
			height="%h%" width="%w%" quality="high" \
			pluginspage="http://www.macromedia.com/go/getflashplayer" \
			type="application/x-shockwave-flash" scale="exactfit"> \
		</embed> \
	</object>';
//}}}
<<tiddler [[include_tiddlers/Plotkin Construction.html#"Plotkin Construction"]]>>
/***

''Inspired by [[TiddlyPom|http://www.warwick.ac.uk/~tuspam/tiddlypom.html]]''

|Name|SplashScreenPlugin|
|Created by|SaqImtiaz|
|Location|http://lewcid.googlepages.com/lewcid.html#SplashScreenPlugin|
|Version|0.21 |
|Requires|~TW2.08+|
!Description:
Provides a simple splash screen that is visible while the TW is loading.

!Installation
Copy the source text of this tiddler to your TW in a new tiddler, tag it with systemConfig and save and reload. The SplashScreen will now be installed and will be visible the next time you reload your TW.

!Customizing
Once the SplashScreen has been installed and you have reloaded your TW, the splash screen html will be present in the MarkupPreHead tiddler. You can edit it and customize to your needs.

!History
* 20-07-06 : version 0.21, modified to hide contentWrapper while SplashScreen is displayed.
* 26-06-06 : version 0.2, first release

!Code
***/
//{{{
var old_lewcid_splash_restart=restart;

restart = function()
{   if (document.getElementById("SplashScreen"))
        document.getElementById("SplashScreen").style.display = "none";
      if (document.getElementById("contentWrapper"))
        document.getElementById("contentWrapper").style.display = "block";

    old_lewcid_splash_restart();

    if (splashScreenInstall)
       {if(config.options.chkAutoSave)
			{saveChanges();}
        displayMessage("TW SplashScreen has been installed, please save and refresh your TW.");
        }
}


var oldText = store.getTiddlerText("MarkupPreHead");
if (oldText.indexOf("SplashScreen")==-1)
   {var siteTitle = store.getTiddlerText("SiteTitle");
   var splasher='\n\n<style type="text/css">#contentWrapper {display:none;}</style><div id="SplashScreen" style="border: 3px solid #ccc; display: block; text-align: center; width: 320px; margin: 100px auto; padding: 50px; color:#000; font-size: 28px; font-family:Tahoma; background-color:#eee;"><b>'+siteTitle +'</b> is loading<blink> ...</blink><br><br><span style="font-size: 14px; color:red;">Requires Javascript.</span></div>';
   if (! store.tiddlerExists("MarkupPreHead"))
       {var myTiddler = store.createTiddler("MarkupPreHead");}
   else
      {var myTiddler = store.getTiddler("MarkupPreHead");}
      myTiddler.set(myTiddler.title,oldText+splasher,config.options.txtUserName,null,null);
      store.setDirty(true);
      var splashScreenInstall = true;
}
//}}}
/***
|''Name''|PluginMathJax|
|''Description''|Displays TeX math using MathJax|
|''Author''|[[Canada East|http://tiddlywiki.canada-east.ca/]]|
|''Version''|1.3|
|''Date''|2010-10-07|
|''CodeRepository''|[[PluginMathJax|http://tiddlywiki.canada-east.ca/#PluginMathJax]]|
|''CoreVersion''|[[2.6.1|http://www.tiddlywiki.com]]|
|''Requires''|[[MathJax v1.01|http://www.mathjax.org/]]|
|''Feedback''|[[Contact|https://spreadsheets.google.com/viewform?formkey=dGg2RkpxZW5zWTh6QjZxOXgzZUlfakE6MQ]]|
|''Tweaks''|~MathJax location and default HTML-CSS scale changed by Gareth Davies.  I've also added a hook to a ~MathJax extension for managing local definitions written by Richard Lupton and I've set the newcommand extension to load on start-up.|
!Description
This plugin uses [[MathJax|http://www.mathjax.org/]] to typeset ([[AMS|http://www.ams.org/publications/authors/tex/amslatex]]) [[LaTeX|http://www.latex-project.org/]]  math. It can also be configured to use additional MathJax functionality.
>"MathJax is an open source JavaScript display engine for mathematics that works in all modern browsers."
!Notes
Right click any math display for a MathJax menu. The user can select the renderer and zoom settings. It performs best in [[Webkit|http://en.wikipedia.org/wiki/List_of_web_browsers#WebKit-based_browsers]] based browsers. Larger math displays such as the additional examples tiddler below can put quite a load on IE. PluginMathJax is based on: [[Plugin: jsMath|http://bob.mcelrath.org/tiddlyjsmath.html]]
!Installation
#''Backup'' your TiddlyWiki!
#It is required that the MathJax directory is installed in '''js/MathJax/''' in the same location as the TiddlyWiki html file.<br>(Or edit the script source where commented in the plugin code below after installation to match the location of your MathJax install.)
#Install this plugin (and examples tiddler linked below if desired).
!Usage
|!Source|!Output|h
|{{{The variable $x$ is real.}}}|The variable $x$ is real.|
|{{{The variable \(y\) is complex.}}}|The variable \(y\) is complex.|
|{{{This \[\int_a^b x = \frac{1}{2}(b^2-a^2)\] is an easy integral.}}}|This \[\int_a^b x = \frac{1}{2}(b^2-a^2)\] is an easy integral.|
|{{{This $$\int_a^b \sin x = -(\cos b - \cos a)$$ is another easy integral.}}}|This $$\int_a^b \sin x = -(\cos b - \cos a)$$ is another easy integral.|
|{{{Block formatted equations may also use the 'equation' environment \begin{equation}  \int \tan x = -\ln \cos x \end{equation} }}}|Block formatted equations may also use the 'equation' environment \begin{equation}  \int \tan x = -\ln \cos x \end{equation}|
|{{{Equation arrays are also supported \begin{eqnarray} a &=& b \\ c &=& d \end{eqnarray} }}}|Equation arrays are also supported \begin{eqnarray} a &=& b \\ c &=& d \end{eqnarray} |
|{{{I spent \$7.38 on lunch.}}}|I spent \$7.38 on lunch.|
|{{{I had to insert a backslash (\\) into my document}}}|I had to insert a backslash (\\) into my document|
| <br>[[Complete list of supported LaTeX commands|http://www.mathjax.org/resources/docs/?tex.html#supported-latex-commands]] |>|
!Examples
[[Additional MathJax Examples|MathJax Examples]]
!Configuration
MathJax can be manually configured if desired by editing the code below (advanced). See the [[MathJax documentation|http://www.mathjax.org/resources/docs/?configuration.html#configuration-options-by-component]] for details.
!Revision History
*v1.3, 2010-10-07, returned to original formatters design, kept modified wikify and recommended way of loading MathJax dynamically, removed the tex2jax extension and corrected several browser compatibility issues (InnerHTML for Opera and IE9).
*v1.2, 2010-10-05, removed some redundant MathJax config entries, moved modified wikify and MathJax.Hub.Queue call.
*v1.1, 2010-10-03, autoLinkWikiWords disabled in absence of DisableWikiLinksPlugin, modifed wikify.
*v1.0, 2010-09-26, Initial Release
!Code
***/
//{{{

if(!version.extensions.PluginMathJax) {
    version.extensions.PluginMathJax = { installed: true };

    config.extensions.PluginMathJax = {

        install: function() {

            var script = document.createElement("script");
            script.type = "text/javascript";

            // *** Use the location of your MathJax! *** :
            /*
             * Gareth: The following line assumes you have
             * MathJax installed on your server in a sensible
             * location.  I've commented this out.
             */
            //script.src = "js/MathJax/MathJax.js";

            /*
             * Because this tiddlywiki is currently hosted on
             * tiddlyspace.com I've had to point to the 'MathJax
             * Content Delivery Network' instead.
             */
            script.src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"
            // EndGareth

            /*
             * Gareth: Richard's local definition ~MathJax
             * extension (implementation of TeXs \let command)
             * has been added to the list of extensions along
             * with the newcommand extension upon which it
             * depends.  Also, the scale option for HTML-CSS was
             * changed from 115 to 100.
             */
            var mjconfig = 'MathJax.Hub.Config({' +
            'jax: ["input/TeX","output/HTML-CSS"],' +
            'extensions: ["TeX/AMSmath.js", "TeX/AMSsymbols.js", "TeX/newcommand.js", "http://www.markus-maute.com/trajectory/localTeX.js"],' +
            '"HTML-CSS": {' +
                'scale: 85' +
                '}' +
            '});' +

            'MathJax.Hub.Startup.onload();';

            var ie9RegExp = /^9\./;
            var UseInnerHTML = (config.browser.isOpera || config.browser.isIE && ie9RegExp.test(config.browser.ieVersion[1]));

            if (UseInnerHTML) {script.innerHTML = mjconfig;}
                else {script.text = mjconfig;}

            script.text = mjconfig;

            document.getElementsByTagName("head")[0].appendChild(script);

            // Define wikifers for latex
            config.formatterHelpers.mathFormatHelper = function(w) {
                var e = document.createElement(this.element);
                e.type = this.type;
                var endRegExp = new RegExp(this.terminator, "mg");
                endRegExp.lastIndex = w.matchStart+w.matchLength;
                var matched = endRegExp.exec(w.source);
                if(matched) {
                    var txt = w.source.substr(w.matchStart+w.matchLength,
                        matched.index-w.matchStart-w.matchLength);
                    if(this.keepdelim) {
                      txt = w.source.substr(w.matchStart, matched.index+matched[0].length-w.matchStart);
                    }
                    if (UseInnerHTML) {
                        e.innerHTML = txt;
                    } else {
                        e.text = txt;
                    }
                    w.output.appendChild(e);
                    w.nextMatch = endRegExp.lastIndex;
                }
            }

            config.formatters.push({
              name: "displayMath1",
              match: "\\\$\\\$",
              terminator: "\\\$\\\$\\n?",
              termRegExp: "\\\$\\\$\\n?",
              element: "script",
              type: "math/tex; mode=display",
              handler: config.formatterHelpers.mathFormatHelper
            });

            config.formatters.push({
              name: "inlineMath1",
              match: "\\\$",
              terminator: "\\\$",
              termRegExp: "\\\$",
              element: "script",
              type: "math/tex",
              handler: config.formatterHelpers.mathFormatHelper
            });

            var backslashformatters = new Array(0);

            backslashformatters.push({
              name: "inlineMath2",
              match: "\\\\\\\(",
              terminator: "\\\\\\\)",
              termRegExp: "\\\\\\\)",
              element: "script",
              type: "math/tex",
              handler: config.formatterHelpers.mathFormatHelper
            });

            backslashformatters.push({
              name: "displayMath2",
              match: "\\\\\\\[",
              terminator: "\\\\\\\]\\n?",
              termRegExp: "\\\\\\\]\\n?",
              element: "script",
              type: "math/tex; mode=display",
              handler: config.formatterHelpers.mathFormatHelper
            });

            backslashformatters.push({
              name: "displayMath3",
              match: "\\\\begin\\{equation\\}",
              terminator: "\\\\end\\{equation\\}\\n?",
              termRegExp: "\\\\end\\{equation\\}\\n?",
              element: "script",
              type: "math/tex; mode=display",
              handler: config.formatterHelpers.mathFormatHelper
            });

            // These can be nested.  e.g. \begin{equation} \begin{array}{ccc} \begin{array}{ccc} ...
            backslashformatters.push({
              name: "displayMath4",
              match: "\\\\begin\\{eqnarray\\}",
              terminator: "\\\\end\\{eqnarray\\}\\n?",
              termRegExp: "\\\\end\\{eqnarray\\}\\n?",
              element: "script",
              type: "math/tex; mode=display",
              keepdelim: true,
              handler: config.formatterHelpers.mathFormatHelper
            });

            // The escape must come between backslash formatters and regular ones.
            // So any latex-like \commands must be added to the beginning of
            // backslashformatters here.
            backslashformatters.push({
                name: "escape",
                match: "\\\\.",
                handler: function(w) {
                    w.output.appendChild(document.createTextNode(w.source.substr(w.matchStart+1,1)));
                    w.nextMatch = w.matchStart+2;
                }
            });

          config.formatters=backslashformatters.concat(config.formatters);

          old_wikify = wikify;
          wikify = function(source,output,highlightRegExp,tiddler)
          {
              old_wikify.apply(this,arguments);
              if (window.MathJax) {MathJax.Hub.Queue(["Typeset",MathJax.Hub,output])}
          };

        }
    };

  config.extensions.PluginMathJax.install();

}

//}}}
<<tiddler [[include_tiddlers/Podkletnov Experiments.html#"Podkletnov Experiments"]]>>
<<tiddler [[include_tiddlers/Poincaré Conjecture.html#"Poincaré Conjecture"]]>>
<<tiddler [[include_tiddlers/Poincaré Duality.html#"Poincaré Duality"]]>>
<<tiddler [[include_tiddlers/Poincaré Gauge Theory of Gravity.html#"Poincaré Gauge Theory of Gravity"]]>>
<<tiddler [[include_tiddlers/Poincaré Homology Sphere.html#"Poincaré Homology Sphere"]]>>
<<tiddler [[include_tiddlers/Poincaré Transformation.html#"Poincaré Transformation"]]>>
<<tiddler [[include_tiddlers/Poincaré-Birkhoff-Witt Theorem.html#"Poincaré-Birkhoff-Witt Theorem"]]>>
<<tiddler [[include_tiddlers/Point Defect.html#"Point Defect"]]>>
<<tiddler [[include_tiddlers/Point Group.html#"Point Group"]]>>
<<tiddler [[include_tiddlers/Poisson Bracket.html#"Poisson Bracket"]]>>
<<tiddler [[include_tiddlers/Poisson Distribution.html#"Poisson Distribution"]]>>
<<tiddler [[include_tiddlers/Poisson Manifold.html#"Poisson Manifold"]]>>
<<tiddler [[include_tiddlers/Polyphenol.html#"Polyphenol"]]>>
<<tiddler [[include_tiddlers/Polytope.html#"Polytope"]]>>
The ''Polyvector Action Principle'' or ''Principle of least Polyvector Action'' is the conventional action principle applied to a [[polyvector action |Polyvector Action]] $S[\mb{X}]$:
\[
\delta S[\mb{X}] = 0
\]
The (minimum-) solutions of this equation are the [[Polyvector Euler-Lagrange equations|Polyvector Euler-Lagrange Equations]].

The minimum of the polyvector action represents the "shortest path" in polyvector space (polyvector geodesic). Contrary to a classical vector space this path is represented by a linear combination of points, curves, surfaces, volumes, etc.
This generalizes the concept of a [[geodesic|Geodesic Equation]] of a point particle in a curved spacetime background.
If one also considers second grades in the polyvector action (as is done in the Clifford algebra approach) one gets the description of a spinning particle in a curved background. Such particles obey an extended equivalence principle, the [[polyvector equivalence principle|Polyvector Equivalence Principle]]. Their trajectory can be interpreted as one of an extended object and might be seen as an alternative to string theory where the Polyakov and [[Nambu-Goto actions|Dirac-Nambu-Goto Action]] for example describe the trajectory of a string that sweeps out a worldsheet.
If one furthermore takes into account all relevant grades in $4$ spacetime dimensions, i.e. grades up to order $4$, one gets the geodesic of the whole universe including the "subgeodesics" of matter within it: ''The Trajectory of the Universe''. This opens up the possibility of a description of the dynamics of the whole universe which is somewhat reminiscent of the Hartle Hawking [[wavefunction of the universe|Wavefunction of the Universe]] description.
<<tiddler [[include_tiddlers/Polyvector Analysis.html#"Polyvector Analysis"]]>>
<<tiddler [[include_tiddlers/Polyvector Autoparallelity.html#"Polyvector Autoparallelity"]]>>
<<tiddler [[include_tiddlers/Polyvector Canonical Quantization.html#"Polyvector Canonical Quantization"]]>>
<<tiddler [[include_tiddlers/Polyvector Derivative.html#"Polyvector Derivative"]]>>
<<tiddler [[include_tiddlers/Polyvector Dirac Equation.html#"Polyvector Dirac Equation"]]>>
<<tiddler [[include_tiddlers/Polyvector Equivalence Principle.html#"Polyvector Equivalence Principle"]]>>
<<tiddler [[include_tiddlers/Polyvector Fourier Transform.html#"Polyvector Fourier Transform"]]>>
<<tiddler [[include_tiddlers/Polyvector Gauge Fields.html#"Polyvector Gauge Fields"]]>>
<<tiddler [[include_tiddlers/Polyvector Hilbert Space.html#"Polyvector Hilbert Space"]]>>
<<tiddler [[include_tiddlers/Polyvector Invariant Mass.html#"Polyvector Invariant Mass"]]>>
<<tiddler [[include_tiddlers/Polyvector Jacobi Matrix.html#"Polyvector Jacobi Matrix"]]>>
<<tiddler [[include_tiddlers/Polyvector Klein-Gordon Equation.html#"Polyvector Klein-Gordon Equation"]]>>
<<tiddler [[include_tiddlers/Polyvector Lagrange Function.html#"Polyvector Lagrange Function"]]>>
<<tiddler [[include_tiddlers/Polyvector Partial Derivative.html#"Polyvector Partial Derivative"]]>>
<<tiddler [[include_tiddlers/Polyvector Path Length.html#"Polyvector Path Length"]]>>
<<tiddler [[include_tiddlers/Polyvector Proper Time.html#"Polyvector Proper Time"]]>>
<<tiddler [[include_tiddlers/Polyvector Space.html#"Polyvector Space"]]>>
<<tiddler [[include_tiddlers/Polyvector Supersymmetry.html#"Polyvector Supersymmetry"]]>>
<<tiddler [[include_tiddlers/Polyvector Total Differential.html#"Polyvector Total Differential"]]>>
<<tiddler [[include_tiddlers/Polyvector Vielbein.html#"Polyvector Vielbein"]]>>
<<tiddler [[include_tiddlers/Polyvector Worldline.html#"Polyvector Worldline"]]>>
The ''Pomeransky\-Khriplovich Equations'' describe the motion of a spinning particle in an electromagnetic and gravitational field. Spin is considered as being linear and quadratic (pole-dipole approximation).

Papers:
* [[Spinning Relativistic Particles in External Fields - I.B. Khriplovich|http://arxiv.org/PS_cache/arxiv/pdf/0801/0801.1881v1.pdf]]
* [[Classical and Quantum Spins in Curved Spacetimes - A. J. Silenko|http://th-www.if.uj.edu.pl/acta/sup1/pdf/s1p0087.pdf]]   [[Transparencies|http://www.fuw.edu.pl/~krp/mathisson/wyklady/Silenko.ppt]] 
<<tiddler [[include_tiddlers/Positive Energy Theorem.html#"Positive Energy Theorem"]]>>
<<tiddler [[include_tiddlers/Power Associative Algebra.html#"Power Associative Algebra"]]>>
<<tiddler [[include_tiddlers/Power Law.html#"Power Law"]]>>
The ''Poynting Theorem'' in electrodynamics states:
\begin{equation}
\vec E \cdot \vec  j + \vec \nabla \cdot \vec  S  -\frac{\partial \rho}{\partial t} = 0
\end{equation}
With $\rho$ the energy density of the electromagnetic field:
\begin{equation}
\rho = \frac{1}{2} ( \vec E  \cdot \vec D + \vec B \cdot \vec H)
\end{equation}
and $\vec S$ the Poynting vector
\begin{equation}
\vec S = \vec E \times \vec H
\end{equation}
<<tiddler [[include_tiddlers/Precognition.html#"Precognition"]]>>
<<tiddler [[include_tiddlers/Pregeometry.html#"Pregeometry"]]>>
<<tiddler [[include_tiddlers/Prime Number.html#"Prime Number"]]>>
<<tiddler [[include_tiddlers/Primitive Element.html#"Primitive Element"]]>>
<<tiddler [[include_tiddlers/Primordial Fluctuations.html#"Primordial Fluctuations"]]>>
<<tiddler [[include_tiddlers/Primordial Magnetic Fields.html#"Primordial Magnetic Fields"]]>>
Papers:
* [[This Time - What a Strange Turn of Events! (2010)- P. E. Gibbs|http://prespacetime.com/index.php/pst/article/viewFile/12/9]] [[local|papers/9.pdf]] pct. 0

Links:
* [[WIKIPEDIA - Event Symmetry|http://en.wikipedia.org/wiki/Event_symmetry]]
<<tiddler [[include_tiddlers/Principle of Local Polyvector Invariance.html#"Principle of Local Polyvector Invariance"]]>>
<<tiddler [[include_tiddlers/Proanthocyanidins.html#"Proanthocyanidins"]]>>
<<tiddler [[include_tiddlers/Probabilistic Cellular Automaton.html#"Probabilistic Cellular Automaton"]]>>
<<tiddler [[include_tiddlers/Problems in Physics.html#"Problems in Physics"]]>>
<<tiddler [[include_tiddlers/Process Physics.html#"Process Physics"]]>>
<<tiddler [[include_tiddlers/Projective General Linear Group.html#"Projective General Linear Group"]]>>
<<tiddler [[include_tiddlers/Projective Lattice.html#"Projective Lattice"]]>>
<<tiddler [[include_tiddlers/Projective Plane.html#"Projective Plane"]]>>
Papers:
* [[Projective Relativity: Present Status and Outlook - B. Fauser|http://arxiv.org/PS_cache/gr-qc/pdf/0011/0011015v1.pdf]]
<<tiddler [[include_tiddlers/Prokaryote.html#"Prokaryote"]]>>
<<tiddler [[include_tiddlers/Proper Time.html#"Proper Time"]]>>
<<tiddler [[include_tiddlers/Proteasome.html#"Proteasome"]]>>
<<tiddler [[include_tiddlers/Protein Folding.html#"Protein Folding"]]>>
<<tiddler [[include_tiddlers/Protein Turnover.html#"Protein Turnover"]]>>
<<tiddler [[include_tiddlers/Proteome.html#"Proteome"]]>>
<<tiddler [[include_tiddlers/Proton Decay.html#"Proton Decay"]]>>
<<tiddler [[include_tiddlers/Proton Radius Puzzle.html#"Proton Radius Puzzle"]]>>
<<tiddler [[include_tiddlers/Pseudo-Riemannian Space.html#"Pseudo-Riemannian Space"]]>>
<<tiddler [[include_tiddlers/Psychokinesiology.html#"Psychokinesiology"]]>>
<<tiddler [[include_tiddlers/Psychokinesis.html#"Psychokinesis"]]>>
<<tiddler [[include_tiddlers/Pterostilbene.html#"Pterostilbene"]]>>
<<tiddler [[include_tiddlers/Pulsar Kick.html#"Pulsar Kick"]]>>
Given a [[linear code|Linear Blockcode]] $C = [n,k,d]_q$ with $d>1$, a ''Punctured Code'' is defined as a $[n-1,k,d-1]_q \equiv C^*$-code and is obtained from $C$ by what is called a ''Projection'' of $C$. This means that the codewords of $C$ are projected onto the first $n-1$ coordinates, i.e.
\[
C^* = \{(x_1,\ldots,x_{n-1}) :  (x_1,\ldots,x_{n-1}, x) \in C\}
\]
The converse is called ''Extension'' of a code. This is not generally possible, yet for odd binary codes (i.e. $q=2$ and $d$ odd) it always is.
The entry in the extra coordinate is known as ''Parity Check Bit''.

Examples of punctured codes are [[punctured Reed-Muller codes|Reed-Muller Code]].
Examples for extensions are the [[extended Hamming codes|Hamming Code]] and the extended binary [[Golay code|Golay Code]]. 
<<tiddler [[include_tiddlers/Pure Element.html#"Pure Element"]]>>
A ''Pyrochlore Lattice'' is built up of tetrahedra.
<html><center><img src="images/pyrochlore.jpg" style="width: 265px; "/></center></html>
<<tiddler [[include_tiddlers/QBism.html#"QBism"]]>>
<<tiddler [[include_tiddlers/QCD.html#"QCD"]]>>
A [[nonassociative algebra|Nonassociative Algebra]] $\mathcal A$ (with unity $\mb e$ over a commutative ring) is called a ''Quadratic Algebra'' if $\mathcal A$ admits a [[quadratic form|Quadratic Form]] $\mathcal Q$ such that
#$\mathcal Q (\mb e) = 1$,
#the ''Quadratic Equation'' $\mb A^2 + b(\mb A, \mb e) \mb A + \mathcal Q(\mb A) \mb e = \mb 0$  is satisfied by all $\mb A \in \mathcal A$ where $b$ is the associated [[symmetric bilinear form|Scalar Product]] given by $b(\mb A, \mb B) \equiv \mathcal Q(\mb A + \mb B) - \mathcal Q(\mb A) - \mathcal Q( \mb B)$.
Videos:
* [[Arizona Winter School 2009: Quadratic Forms|http://swc.math.arizona.edu/aws/09/index.html]]
<<tiddler [[include_tiddlers/Qualia.html#"Qualia"]]>>
<<tiddler [[include_tiddlers/Quantization.html#"Quantization"]]>>
<<tiddler [[include_tiddlers/Quantization and Astromomy.html#"Quantization and Astromomy"]]>>
<<tiddler [[include_tiddlers/Quantized Time.html#"Quantized Time"]]>>
<<tiddler [[include_tiddlers/Quantum Absorber Theory.html#"Quantum Absorber Theory"]]>>
<<tiddler [[include_tiddlers/Quantum Anharmonic Oscillator.html#"Quantum Anharmonic Oscillator"]]>>
<<tiddler [[include_tiddlers/Quantum Biology.html#"Quantum Biology"]]>>
<<tiddler [[include_tiddlers/Quantum Black Hole.html#"Quantum Black Hole"]]>>
<<tiddler [[include_tiddlers/Quantum Brain Dynamics.html#"Quantum Brain Dynamics"]]>>
<<tiddler [[include_tiddlers/Quantum Chaos.html#"Quantum Chaos"]]>>
<<tiddler [[include_tiddlers/Quantum Cognition.html#"Quantum Cognition"]]>>
<<tiddler [[include_tiddlers/Quantum Computer.html#"Quantum Computer"]]>>
<<tiddler [[include_tiddlers/Quantum Computing.html#"Quantum Computing"]]>>
<<tiddler [[include_tiddlers/Quantum Conformal Gravity.html#"Quantum Conformal Gravity"]]>>
<<tiddler [[include_tiddlers/Quantum Consciousness.html#"Quantum Consciousness"]]>>
<<tiddler [[include_tiddlers/Quantum Contextuality.html#"Quantum Contextuality"]]>>
<<tiddler [[include_tiddlers/Quantum Cosmology.html#"Quantum Cosmology"]]>>
<<tiddler [[include_tiddlers/Quantum Critical Point.html#"Quantum Critical Point"]]>>
<<tiddler [[include_tiddlers/Quantum Damped Harmonic Oscillator.html#"Quantum Damped Harmonic Oscillator"]]>>
<<tiddler [[include_tiddlers/Quantum Darwinism.html#"Quantum Darwinism"]]>>
<<tiddler [[include_tiddlers/Quantum Discord.html#"Quantum Discord"]]>>
<<tiddler [[include_tiddlers/Quantum Electrodynamics.html#"Quantum Electrodynamics"]]>>
<<tiddler [[include_tiddlers/Quantum Entanglement.html#"Quantum Entanglement"]]>>
<<tiddler [[include_tiddlers/Quantum Equivalence Principle.html#"Quantum Equivalence Principle"]]>>
<<tiddler [[include_tiddlers/Quantum Field Biology.html#"Quantum Field Biology"]]>>
<<tiddler [[include_tiddlers/Quantum Field Cellular Automaton.html#"Quantum Field Cellular Automaton"]]>>
<<tiddler [[include_tiddlers/Quantum Field Computer.html#"Quantum Field Computer"]]>>
<<tiddler [[include_tiddlers/Quantum Field Theory.html#"Quantum Field Theory"]]>>
<<tiddler [[include_tiddlers/Quantum Field Theory in Curved Spacetime.html#"Quantum Field Theory in Curved Spacetime"]]>>
<<tiddler [[include_tiddlers/Quantum Graphity.html#"Quantum Graphity"]]>>
<<tiddler [[include_tiddlers/Quantum Gravity.html#"Quantum Gravity"]]>>
<<tiddler [[include_tiddlers/Quantum Gravity Experiments.html#"Quantum Gravity Experiments"]]>>
<<tiddler [[include_tiddlers/Quantum Gravity from Dark Energy.html#"Quantum Gravity from Dark Energy"]]>>
See [[Hopf Algebra]]. 
<<tiddler [[include_tiddlers/Quantum Hall Effect.html#"Quantum Hall Effect"]]>>
<<tiddler [[include_tiddlers/Quantum Harmonic Oscillator.html#"Quantum Harmonic Oscillator"]]>>
<<tiddler [[include_tiddlers/Quantum Information.html#"Quantum Information"]]>>
<<tiddler [[include_tiddlers/Quantum Mechanics.html#"Quantum Mechanics"]]>>
<<tiddler [[include_tiddlers/Quantum Mechanics Explained.html#"Quantum Mechanics Explained"]]>>
<<tiddler [[include_tiddlers/Quantum Mechanics from General Relativity.html#"Quantum Mechanics from General Relativity"]]>>
<<tiddler [[include_tiddlers/Quantum Neuronal Network.html#"Quantum Neuronal Network"]]>>
<<tiddler [[include_tiddlers/Quantum Phase Transition.html#"Quantum Phase Transition"]]>>
<<tiddler [[include_tiddlers/Quantum Polyvector Geometrodynamics.html#"Quantum Polyvector Geometrodynamics"]]>>
<<tiddler [[include_tiddlers/Quantum Solid.html#"Quantum Solid"]]>>
<br><<tiddler [[include_tiddlers/Quantum State.html#"Quantum State"]]>>
<<tiddler [[include_tiddlers/Quantum Statistical Mechanics.html#"Quantum Statistical Mechanics"]]>>
<<tiddler [[include_tiddlers/Quantum Statistics.html#"Quantum Statistics"]]>>
<<tiddler [[include_tiddlers/Quantum Teleportation.html#"Quantum Teleportation"]]>>
<<tiddler [[include_tiddlers/Quantum Tunneling.html#"Quantum Tunneling"]]>>
<<tiddler [[include_tiddlers/Quantum Zeno Effect.html#"Quantum Zeno Effect"]]>>
<<tiddler [[include_tiddlers/Quark.html#"Quark"]]>>
<<tiddler [[include_tiddlers/Quartic Interaction.html#"Quartic Interaction"]]>>
A ''Quasi\-Hopf Algebra'' is a generalization of a [[Hopf algebra|Hopf Algebra]], which was introduced by Vladimir Drinfeld in 1989.
Quasi\-Hopf algebras have coproducts which are coassociative only up to a [[3-cocycle|3-Cocycle]].

Papers:
* [[Quasialgebra Structure of the Octonions - H. Albuquerque, S. Majid|http://arxiv.org/PS_cache/math/pdf/9802/9802116v1.pdf]] [[pct. 49|http://scholar.google.de/scholar?cites=15697425696231458277&hl=de&as_sdt=2000]]
<<tiddler [[include_tiddlers/Quasi-Steady State Cosmology.html#"Quasi-Steady State Cosmology"]]>>
<<tiddler [[include_tiddlers/Quasicrystal.html#"Quasicrystal"]]>>
<<tiddler [[include_tiddlers/Quasigroup.html#"Quasigroup"]]>>
<<tiddler [[include_tiddlers/Quasigroup Isotopy.html#"Quasigroup Isotopy"]]>>
<<tiddler [[include_tiddlers/Quasigroup Manifold.html#"Quasigroup Manifold"]]>>
<<tiddler [[include_tiddlers/Quasiparticle.html#"Quasiparticle"]]>>
<<tiddler [[include_tiddlers/Quaternator.html#"Quaternator"]]>>
<<tiddler [[include_tiddlers/Quaternator Identities.html#"Quaternator Identities"]]>>
<<tiddler [[include_tiddlers/Quaternion.html#"Quaternion"]]>>
<<tiddler [[include_tiddlers/Quaternion Differential.html#"Quaternion Differential"]]>>
<<tiddler [[include_tiddlers/Quaternion Group.html#"Quaternion Group"]]>>
<<tiddler [[include_tiddlers/Quaternion Multiplication Tables.html#"Quaternion Multiplication Tables"]]>>
<<tiddler [[include_tiddlers/Quaternionic Analysis.html#"Quaternionic Analysis"]]>>
<br><<tiddler [[include_tiddlers/Quaternionic Quantum Mechanics.html#"Quaternionic Quantum Mechanics"]]>>
<<tiddler [[include_tiddlers/Quateroctonion.html#"Quateroctonion"]]>>
<<tiddler [[include_tiddlers/Prime Number.html#"Prime Number"]]>>
<<tiddler [[include_tiddlers/Qubit.html#"Qubit"]]>>
<<tiddler [[include_tiddlers/Query UI.html#"Query UI"]]>>
<<tiddler [[include_tiddlers/Quintessence.html#"Quintessence"]]>>
Verba volant, scripta manent.
----
A good part of science is distinguishing between useful crazy ideas and those that are just plain nutty. - Princeton University book advertisement -
----
An expert is a person who has made all the mistakes that can be made in a very narrow field. - Niels Bohr -
----
The ordinary man wonders at marvellous things; the wise man wonders at the commonplace. - Confucius -
----
I hear and I forget. I see and I remember. I do and I understand - Confucius -
----
One who learns but does not think, is lost. One who thinks but does not learn is in great danger. - Confucius -
----
Nothing in life is to be feared, it is only to be understood. Now is the time to understand more, so that we may fear less. - Marie Curie -
----
Any fool can know. The point is to understand. - Albert Einstein -
----
Not everything that can be counted counts, and not everything that counts can be counted. - Albert Einstein -
----
Everything should be made as simple as possible, but not simpler. - Albert Einstein -
----
Solange man jung ist, gehören alle Gedanken der Liebe - später gehört alle Liebe den Gedanken. - Albert Einstein -
----
Geniale Menschen sind selten ordentlich, ordentliche selten genial. - Albert Einstein -
----
Reading, after a certain age, diverts the mind too much from its creative pursuits. Any man who reads too much and uses his own brain too little falls into lazy habits of thinking. - Albert Einstein -
----
Only two things are infinite, the universe and human stupidity, and I'm not sure about the former. - Albert Einstein -
----
Strive not to be a success, but rather to be of value. - Albert Einstein -
----
Only those who will risk going too far can possibly find out how far one can go. - Thomas Stearns Eliot -
----
I wonder why I wonder why. I wonder why I wonder. I wonder why I wonder why I wonder why I wonder! - Richard Feynman -
----
I think I can safely say that nobody understands quantum mechanics. - Richard Feynman -
----
Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? - Stephen Hawking -
----
When I hear of Schrödinger's cat, I reach for my gun. - Stephen Hawking -
----
Der Horizont vieler Menschen ist ein Kreis mit Radius Null - und das nennen sie ihren Standpunkt. - David Hilbert -
----
Physics is much too hard for physicists. - David Hilbert -
----
The whole purpose of physics is to find a number, with decimal points, etc! Otherwise you haven't done anything. - Richard Feynman -
----
Every scientific statement must remain tentative for ever. - Karl Raimund Popper -
----
When all else fails, you can always tell the truth. - Abdus Salam -
----
Die Grenzen meiner Sprache bedeuten die Grenzen meiner Welt. - Ludwig Wittgenstein -
----
Nicht wie die Welt ist, ist das Mystische, sondern dass sie ist. - Ludwig Wittgenstein -
----
Wovon man nicht sprechen kann, darüber muss man schweigen. - Ludwig Wittgenstein -
----
The effort to understand the universe is one of the very few things that lifts human life a little above the level of farce, and gives it some of the grace of tragedy. - Steven Weinberg -
----
We're all lying in the gutter, but some of us are gazing at the stars. - Oscar Wilde -
----
The whole purpose of science is to find meaningful simplicity in the midst of disorderly complexity - Herbert Simon -
----
The unreasonable effectiveness of mathematics in the natural sciences - Eugene Wigner -
----
Science is a wonderful thing if one does not have to earn one's living at it - Albert Einstein -
----
As far as I see, all a priori statements in physics have their origin in symmetry. - Hermann Weyl -
----
Number rules the universe. - Pythagoras -
----
Use examples; that such as thou teachest may understand thee the better! - Pythagoras -
----
A fool is known by his Speech; and a wise man by Silence. - Pythagoras -
----
Time is the soul of this world. - Pythagoras -
----
The oldest, shortest words - "yes" and "no" - are those which require the most thought. - Pythagoras -
----
You cannot teach a man anything; you can only help him discover it in himself. - Galileo Galilei -
----
The great book of nature can be read only by those who know the language in which it was written. And this language is mathematics. - Galileo Galilei -
----
All is flux, nothing stays still. - Heraclit -
----
Nothing endures but change. - Heraklit -
----
Enthusiasm is followed by disappointment and even depression, and then by renewed enthusiasm. - Murray Gell-Mann -
----
Available energy is the main object at stake in the struggle for existence and the evolution of the world. - Ludwig Boltzmann -
----
What then is time? If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know. - Saint Augustine -
----
Omnibus ex nihil decendis sufficit unum. (One suffices to derive all out of nothing.) - Gottfried Leibniz
----
Never make a calculation until you know the answer. - John Archibald Wheeler
----
The most powerful method of advance is to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics. - Paul Adrien Maurice Dirac -
----
The notion of existence is one of the primitive concepts with which we must begin as given. It is the clearest concept we have. - Kurt Gödel
----
Numquam ponenda est pluralitas sine necessitate. - William of Ockham
----
Wer nicht denken kann muss glauben. - Anonymous - 
----
The awareness of our own strength makes us modest. - Paul Cezanne -
----
If A equals success, then the formula is A = X + Y + Z. X is work. Y is play. Z is keep your mouth shut. - Albert Einstein -
----
Ich behaupte aber, dass in jeder besonderen Naturlehre nur so viel eigentliche Wissenschaft angetroffen werden könne, als darin Mathematik anzutreffen ist. - Immanuel Kant -
----
Mathemata mathematicis scribuntur (mathematics is written for mathematicians) - Nicolaus Copernicus -
----
The art of doing mathematics consists in finding that special case which contains all the germs of generality. - David Hilbert -
----
Fundamental concepts are rare. - Shiing Shen Chern -
----
If I could explain it to the average person, I wouldn't have been worth the Nobel Prize. - Richard Feynman -
----
If you haven't found something strange during the day, it hasn't been much of a day. - Archibald Wheeler -
----
Publish or perish. - J. C. Polkinghorne -
----
Shut up and calculate! - David Mermin -
----
The most incomprehensible thing about the world is that it is comprehensible. - Albert Einstein -
----
The superior man is modest in his speech, but exceeds in his actions. - Confucius -
----
Ich brauche mehr Informationen - Dieter Hallervorden -
----
Basic research is what I am doing when I don't know what I am doing. - Wernher von Braun -
----
Life is good for only two things, discovering mathematics and teaching mathematics. - Siméon Poisson -
----
Life's a piece of shit when you look at it. - Monthy Python -
----
All science is either physics or stamp collecting. - Ernest Rutherford -
----
Modest doubt is called the beacon of the wise. - William Shakespeare -
----
Nothing is worth doing except what the world says is impossible. - Oscar Wilde -
----
The great tragedy of science - the slaying of a beautiful hypothesis by an ugly fact. - Thomas Henry Huxley -
----
The unexamined life is not worth living - Sokrates -
----
What can be said at all can be said clearly. -  Ludwig Wittgenstein -
----
Never forget that only dead fish swim with the stream. - Malcolm Muggeridge -
----
Experiment escorts us last - his pungent company will not allow an axiom an opportunity. - Emily Dickinson -
/***
|Name|QuoteOfTheDayPlugin|
|Source|http://www.TiddlyTools.com/#QuoteOfTheDayPlugin|
|Documentation|http://www.TiddlyTools.com/#QuoteOfTheDayPluginInfo|
|Version|1.4.1|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements <br>and [[Creative Commons Attribution-ShareAlike 2.5 License|http://creativecommons.org/licenses/by-sa/2.5/]]|
|~CoreVersion|2.1|
|Type|plugin|
|Requires||
|Overrides||
|Description|Display a randomly selected "quote of the day" from a list defined in a separate tiddler|

!!!!!Documentation
>see [[QuoteOfTheDayPluginInfo]]
!!!!!Revisions
<<<
2008.03.21 [1.4.1] in showNextItem(), corrected handling for random selection so that //initial// index value will randomized correctly instead of always showing first item, even when randomizing.  Thanks to Riccardo Gherardi for finding this.
| Please see [[QuoteOfTheDayPluginInfo]] for previous revision details |
2005.10.21 [1.0.0] Initial Release.  Based on a suggestion by M.Russula
<<<
!!!!!Code
***/
//{{{
version.extensions.QuoteOfTheDayPlugin= {major: 1, minor: 4, revision: 1, date: new Date(2008,3,21)};
config.macros.QOTD = {
	clickTooltip: "click to view another item",
	timerTooltip: "auto-timer stopped...  'mouseout' to restart timer",
	timerClickTooltip: "auto-timer stopped...  click to view another item, or 'mouseout' to restart timer",
	handler:
	function(place,macroName,params) {
		var tid=params.shift(); // source tiddler containing HR-separated quotes
		var p=params.shift();
		var click=true; // allow click for next item
		var inline=false; // wrap in slider for animation effect
		var random=true; // pick an item at random (default for "quote of the day" usage)
		var folder=false; // use local filesystem folder list
		var cookie=""; // default to no cookie
		var next=0; // default to first item (or random item)
		while (p) {
			if (p.toLowerCase()=="noclick") var click=false;
			if (p.toLowerCase()=="inline") var inline=true;
			if (p.toLowerCase()=="norandom") var random=false;
			if (p.toLowerCase().substr(0,7)=="cookie:") var cookie=p.substr(8);
			if (!isNaN(p)) var delay=p;
			p=params.shift();
		}
		if ((click||delay) && !inline) {
			var panel = createTiddlyElement(null,"div",null,"sliderPanel",null,{style:"font-family:sans-serif;background-color: #DEDdDd"});
			panel.style.display="inherit";
			place.appendChild(panel);
			var here=createTiddlyElement(panel,click?"a":"span",null,"QOTD");
		}
		else
			var here=createTiddlyElement(place,click?"a":"span",null,"QOTD");
		here.id=(new Date()).convertToYYYYMMDDHHMMSSMMM()+Math.random().toString(); // unique ID
		// get items from tiddler or file list
		var list=store.getTiddlerText(tid,"");
		if (!list||!list.length) { // not a tiddler... maybe an image directory?
			var list=this.getImageFileList(tid);
			if (!list.length) { // maybe relative path... fixup and try again
				var h=document.location.href;
				var p=getLocalPath(decodeURIComponent(h.substr(0,h.lastIndexOf("/")+1)));
				var list=this.getImageFileList(p+tid);
			}
		}
		if (!list||!list.length) return false; // no contents... nothing to display!
		here.setAttribute("list",list);
		if (delay) here.setAttribute("delay",delay);
		here.setAttribute("random",random);
		here.setAttribute("cookie",cookie);
		if (click) {
			here.title=this.clickTooltip
			if (!inline) here.style.display="block";
			here.setAttribute("href","javascript:;");
			here.onclick=function(event)
				{ config.macros.QOTD.showNextItem(this); }
		}
		if (config.options["txtQOTD_"+cookie]!=undefined) next=parseInt(config.options["txtQOTD_"+cookie]);
		here.setAttribute("nextItem",next);
		config.macros.QOTD.showNextItem(here);
		if (delay) {
			here.title=click?this.timerClickTooltip:this.timerTooltip
			here.onmouseover=function(event)
				{ clearTimeout(this.ticker); };
			here.onmouseout=function(event)
				{ this.ticker=setTimeout("config.macros.QOTD.tick('"+this.id+"')",this.getAttribute("delay")); };
			here.ticker=setTimeout("config.macros.QOTD.tick('"+here.id+"')",delay);
		}
	},
	tick: function(id) {
		var here=document.getElementById(id); if (!here) return;
		config.macros.QOTD.showNextItem(here);
		here.ticker=setTimeout("config.macros.QOTD.tick('"+id+"')",here.getAttribute("delay"));
	},
	showNextItem:
	function (here) {
		// hide containing slider panel (if any)
		var p=here.parentNode;
		if (p.className=="sliderPanel") p.style.display = "inherit"
		// get a new quote
		var index=here.getAttribute("nextItem");
		var items=here.getAttribute("list").split("\n----\n");
		if (index<0||index>=items.length) index=0;
		if (here.getAttribute("random")=="true") index=Math.floor(Math.random()*items.length);
		var txt=items[index];
		// re-render quote display element, and advance index counter
		removeChildren(here); wikify(txt,here);
		index++; here.setAttribute("nextItem",index);
		var cookie=here.getAttribute("cookie");
		if (cookie.length) {
			config.options["txtQOTD_"+cookie]=index.toString();
			saveOptionCookie("txtQOTD_"+cookie);
		}
		// redisplay slider panel (if any)
		if (p.className=="sliderPanel") {
			if(anim && config.options.chkAnimate)
				anim.startAnimating(new Slider(p,true,false,"none"));
			else p.style.display="block";
		}
	},
	getImageFileList: function(cwd) { // returns HR-separated list of image files
		function isImage(fn) {
			var ext=fn.substr(fn.length-3,3).toLowerCase();
			return ext=="jpg"||ext=="gif"||ext=="png";
		}
		var files=[];
		if (config.browser.isIE) {
			cwd=cwd.replace(/\//g,"\\");
			// IE uses ActiveX to read filesystem info
			var fso = new ActiveXObject("Scripting.FileSystemObject");
			if(!fso.FolderExists(cwd)) return [];
			var dir=fso.GetFolder(cwd);
			for(var f=new Enumerator(dir.Files); !f.atEnd(); f.moveNext())
				if (isImage(f.item().path)) files.push("[img[%0]]".format(["file:///"+f.item().path.replace(/\\/g,"/")]));
		} else {
			// FireFox (mozilla) uses "components" to read filesystem info
			// get security access
			if(!window.Components) return;
			try { netscape.security.PrivilegeManager.enablePrivilege("UniversalXPConnect"); }
			catch(e) { alert(e.description?e.description:e.toString()); return []; }
			// open/validate directory
			var file=Components.classes["@mozilla.org/file/local;1"].createInstance(Components.interfaces.nsILocalFile);
			try { file.initWithPath(cwd); } catch(e) { return []; }
			if (!file.exists() || !file.isDirectory()) { return []; }
			var folder=file.directoryEntries;
			while (folder.hasMoreElements()) {
				var f=folder.getNext().QueryInterface(Components.interfaces.nsILocalFile);
				if (f instanceof Components.interfaces.nsILocalFile)
					if (isImage(f.path)) files.push("[img[%0]]".format(["file:///"+f.path.replace(/\\/g,"/")]));
			}
		}
		return files.join("\n----\n");
	}
}
//}}}
<<tiddler [[include_tiddlers/Quotient Group.html#"Quotient Group"]]>>
<<tiddler [[include_tiddlers/Radiation Reaction.html#"Radiation Reaction"]]>>
<<tiddler [[include_tiddlers/Randall-Sundrum Model.html#"Randall-Sundrum Model"]]>>
<<tiddler [[include_tiddlers/Random Number Generator.html#"Random Number Generator"]]>>
<<tiddler [[include_tiddlers/Random Walk.html#"Random Walk"]]>>
<<tiddler [[include_tiddlers/Rank.html#"Rank"]]>>
<<tiddler [[include_tiddlers/Rate of Living Theory.html#"Rate of Living Theory"]]>>
<<tiddler [[include_tiddlers/Ray.html#"Ray"]]>>
Links:
* [[WIKIPEDIA - Raychaudhuri Equation|http://en.wikipedia.org/wiki/Raychaudhuri_equation]]
<<tiddler [[include_tiddlers/Rayleigh-Bénard Convection.html#"Rayleigh-Bénard Convection"]]>>
<<tiddler [[include_tiddlers/Reactive Carbonyl Species.html#"Reactive Carbonyl Species"]]>>
<<tiddler [[include_tiddlers/Reactive Oxygen Species.html#"Reactive Oxygen Species"]]>>
<<tiddler [[include_tiddlers/Reactive Species.html#"Reactive Species"]]>>
<<tiddler [[include_tiddlers/Real Spectral Triple.html#"Real Spectral Triple"]]>>
<<tiddler [[include_tiddlers/Real Structure.html#"Real Structure"]]>>
<<tiddler [[include_tiddlers/Reconstruction Theorem.html#"Reconstruction Theorem"]]>>
<<tiddler [[include_tiddlers/Redshift Quantization.html#"Redshift Quantization"]]>>
<<tiddler [[include_tiddlers/Reductio Ad Absurbum.html#"Reductio Ad Absurbum"]]>>
A (locally) ''Reductive Space'' is a generalisation of a [[symmetric space|Symmetric Space]] and a [[Lie group manifold|Lie Group Manifold]]. Given an affine connection and a covariant derivative $\nabla$, one distinguishes

$\quad\quad$ Lie Group: $\;\quad \quad \quad \;\,\nabla T = 0, \quad \;R = 0$
$\quad\quad$ Symmetric space: $\;\; \,T = 0, \;\, \nabla R = 0$
$\quad\quad$ Reductive space: $\;\nabla T=0, \;\, \nabla R = 0$

with $T$ denoting the [[torsion tensor|Torsion]] and $R$ the [[Riemann tensor|Riemann Tensor]].

Reductive spaces were introduced by K. Nomizu in 1954.

[[Tangent algebras|Tangent Algebra]] of locally reductive spaces are [[Lie triple algebras|Bol Algebra]]. A construction was given by K. Yamaguti.

Papers:
* [[A Generalization of Lie Groups and Symmetric Spaces - N. Hitotsuyanag|http://ir.kagoshima-u.ac.jp/bitstream/10232/7165/1/AN00408518_1986_001.pdf]] [[local|papers/AN00408518_1986_001.pdf]] pct. 0
<<tiddler [[include_tiddlers/Reed-Muller Code.html#"Reed-Muller Code"]]>>
<<tiddler [[include_tiddlers/Reeh-Schlieder Theorem.html#"Reeh-Schlieder Theorem"]]>>
<<tiddler [[include_tiddlers/Regenerative Medicine.html#"Regenerative Medicine"]]>>
<<tiddler [[include_tiddlers/Regge Trajectory.html#"Regge Trajectory"]]>>
<<tiddler [[include_tiddlers/Regularization.html#"Regularization"]]>>
<<tiddler [[include_tiddlers/Reidemeister Move.html#"Reidemeister Move"]]>>
<<tiddler [[include_tiddlers/Reissner-Nordström Metric.html#"Reissner-Nordström Metric"]]>>
<<tiddler [[include_tiddlers/Rejuvenation.html#"Rejuvenation"]]>>
<<tiddler [[include_tiddlers/Relational Quantum Mechanics.html#"Relational Quantum Mechanics"]]>>
<<tiddler [[include_tiddlers/Relativistic Quantum Information.html#"Relativistic Quantum Information"]]>>
<<tiddler [[include_tiddlers/Reliability Theory of Aging.html#"Reliability Theory of Aging"]]>>
<<tiddler [[include_tiddlers/Religiosity and IQ.html#"Religiosity and IQ"]]>>
<<tiddler [[include_tiddlers/Remote Application Platform.html#"Remote Application Platform"]]>>
<<tiddler [[include_tiddlers/Remote Viewing.html#"Remote Viewing"]]>>
<<tiddler [[include_tiddlers/Renormalization.html#"Renormalization"]]>>
<<tiddler [[include_tiddlers/Renormalization Group.html#"Renormalization Group"]]>>
<<tiddler [[include_tiddlers/Representation Theory.html#"Representation Theory"]]>>
<<tiddler [[include_tiddlers/Repulsive Casimir Effect.html#"Repulsive Casimir Effect"]]>>
<<tiddler [[include_tiddlers/Resolvable Design.html#"Resolvable Design"]]>>
<<tiddler [[include_tiddlers/Responsive Web Design.html#"Responsive Web Design"]]>>
A subspace $S$ of $X$ is called a ''Retract'' of $X$ if there is a continuous map $ f: X \rightarrow X$ (called a retraction) such that $\forall x \in X$ and $\forall s \in S$:
# $f(x) \in S$ and
# $f(s)=s$
(i.e. the subspace is preserved under the map and the rest is mapped into the subspace).

Equivalently, a subspace $S$ of $X$ is called a retract of $X$ if there is a continuous map $ f: X \rightarrow S$ (called a retraction) such that $\forall s \in S$: $f(s)=s$. 
<<tiddler [[include_tiddlers/Retrocausality.html#"Retrocausality"]]>>
<<tiddler [[include_tiddlers/Ribosome.html#"Ribosome"]]>>
<<tiddler [[include_tiddlers/Ricci Tensor.html#"Ricci Tensor"]]>>
<<tiddler [[include_tiddlers/Richard Dawkins.html#"Richard Dawkins"]]>>
<<tiddler [[include_tiddlers/Richard Feynman.html#"Richard Feynman"]]>>
<<tiddler [[include_tiddlers/Riemann Hypothesis.html#"Riemann Hypothesis"]]>>
<<tiddler [[include_tiddlers/Riemann Normal Coordinates.html#"Riemann Normal Coordinates"]]>>
<<tiddler [[include_tiddlers/Riemann Space.html#"Riemann Space"]]>>
<<tiddler [[include_tiddlers/Riemann Surface.html#"Riemann Surface"]]>>
<<tiddler [[include_tiddlers/Riemann Tensor.html#"Riemann Tensor"]]>>
<<tiddler [[include_tiddlers/Riemann Zeta Function.html#"Riemann Zeta Function"]]>>
A ''Riemann\-Cartan Space (= Einstein\-Cartan Space)'' is the most general [[metric compatible|Metric Compatibility]], [[metric affine space|Metric Affine Space]], commonly denoted $\mathbb U_4$. It is a special case of a [[Weyl-Cartan space|Weyl Space]].

For such a space a general [[connection|Connection]] simplifies to:
\[
\Gamma_{\mu\nu}^\sigma = \Chr{\sigma}{\mu\nu} + K_{\mu\nu}^\sigma
\]
called ''Riemann\-Cartan Connection'', with $\Chr{\sigma}{\mu\nu} $ the [[Chrisoffel symbols|Christoffel Symbols]] and $K_{\mu\nu}^\sigma$ the [[contorsion|Contorsion]] tensor.

Thus the differential geometry of $\mathbb U_4$ is completely determined by two tensor fields:
# the [[metric tensor|Metric Tensor]] $g_{\mu\nu}$ and,
# the contorsion tensor $K_{\mu\nu}^\sigma$, or equivalently the torsion tensor $T_{\mu\nu}^\sigma$.

For vanishing torsion and hence contorsion, one arrives at a [[Riemann Space]].

!!!! Applications
* [[Einstein-Cartan theory of gravity|Einstein-Cartan Theory]].
* Plastic flow and material fatigue can be described by Riemann\-Cartan spaces as they require [[curvature|Curvature]] and [[torsion|Torsion]] due to [[disclinations|Disclination]] and [[dislocations|Dislocation]].
<<tiddler [[include_tiddlers/Rigged Hilbert Space.html#"Rigged Hilbert Space"]]>>
<<tiddler [[include_tiddlers/Ring.html#"Ring"]]>>
<<tiddler [[include_tiddlers/Robert Betts Laughlin.html#"Robert Betts Laughlin"]]>>
<<tiddler [[include_tiddlers/Robotic Telescopes.html#"Robotic Telescopes"]]>>
<<tiddler [[include_tiddlers/Roger Penrose.html#"Roger Penrose"]]>>
<<tiddler [[include_tiddlers/Root Lattice.html#"Root Lattice"]]>>
<<tiddler [[include_tiddlers/Root Vector.html#"Root Vector"]]>>
<<tiddler [[include_tiddlers/Rotation of the Universe.html#"Rotation of the Universe"]]>>
<<tiddler [[include_tiddlers/Running Cosmological Constant.html#"Running Cosmological Constant"]]>>
<<tiddler [[include_tiddlers/Rényi Entropy.html#"Rényi Entropy"]]>>
A [[Steiner system|Steiner System]] ''$S(5, 6, 12)$'' is a collection of special six point subsets (hexads) of a twelve point set $\Omega$, such that every five point subset of $\Omega$ is contained in exactly one special hexad. There are $132$ such hexads.
<<tiddler [[include_tiddlers/S(5,8,24).html#"S(5,8,24)"]]>>
<<tiddler [[include_tiddlers/S-Matrix.html#"S-Matrix"]]>>
<<tiddler [[include_tiddlers/SL(2,C).html#"SL(2,C)"]]>>
Papers:
* [[A New Family Symmetry for SO(10) GUTs (2009) - S. F. King, C. Luhn|http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.1686v1.pdf]] [[local|papers/0905.1686v1.pdf]] [[pct. 15|http://scholar.google.de/scholar?cites=17881056879842397469&as_sdt=2005&sciodt=2000&hl=de]]
* [[A Supersymmetric Grand Unified Theory of Flavour with PSL2(7) ̉ SO(10) (2010) - S. F. King, C. Luhn|http://arxiv.org/PS_cache/arxiv/pdf/0912/0912.1344v3.pdf]] [[local|papers/0912.1344v3.pdf]] [[pct. 6|http://scholar.google.de/scholar?cites=8133791727243503799&as_sdt=2005&sciodt=2000&hl=de]]
* [[Renormalisable SO (10) Models and Neutrino Masses and Mixing (2007) - W. Grimus, H. Kühböck|http://th-www.if.uj.edu.pl/acta/vol38/pdf/v38p3373.pdf]] [[local|papers/v38p3373.pdf]] pct. 0
<<tiddler [[include_tiddlers/SO(10) Unification.html#"SO(10) Unification"]]>>
Papers:
* [[Going down from a 3-form in 16 Dimensions - L. Baulieu |http://arxiv.org/PS_cache/hep-th/pdf/0207/0207184v1.pdf]]
<<tiddler [[include_tiddlers/SO(32).html#"SO(32)"]]>>
See [[conformal group|Conformal Group]]. 
Papers:
* [[Representation Theory and Projective Geometry (2004) - J. M. Landsberg, L. Manivel|http://www.math.tamu.edu/~jml/LMpopreprint.pdf]] [[local|papers/LMpopreprint.pdf]] [[pct. 25|http://scholar.google.de/scholar?cites=12280133377652367576&hl=de&as_sdt=2000]]
<<tiddler [[include_tiddlers/SO(8).html#"SO(8)"]]>>
''STEP'' (the ''S''atellite ''T''est of the ''E''quivalence ''P''rinciple) will advance experimental limits on violations of [[Einsteinââ¬â¢s equivalence principle|Equivalence Principles]] from their present sensitivity of $2$ parts in $10^{13}$ to $1$ part in $10^{18}$ through multiple comparison of the motions of four pairs of test masses of different compositions in an earth-orbiting drag-free satellite.

Papers:
* [[The Science Case for STEP - J. Overduin, F. Everitt, J. Mester, P. Worden|http://arxiv.org/PS_cache/arxiv/pdf/0902/0902.2247v2.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=10364899760222302819&hl=de]]
<<tiddler [[include_tiddlers/STS(15).html#"STS(15)"]]>>
<<tiddler [[include_tiddlers/SU(1,1).html#"SU(1,1)"]]>>
<<tiddler [[include_tiddlers/SU(2,2).html#"SU(2,2)"]]>>
<<tiddler [[include_tiddlers/SU(3).html#"SU(3)"]]>>
<<tiddler [[include_tiddlers/SU(5).html#"SU(5)"]]>>
<<tiddler [[include_tiddlers/SUq(1,1).html#"SUq(1,1)"]]>>
<<tiddler [[include_tiddlers/Sabinin Algebra.html#"Sabinin Algebra"]]>>
<<tiddler [[include_tiddlers/Sabinin Algebra - Construction.html#"Sabinin Algebra - Construction"]]>>
<<tiddler [[include_tiddlers/Saccharomyces Cerevisiae.html#"Saccharomyces Cerevisiae"]]>>
<<tiddler [[include_tiddlers/Sage.html#"Sage"]]>>
<<tiddler [[include_tiddlers/Sagle Identity.html#"Sagle Identity"]]>>
<<tiddler [[include_tiddlers/Satake Diagram.html#"Satake Diagram"]]>>
<<tiddler [[include_tiddlers/Saturn's Hexagon.html#"Saturn's Hexagon"]]>>
<<tiddler [[include_tiddlers/Scalar Product.html#"Scalar Product"]]>>
<<tiddler [[include_tiddlers/Scalar-tensor Theory.html#"Scalar-tensor Theory"]]>>
<<tiddler [[include_tiddlers/Scale Invariance.html#"Scale Invariance"]]>>
<<tiddler [[include_tiddlers/Scale Relativity.html#"Scale Relativity"]]>>
<<tiddler [[include_tiddlers/Schläfli Symbol.html#"Schläfli Symbol"]]>>
<<tiddler [[include_tiddlers/Schreier Variety.html#"Schreier Variety"]]>>
<<tiddler [[include_tiddlers/Schrödinger Equation.html#"Schrödinger Equation"]]>>
<<tiddler [[include_tiddlers/Schrödinger Picture.html#"Schrödinger Picture"]]>>
<<tiddler [[include_tiddlers/Schrödinger's Cat.html#"Schrödinger's Cat"]]>>
<<tiddler [[include_tiddlers/Schrödinger-Helmholtz Equation.html#"Schrödinger-Helmholtz Equation"]]>>
<<tiddler [[include_tiddlers/Schwarzschild Metric.html#"Schwarzschild Metric"]]>>
<<tiddler [[include_tiddlers/Schwarzschild Radius.html#"Schwarzschild Radius"]]>>
<<tiddler [[include_tiddlers/Schwarzschild-De Sitter Metric.html#"Schwarzschild-De Sitter Metric"]]>>
<<tiddler [[include_tiddlers/Schwinger Proper Time Representation.html#"Schwinger Proper Time Representation"]]>>
<<tiddler [[include_tiddlers/Schwinger-Keldysh Formalism.html#"Schwinger-Keldysh Formalism"]]>>
<<tiddler [[include_tiddlers/Scientific Calculator.html#"Scientific Calculator"]]>>
The following search keywords are used:

"pct." - paper citations (Google scholar)
"jct." - journal citations (Google scholar)
"bct." - book citations (Google scholar)

"prl." - paper relevance, scale 1,...,10, with 10 the best rating (a personal estimate)
"jrl." -  journal relevance, scale 1,...,10, with 10 the best rating (a personal estimate)
"trl." - presentation/talk relevance, scale 1,...,10, with 10 the best rating (a personal estimate)
"brl." - book relevance, scale 1,...,10, with 10 the best rating (a personal estimate)

/***
|Name|SearchOptionsPlugin|
|Source|http://www.TiddlyTools.com/#SearchOptionsPlugin|
|Documentation|http://www.TiddlyTools.com/#SearchOptionsPluginInfo|
|Version|3.0.6|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|extend core search function with additional user-configurable options|
Adds extra options to core search function including selecting which data items to search, enabling/disabling incremental key-by-key searches, and generating a ''list of matching tiddlers'' instead of immediately displaying all matches.  This plugin also adds syntax for rendering 'search links' within tiddler content to embed one-click searches using pre-defined 'hard-coded' search terms.
!!!!!Documentation
>see [[SearchOptionsPluginInfo]]
!!!!!Configuration
<<<
Search in:
<<option chkSearchTitles>> titles <<option chkSearchText>> text <<option chkSearchTags>> tags <<option chkSearchFields>> fields <<option chkSearchShadows>> shadows
<<option chkSearchHighlight>> Highlight matching text in displayed tiddlers
<<option chkSearchList>> Show list of matches
<<option chkSearchListTiddler>> Write list to [[SearchResults]] tiddler
<<option chkSearchTitlesFirst>> Show title matches first
<<option chkSearchByDate>> Sort matching tiddlers by modification date (most recent first)
<<option chkIncrementalSearch>> Incremental key-by-key search: {{twochar{<<option txtIncrementalSearchMin>>}}} or more characters,  {{threechar{<<option txtIncrementalSearchDelay>>}}} msec delay
<<option chkSearchOpenTiddlers>> Search only in tiddlers that are currently displayed
<<option chkSearchExcludeTags>> Exclude tiddlers tagged with: <<option txtSearchExcludeTags>>
<<<
!!!!!Revisions
<<<
2009.09.22 [3.0.6] in TiddlyWiki.prototype.search, added 'match' param for core compatibility
2009.01.16 [3.0.5] added chkSearchOpenTiddlers option to limit searches to displayed tiddlers only
|please see [[SearchOptionsPluginInfo]] for additional revision details|
2005.10.18 [1.0.0] Initial Release
<<<
!!!!!Code
***/
//{{{
version.extensions.SearchOptionsPlugin= {major: 3, minor: 0, revision: 6, date: new Date(2009,9,22)};

var defaults={
	chkSearchTitles:	true,
	chkSearchText:		true,
	chkSearchTags:		true,
	chkSearchFields:	true,
	chkSearchTitlesFirst:	true,
	chkSearchList:		true,
	chkSearchHighlight:	true,
	chkSearchListTiddler:	false,
	chkSearchByDate:	false,
	chkIncrementalSearch:	true,
	chkSearchShadows:	true,
	chkSearchOpenTiddlers:	false,
	chkSearchExcludeTags:	true,
	txtSearchExcludeTags:	'excludeSearch',
	txtIncrementalSearchDelay:	500,
	txtIncrementalSearchMin:	3
}; for (var id in defaults) if (config.options[id]===undefined)
	config.options[id]=defaults[id];

if (config.macros.search.reportTitle==undefined)
	config.macros.search.reportTitle="SearchResults"; // note: not a cookie!
config.macros.search.label+="\xa0"; // a little bit of space just because it looks better
//}}}
// // searchLink: {{{[search[text to find]] OR [search[text to display|text to find]]}}}
//{{{
config.formatters.push( {
	name: "searchLink",
	match: "\\[search\\[",
	lookaheadRegExp: /\[search\[(.*?)(?:\|(.*?))?\]\]/mg,
	prompt: "search for: '%0'",
	handler: function(w)
	{
		this.lookaheadRegExp.lastIndex = w.matchStart;
		var lookaheadMatch = this.lookaheadRegExp.exec(w.source);
		if(lookaheadMatch && lookaheadMatch.index == w.matchStart) {
			var label=lookaheadMatch[1];
			var text=lookaheadMatch[2]||label;
			var prompt=this.prompt.format([text]);
			var btn=createTiddlyButton(w.output,label,prompt,
				function(){story.search(this.getAttribute("searchText"))},"searchLink");
			btn.setAttribute("searchText",text);
			w.nextMatch = this.lookaheadRegExp.lastIndex;
		}
	}
});
//}}}
// // incremental search uses option settings instead of hard-coded delay and minimum input values
//{{{
var fn=config.macros.search.onKeyPress;
fn=fn.toString().replace(/500/g, "config.options.txtIncrementalSearchDelay||500");
fn=fn.toString().replace(/> 2/g, ">=(config.options.txtIncrementalSearchMin||3)");
eval("config.macros.search.onKeyPress="+fn);
//}}}
// // REPLACE story.search() for option to "show search results in a list"
//{{{
Story.prototype.search = function(text,useCaseSensitive,useRegExp)
{
	var co=config.options; // abbrev
	var re=new RegExp(useRegExp ? text : text.escapeRegExp(),useCaseSensitive ? "mg" : "img");
	if (config.options.chkSearchHighlight) highlightHack=re;
	var matches = store.search(re,co.chkSearchByDate?"modified":"title","");
	if (co.chkSearchByDate) matches=matches.reverse(); // most recent first
	var q = useRegExp ? "/" : "'";
	clearMessage();
	if (!matches.length) {
		if (co.chkSearchListTiddler) discardSearchResults();
		displayMessage(config.macros.search.failureMsg.format([q+text+q]));
	} else {
		if (co.chkSearchList||co.chkSearchListTiddler)
			reportSearchResults(text,matches);
		else {
			var titles = []; for(var t=0; t<matches.length; t++) titles.push(matches[t].title);
			this.closeAllTiddlers(); story.displayTiddlers(null,titles);
			displayMessage(config.macros.search.successMsg.format([matches.length, q+text+q]));
		}
	}
	highlightHack = null;
}
//}}}
// // REPLACE store.search() for enhanced searching/sorting options
//{{{
TiddlyWiki.prototype.search = function(searchRegExp,sortField,excludeTag,match)
{
	var co=config.options; // abbrev
	var tids = this.reverseLookup("tags",excludeTag,!!match,sortField);
	var opened=[]; story.forEachTiddler(function(tid,elem){opened.push(tid);});

	// eliminate tiddlers tagged with excluded tags
	if (co.chkSearchExcludeTags&&co.txtSearchExcludeTags.length) {
		var ex=co.txtSearchExcludeTags.readBracketedList();
		var temp=[]; for(var t=tids.length-1; t>=0; t--)
			if (!tids[t].tags.containsAny(ex)) temp.push(tids[t]);
		tids=temp;
	}

	// scan for matching titles first...
	var results = [];
	if (co.chkSearchTitles) {
		for(var t=0; t<tids.length; t++) {
			if (co.chkSearchOpenTiddlers && !opened.contains(tids[t].title)) continue;
			if(tids[t].title.search(searchRegExp)!=-1) results.push(tids[t]);
		}
		if (co.chkSearchShadows)
			for (var t in config.shadowTiddlers) {
				if (co.chkSearchOpenTiddlers && !opened.contains(t)) continue;
				if ((t.search(searchRegExp)!=-1) && !store.tiddlerExists(t))
					results.push((new Tiddler()).assign(t,config.shadowTiddlers[t]));
			}
	}
	// then scan for matching text, tags, or field data
	for(var t=0; t<tids.length; t++) {
		if (co.chkSearchOpenTiddlers && !opened.contains(tids[t].title)) continue;
		if (co.chkSearchText && tids[t].text.search(searchRegExp)!=-1)
			results.pushUnique(tids[t]);
		if (co.chkSearchTags && tids[t].tags.join(" ").search(searchRegExp)!=-1)
			results.pushUnique(tids[t]);
		if (co.chkSearchFields && store.forEachField!=undefined)
			store.forEachField(tids[t],
				function(tid,field,val) {
					if (val.search(searchRegExp)!=-1) results.pushUnique(tids[t]);
				},
				true); // extended fields only
	}
	// then check for matching text in shadows
	if (co.chkSearchShadows)
		for (var t in config.shadowTiddlers) {
			if (co.chkSearchOpenTiddlers && !opened.contains(t)) continue;
			if ((config.shadowTiddlers[t].search(searchRegExp)!=-1) && !store.tiddlerExists(t))
				results.pushUnique((new Tiddler()).assign(t,config.shadowTiddlers[t]));
		}

	// if not 'titles first', or sorting by modification date,
	// re-sort results to so titles, text, tag and field matches are mixed together
	if(!sortField) sortField = "title";
	var bySortField=function(a,b){
		if(a[sortField]==b[sortField])return(0);else return(a[sortField]<b[sortField])?-1:+1;
	}
	if (!co.chkSearchTitlesFirst || co.chkSearchByDate) results.sort(bySortField);

	return results;
}
//}}}
// // HIJACK core {{{<<search>>}}} macro to add "report" and "simple inline" output
//{{{
config.macros.search.SOP_handler=config.macros.search.handler;
config.macros.search.handler = function(place,macroName,params)
{
	// if "report", use SearchOptionsPlugin report generator for inline output
	if (params[1]&&params[1].substr(0,6)=="report") {
		var keyword=params[0];
		var options=params[1].split("=")[1]; // split "report=option+option+..."
		var heading=params[2]?params[2].unescapeLineBreaks():"";
		var matches=store.search(new RegExp(keyword.escapeRegExp(),"img"),"title","excludeSearch");
		if (matches.length) wikify(heading+window.formatSearchResults(keyword,matches,options),place);
	} else if (params[1]) {
		var keyword=params[0];
		var heading=params[1]?params[1].unescapeLineBreaks():"";
		var seperator=params[2]?params[2].unescapeLineBreaks():", ";
		var matches=store.search(new RegExp(keyword.escapeRegExp(),"img"),"title","excludeSearch");
		if (matches.length) {
			var out=[];
			for (var m=0; m<matches.length; m++) out.push("[["+matches[m].title+"]]");
			wikify(heading+out.join(seperator),place);
		}
	} else
		config.macros.search.SOP_handler.apply(this,arguments);
};
//}}}
// // SearchResults panel handling
//{{{
setStylesheet(".searchResults { padding:1em 1em 0 1em; }","searchResults"); // matches std tiddler padding

config.macros.search.createPanel=function(text,matches,body) {

	function getByClass(e,c) { var d=e.getElementsByTagName("div");
		for (var i=0;i<d.length;i++) if (hasClass(d[i],c)) return d[i]; }
	var panel=createTiddlyElement(null,"div","searchPanel","searchPanel");
	this.renderPanel(panel,text,matches,body);
	var oldpanel=document.getElementById("searchPanel");
	if (!oldpanel) { // insert new panel just above tiddlers
		var da=document.getElementById("displayArea");
		da.insertBefore(panel,da.firstChild);
	} else { // if panel exists
		var oldwrap=getByClass(oldpanel,"searchResults");
		var newwrap=getByClass(panel,"searchResults");
		// if no prior content, just insert new content
		if (!oldwrap) oldpanel.insertBefore(newwrap,null);
		else {	// swap search results content but leave containing panel intact
			oldwrap.style.display='block'; // unfold wrapper if needed
			var i=oldwrap.getElementsByTagName("input")[0]; // get input field
			if (i) { var pos=this.getCursorPos(i); i.onblur=null; } // get cursor pos, ignore blur
			oldpanel.replaceChild(newwrap,oldwrap);
			panel=oldpanel; // use existing panel
		}
	}
	this.showPanel(true,pos);
	return panel;
}

config.macros.search.renderPanel=function(panel,text,matches,body) {

	var wrap=createTiddlyElement(panel,"div",null,"searchResults");
	wrap.onmouseover = function(e){ addClass(this,"selected"); }
	wrap.onmouseout = function(e){ removeClass(this,"selected"); }
	// create toolbar: "open all", "fold/unfold", "close"
	var tb=createTiddlyElement(wrap,"div",null,"toolbar");
	var b=createTiddlyButton(tb, "open all", "open all matching tiddlers", function() {
		story.displayTiddlers(null,this.getAttribute("list").readBracketedList()); return false; },"button");
	var list=""; for(var t=0;t<matches.length;t++) list+='[['+matches[t].title+']] ';
	b.setAttribute("list",list);
	var b=createTiddlyButton(tb, "fold", "toggle display of search results", function() {
		config.macros.search.foldPanel(this); return false; },"button");
	var b=createTiddlyButton(tb, "close", "dismiss search results",	function() {
		config.macros.search.showPanel(false); return false; },"button");
	createTiddlyText(createTiddlyElement(wrap,"div",null,"title"),"Search for: "+text); // title
	wikify(body,createTiddlyElement(wrap,"div",null,"viewer")); // report
	return panel;
}

config.macros.search.showPanel=function(show,pos) {
	var panel=document.getElementById("searchPanel");
	var i=panel.getElementsByTagName("input")[0];
	i.onfocus=show?function(){config.macros.search.stayFocused(true);}:null;
	i.onblur=show?function(){config.macros.search.stayFocused(false);}:null;
	if (show && panel.style.display=="block") { // if shown, grab focus, restore cursor
		if (i&&this.stayFocused()) { i.focus(); this.setCursorPos(i,pos); }
		return;
	}
	if(!config.options.chkAnimate) {
		panel.style.display=show?"block":"none";
		if (!show) { removeChildren(panel); config.macros.search.stayFocused(false); }
	} else {
		var s=new Slider(panel,show,false,show?"none":"children");
		s.callback=function(e,p){e.style.overflow="visible";}
		anim.startAnimating(s);
	}
	return panel;
}

config.macros.search.foldPanel=function(button) {
	var d=document.getElementById("searchPanel").getElementsByTagName("div");
	for (var i=0;i<d.length;i++) if (hasClass(d[i],"viewer")) var v=d[i]; if (!v) return;
	var show=v.style.display=="none";
	if(!config.options.chkAnimate)
		v.style.display=show?"block":"none";
	else {
		var s=new Slider(v,show,false,"none");
		s.callback=function(e,p){e.style.overflow="visible";}
		anim.startAnimating(s);
	}
	button.innerHTML=show?"fold":"unfold";
	return false;
}

config.macros.search.stayFocused=function(keep) { // TRUE/FALSE=set value, no args=get value
	if (keep===undefined) return this.keepReportInFocus;
	this.keepReportInFocus=keep;
	return keep
}

config.macros.search.getCursorPos=function(i) {
	var s=0; var e=0; if (!i) return { start:s, end:e };
	try {
		if (i.setSelectionRange) // FF
			{ s=i.selectionStart; e=i.selectionEnd; }
		if (document.selection && document.selection.createRange) { // IE
			var r=document.selection.createRange().duplicate();
			var len=r.text.length; s=0-r.moveStart('character',-100000); e=s+len;
		}
	}catch(e){};
	return { start:s, end:e };
}
config.macros.search.setCursorPos=function(i,pos) {
	if (!i||!pos) return; var s=pos.start; var e=pos.end;
	if (i.setSelectionRange) //FF
		i.setSelectionRange(s,e);
	if (i.createTextRange) // IE
		{ var r=i.createTextRange(); r.collapse(true); r.moveStart("character",s); r.select(); }
}
//}}}
// // SearchResults report generation
// note: these functions are defined globally, so they can be more easily redefined to customize report formats//
//{{{
if (!window.reportSearchResults) window.reportSearchResults=function(text,matches)
{
	var cms=config.macros.search; // abbrev
	var body=window.formatSearchResults(text,matches);
	if (!config.options.chkSearchListTiddler) // show #searchResults panel
		window.scrollTo(0,ensureVisible(cms.createPanel(text,matches,body)));
	else { // write [[SearchResults]] tiddler
		var title=cms.reportTitle;
		var who=config.options.txtUserName;
		var when=new Date();
		var tags="excludeLists excludeSearch temporary";
		var tid=store.getTiddler(title); if (!tid) tid=new Tiddler();
		tid.set(title,body,who,when,tags);
		store.addTiddler(tid);
		story.closeTiddler(title);
		story.displayTiddler(null,title);
	}
}

if (!window.formatSearchResults) window.formatSearchResults=function(text,matches,opt)
{
	var body='';
	var title=config.macros.search.reportTitle
	var q = config.options.chkRegExpSearch ? "/" : "'";
	if (!opt) var opt="all";
	var parts=opt.split("+");
	for (var i=0; i<parts.length; i++) { var p=parts[i].toLowerCase();
		if (p=="again"||p=="all")   body+=window.formatSearchResults_again(text,matches);
		if (p=="summary"||p=="all") body+=window.formatSearchResults_summary(text,matches);
		if (p=="list"||p=="all")    body+=window.formatSearchResults_list(text,matches);
		if (p=="buttons"||p=="all") body+=window.formatSearchResults_buttons(text,matches);
	}
	return body;
}

if (!window.formatSearchResults_again) window.formatSearchResults_again=function(text,matches)
{
	var title=config.macros.search.reportTitle
	var body='';
	// search again
	body+='{{span{<<search "'+text.replace(/"/g,'&#x22;')+'">> /%\n';
	body+='%/<html><input type="button" value="search again"';
	body+=' onclick="var t=this.parentNode.parentNode.getElementsByTagName(\'input\')[0];';
	body+=' config.macros.search.doSearch(t); return false;">';
	body+=' <a href="javascript:;" onclick="';
	body+=' var e=this.parentNode.nextSibling;';
	body+=' var show=e.style.display!=\'block\';';
	body+=' if(!config.options.chkAnimate) e.style.display=show?\'block\':\'none\';';
	body+=' else anim.startAnimating(new Slider(e,show,false,\'none\'));';
	body+=' return false;">options...</a>';
	body+='</html>@@display:none;border-left:1px dotted;margin-left:1em;padding:0;padding-left:.5em;font-size:90%;/%\n';
	body+='	%/<<option chkSearchTitles>>titles /%\n';
	body+='	%/<<option chkSearchText>>text /%\n';
	body+='	%/<<option chkSearchTags>>tags /%\n';
	body+='	%/<<option chkSearchFields>>fields /%\n';
	body+='	%/<<option chkSearchShadows>>shadows\n';
	body+='	<<option chkCaseSensitiveSearch>>case-sensitive /%\n';
	body+='	%/<<option chkRegExpSearch>>text patterns /%\n';
	body+='	%/<<option chkSearchByDate>>sorted by date\n';
	body+='	<<option chkSearchHighlight>> highlight matching text in displayed tiddlers\n';
	body+='	<<option chkIncrementalSearch>>incremental key-by-key search: /%\n';
	body+='	%/{{twochar{<<option txtIncrementalSearchMin>>}}} or more characters, /%\n';
	body+='	%/{{threechar{<<option txtIncrementalSearchDelay>>}}} msec delay\n';
	body+='	<<option chkSearchOpenTiddlers>> search only in tiddlers that are currently displayed\n';
	body+='	<<option chkSearchExcludeTags>>exclude tiddlers tagged with:\n';
	body+='	{{editor{<<option txtSearchExcludeTags>>}}}/%\n';
	body+='%/@@}}}\n\n';
	return body;
}

if (!window.formatSearchResults_summary) window.formatSearchResults_summary=function(text,matches)
{
	// summary: nn tiddlers found matching '...', options used
	var body='';
	var co=config.options; // abbrev
	var title=config.macros.search.reportTitle
	var q = co.chkRegExpSearch ? "/" : "'";
	body+="''"+config.macros.search.successMsg.format([matches.length,q+"{{{"+text+"}}}"+q])+"''\n";
	var opts=[];
	if (co.chkSearchTitles) opts.push("titles");
	if (co.chkSearchText) opts.push("text");
	if (co.chkSearchTags) opts.push("tags");
	if (co.chkSearchFields) opts.push("fields");
	if (co.chkSearchShadows) opts.push("shadows");
	if (co.chkSearchOpenTiddlers) body+="^^//search limited to displayed tiddlers only//^^\n";
	body+="~~&nbsp; searched in "+opts.join(" + ")+"~~\n";
	body+=(co.chkCaseSensitiveSearch||co.chkRegExpSearch?"^^&nbsp; using ":"")
		+(co.chkCaseSensitiveSearch?"case-sensitive ":"")
		+(co.chkRegExpSearch?"pattern ":"")
		+(co.chkCaseSensitiveSearch||co.chkRegExpSearch?"matching^^\n":"");
	return body;
}

if (!window.formatSearchResults_list) window.formatSearchResults_list=function(text,matches)
{
	// bullet list of links to matching tiddlers
	var body='';
	var pattern=co.chkRegExpSearch?text:text.escapeRegExp();
	var sensitive=co.chkCaseSensitiveSearch?"mg":"img";
	var link='{{tiddlyLinkExisting{<html><nowiki><a href="javascript:;" onclick="'
		+'if(config.options.chkSearchHighlight)'
		+'	highlightHack=new RegExp(\x27'+pattern+'\x27.escapeRegExp(),\x27'+sensitive+'\x27);'
		+'story.displayTiddler(null,\x27%0\x27);'
		+'highlightHack = null; return false;'
		+'" title="%2">%1</a></html>}}}';
	for(var t=0;t<matches.length;t++) {
		body+="* ";
		if (config.options.chkSearchByDate)
			body+=matches[t].modified.formatString('YYYY.0MM.0DD 0hh:0mm')+" ";
		var title=matches[t].title;
		var fixup=title.replace(/'/g,"\\x27").replace(/"/g,"\\x22");
		var tid=store.getTiddler(title);
		var tip=tid?tid.getSubtitle():''; tip=tip.replace(/"/g,"&quot;");
		body+=link.format([fixup,title,tip])+'\n';
	}
	return body;
}

if (!window.formatSearchResults_buttons) window.formatSearchResults_buttons=function(text,matches)
{
	// embed buttons only if writing SearchResults to tiddler
	if (!config.options.chkSearchListTiddler) return "";
	// "open all" button
	var title=config.macros.search.reportTitle;
	var body="";
	body+="@@diplay:block;<html><input type=\"button\" href=\"javascript:;\" "
		+"onclick=\"story.displayTiddlers(null,[";
	for(var t=0;t<matches.length;t++)
		body+="'"+matches[t].title.replace(/\'/mg,"\\'")+"'"+((t<matches.length-1)?", ":"");
	body+="],1);\" accesskey=\"O\" value=\"open all matching tiddlers\"></html> ";
	// "discard SearchResults" button
	body+="<html><input type=\"button\" href=\"javascript:;\" "
		+"onclick=\"discardSearchResults()\" value=\"discard "+title+"\"></html>";
	body+="@@\n";
	return body;
}

if (!window.discardSearchResults) window.discardSearchResults=function()
{
	// remove the tiddler
	story.closeTiddler(config.macros.search.reportTitle);
	store.deleteTiddler(config.macros.search.reportTitle);
	store.notify(config.macros.search.reportTitle,true);
}
//}}}
<<tiddler [[include_tiddlers/Second Law of Thermodynamics.html#"Second Law of Thermodynamics"]]>>
''Second\-Order Arithmetic'' is a collection of axiomatic systems that formalize the natural numbers and sets thereof.
Pairs of natural numbers can be coded in the usual way as natural numbers. In this way arbitrary integers or rational numbers can be represented.
Real numbers can be defined as Cauchy sequences of rational numbers. Real functions are legitimate objects of study, since they are defined by their values on the rationals.
<<tiddler [[include_tiddlers/Sedenion.html#"Sedenion"]]>>
<<tiddler [[include_tiddlers/Sedenion Loop.html#"Sedenion Loop"]]>>
<<tiddler [[include_tiddlers/Sedenion Multiplication Tables.html#"Sedenion Multiplication Tables"]]>>
<<tiddler [[include_tiddlers/Sedenion Projective Geometry.html#"Sedenion Projective Geometry"]]>>
<<tiddler [[include_tiddlers/Sedenion Subalgebras.html#"Sedenion Subalgebras"]]>>
<<tiddler [[include_tiddlers/Sedenion Zero Divisor.html#"Sedenion Zero Divisor"]]>>
<<tiddler [[include_tiddlers/Sedenionic World Model.html#"Sedenionic World Model"]]>>
<<tiddler [[include_tiddlers/Seiberg-Witten Invariant.html#"Seiberg-Witten Invariant"]]>>
Seiberg and Witten have argued that certain noncommutative [[gauge theories|Gauge Theory]] are equivalent to commutative ones and in particular that there exists a map from a commutative gauge field to a noncommutative one, which is compatible with the gauge structure of each. This map has become known as the ''Seiberg\-Witten (SW) Map''.
<<tiddler [[include_tiddlers/Selberg Trace Formula.html#"Selberg Trace Formula"]]>>
<<tiddler [[include_tiddlers/Selenium.html#"Selenium"]]>>
''Self Organized Criticality'' (= ''SOC'').

Links:
* [[WIKIPEDIA - Self-organized Criticality|http://en.wikipedia.org/wiki/Self-organized_criticality]]

Papers:
* [[Phase Transitions and Complex Systems - R. V. Solé, S. C. Manrubia, B. Luque, J. Delgado|http://complex.upf.es/~ricard/COMPLEXITY-96.pdf]] [[pct. 66|http://scholar.google.de/scholar?cites=15790333541130288044&hl=de]]
* [[Cellular Automata and Self-organized Criticality - M. Creutz|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.50.5288&rep=rep1&type=pdf]] [[pct. 8|http://scholar.google.com/scholar?cites=9131077436801048589&as_sdt=2005&sciodt=2000&hl=de]] TRD
* [[Symmetropy and Self-Organized Criticality - K. Nanjo, H. Nagahama, E. Yodogawa|http://www.scipress.org/journals/forma/pdf/1603/16030213.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=2013446146386239124&hl=de]]
* [[Y-Bias and Angularity: The Dynamics of Self-Organizing Criticality From the Zero Point to Infinity - D. G. Yurth, D. Ayres|http://www.pureenergysystems.com/academy/papers/Y-Bias_and_Angularity/Y-Bias_Monograph_Preview_Edition_25Apr06c.doc]] pct. 0
<<tiddler [[include_tiddlers/Self-Reproducing Automaton.html#"Self-Reproducing Automaton"]]>>
<<tiddler [[include_tiddlers/Self-energy.html#"Self-energy"]]>>
<<tiddler [[include_tiddlers/Self-gravitating Bose-Einstein Condensate.html#"Self-gravitating Bose-Einstein Condensate"]]>>
<<tiddler [[include_tiddlers/Self-organization.html#"Self-organization"]]>>
<<tiddler [[include_tiddlers/Self-organized Criticality.html#"Self-organized Criticality"]]>>
<<tiddler [[include_tiddlers/Semiclassical Einstein Equation.html#"Semiclassical Einstein Equation"]]>>
<<tiddler [[include_tiddlers/Semiclassicality.html#"Semiclassicality"]]>>
A ''Semigroup'' is an algebraic structure consisting of a nonempty set together with an associative binary operation. A semigroup differs from a [[group|Group]] in that for each of its elements there may not exist an inverse and an identity element.
<<tiddler [[include_tiddlers/Semisimple Group.html#"Semisimple Group"]]>>
<<tiddler [[include_tiddlers/Seneca.html#"Seneca"]]>>
<<tiddler [[include_tiddlers/Senescence.html#"Senescence"]]>>
<<tiddler [[include_tiddlers/Senolytic.html#"Senolytic"]]>>
<<tiddler [[include_tiddlers/Separable Space.html#"Separable Space"]]>>
<<tiddler [[include_tiddlers/Separable State.html#"Separable State"]]>>
<<tiddler [[include_tiddlers/Sextenion.html#"Sextenion"]]>>
<<tiddler [[include_tiddlers/Shahn Majid.html#"Shahn Majid"]]>>
<<tiddler [[include_tiddlers/Shannon Entropy.html#"Shannon Entropy"]]>>
<<tiddler [[include_tiddlers/Shear.html#"Shear"]]>>
<<tiddler [[include_tiddlers/Ship of Theseus.html#"Ship of Theseus"]]>>
The ''Shuffle Product'' $\mb A \sqcup \! \sqcup \mb B$ of two words $\mb A = a_1 \ldots a_m$ and $\mb B = b_{m+1} \ldots b_{m+n}$ is defined as the sum of all words it is possible to construct from $\mb A$ and $\mb B$ by preserving the order of all the letters in each of the words.
I.e.
\[
\mb A \sqcup \! \sqcup \mb B \equiv \sum_\sigma  a_{\sigma(1)} a_{\sigma(2)}\ldots  a_{\sigma(m)}  b_{\sigma(m+1)} b_{\sigma(m+2)}\ldots b_{\sigma(m+n)}
\] with $\sigma(1) < \sigma(2) < \ldots < \sigma(m)$ and $\sigma(m + 1) < \sigma(m + 2) < \ldots < \sigma(m + n)$, where $\sigma$ is the operation of permutation.

The product is also called a ''(m, n)-Shuffle''.

!!!!Examples
# $12 \sqcup\!\sqcup ab= 12ab + 1a2b + 1ab2 + a12b + a1b2 + ab12 $.  <br><br>
# $123 \sqcup\!\sqcup ab = 123ab + 12a3b + 12ab3 + 1a23b + 1a2b3 + 1ab23 + a123b + a12b3 + a1b23 + ab123$.

!!!!Properties
* Commutative, i.e. $\mb A \sqcup \! \sqcup \mb B = \mb B \sqcup \! \sqcup \mb A$. <br><br>
* The number of terms of the product is given by ${m + n \choose m} = {m + n \choose n}$.
<br><br><br><<closeAll>> <br> &nbsp; {{button{ font size}}}<<fontSize>> <br><br><<search>> <br>{{ button{&nbsp;  goto}}} <br><<gotoTiddler>><br>
<<tabs txtMainTab Latest 'Recently modified notes' TabTimeline Topics 'Selected tags' TabContents Alphabetical 'Selected tags' TabContents2  All 'List all notes' TabAll>>
<<tiddler [[include_tiddlers/Sign Tables.html#"Sign Tables"]]>>
<<tiddler [[include_tiddlers/Signature.html#"Signature"]]>>
<<tiddler [[include_tiddlers/Signature Change.html#"Signature Change"]]>>
<<tiddler [[include_tiddlers/Simon Donaldson.html#"Simon Donaldson"]]>>
<<tiddler [[include_tiddlers/Simple Root.html#"Simple Root"]]>>
<<tiddler [[include_tiddlers/Simplicial Complex.html#"Simplicial Complex"]]>>
<<tiddler [[include_tiddlers/Simply Connected Space.html#"Simply Connected Space"]]>>
<<tiddler [[include_tiddlers/Single Value Decomposition.html#"Single Value Decomposition"]]>>
<<tiddler [[include_tiddlers/Single-photon Emission Computed Tomography.html#"Single-photon Emission Computed Tomography"]]>>
/***
|Name|SinglePageModePlugin|
|Source|http://www.TiddlyTools.com/#SinglePageModePlugin|
|Documentation|http://www.TiddlyTools.com/#SinglePageModePluginInfo|
|Version|2.9.6|
|Author|Eric Shulman - ELS Design Studios|
|License|http://www.TiddlyTools.com/#LegalStatements <br>and [[Creative Commons Attribution-ShareAlike 2.5 License|http://creativecommons.org/licenses/by-sa/2.5/]]|
|~CoreVersion|2.1|
|Type|plugin|
|Requires||
|Overrides|Story.prototype.displayTiddler(), Story.prototype.displayTiddlers()|
|Options|##Configuration|
|Description|Show tiddlers one at a time with automatic permalink, or always open tiddlers at top/bottom of page.|
This plugin allows you to configure TiddlyWiki to navigate more like a traditional multipage web site with only one tiddler displayed at a time.
!!!!!Documentation
>see [[SinglePageModePluginInfo]]
!!!!!Configuration
<<<
<<option chkSinglePageMode>> Display one tiddler at a time
><<option chkSinglePagePermalink>> Automatically permalink current tiddler
><<option chkSinglePageKeepFoldedTiddlers>> Don't close tiddlers that are folded
><<option chkSinglePageKeepEditedTiddlers>> Don't close tiddlers that are being edited
<<option chkTopOfPageMode>> Open tiddlers at the top of the page
<<option chkBottomOfPageMode>> Open tiddlers at the bottom of the page
<<option chkSinglePageAutoScroll>> Automatically scroll tiddler into view (if needed)

Notes:
* The "display one tiddler at a time" option can also be //temporarily// set/reset by including a 'paramifier' in the document URL: {{{#SPM:true}}} or {{{#SPM:false}}}.
* If more than one display mode is selected, 'one at a time' display takes precedence over both 'top' and 'bottom' settings, and if 'one at a time' setting is not used, 'top of page' takes precedence over 'bottom of page'.
* When using Apple's Safari browser, automatically setting the permalink causes an error and is disabled.
<<<
!!!!!Revisions
<<<
2008.10.17 [2.9.6] changed chkSinglePageAutoScroll default to false
| Please see [[SinglePageModePluginInfo]] for previous revision details |
2005.08.15 [1.0.0] Initial Release.  Support for BACK/FORWARD buttons adapted from code developed by Clint Checketts.
<<<
!!!!!Code
***/
//{{{
version.extensions.SinglePageModePlugin= {major: 2, minor: 9, revision: 6, date: new Date(2008,10,17)};
//}}}
//{{{
config.paramifiers.SPM = { onstart: function(v) {
	config.options.chkSinglePageMode=eval(v);
	if (config.options.chkSinglePageMode && config.options.chkSinglePagePermalink && !config.browser.isSafari) {
		config.lastURL = window.location.hash;
		if (!config.SPMTimer) config.SPMTimer=window.setInterval(function() {checkLastURL();},1000);
	}
} };
//}}}
//{{{
if (config.options.chkSinglePageMode==undefined)
	config.options.chkSinglePageMode=true;
if (config.options.chkSinglePagePermalink==undefined)
	config.options.chkSinglePagePermalink=true;
if (config.options.chkSinglePageKeepFoldedTiddlers==undefined)
	config.options.chkSinglePageKeepFoldedTiddlers=false;
if (config.options.chkSinglePageKeepEditedTiddlers==undefined)
	config.options.chkSinglePageKeepEditedTiddlers=false;
if (config.options.chkTopOfPageMode==undefined)
	config.options.chkTopOfPageMode=true;
if (config.options.chkBottomOfPageMode==undefined)
	config.options.chkBottomOfPageMode=false;
if (config.options.chkSinglePageAutoScroll==undefined)
	config.options.chkSinglePageAutoScroll=false;
//}}}
//{{{
config.SPMTimer = 0;
config.lastURL = window.location.hash;
function checkLastURL()
{
	if (!config.options.chkSinglePageMode)
		{ window.clearInterval(config.SPMTimer); config.SPMTimer=0; return; }
	if (config.lastURL == window.location.hash) return; // no change in hash
	var tids=decodeURIComponent(window.location.hash.substr(1)).readBracketedList();
	if (tids.length==1) // permalink (single tiddler in URL)
		story.displayTiddler(null,tids[0]);
	else { // restore permaview or default view
		config.lastURL = window.location.hash;
		if (!tids.length) tids=store.getTiddlerText("DefaultTiddlers").readBracketedList();
		story.closeAllTiddlers();
		story.displayTiddlers(null,tids);
	}
}


if (Story.prototype.SPM_coreDisplayTiddler==undefined)
	Story.prototype.SPM_coreDisplayTiddler=Story.prototype.displayTiddler;
Story.prototype.displayTiddler = function(srcElement,tiddler,template,animate,slowly)
{
	var title=(tiddler instanceof Tiddler)?tiddler.title:tiddler;
	var tiddlerElem=document.getElementById(story.idPrefix+title); // ==null unless tiddler is already displayed
	var opt=config.options;
	var single=opt.chkSinglePageMode && !startingUp;
	var top=opt.chkTopOfPageMode && !startingUp;
	var bottom=opt.chkBottomOfPageMode && !startingUp;
	if (single) {
		story.forEachTiddler(function(tid,elem) {
			// skip current tiddler and, optionally, tiddlers that are folded.
			if (	tid==title
				|| (opt.chkSinglePageKeepFoldedTiddlers && elem.getAttribute("folded")=="true"))
				return;
			// if a tiddler is being edited, ask before closing
			if (elem.getAttribute("dirty")=="true") {
				if (opt.chkSinglePageKeepEditedTiddlers) return;
				// if tiddler to be displayed is already shown, then leave active tiddler editor as is
				// (occurs when switching between view and edit modes)
				if (tiddlerElem) return;
				// otherwise, ask for permission
				var msg="'"+tid+"' is currently being edited.\n\n";
				msg+="Press OK to save and close this tiddler\nor press Cancel to leave it opened";
				if (!confirm(msg)) return; else story.saveTiddler(tid);
			}
			story.closeTiddler(tid);
		});
	}
	else if (top)
		arguments[0]=null;
	else if (bottom)
		arguments[0]="bottom";
	if (single && opt.chkSinglePagePermalink && !config.browser.isSafari) {
		window.location.hash = encodeURIComponent(String.encodeTiddlyLink(title));
		config.lastURL = window.location.hash;
		document.title = wikifyPlain("SiteTitle") + " - " + title;
		if (!config.SPMTimer) config.SPMTimer=window.setInterval(function() {checkLastURL();},1000);
	}
	if (tiddlerElem && tiddlerElem.getAttribute("dirty")=="true") { // editing... move tiddler without re-rendering
		var isTopTiddler=(tiddlerElem.previousSibling==null);
		if (!isTopTiddler && (single || top))
			tiddlerElem.parentNode.insertBefore(tiddlerElem,tiddlerElem.parentNode.firstChild);
		else if (bottom)
			tiddlerElem.parentNode.insertBefore(tiddlerElem,null);
		else this.SPM_coreDisplayTiddler.apply(this,arguments); // let CORE render tiddler
	} else
		this.SPM_coreDisplayTiddler.apply(this,arguments); // let CORE render tiddler
	var tiddlerElem=document.getElementById(story.idPrefix+title);
	if (tiddlerElem&&opt.chkSinglePageAutoScroll) {
		// scroll to top of page or top of tiddler
		var isTopTiddler=(tiddlerElem.previousSibling==null);
		var yPos=isTopTiddler?0:ensureVisible(tiddlerElem);
		// if animating, defer scroll until after animation completes
		var delay=opt.chkAnimate?config.animDuration+10:0;
		setTimeout("window.scrollTo(0,"+yPos+")",delay);
	}
}

if (Story.prototype.SPM_coreDisplayTiddlers==undefined)
	Story.prototype.SPM_coreDisplayTiddlers=Story.prototype.displayTiddlers;
Story.prototype.displayTiddlers = function() {
	// suspend single/top/bottom modes when showing multiple tiddlers
	var opt=config.options;
	var saveSPM=opt.chkSinglePageMode; opt.chkSinglePageMode=false;
	var saveTPM=opt.chkTopOfPageMode; opt.chkTopOfPageMode=false;
	var saveBPM=opt.chkBottomOfPageMode; opt.chkBottomOfPageMode=false;
	this.SPM_coreDisplayTiddlers.apply(this,arguments);
	opt.chkBottomOfPageMode=saveBPM;
	opt.chkTopOfPageMode=saveTPM;
	opt.chkSinglePageMode=saveSPM;
}
//}}}
<<tiddler [[include_tiddlers/Singlet Oxygen.html#"Singlet Oxygen"]]>>
<<tiddler [[include_tiddlers/Singular Homology.html#"Singular Homology"]]>>
''Singular Value Decomposition (SVD)'' is the decomposition of a given matrix $\mb M$ (not necessarily a square matrix) into a product of three special matrices:
\begin{equation}
\mb M = \mb U \mb D \mb V
\end{equation}
with $\mb D$ a diagonal matrix and $\mb U$ and $\mb V$ unitary matrices.
The diagonal entries of $\mb D$ are called ''singular values''. For certain matrices $\mb M$ the singular values coincide with the eigenvalues of the matrix.  
<<tiddler [[include_tiddlers/Singularity.html#"Singularity"]]>>
<<tiddler [[include_tiddlers/Singularity Theorems.html#"Singularity Theorems"]]>>
<<tiddler [[include_tiddlers/Sirtuin.html#"Sirtuin"]]>>

<html><img src="images/PTrajectory3.png" style="width: 470px; height: 29px;"/><img src="images/Subtitle3.png" style="width: 519px; height: 21px;"/></html>
<<tiddler [[include_tiddlers/Skew Field.html#"Skew Field"]]>>
<<tiddler [[include_tiddlers/Skylight Model.html#"Skylight Model"]]>>
<<tiddler [[include_tiddlers/Skyrmion.html#"Skyrmion"]]>>
<<tiddler [[include_tiddlers/Sleep.html#"Sleep"]]>>
<<tiddler [[include_tiddlers/Sloan Digital Sky Survey.html#"Sloan Digital Sky Survey"]]>>
<<tiddler [[include_tiddlers/Smooth Manifold.html#"Smooth Manifold"]]>>
<<tiddler [[include_tiddlers/Smooth Spacetime.html#"Smooth Spacetime"]]>>
<<tiddler [[include_tiddlers/Snow Crystal.html#"Snow Crystal"]]>>
<<tiddler [[include_tiddlers/Soft Graviton.html#"Soft Graviton"]]>>
<<tiddler [[include_tiddlers/Soft Photon.html#"Soft Photon"]]>>
<<tiddler [[include_tiddlers/Software.html#"Software"]]>>
<<tiddler [[include_tiddlers/Software Design Pattern.html#"Software Design Pattern"]]>>
Links:
* [[WIKIPEDIA - Solder Form|http://en.wikipedia.org/wiki/Solder_form]]
 
<<tiddler [[include_tiddlers/Soliton.html#"Soliton"]]>>
<<tiddler [[include_tiddlers/Soy.html#"Soy"]]>>
<<tiddler [[include_tiddlers/Space Art.html#"Space Art"]]>>
<<tiddler [[include_tiddlers/Space Exploration.html#"Space Exploration"]]>>
<<tiddler [[include_tiddlers/Spacetime Algebra.html#"Spacetime Algebra"]]>>
<<tiddler [[include_tiddlers/Spacetime Algebra Zero Divisor.html#"Spacetime Algebra Zero Divisor"]]>>
<<tiddler [[include_tiddlers/Spacetime Condensate.html#"Spacetime Condensate"]]>>
<<tiddler [[include_tiddlers/Spacetime Entropy.html#"Spacetime Entropy"]]>>
<<tiddler [[include_tiddlers/Spacetime Geometry and Quantum Mechanics.html#"Spacetime Geometry and Quantum Mechanics"]]>>
<<tiddler [[include_tiddlers/Spacetime and Criticality.html#"Spacetime and Criticality"]]>>
<<tiddler [[include_tiddlers/Spaghettification.html#"Spaghettification"]]>>
<<tiddler [[include_tiddlers/Special Conformal Transformation.html#"Special Conformal Transformation"]]>>
The ''Special Orthogonal Group $SO(r, s)$'' is defined by:
\[
SO(r, s) \equiv \{\mb A \in O(r, s) : \det(\mb A) = 1\}
\]
with $O(r, s)$ denoting the [[orthogonal group|Orthogonal Group]].
''Special Relativity'' (short ''SR'') is synonymous with [[Poincaré invariance|Poincaré Transformation]] of spacetime, which includes the Noether conservation of angular momentum.

Papers:
* [[Alternative-Algebra Methods in the Special Theory of Relativity - E. K. Loginov|http://www.springerlink.com/content/uqq32882562804h2/fulltext.pdf]] pct. 0
<<tiddler [[include_tiddlers/Speciality of the World.html#"Speciality of the World"]]>>
<<tiddler [[include_tiddlers/Spectral Action.html#"Spectral Action"]]>>
<<tiddler [[include_tiddlers/Spectral Action and Torsion.html#"Spectral Action and Torsion"]]>>
<<tiddler [[include_tiddlers/Spectral Distance.html#"Spectral Distance"]]>>
<<tiddler [[include_tiddlers/Spectral Geometry.html#"Spectral Geometry"]]>>
<<tiddler [[include_tiddlers/Spectral Theorem.html#"Spectral Theorem"]]>>
A ''Spectral Triple $(\mathcal A, H, D)$'' (a.k.a. ''Unbounded K\-Cycle'') consists of an algebra $\mathcal A$ (not necessarily commutative), a [[Hilbert space|Hilbert Space]] $H$ and a [[Dirac operator|Dirac Operator]] $D$ and defines a [[noncommutative space|Noncommutative Manifold]] or geometry.

Spectral triples allow for a (noncommutative) generalization of [[Riemannian geometry|Riemann Space]].

There are not many examples of spectral triples: so far all known are classical (commutative geometry), finite dimensional (over finite algebras) or based on noncommutative tori (including the recently introduced "isospectral deformations").

Papers:
* [[Equivariant Lorentzian Spectral Triples - M. Paschke, A. Sitarz|http://arxiv.org/PS_cache/math-ph/pdf/0611/0611029v1.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=6447584271726142251&hl=de]]
<<tiddler [[include_tiddlers/Speed Reading.html#"Speed Reading"]]>>
<<tiddler [[include_tiddlers/Speed of Gravity.html#"Speed of Gravity"]]>>
<<tiddler [[include_tiddlers/Speed of Light.html#"Speed of Light"]]>>
<<tiddler [[include_tiddlers/Spermidine.html#"Spermidine"]]>>
<<tiddler [[include_tiddlers/Sphere.html#"Sphere"]]>>
<<tiddler [[include_tiddlers/Spin Connection.html#"Spin Connection"]]>>
<<tiddler [[include_tiddlers/Spin Foam.html#"Spin Foam"]]>>
<<tiddler [[include_tiddlers/Spin Network.html#"Spin Network"]]>>
>... I wish to recall what I still consider to be a fundamental open question of path integration. This is the providing of a clean, intellectually satisfying relativistic path integral for particles with spin.
> - L. S. Schulman [1] -

See also:
* [[Feynman checkerboard|Feynman Checkerboard]]


Papers:
* [[A Path Integral for Spin (1968) - L. Schulman|http://people.clarkson.edu/~lschulma/1968PISpinPR.pdf]] [[local|papers/1968PISpinPR.pdf]] {{t100Cite{[[pct. 278|http://scholar.google.de/scholar?hl=de&lr=&cites=8352899487690416685&um=1&ie=UTF-8&ei=yiPdTZLJEIj5sgalxajHDg&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCQQzgIwAA]]}}}
* [[Path Integrals and Pseudoclassical Description for Spinning Particles in Arbitrary Dimensions (1997) - D. M. Gitman|http://www.dfn.if.usp.br/pesq/tqr/gitman/10.pdf]] [[local|papers/10.pdf]] [[pct. 29|http://scholar.google.de/scholar?hl=de&lr=&cites=15081806253202322783&um=1&ie=UTF-8&ei=JG5PTpDaIeqQ4gT147DrBw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CCIQzgIwAA]]
* [[A Rigorous Path Integral for Quantum Spin using Flat-space Wiener Regularization - B. Bodmann, H. Leschke, S. Warzel|http://www.theorie1.physik.uni-erlangen.de/leschke/papers/JMP02549.pdf]] [[local|papers/JMP02549.pdf]] [[pct. 10|http://scholar.google.de/scholar?hl=de&lr=&cites=8992519366958274455&um=1&ie=UTF-8&ei=LLFPTpDTE-T04QS0uqHPBw&sa=X&oi=science_links&ct=sl-citedby&resnum=1&ved=0CB8QzgIwAA]]
* [[Spin in Quantum Field Theory (2005) - S. Forte|http://arxiv.org/pdf/hep-th/0507291]] [[local|papers/0507291v4.pdf]] [[pct. 4|http://scholar.google.at/scholar?cites=1181921085743274482&as_sdt=2005&sciodt=0,5&hl=de]]
* [[[1] Frontiers of Path Integration (1994) - L. S. Schulman|http://people.clarkson.edu/~lschulma/1994FrontiersPathIntegration.pdf]] [[local|papers/1994FrontiersPathIntegration.pdf]] pct. 0

Links:
* [[L. S. Schulman|http://people.clarkson.edu/~lschulma/]]
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<<tiddler [[include_tiddlers/Spinor Representation.html#"Spinor Representation"]]>>
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<<tiddler [[include_tiddlers/Split Algebra Zero Divisor.html#"Split Algebra Zero Divisor"]]>>
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<<tiddler [[include_tiddlers/Split Complex Number.html#"Split Complex Number"]]>>
<<tiddler [[include_tiddlers/Split G2.html#"Split G2"]]>>
<<tiddler [[include_tiddlers/Split Octonion.html#"Split Octonion"]]>>
<<tiddler [[include_tiddlers/Split Octonion Zero Divisor.html#"Split Octonion Zero Divisor"]]>>
<<tiddler [[include_tiddlers/Split Quaternion.html#"Split Quaternion"]]>>
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<<tiddler [[include_tiddlers/Split Sedenion Zero Divisor.html#"Split Sedenion Zero Divisor"]]>>
<<tiddler [[include_tiddlers/Split-Biquaternion.html#"Split-Biquaternion"]]>>
<<tiddler [[include_tiddlers/Spontaneous Symmetry Breaking.html#"Spontaneous Symmetry Breaking"]]>>
<<tiddler [[include_tiddlers/Sporadic Group.html#"Sporadic Group"]]>>
A ''Spread'' in a three-dimensional [[projective space|Projective Space]] $PG(3, q)$ is a set of lines such that each point of the space is incident with exactly one line. (In other words: the lines partition the set of points). A spread in $PG(3, q)$ exists for any prime power $q$.

!!!!Example
[[PG(3,2)]] contains $56$ spreads with $5$ elements each.

<html><center><img src="images/spread.jpg" style="width: 280px; "/></center></html>
<<tiddler [[include_tiddlers/Squeezed State.html#"Squeezed State"]]>>
Papers:
* [[Incompatibility and Instability Based Size Eects in Crystals and Composites at Finite Elastoplastic Strains - M. Becker|http://elib.uni-stuttgart.de/opus/volltexte/2006/2610/pdf/Becker06.pdf]]  [[local|papers/Becker06.pdf]]
<<tiddler [[include_tiddlers/Stability of De Sitter Space.html#"Stability of De Sitter Space"]]>>
<<tiddler [[include_tiddlers/Stability of the Solar System.html#"Stability of the Solar System"]]>>
Given a group $G$ and a manifold $\mathcal M$, the ''Stabilizer'' (''Stationary Subgroup'' or ''Little Group'') of a point $x \in \mathcal M$ is the subgroup $G_x \subset G$ which contains only those elements $g \subset G$ which leave the point $x$ fixed
\begin{equation}
G_x \equiv \{g \in G | gx = x \}
\end{equation}
<<tiddler [[include_tiddlers/Standard Model.html#"Standard Model"]]>>
<<tiddler [[include_tiddlers/Standard Model Lagrangian.html#"Standard Model Lagrangian"]]>>
<<tiddler [[include_tiddlers/Standard Model Scalar Singlet.html#"Standard Model Scalar Singlet"]]>>
<<tiddler [[include_tiddlers/Star Product.html#"Star Product"]]>>
<<tiddler [[include_tiddlers/Stasheff Polytope.html#"Stasheff Polytope"]]>>
<<tiddler [[include_tiddlers/Static Universe.html#"Static Universe"]]>>
<<tiddler [[include_tiddlers/Stationary Spacetime.html#"Stationary Spacetime"]]>>
<<tiddler [[include_tiddlers/Statistical Operator.html#"Statistical Operator"]]>>
<<tiddler [[include_tiddlers/Steady State.html#"Steady State"]]>>
<<tiddler [[include_tiddlers/Stein Manifold.html#"Stein Manifold"]]>>
<<tiddler [[include_tiddlers/Steiner Quadruple System.html#"Steiner Quadruple System"]]>>
<<tiddler [[include_tiddlers/Steiner System.html#"Steiner System"]]>>
<<tiddler [[include_tiddlers/Steiner Triple System.html#"Steiner Triple System"]]>>
<<tiddler [[include_tiddlers/Stem Cell.html#"Stem Cell"]]>>
<<tiddler [[include_tiddlers/Stephen Hawking.html#"Stephen Hawking"]]>>
<<tiddler [[include_tiddlers/Sterile Neutrino.html#"Sterile Neutrino"]]>>
<<tiddler [[include_tiddlers/Stern-Gerlach Experiment.html#"Stern-Gerlach Experiment"]]>>
<<tiddler [[include_tiddlers/Steven Weinberg.html#"Steven Weinberg"]]>>
<<tiddler [[include_tiddlers/Stiefel Manifold.html#"Stiefel Manifold"]]>>
<<tiddler [[include_tiddlers/Stilbene.html#"Stilbene"]]>>
<<tiddler [[include_tiddlers/Stilbenoids.html#"Stilbenoids"]]>>
<<tiddler [[include_tiddlers/Stimulated Emission.html#"Stimulated Emission"]]>>
<<tiddler [[include_tiddlers/Stochastic Quantization.html#"Stochastic Quantization"]]>>
<<tiddler [[include_tiddlers/Stochastic Quantum Mechanics.html#"Stochastic Quantum Mechanics"]]>>
<<tiddler [[include_tiddlers/Stone-Von Neumann Theorem.html#"Stone-Von Neumann Theorem"]]>>
<<tiddler [[include_tiddlers/Strain.html#"Strain"]]>>
<br><<tiddler [[include_tiddlers/Strangeness.html#"Strangeness"]]>>
<<tiddler [[include_tiddlers/Strategies for Engineered Negligible Senescence.html#"Strategies for Engineered Negligible Senescence"]]>>
<br>@@display:block;text-align:center;[img[images/stress.gif]]@@

See also:
* [[Strain|Strain]]
<<tiddler [[include_tiddlers/Stress-energy-momentum Pseudotensor.html#"Stress-energy-momentum Pseudotensor"]]>>
<<tiddler [[include_tiddlers/Stress-energy-momentum Tensor.html#"Stress-energy-momentum Tensor"]]>>
<<tiddler [[include_tiddlers/Strict Proofs and the Natural Sciences.html#"Strict Proofs and the Natural Sciences"]]>>
<<tiddler [[include_tiddlers/String Cosmology.html#"String Cosmology"]]>>
<<tiddler [[include_tiddlers/String Field Theory.html#"String Field Theory"]]>>
<<tiddler [[include_tiddlers/Strong CP Problem.html#"Strong CP Problem"]]>>
<<tiddler [[include_tiddlers/Strong Gravity.html#"Strong Gravity"]]>>
<<tiddler [[include_tiddlers/Strong Interactions.html#"Strong Interactions"]]>>
<<tiddler [[include_tiddlers/Structure Formation.html#"Structure Formation"]]>>
<<tiddler [[include_tiddlers/Structures on Manifolds.html#"Structures on Manifolds"]]>>
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<<tiddler [[include_tiddlers/Sub-Planckian Physics.html#"Sub-Planckian Physics"]]>>
<<tiddler [[include_tiddlers/Subalgebra.html#"Subalgebra"]]>>
<<tiddler [[include_tiddlers/Subitizing.html#"Subitizing"]]>>
<<tiddler [[include_tiddlers/Superalgebra.html#"Superalgebra"]]>>
<<tiddler [[include_tiddlers/Supercentenarian.html#"Supercentenarian"]]>>
<<tiddler [[include_tiddlers/Superconductor.html#"Superconductor"]]>>
<<tiddler [[include_tiddlers/Superconductor Gravity Interaction.html#"Superconductor Gravity Interaction"]]>>
<<tiddler [[include_tiddlers/Superconformal Algebra.html#"Superconformal Algebra"]]>>
<<tiddler [[include_tiddlers/Superfluidity.html#"Superfluidity"]]>>
Papers:
* [[Introduction to Supergravity - H. Samtleben|http://www.itp.uni-hannover.de/~lechtenf/Events/Lectures/samtleben.pdf]]
* [[Supergravities without an Action: Gauging the Trombone - A. L. Dion, H. Samtleben|http://prunel.ccsd.cnrs.fr/docs/00/32/54/44/PDF/trombone_Sep29.pdf]]
<<tiddler [[include_tiddlers/Superluminal Neutrino.html#"Superluminal Neutrino"]]>>
<<tiddler [[include_tiddlers/Superoxide Dismutase.html#"Superoxide Dismutase"]]>>
<<tiddler [[include_tiddlers/Supersolid.html#"Supersolid"]]>>
Many complex systems in various areas of science exhibit a spatio-temporal dynamics that is inhomogeneous and can be effectively described by a superposition of  several statistics on different scales, called ''Superstatistics''. The notion was introduced in 2003 by Christian Beck.

Papers:
* [[Recent Developments in Superstatistics (2009) - C. Beck|http://arxiv.org/PS_cache/arxiv/pdf/0811/0811.4363v2.pdf]] [[local|papers/0811.4363v2.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=11128355148008017349&as_sdt=2005&sciodt=2000&hl=de]]
<<tiddler [[include_tiddlers/Superstring Theory.html#"Superstring Theory"]]>>
<<tiddler [[include_tiddlers/Supersymmetric Quantum Mechanics.html#"Supersymmetric Quantum Mechanics"]]>>
<<tiddler [[include_tiddlers/Supersymmetry.html#"Supersymmetry"]]>>
<<tiddler [[include_tiddlers/Supertask.html#"Supertask"]]>>
<<tiddler [[include_tiddlers/Surface Gravity.html#"Surface Gravity"]]>>
<<tiddler [[include_tiddlers/Surreal Number.html#"Surreal Number"]]>>
<<tiddler [[include_tiddlers/Survival Function.html#"Survival Function"]]>>
A ''Sylvester Matrix'' is a special representation of a [[Hadamard Matrix]], based on the following construction (a.k.a. ''Sylvester Construction''):

If $H_n$ is a Hadamard matrix of order $n$ then
\[
H_{2n}  = \begin{pmatrix} H_n & H_n \\ H_n & -H_n \end{pmatrix}
\]
is a Hadamard matrix of order $2n$ (starting from $H_1 = (1))$.

!!!![[Block Designs|Balanced Incomplete Block Design]]
* The [[Hadamard 3-design|Hadamard Design]] obtained from a Sylvester matrix is the point-hyperplane design of an [[affine geometry|Affine Geometry]] over $GF(2)$.
* Its derived [[Hadamard 2-design|Hadamard Design]] is the point-hyperplane design of a [[projective geometry|Projective Geometry]] over $GF(2)$.
<<tiddler [[include_tiddlers/Symbolic Dynamics.html#"Symbolic Dynamics"]]>>
<<tiddler [[include_tiddlers/Symmetric Design.html#"Symmetric Design"]]>>
<<tiddler [[include_tiddlers/Symmetric Group.html#"Symmetric Group"]]>>
<<tiddler [[include_tiddlers/Symmetric Space.html#"Symmetric Space"]]>>
<<tiddler [[include_tiddlers/Symmetries of the Riemann Tensor.html#"Symmetries of the Riemann Tensor"]]>>
<<tiddler [[include_tiddlers/Symmetry Breaking.html#"Symmetry Breaking"]]>>
<<tiddler [[include_tiddlers/Synchronicity.html#"Synchronicity"]]>>
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<<tiddler [[include_tiddlers/Systems Biology.html#"Systems Biology"]]>>
<<tiddler [[include_tiddlers/T-Design.html#"T-Design"]]>>
<<tiddler [[include_tiddlers/T-Duality.html#"T-Duality"]]>>
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    essay
    gauge
    genetics
    geometry
    gravitation
    group
    highlight
    informatics
    java
    lattice
    manifold
    mathematics
    medicine
    music
    network
    nonassociative
    noncommutative
    octonion
    parapsychology
    people
    philosophy
    physics
    polyvector
    programming
    psychology
    qft
    quantum
    quaternion
    sedenion
    software
    standardModel
    symmetry
    systemConfigX
    thermodynamics
    topology
    trigintaduonion

   excludeLists excludeSearch mjax search systemConfig systemNotes>>


  
/***
|Name|TagCloudPlugin|
|Source|http://www.TiddlyTools.com/#TagCloudPlugin|
|Version|1.7.0|
|Author|Eric Shulman|
|Original Author|Clint Checketts|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|present a 'cloud' of tags (or links) using proportional font display|
!Usage
<<<
{{{
<<cloud type action:... limit:... tag tag tag ...>>
<<cloud type action:... limit:... +TiddlerName>>
<<cloud type action:... limit:... -TiddlerName>>
<<cloud type action:... limit:... =tagvalue>>
}}}
where:
* //type// is a keyword, one of:
** ''tags'' (default) - displays a cloud of tags, based on frequency of use
** ''links'' - displays a cloud of tiddlers, based on number of links //from// each tiddler
** ''references'' - displays a cloud of tiddlers, based on number of links //to// each tiddler
* ''action:popup'' (default) - clicking a cloud item shows a popup with links to related tiddlers<br>//or//<br> ''action:goto'' - clicking a cloud item immediately opens the tiddler corresponding to that item
* ''limit:N'' (optional) - restricts the cloud display to only show the N most popular tags/links
* ''tag tag tag...'' (or ''title title title'' if ''links''/''references'' is used)<br>shows all tags/links in the document //except// for those listed as macro parameters
* ''+TiddlerName''<br>show only tags/links read from a space-separated, bracketed list stored in a separate tiddler.
* ''-TiddlerName''<br>show all tags/links //except// those read from a space-separated, bracketed list stored in a separate tiddler.
* ''=tagvalue'' (//only if type=''tags''//)<br>shows only tags that are themselves tagged with the indicated tag value (i.e., ~TagglyTagging usage)
//note: for backward-compatibility, you can also use the macro {{{<<tagCloud ...>>}}} in place of {{{<<cloud ...>>}}}//
<<<
!Examples
<<<
//all tags excluding<<tag systemConfig>>, <<tag excludeMissing>> and <<tag script>>//
{{{<<cloud systemConfig excludeMissing script>>}}}
{{groupbox{<<cloud systemConfig excludeMissing script>>}}}
//top 10 tags excluding<<tag systemConfig>>, <<tag excludeMissing>> and <<tag script>>//
{{{<<cloud limit:10 systemConfig excludeMissing script>>}}}
{{groupbox{<<cloud limit:10 systemConfig excludeMissing script>>}}}
//tags listed in// [[FavoriteTags]]
{{{<<cloud +FavoriteTags>>}}}
{{groupbox{<<cloud +FavoriteTags>>}}}
//tags NOT listed in// [[FavoriteTags]]
{{{<<cloud -FavoriteTags>>}}}
{{groupbox{<<cloud -FavoriteTags>>}}}
//links to tiddlers tagged with 'package'//
{{{<<cloud action:goto =package>>}}}
{{groupbox{<<cloud action:goto =package>>}}}
//top 20 most referenced tiddlers//
{{{<<cloud references limit:20>>}}}
{{groupbox{<<cloud references limit:20>>}}}
//top 20 tiddlers that contain the most links//
{{{<<cloud links limit:20>>}}}
{{groupbox{<<cloud links limit:20>>}}}
<<<
!Revisions
<<<
2009.07.17 [1.7.0] added {{{-TiddlerName}}} parameter to exclude tags that are listed in the indicated tiddler
2009.02.26 [1.6.0] added {{{action:...}}} parameter to apply popup vs. goto action when clicking cloud items
2009.02.05 [1.5.0] added ability to show links or back-links (references) instead of tags and renamed macro to {{{<<cloud>>}}} to reflect more generalized usage.
2008.12.16 [1.4.2] corrected group calculation to prevent 'group=0' error
2008.12.16 [1.4.1] revised tag filtering so excluded tags don't affect calculations
2008.12.15 [1.4.0] added {{{limit:...}}} parameter to restrict the number of tags displayed to the top N most popular
2008.11.15 [1.3.0] added {{{+TiddlerName}}} parameter to include only tags that are listed in the indicated tiddler
2008.09.05 [1.2.0] added '=tagname' parameter to include only tags that are themselves tagged with the specified value (i.e., ~TagglyTagging usage)
2008.07.03 [1.1.0] added 'segments' property to macro object.  Extensive code cleanup
<<<
!Code
***/
//{{{
version.extensions.TagCloudPlugin= {major: 1, minor: 7 , revision: 0, date: new Date(2009,7,17)};
//Originally created by Clint Checketts, contributions by Jonny Leroy and Eric Shulman
//Currently maintained and enhanced by Eric Shulman
//}}}
//{{{
config.macros.cloud = {
	tagstip: "%1 tiddlers tagged with '%0'",
	refslabel: " (%0 references)",
	refstip: "%1 tiddlers have links to '%0'",
	linkslabel: " (%0 links)",
	linkstip: "'%0' has links to %1 other tiddlers",
	groups: 9,
	init: function() {
		config.macros.tagCloud=config.macros.cloud; // for backward-compatibility
		config.shadowTiddlers.TagCloud='<<cloud>>';
		config.shadowTiddlers.StyleSheetTagCloud=
			'/*{{{*/\n'
			+'.tagCloud span {line-height: 3.5em; margin:3px;}\n'
			+'.tagCloud1{font-size: 80%;}\n'
			+'.tagCloud2{font-size: 100%;}\n'
			+'.tagCloud3{font-size: 120%;}\n'
			+'.tagCloud4{font-size: 140%;}\n'
			+'.tagCloud5{font-size: 160%;}\n'
			+'.tagCloud6{font-size: 180%;}\n'
			+'.tagCloud7{font-size: 200%;}\n'
			+'.tagCloud8{font-size: 220%;}\n'
			+'.tagCloud9{font-size: 240%;}\n'
			+'/*}}}*/\n';
		setStylesheet(store.getTiddlerText('StyleSheetTagCloud'),'tagCloudsStyles');
	},
	getLinks: function(tiddler) { // get list of links to existing tiddlers and shadows
		if (!tiddler.linksUpdated) tiddler.changed();
		var list=[]; for (var i=0; i<tiddler.links.length; i++) {
			var title=tiddler.links[i];
			if (store.isShadowTiddler(title)||store.tiddlerExists(title))
				list.push(title);
		}
		return list;
	},
	handler: function(place,macroName,params) {
		// unpack params
		var inc=[]; var ex=[]; var limit=0; var action='popup';
		var links=(params[0]&&params[0].toLowerCase()=='links'); if (links) params.shift();
		var refs=(params[0]&&params[0].toLowerCase()=='references'); if (refs) params.shift();
		if (params[0]&&params[0].substr(0,7).toLowerCase()=='action:')
			action=params.shift().substr(7).toLowerCase();
		if (params[0]&&params[0].substr(0,6).toLowerCase()=='limit:')
			limit=parseInt(params.shift().substr(6));
		while (params.length) {
			if (params[0].substr(0,1)=='+') { // read taglist from tiddler
				inc=inc.concat(store.getTiddlerText(params[0].substr(1),'').readBracketedList());
			} else if (params[0].substr(0,1)=='-') { // exclude taglist from tiddler
				ex=ex.concat(store.getTiddlerText(params[0].substr(1),'').readBracketedList());
			} else if (params[0].substr(0,1)=='=') { // get tag list using tagged tags
				var tagged=store.getTaggedTiddlers(params[0].substr(1));
				for (var t=0; t<tagged.length; t++) inc.push(tagged[t].title);
			} else ex.push(params[0]); // exclude params
			params.shift();
		}
		// get all items, include/exclude specific items
		var items=[];
		var list=(links||refs)?store.getTiddlers('title','excludeLists'):store.getTags();
		for (var t=0; t<list.length; t++) {
			var title=(links||refs)?list[t].title:list[t][0];
			if (links)	var count=this.getLinks(list[t]).length;
			else if (refs)	var count=store.getReferringTiddlers(title).length;
			else 		var count=list[t][1];
			if ((!inc.length||inc.contains(title))&&(!ex.length||!ex.contains(title)))
				items.push({ title:title, count:count });
		}
		if(!items.length) return;
		// sort by decending count, limit results (optional)
		items=items.sort(function(a,b){return(a.count==b.count)?0:(a.count>b.count?-1:1);});
		while (limit && items.length>limit) items.pop();
		// find min/max and group size
		var most=items[0].count;
		var least=items[items.length-1].count;
		var groupSize=(most-least+1)/this.groups;
		// sort by title and draw the cloud of items
		items=items.sort(function(a,b){return(a.title==b.title)?0:(a.title>b.title?1:-1);});
		var cloudWrapper = createTiddlyElement(place,'div',null,'tagCloud',null);
		for (var t=0; t<items.length; t++) {
			cloudWrapper.appendChild(document.createTextNode(' '));
			var group=Math.ceil((items[t].count-least)/groupSize)||1;
			var className='tagCloudtag tagCloud'+group;
			var tip=refs?this.refstip:links?this.linkstip:this.tagstip;
			tip=tip.format([items[t].title,items[t].count]);
			if (action=='goto') { // TAG/LINK/REFERENCES GOTO
				var btn=createTiddlyLink(cloudWrapper,items[t].title,true,className);
				btn.title=tip;
				btn.style.fontWeight='normal';
			} else if (!links&&!refs) { // TAG POPUP
				var btn=createTiddlyButton(cloudWrapper,items[t].title,tip,onClickTag,className);
				btn.setAttribute('tag',items[t].title);
			} else { // LINK/REFERENCES POPUP
				var btn=createTiddlyButton(cloudWrapper,items[t].title,tip,
					function(ev) { var e=ev||window.event; var cmt=config.macros.cloud;
						var popup = Popup.create(this);
						var title = this.getAttribute('tiddler');
						var count = this.getAttribute('count');
						var refs  = this.getAttribute('refs')=='T';
						var links = this.getAttribute('links')=='T';
						var label = (refs?cmt.refslabel:cmt.linkslabel).format([count]);
						createTiddlyLink(popup,title,true);
						createTiddlyText(popup,label);
						createTiddlyElement(popup,'hr');
						if (refs) {
							popup.setAttribute('tiddler',title);
							config.commands.references.handlePopup(popup,title);
						}
						if (links) {
							var tiddler = store.fetchTiddler(title);
							var links=config.macros.cloud.getLinks(tiddler);
							for(var i=0;i<links.length;i++)
								createTiddlyLink(createTiddlyElement(popup,'li'),
									links[i],true);
						}
						Popup.show();
						e.cancelBubble=true; if(e.stopPropagation) e.stopPropagation();
						return false;
					}, className);
				btn.setAttribute('tiddler',items[t].title);
				btn.setAttribute('count',items[t].count);
				btn.setAttribute('refs',refs?'T':'F');
				btn.setAttribute('links',links?'T':'F');
				btn.title=tip;
			}
		}
	}
};
//}}}
<<tiddler [[include_tiddlers/Tangent Algebra.html#"Tangent Algebra"]]>>
Let $x^\mu = x^\mu(t)$ be a smooth curve passing through a point $\mb x$ of an $n$-dimensional differentiable manifold $\mathcal M$. One can parametrize this curve such that $x^\mu(0) = 0$. The quantities $ \frac{dx^\mu(\lambda)}{d\lambda} |_{\lambda =0} = \xi^\mu$ are called the coordinates of the tangent vector $\bs \xi$ to the  curve in the point $\mb x$. The parametric equations of the curve can be written as $x^\mu(\lambda) = \xi^\mu \lambda + o(\lambda)$, where $o(\lambda)$ are infinitesimals of orders higher than $\lambda$.

The set of tangent vectors to all curves passing through a point $\mb x \in \mathcal M$ forms a $n$-dimensional vector space. This space is called the ''Tangent Space'' of the manifold $\mathcal M$ in the point $\mb x$ and is denoted by $T_{\mb x}(\mathcal M)$.

The set of all tangent spaces of the manifold $\mathcal M$ is called its ''Tangent Bundle'' and is denoted by $T(\mathcal M)$. An element of the tangent bundle is a pair $(\mb x,\bs \xi)$, where $\mb x \in \mathcal M$ and $\bs \xi \in T_{\mb x}(\mathcal M)$. This explains why the tangent bundle is also a differentiable manifold, having dimension $2n$.
<<tiddler [[include_tiddlers/Tannins.html#"Tannins"]]>>
<<tiddler [[include_tiddlers/Taoism.html#"Taoism"]]>>
<<tiddler [[include_tiddlers/Tardigrade.html#"Tardigrade"]]>>
<<tiddler [[include_tiddlers/Taylor Series.html#"Taylor Series"]]>>
<<tiddler [[include_tiddlers/Technological Singularity.html#"Technological Singularity"]]>>
<<tiddler [[include_tiddlers/Teleparallel Gravity.html#"Teleparallel Gravity"]]>>
<<tiddler [[include_tiddlers/Telomere.html#"Telomere"]]>>
<<tiddler [[include_tiddlers/Telomere Length Testing.html#"Telomere Length Testing"]]>>
<<tiddler [[include_tiddlers/Telomere Length and Cancer.html#"Telomere Length and Cancer"]]>>
<<tiddler [[include_tiddlers/Temperley-Lieb Algebra.html#"Temperley-Lieb Algebra"]]>>
<<tiddler [[include_tiddlers/Template Method Design Pattern.html#"Template Method Design Pattern"]]>>
<<tiddler [[include_tiddlers/Tensor.html#"Tensor"]]>>
<<tiddler [[include_tiddlers/Tensor Algebra.html#"Tensor Algebra"]]>>
The ''Tensor Product'', denoted by $\otimes$, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general bilinear operation. In some contexts, this product is also referred to as ''Outer Product''.

Links:
* [[WIKIPEDIA - Tensor Product|http://en.wikipedia.org/wiki/Tensor_product]]
* [[WIKIPEDIA - Kronecker Product|http://en.wikipedia.org/wiki/Kronecker_product]]
<<tiddler [[include_tiddlers/Tensor-vector-scalar Theory of Gravity.html#"Tensor-vector-scalar Theory of Gravity"]]>>
<<tiddler [[include_tiddlers/Ternary Algebra.html#"Ternary Algebra"]]>>
A ''Tessellation'' or ''Tiling'' of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. Generalizations to higher dimensions are possible.

Links:
* [[Tilings - J. Baez|http://math.ucr.edu/home/baez/tilings/]]
* [[The Edges of Plausibility - Exploring the Boundaries of Filters, Animation, Gradients, Patterns and Script in SVG - D. Dailey|http://srufaculty.sru.edu/david.dailey/svg/SVGOpen2008/edges_of_plausibility.htm]]
<<tiddler [[include_tiddlers/Tests of General Relativity.html#"Tests of General Relativity"]]>>
A ''Tetracategory'' is related to the [[associahedron|Associahedron]] $\mathcal K(6)$.

Links:
* [[WIKIPEDIA - Tetracategory|http://en.wikipedia.org/wiki/Tetracategory]]
The ''Tetracode'' is a [[self-dual|Dual Code]] $(4,2,3)$-[[code|Blockcode]] over $GF(3)$.

Papers:
* [[Kittens, S(5, 6, 12), and Mathematical Blackjack in SAGE - D. Joyner, A. Casey|http://sage.math.washington.edu/home/wdj/expository/hexads_sage.pdf]] pct. 0

Links:
* [[Applied Abstract Algebra, The Tetracode Construction - D. Joyner, R. Kreminski, J. Turisco|http://www.usna.edu/Users/math/wdj/book/node224.html]]
<<tiddler [[include_tiddlers/Tetrad.html#"Tetrad"]]>>
<<tiddler [[include_tiddlers/Tetrad Postulate.html#"Tetrad Postulate"]]>>
<<tiddler [[include_tiddlers/Tetrahedral Group.html#"Tetrahedral Group"]]>>
<<tiddler [[include_tiddlers/Tetraquark.html#"Tetraquark"]]>>
<<tiddler [[include_tiddlers/Tetron Model.html#"Tetron Model"]]>>
<html><br><center><img src="images/Author.jpg" style="width: 137px; "/></a></center></html>
>The effort to understand the universe is one of the very few things that lifts human life a little above the level of farce, and gives it some of the grace of tragedy.
> - [[Steven Weinberg]] -
<<tiddler [[include_tiddlers/The Coping with Death Hypothesis.html#"The Coping with Death Hypothesis"]]>>
<<tiddler [[include_tiddlers/The Landscape.html#"The Landscape"]]>>
<<tiddler [[include_tiddlers/The Simulation Hypothesis.html#"The Simulation Hypothesis"]]>>
<<tiddler [[include_tiddlers/The Smallest Particle in the Universe.html#"The Smallest Particle in the Universe"]]>>
<<tiddler [[include_tiddlers/The Ultimate Principle of Physical Reality.html#"The Ultimate Principle of Physical Reality"]]>>
<<tiddler [[include_tiddlers/Theorema Egregium.html#"Theorema Egregium"]]>>
<<tiddler [[include_tiddlers/Theory of Everything.html#"Theory of Everything"]]>>
<<tiddler [[include_tiddlers/Theory of Quasi Synchronous Aging.html#"Theory of Quasi Synchronous Aging"]]>>
<<tiddler [[include_tiddlers/Thermal Quantum Field Theory.html#"Thermal Quantum Field Theory"]]>>
<<tiddler [[include_tiddlers/Thermal Time.html#"Thermal Time"]]>>
<<tiddler [[include_tiddlers/Thermal de Broglie Wavelength.html#"Thermal de Broglie Wavelength"]]>>
<<tiddler [[include_tiddlers/Thermo Field Dynamics.html#"Thermo Field Dynamics"]]>>
<<tiddler [[include_tiddlers/Thermodynamics.html#"Thermodynamics"]]>>
<<tiddler [[include_tiddlers/Thermodynamics of Biological Aging.html#"Thermodynamics of Biological Aging"]]>>
The ''Theta Series $\theta_L (z)$'' of a [[lattice|Lattice]] $L$ is a generating function for the [[norms|Norm]] $||\vec x||$ of lattice vectors $\vec x$.
It is defined as
\[
\theta_L (z) \equiv \sum_{\vec x \in L}N(||\vec x||^2) \exp(\pi iz)^{\langle \vec x | \vec x \rangle}
\]
where $N(||\vec x||^2)$ is the number of lattice points of $L$ at squared distance $||\vec x||^2 = \langle \vec x | \vec x \rangle$ from the origin.

The series does not contain the information about the distribution of points in each concentric shell. Indeed, there are several examples of pairs of distinct lattices with the same theta series. But these lattices have different [[Voronoi polytopes|Voronoi Polytope]].

The theta series of an integral lattice is a [[modular form|Modular Form]].
<<tiddler [[include_tiddlers/Thoughts.html#"Thoughts"]]>>
<<tiddler [[include_tiddlers/Thoughts About Problems in Physics.html#"Thoughts About Problems in Physics"]]>>
<<tiddler [[include_tiddlers/Thoughts about Algebra and Differential Geometry.html#"Thoughts about Algebra and Differential Geometry"]]>>
<<tiddler [[include_tiddlers/Thoughts about a Discrete Structure of Spacetime.html#"Thoughts about a Discrete Structure of Spacetime"]]>>
<<tiddler [[include_tiddlers/Three.js.html#"Three.js"]]>>
/***
|Name|TiddlerTweakerPlugin|
|Source|http://www.TiddlyTools.com/#TiddlerTweakerPlugin|
|Version|2.4.4|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|select multiple tiddlers and modify author, created, modified and/or tag values|
~TiddlerTweaker is a 'power tool' for TiddlyWiki authors.  Select multiple tiddlers from a listbox and 'bulk modify' the creator, author, created, modified and/or tag values of those tiddlers using a compact set of form fields.  The values you enter into the fields simultaneously overwrite the existing values in all tiddlers you have selected.
!!!!!Usage
<<<
{{{<<tiddlerTweaker>>}}}
{{smallform{<<tiddlerTweaker>>}}}
By default, any tags you enter into the TiddlerTweaker will //replace// the existing tags in all the tiddlers you have selected.  However, you can also use TiddlerTweaker to quickly filter specified tags from the selected tiddlers, while leaving any other tags assigned to those tiddlers unchanged:
>Any tag preceded by a '+' (plus) or '-' (minus), will be added or removed from the existing tags //instead of replacing the entire tag definition// of each tiddler (e.g., enter '-excludeLists' to remove that tag from all selected tiddlers.  When using this syntax, care should be taken to ensure that //every// tag is preceded by '+' or '-', to avoid inadvertently overwriting any other existing tags on the selected tiddlers.  (note: the '+' or '-' prefix on each tag value is NOT part of the tag value, and is only used by TiddlerTweaker to control how that tag value is processed)
Important Notes:
* TiddlerTweaker is a 'power user' tool that can make changes to many tiddlers at once.  ''You should always have a recent backup of your document (or 'save changes' just *before* tweaking the tiddlers), just in case you accidentally 'shoot yourself in the foot'.''
* The date and author information on any tiddlers you tweak will ONLY be updated if the corresponding checkboxes have been selected.  As a general rule, after using TiddlerTweaker, always ''//remember to save your document//'' when you are done, even though the tiddler timeline tab may not show any recently modified tiddlers.
* Selecting and updating all tiddlers in a document can take a while.  Your browser may warn about an 'unresponsive script'.  Usually, if you allow it to continue, it should complete the processing... eventually.  Nonetheless, be sure to save your work before you begin tweaking lots of tiddlers, just in case something does get stuck.
<<<
!!!!!Revisions
<<<
2009.09.15 2.4.4 added 'edit' button. moved html definition to separate section
2009.09.13 2.4.3 in settiddlers(), convert backslashed chars (\n\b\s\t) in replacement text
2009.06.26 2.4.2 only add brackets around tags containing spaces
2009.06.22 2.4.1 in setFields(), add brackets around all tags shown tweaker edit field
2009.03.30 2.4.0 added 'sort by modifier'
2009.01.22 2.3.0 added support for text pattern find/replace
2008.10.27 2.2.3 in setTiddlers(), fixed Safari bug by replacing static Array.concat(...) with new Array().concat(...)
2008.09.07 2.2.2 added removeCookie() function for compatibility with [[CookieManagerPlugin]]
2008.05.12 2.2.1 replace built-in backstage tweak task with tiddler tweaker control panel (moved from BackstageTweaks)
2008.01.13 2.2.0 added 'auto-selection' links: all, changed, tags, title, text
2007.12.26 2.1.0 added support for managing 'creator' custom field (see [[CoreTweaks]])
2007.11.01 2.0.3 added config.options.txtTweakerSortBy for cookie-based persistence of list display order preference setting.
2007.09.28 2.0.2 in settiddlers() and deltiddlers(), added suspend/resume notification handling (improves performance when operating on multiple tiddlers)
2007.08.03 2.0.1 added shadow definition for [[TiddlerTweaker]] tiddler for use as parameter references with {{{<<tiddler>>, <<slider>> or <<tabs>>}}} macros.
2007.08.03 2.0.0 converted from inline script
2006.01.01 1.0.0 initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.TiddlerTweakerPlugin= {major: 2, minor: 4, revision: 4, date: new Date(2009,9,15)};

// shadow tiddler
config.shadowTiddlers.TiddlerTweaker='<<tiddlerTweaker>>';

// defaults
if (config.options.txtTweakerSortBy==undefined) config.options.txtTweakerSortBy='modified';

// backstage task
if (config.tasks) { // for TW2.2b3 or above
	config.tasks.tweak.tooltip='review/modify tiddler internals: dates, authors, tags, etc.';
	config.tasks.tweak.content='{{smallform small groupbox{<<tiddlerTweaker>>}}}';
}

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

config.macros.tiddlerTweaker = {
	handler: function(place,macroName,params,wikifier,paramString,tiddler) {
		var span=createTiddlyElement(place,'span');
		span.innerHTML=store.getTiddlerText('TiddlerTweakerPlugin##html');
		this.init(span.getElementsByTagName('form')[0],config.options.txtTweakerSortBy);
	},
	init: function(f,sortby) { // initialize form controls
		if (!f) return; // form might not be rendered yet...
		while (f.list.options[0]) f.list.options[0]=null; // empty current list content
		var tids=store.getTiddlers(sortby);
		if (sortby=='size') // descending order
			tids.sort(function(a,b) {return a.text.length > b.text.length ? -1 : (a.text.length == b.text.length ? 0 : +1);});
		var who='';
		for (i=0; i<tids.length; i++) { var t=tids[i];
			var label=t.title; var value=t.title;
			switch (sortby) {
				case 'modified':
				case 'created':
					var t=tids[tids.length-i-1]; // reverse order
					var when=t[sortby].formatString('YY.0MM.0DD 0hh:0mm ');
					label=when+t.title;
					value=t.title;
					break;
				case 'size':
					label='['+t.text.length+'] '+label;
					break;
				case 'modifier':
				case 'creator':
					if (who!=t[sortby]) {
						who=t[sortby];
						f.list.options[f.list.length]=new Option('by '+who+':','',false,false);
					}
					label='\xa0\xa0\xa0'+label; // indent
					break;
			}
			f.list.options[f.list.length]=new Option(label,value,false,false);
		}
		f.title.value=f.who.value=f.creator.value=f.tags.value='';
		f.cm.value=f.cd.value=f.cy.value=f.ch.value=f.cn.value='';
		f.mm.value=f.md.value=f.my.value=f.mh.value=f.mn.value='';
		f.stats.disabled=f.set.disabled=f.del.disabled=f.edit.disabled=f.display.disabled=true;
		f.settitle.disabled=false;
		config.options.txtTweakerSortBy=sortby;
		f.sortby.value=sortby; // sync droplist
		if (sortby!='modified') saveOptionCookie('txtTweakerSortBy');
		else removeCookie('txtTweakerSortBy');
	},
	selecttiddlers: function(here) { // enables/disables inputs based on #items selected
		var f=here.form; var list=f.list;
		var c=0; for (i=0;i<list.length;i++) if (list.options[i].selected) c++;
		if (c>1) f.title.disabled=true;
		if (c>1) f.settitle.checked=false;
		f.set.disabled=(c==0);
		f.del.disabled=(c==0);
		f.edit.disabled=(c==0);
		f.display.disabled=(c==0);
		f.settitle.disabled=(c>1);
		f.stats.disabled=(c==0);
		var msg=(c==0)?'select tiddlers':(c+' tiddler'+(c!=1?'s':'')+' selected');
		here.previousSibling.firstChild.firstChild.nextSibling.innerHTML=msg;
		if (c) clearMessage(); else displayMessage('no tiddlers selected');
	},
	setfields: function(here) { // set fields from first selected tiddler
		var f=here.form;
		if (!here.value.length) {
			f.title.value=f.who.value=f.creator.value=f.tags.value='';
			f.cm.value=f.cd.value=f.cy.value=f.ch.value=f.cn.value='';
			f.mm.value=f.md.value=f.my.value=f.mh.value=f.mn.value='';
			return;
		}
		var tid=store.getTiddler(here.value); if (!tid) return;
		f.title.value=tid.title;
		f.who.value=tid.modifier;
		f.creator.value=tid.fields['creator']||''; // custom field - might not exist
		f.tags.value=tid.tags.map(function(t){return String.encodeTiddlyLink(t)}).join(' ');
		var c=tid.created; var m=tid.modified;
		f.cm.value=c.getMonth()+1;
		f.cd.value=c.getDate();
		f.cy.value=c.getFullYear();
		f.ch.value=c.getHours();
		f.cn.value=c.getMinutes();
		f.mm.value=m.getMonth()+1;
		f.md.value=m.getDate();
		f.my.value=m.getFullYear();
		f.mh.value=m.getHours();
		f.mn.value=m.getMinutes();
	},
	settiddlers: function(here) {
		var f=here.form; var list=f.list;
		var tids=[];
		for (i=0;i<list.length;i++) if (list.options[i].selected) tids.push(list.options[i].value);
		if (!tids.length) { alert('please select at least one tiddler'); return; }
		var cdate=new Date(f.cy.value,f.cm.value-1,f.cd.value,f.ch.value,f.cn.value);
		var mdate=new Date(f.my.value,f.mm.value-1,f.md.value,f.mh.value,f.mn.value);
		if (tids.length>1 && !confirm('Are you sure you want to update these tiddlers:\n\n'+tids.join(', '))) return;
		store.suspendNotifications();
		for (t=0;t<tids.length;t++) {
			var tid=store.getTiddler(tids[t]); if (!tid) continue;
			var title=!f.settitle.checked?tid.title:f.title.value;
			var who=!f.setwho.checked?tid.modifier:f.who.value;
			var text=tid.text;
			if (f.replacetext.checked) {
				var r=f.replacement.value.replace(/\\t/mg,'\t').unescapeLineBreaks();
				text=text.replace(new RegExp(f.pattern.value,'mg'),r);
			}
			var tags=tid.tags;
			if (f.settags.checked) {
				var intags=f.tags.value.readBracketedList();
				var addtags=[]; var deltags=[]; var reptags=[];
				for (i=0;i<intags.length;i++) {
					if (intags[i].substr(0,1)=='+')
						addtags.push(intags[i].substr(1));
					else if (intags[i].substr(0,1)=='-')
						deltags.push(intags[i].substr(1));
					else
						reptags.push(intags[i]);
				}
				if (reptags.length)
					tags=reptags;
				if (addtags.length)
					tags=new Array().concat(tags,addtags);
				if (deltags.length)
					for (i=0;i<deltags.length;i++)
						{ var pos=tags.indexOf(deltags[i]); if (pos!=-1) tags.splice(pos,1); }
			}
			if (!f.setcdate.checked) cdate=tid.created;
			if (!f.setmdate.checked) mdate=tid.modified;
			store.saveTiddler(tid.title,title,text,who,mdate,tags,tid.fields);
			if (f.setcreator.checked) store.setValue(tid.title,'creator',f.creator.value); // set creator
			if (f.setcdate.checked) tid.assign(null,null,null,null,null,cdate); // set create date
		}
		store.resumeNotifications();
		this.init(f,f.sortby.value);
	},
	displaytiddlers: function(here,edit) {
		var f=here.form; var list=f.list;
		var tids=[];
		for (i=0; i<list.length;i++) if (list.options[i].selected) tids.push(list.options[i].value);
		if (!tids.length) { alert('please select at least one tiddler'); return; }
		story.displayTiddlers(story.findContainingTiddler(f),tids,edit?DEFAULT_EDIT_TEMPLATE:null);
	},
	deltiddlers: function(here) {
		var f=here.form; var list=f.list;
		var tids=[];
		for (i=0;i<list.length;i++) if (list.options[i].selected) tids.push(list.options[i].value);
		if (!tids.length) { alert('please select at least one tiddler'); return; }
		if (!confirm('Are you sure you want to delete these tiddlers:\n\n'+tids.join(', '))) return;
		store.suspendNotifications();
		for (t=0;t<tids.length;t++) {
			var tid=store.getTiddler(tids[t]); if (!tid) continue;
			if (tid.tags.contains('systemConfig')) {
				var msg=tid.title+' is tagged with systemConfig.'
					+'\n\nRemoving this tiddler may cause unexpected results.  Are you sure?';
				if (!confirm(msg)) continue;
			}
			store.removeTiddler(tid.title);
			story.closeTiddler(tid.title);
		}
		store.resumeNotifications();
		this.init(f,f.sortby.value);
	},
	stats: function(here) {
		var f=here.form; var list=f.list; var tids=[]; var out=''; var tot=0;
		var target=f.nextSibling;
		for (i=0;i<list.length;i++) if (list.options[i].selected) tids.push(list.options[i].value);
		if (!tids.length) { alert('please select at least one tiddler'); return; }
		for (t=0;t<tids.length;t++) {
			var tid=store.getTiddler(tids[t]); if (!tid) continue;
			out+='[['+tid.title+']] '+tid.text.length+'\n'; tot+=tid.text.length;
		}
		var avg=tot/tids.length;
		out=tot+' bytes in '+tids.length+' selected tiddlers ('+avg+' bytes/tiddler)\n<<<\n'+out+'<<<\n';
		removeChildren(target);
		target.innerHTML="<hr><font size=-2><a href='javascript:;' style='float:right' "
			+"onclick='this.parentNode.parentNode.style.display=\"none\"'>close</a></font>";
		wikify(out,target);
		target.style.display='block';
	}
};
//}}}
/***
//{{{
!html
<style>
.tiddlerTweaker table,
.tiddlerTweaker table tr,
.tiddlerTweaker table td
	{ padding:0;margin:0;border:0;white-space:nowrap; }
</style><form class='tiddlerTweaker'><!--
--><table style="width:100%"><tr valign="top"><!--
--><td style="text-align:center;width:99%;"><!--
	--><font size=-2><div style="text-align:left;"><span style="float:right"><!--
	-->&nbsp; <a href="javascript:;"
		title="select all tiddlers"
		onclick="
		var f=this; while (f&&f.nodeName.toLowerCase()!='form')f=f.parentNode;
		for (var t=0; t<f.list.options.length; t++)
			if (f.list.options[t].value.length) f.list.options[t].selected=true;
		config.macros.tiddlerTweaker.selecttiddlers(f.list);
		return false">all</a><!--
	-->&nbsp; <a href="javascript:;"
		title="select tiddlers that are new/changed since the last file save"
		onclick="
		var lastmod=new Date(document.lastModified);
		var f=this; while (f&&f.nodeName.toLowerCase()!='form')f=f.parentNode;
		for (var t=0; t<f.list.options.length; t++) {
			var tid=store.getTiddler(f.list.options[t].value);
			f.list.options[t].selected=tid&&tid.modified>lastmod;
		}
		config.macros.tiddlerTweaker.selecttiddlers(f.list);
		return false">changed</a><!--
	-->&nbsp; <a href="javascript:;"
		title="select tiddlers with at least one matching tag"
		onclick="
		var t=prompt('Enter space-separated tags (match ONE)');
		if (!t||!t.length) return false;
		var tags=t.readBracketedList();
		var f=this; while (f&&f.nodeName.toLowerCase()!='form')f=f.parentNode;
		for (var t=0; t<f.list.options.length; t++) {
			f.list.options[t].selected=false;
			var tid=store.getTiddler(f.list.options[t].value);
			if (tid&&tid.tags.containsAny(tags)) f.list.options[t].selected=true;
		}
		config.macros.tiddlerTweaker.selecttiddlers(f.list);
		return false">tags</a><!--
	-->&nbsp; <a href="javascript:;"
		title="select tiddlers whose titles include matching text"
		onclick="
		var txt=prompt('Enter a title (or portion of a title) to match');
		if (!txt||!txt.length) return false;
		var f=this; while (f&&f.nodeName.toLowerCase()!='form')f=f.parentNode;
		for (var t=0; t<f.list.options.length; t++) {
			f.list.options[t].selected=f.list.options[t].value.indexOf(txt)!=-1;
		}
		config.macros.tiddlerTweaker.selecttiddlers(f.list);
		return false">titles</a><!--
	-->&nbsp; <a href="javascript:;"
		title="select tiddlers containing matching text"
		onclick="
		var txt=prompt('Enter tiddler text (content) to match');
		if (!txt||!txt.length) return false;
		var f=this; while (f&&f.nodeName.toLowerCase()!='form')f=f.parentNode;
		for (var t=0; t<f.list.options.length; t++) {
			var tt=store.getTiddlerText(f.list.options[t].value,'');
			f.list.options[t].selected=(tt.indexOf(txt)!=-1);
		}
		config.macros.tiddlerTweaker.selecttiddlers(f.list);
		return false">text</a> &nbsp;<!--
	--></span><span>select tiddlers</span><!--
	--></div><!--
	--></font><select multiple name=list size="11" style="width:99.99%"
		title="use click, shift-click and/or ctrl-click to select multiple tiddler titles"
		onclick="config.macros.tiddlerTweaker.selecttiddlers(this)"
		onchange="config.macros.tiddlerTweaker.setfields(this)"><!--
	--></select><br><!--
	-->show<input type=text size=1 value="11"
		onchange="this.form.list.size=this.value; this.form.list.multiple=(this.value>1);"><!--
	-->by<!--
	--><select name=sortby size=1
		onchange="config.macros.tiddlerTweaker.init(this.form,this.value)"><!--
	--><option value="title">title</option><!--
	--><option value="size">size</option><!--
	--><option value="modified">modified</option><!--
	--><option value="created">created</option><!--
	--><option value="modifier">modifier</option><!--
	--></select><!--
	--><input type="button" value="refresh"
		onclick="config.macros.tiddlerTweaker.init(this.form,this.form.sortby.value)"<!--
	--> <input type="button" name="stats" disabled value="totals..."
		onclick="config.macros.tiddlerTweaker.stats(this)"><!--
--></td><td style="width:1%"><!--
	--><div style="text-align:left"><font size=-2>&nbsp;modify values</font></div><!--
	--><table style="width:100%;"><tr><!--
	--><td style="padding:1px"><!--
		--><input type=checkbox name=settitle unchecked
			title="allow changes to tiddler title (rename tiddler)"
			onclick="this.form.title.disabled=!this.checked">title<!--
	--></td><td style="padding:1px"><!--
		--><input type=text name=title size=35 style="width:98%" disabled><!--
	--></td></tr><tr><td style="padding:1px"><!--
		--><input type=checkbox name=setcreator unchecked
			title="allow changes to tiddler creator"
			onclick="this.form.creator.disabled=!this.checked">created by<!--
	--></td><td style="padding:1px;"><!--
		--><input type=text name=creator size=35 style="width:98%" disabled><!--
	--></td></tr><tr><td style="padding:1px"><!--
		--><input type=checkbox name=setwho unchecked
			title="allow changes to tiddler author"
			onclick="this.form.who.disabled=!this.checked">modified by<!--
	--></td><td style="padding:1px"><!--
		--><input type=text name=who size=35 style="width:98%" disabled><!--
	--></td></tr><tr><td style="padding:1px"><!--
		--><input type=checkbox name=setcdate unchecked
			title="allow changes to created date"
			onclick="var f=this.form;
				f.cm.disabled=f.cd.disabled=f.cy.disabled=f.ch.disabled=f.cn.disabled=!this.checked"><!--
		-->created on<!--
	--></td><td style="padding:1px"><!--
		--><input type=text name=cm size=2 style="width:2em;padding:0;text-align:center" disabled><!--
		--> / <input type=text name=cd size=2 style="width:2em;padding:0;text-align:center" disabled><!--
		--> / <input type=text name=cy size=4 style="width:3em;padding:0;text-align:center" disabled><!--
		--> at <input type=text name=ch size=2 style="width:2em;padding:0;text-align:center" disabled><!--
		--> : <input type=text name=cn size=2 style="width:2em;padding:0;text-align:center" disabled><!--
	--></td></tr><tr><td style="padding:1px"><!--
		--><input type=checkbox name=setmdate unchecked
			title="allow changes to modified date"
			onclick="var f=this.form;
				f.mm.disabled=f.md.disabled=f.my.disabled=f.mh.disabled=f.mn.disabled=!this.checked"><!--
		-->modified on<!--
	--></td><td style="padding:1px"><!--
		--><input type=text name=mm size=2 style="width:2em;padding:0;text-align:center" disabled><!--
		--> / <input type=text name=md size=2 style="width:2em;padding:0;text-align:center" disabled><!--
		--> / <input type=text name=my size=4 style="width:3em;padding:0;text-align:center" disabled><!--
		--> at <input type=text name=mh size=2 style="width:2em;padding:0;text-align:center" disabled><!--
		--> : <input type=text name=mn size=2 style="width:2em;padding:0;text-align:center" disabled><!--
	--></td></tr><tr><td style="padding:1px"><!--
		--><input type=checkbox name=replacetext unchecked
			title="find/replace matching text"
			onclick="this.form.pattern.disabled=this.form.replacement.disabled=!this.checked">replace text<!--
	--></td><td style="padding:1px"><!--
		--><input type=text name=pattern size=15 value="" style="width:40%" disabled
			title="enter TEXT PATTERN (regular expression)"> with<!--
		--><input type=text name=replacement size=15 value="" style="width:40%" disabled
			title="enter REPLACEMENT TEXT"><!--
	--></td></tr><tr><td style="padding:1px"><!--
		--><input type=checkbox name=settags checked
			title="allow changes to tiddler tags"
			onclick="this.form.tags.disabled=!this.checked">tags<!--
	--></td><td style="padding:1px"><!--
		--><input type=text name=tags size=35 value="" style="width:98%"
			title="enter new tags or use '+tag' and '-tag' to add/remove tags from existing tags"><!--
	--></td></tr></table><!--
	--><div style="text-align:center"><!--
	--><nobr><input type=button name=display disabled style="width:24%" value="display"
		title="show selected tiddlers"
		onclick="config.macros.tiddlerTweaker.displaytiddlers(this,false)"><!--
	--> <input type=button name=edit disabled style="width:23%" value="edit"
		title="edit selected tiddlers"
		onclick="config.macros.tiddlerTweaker.displaytiddlers(this,true)"><!--
	--> <input type=button name=del disabled style="width:24%" value="delete"
		title="remove selected tiddlers"
		onclick="config.macros.tiddlerTweaker.deltiddlers(this)"><!--
	--> <input type=button name=set disabled style="width:24%" value="update"
		title="update selected tiddlers"
		onclick="config.macros.tiddlerTweaker.settiddlers(this)"></nobr><!--
	--></div><!--
--></td></tr></table><!--
--></form><span style="display:none"><!--content replaced by tiddler "stats"--></span>
!end
//}}}
***/
>A \TiddlyWiki is like a blog because it's divided up into neat little chunks (tiddlers), but it encourages you to read it by hyperlinking rather than sequentially: if you like, a non-linear blog analogue that binds the individual microcontent items into a cohesive whole. I think that \TiddlyWiki represents a novel medium for writing, and will promote its own distinctive writing style.
>- Jeremy Ruston -

>Sometimes you come across a piece of software that is so simple and useful that you can't understand how you managed without it. \TiddlyWiki is one such application.
> - LINUX magazine -

My own opinion: First of all a \TiddlyWiki is cool ...

!!!Links
* [[TiddlyTools|http://www.tiddlytools.com/]] - Huge collection of plugins and macros.
* [[TiddlyWikiTutorial Learn the Basics of TiddlyWiki|http://www.blogjones.com/TiddlyWikiTutorial.html]]
* [[TiddlyWiki Cheat Sheet|doku/smb487ytqpsppjsw1y3.pdf]] - Small guide of Wiki commands
* [[TiddlyWiki for the Rest of Us- An Easy-to-use Entry-level TiddlyWiki Tutorial|http://www.giffmex.org/twfortherestofus.html]]
* [[Hier geht's zum deutschen Handbuch | doku\tiddlywikihandbuch.html]]
* [[Tiddlywikitips|http://tiddlywikitips.com/]] srl. 10
* [[FiddlyWIKI|http://way.net/FiddlyWiki/FiddlyWiki.html]]
* [[AbegoExtensions|http://tiddlywiki.abego-software.de/]] srl. 10
* [[TW Help|http://tiddlyspot.com/twhelp/]] srl. 10
* [[WIKIPEDIA - TiddlyWiki (old version)|http://en.wikipedia.org/w/index.php?title=TiddlyWiki&oldid=298131230]] [[local|html/TiddlyWiki.htm]]
* [[Server-Side Implementations|http://tiddlywiki.org/wiki/Server-Side_Solutions]]
* [[Server Side TiddlyWiki with PHP - B. Masis|http://www.borism.net/2009/03/18/server-side-tiddlywiki-with-php-easy/#comments]]
* [[TiddlyVault - An Index of TiddlyWiki Extensions from http://www.giffmex.org|http://tiddlyvault.tiddlyspot.com/]]
* [[Installing Disqus on TiddlySpace|http://jon.tiddlyspace.com/Installing%20Disqus%20on%20TiddlySpace]]
* [[Climate Change 2.0|http://www.climate-change-two.net/]]
* [[Bauwe Bijl TiddlyWiki|http://bauwebijl-tiddlywiki.appspot.com/]] - Nice templates

Some other physics and maths oriented linkes, that particularly demonstrate the use of [[jsmath|http://www.math.union.edu/~dpvc/jsMath/welcome.html]] with a Tiddly Wiki:
* [[Deferential Geometry|http://deferentialgeometry.org/]]
* [[Ferrari Beta|http://www.physics.mcgill.ca/~adriferr/]]
* [[Collection of useless thoughts - some notes on my research|http://internal.physics.uwa.edu.au/~styler/wiki/MainWiki.html]]

!!!Inline Formatting
|!Option|!Syntax|!Output|
|bold font|{{{''bold''}}}|''bold''|
|italic type|{{{//italic//}}}|//italic//|
|underlined text|{{{__underlined__}}}|__underlined__|
|strikethrough text|{{{--strikethrough--}}}|--strikethrough--|
|superscript text|{{{^^super^^script}}}|^^super^^script|
|subscript text|{{{~~sub~~script}}}|~~sub~~script|
|highlighted text|{{{@@highlighted@@}}}|@@highlighted@@|
|preformatted text|<html><code>{{{preformatted}}}</code></html>|{{{preformatted}}}|
!!!Block Elements
!!!!Headings
{{{
!Heading 1
!!Heading 2
!!!Heading 3
!!!!Heading 4
!!!!!Heading 5
}}}
<<<
!Heading 1
!!Heading 2
!!!Heading 3
!!!!Heading 4
!!!!!Heading 5
<<<
!!!!Lists
{{{
* unordered list, level 1
** unordered list, level 2
*** unordered list, level 3

# ordered list, level 1
## ordered list, level 2
### unordered list, level 3

; definition list, term
: definition list, description
}}}
<<<
* unordered list, level 1
** unordered list, level 2
*** unordered list, level 3

# ordered list, level 1
## ordered list, level 2
### unordered list, level 3

; definition list, term
: definition list, description
<<<
!!!!Blockquotes
{{{
> blockquote, level 1
>> blockquote, level 2
>>> blockquote, level 3

<<<
blockquote
<<<
}}}
<<<
> blockquote, level 1
>> blockquote, level 2
>>> blockquote, level 3

> blockquote
<<<
!!!!Preformatted Text
<html><pre>
{{{
preformatted (e.g. code)
}}}
</pre></html>
<<<
{{{
preformatted (e.g. code)
}}}
<<<
!!!!Tables
{{{
|CssClass|k
|!heading column 1|!heading column 2|
|row 1, column 1|row 1, column 2|
|row 2, column 1|row 2, column 2|
|>|COLSPAN|
|ROWSPAN| ⬦ |
|~| ⬦ |
|CssProperty:value;⬦| ⬦ |
|caption|c
}}}
''Annotation:''
* The {{{>}}} marker creates a "colspan", causing the current cell to merge with the one to the right.
* The {{{~}}} marker creates a "rowspan", causing the current cell to merge with the one above.
<<<
|CssClass|k
|!heading column 1|!heading column 2|
|row 1, column 1|row 1, column 2|
|row 2, column 1|row 2, column 2|
|>|COLSPAN|
|ROWSPAN| ⬦ |
|~| ⬦ |
|CssProperty:value;⬦| ⬦ |
|caption|c
<<<
!!!!Images
See [[TiddlyWiki.com|http://www.tiddlywiki.com/#EmbeddedImages]]

Example:
|<html><a href="http://www.hyperfract.com/gallery3/s4_1.jpg" title="Example picture"><img src="http://www.hyperfract.com/gallery3/s4_1.jpg" width="375" height="500" alt="..." /></a></html>|<html><a href="http://www.hyperfract.com/gallery3/q5_1.jpg" title="Example"><img src="http://www.hyperfract.com/gallery3/q5_1.jpg" width="500" height="333" alt="100_1400" /></a></html>|

!!!Hyperlinks
* [[WikiWords|WikiWord]] are automatically transformed to hyperlinks to the respective tiddler
** the automatic transformation can be suppressed by preceding the respective WikiWord with a tilde ({{{~}}}): {{{~WikiWord}}}
* [[PrettyLinks]] are enclosed in square brackets and contain the desired tiddler name: {{{[[tiddler name]]}}}
** optionally, a custom title or description can be added, separated by a pipe character ({{{|}}}): {{{[[title|target]]}}}<br>'''N.B.:''' In this case, the target can also be any website (i.e. URL).
!!!Custom Styling
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''Transitivity'' means that for any two elements $\mb A$ and $\mb B$ of a space (or a set) there exists an element $\mb g$ of a group $\mathcal G$ such that
\[
\mb B = \mb{g\mb A}
\]
More generally one sais that $\mathcal G$ acts ''n-transitively'' on a set $X$ with $ord(X) \ge n$, if for any pairwise distinct $x_1, \ldots , x_n$ and any pairwise distinct $y_1,\ldots ,y_n$ there is a $g \in \mathcal G$, such that $g x_i = y_i$, $\forall i =1,\ldots,n$.
Transitive defined above thus is equal to 1-transitive. 
<<tiddler [[include_tiddlers/Tri-bimaximal Mixing.html#"Tri-bimaximal Mixing"]]>>
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<<tiddler [[include_tiddlers/Trigintaduonion Multiplication Tables.html#"Trigintaduonion Multiplication Tables"]]>>
<br><<tiddler [[include_tiddlers/Trotter Formula.html#"Trotter Formula"]]>>
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Videos:
* [[Michio Kaku on UFOs|http://www.youtube.com/watch?v=zCe4WvdMgzM&feature=related]]
* [[Interview: Illobrand von Ludwiger über wissenschaftliche Beweise für UFOs|http://video.google.com/videoplay?docid=-7486657960591754798#]]
<<tiddler [[include_tiddlers/UV Completion.html#"UV Completion"]]>>
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<<tiddler [[include_tiddlers/VST Plugins.html#"VST Plugins"]]>>
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<<tiddler [[include_tiddlers/Vacuum Angle.html#"Vacuum Angle"]]>>
<<tiddler [[include_tiddlers/Vacuum Einstein Field Equations.html#"Vacuum Einstein Field Equations"]]>>
<<tiddler [[include_tiddlers/Vacuum Entanglement.html#"Vacuum Entanglement"]]>>
<<tiddler [[include_tiddlers/Vacuum Fluctuations.html#"Vacuum Fluctuations"]]>>
<<tiddler [[include_tiddlers/Vacuum Polarization.html#"Vacuum Polarization"]]>>
<<tiddler [[include_tiddlers/Vacuum Stability.html#"Vacuum Stability"]]>>
<<tiddler [[include_tiddlers/Vacuum State.html#"Vacuum State"]]>>
<<tiddler [[include_tiddlers/Variable Cosmological Constant.html#"Variable Cosmological Constant"]]>>
<<tiddler [[include_tiddlers/Variational Principle for Algebras.html#"Variational Principle for Algebras"]]>>
<<tiddler [[include_tiddlers/Varying Physical Constants.html#"Varying Physical Constants"]]>>
<<tiddler [[include_tiddlers/Varying Speed of Light.html#"Varying Speed of Light"]]>>
<<tiddler [[include_tiddlers/Vegetarism.html#"Vegetarism"]]>>
<<tiddler [[include_tiddlers/Vertex Operator Algebra.html#"Vertex Operator Algebra"]]>>
Links:
*[[The Ultimate Video Feedback Page|http://www.videofeedback.dk/World/]]
<<tiddler [[include_tiddlers/Videos.html#"Videos"]]>>
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<<tiddler [[include_tiddlers/Virasoro Algebra.html#"Virasoro Algebra"]]>>
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Links:
* [[Von Neumann Integer Generator|http://www.uni-bonn.de/~manfear/von-neumann-integer-gen.php]]
<<tiddler [[include_tiddlers/Voronin Universality Theorem.html#"Voronin Universality Theorem"]]>>
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<<tiddler [[include_tiddlers/WKB Approximation.html#"WKB Approximation"]]>>
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<<tiddler [[include_tiddlers/Wedge Product.html#"Wedge Product"]]>>
''Weighing Matrices'' are a generalization of [[Hadamard matrices|Hadamard Matrix]]. Their properties are quite similar to those of Hadamard matrices, and they are important for some construction methods of the latter.

A weighing matrix $W(m, n)$ is defined as a $m \times n$ matrix $H$ with entries $?1$, $0$ and $1$ such that
\[
HH^T = n I
\]
for some nonnegative integer $n$.
$n$ is called the ''Weight'' of $H$.
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~~Version: 2017_12_03 with 2096 out of 4273 (49%) tiddlers online. ~~
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__ Video of the week:__ [[Quantum equivalence principle|Quantum Equivalence Principle]], [[quantum gravity at cosmological scales|Quantum Gravity from Dark Energy]], etc. Lee Smolin presents some ideas which bear striking similarities to the ones I had before.

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''Welcome'' to this mathematics, sciences, philosophy & software programming notebook.

For quite some time I have been maintaining a notebook locally where I collect facts that seem to be relevant to me and where I write down own thoughts, ideas, results, etc. Some of the things are probably not interesting for the general public as they are too personal, unfinished or preliminary. Some might be and I will contain them in this Wiki. Furthermore I have some new results in theoretical physics that took me quite some effort to work out over the years and I want to hold them back as I consider publishing some of them one day.
I should really stress that I have not made a distinction between things that are part of general knowledge and things that I have "invented" - therefore be warned! If you are looking for an encyclopedia, you'd probably better consult some other site. (In particular [[P-theory|P-Theory]], the theory of [[polyvector spaces|Polyvector Space]] the way represented here is due to myself. If you don't like it, just ignore the [[t