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

* [[WIKIPEDIA - *-Algebra|*-algebra]]
''​1⁄f [[Noise]]'' refers to any noise with a power spectral density (power per frequency interval) of the form
S(f) \propto \frac 1 {f^\alpha }
where $f$ is frequency, and $0 < α < 2$, with exponent $\alpha$ usually close to 1. 

The canonical case with $\alpha = 1$ is called ''Pink Noise''. The name "pink noise" arises from the pink appearance of visible light with this power spectrum. Pink noise sources include ''Flicker Noise'' in electronic devices. Pink noise is one of the most common signals in biological systems. 

1⁄f noise varies with a predictability intermediate between [[white noise|White Noise]] (no correlation in time, $S(f) \propto 1/f^0$) and [[random walk (Brownian motion) noise|Brown Noise]] (no correlation between increments, $S(f) \propto 1/f^2$). 

{{center{[img(409px+, )[images6/TypesOfNoise.jpg]]}}}
The low-frequency 1⁄f noise is an ubiquitous phenomenon, found everywhere from fluctuations of sea level in Bermuda and the intensity of a recording of Bach's Brandenburg Concerto, to human heart rates and electrical currents in semiconductor devices.

{{center{[img(607px+, )[images6/PinkNoise.jpg]]}}}
Noises with $\alpha$ near 1 generally come from condensed-matter systems in quasi-equilibrium. Noises with a broad range of $\alpha$ generally correspond to a wide range of non-equilibrium driven dynamical systems. 

There are no simple, even linear stochastic differential equations generating signals with 1⁄f  noise. The widespread occurrence of signals exhibiting such behavior suggests that a generic mathematical explanation might exist. Except for some formal mathematical descriptions like fractional [[Brownian motion|Brownian Motion]] (half-integral of a white noise signal), however, no generally recognized physical explanation of 1⁄f noise has been proposed. Consequently, the ubiquity of 1⁄f  noise is one of the oldest puzzles of contemporary physics and science in general. 

* [[Flicker (1/f) Noise in Tunnel Junction DC SQUIDs (1982) - R. H. Koch, J. Clarke, W. M. Goubau, J. M. Martinis, C. M. Pegrum, D. J. Van Harlingen|]] [[local|papers4/qt1zh4d8n5.pdf]] {{t100Cite{[[pct. 269|,5&hl=en]]}}}
* [[Model for 1/f Flux Noise in SQUIDs and Qubits (2007) - R. H. Koch, D. P. DiVincenzo, J. Clarke|]] [[local|papers5/0702025.pdf]] {{t100Cite{[[pct. 254|,5&hl=en]]}}}
* [[1/f Noise: a Pedagogical Review (2002) - E. Milotti|]] [[local|papers5/0204033.pdf]] {{t100Cite{[[pct. 150|,5&hl=en]]}}}
* [[A Theory of 1/f noise in Human Cognition (2009) - P. Grigolini, G. Aquino, M. Bologna, M. Luković, B. J. West|]] [[local|papers5/A_theory_of_1_f_noise_in_human_cognition.pdf]]  [[pct. 85|,5&hl=en]]
* [[Cosmological Flux Noise and Measured Noise Power Spectra in SQUIDs (2016) - C. Beck|]] [[local|papers4/srep28275.pdf]] [[pct. 11|,5&hl=en]]

* [[WIKIPEDIA - Pink Noise|]]
* [[SCHOLARPEDIA - 1/f noise|]]
* [[The mystery of the 1/f noise (2013)|]]
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''5α-reductases'' are enzymes involved in steroid metabolism. They participate in three metabolic pathways: 
* bile acid biosynthesis 
* [[androgen|Androgen]] metabolism 
* estrogen metabolism 
There are three isozymes of 5α-reductase, \SRD5A1, \SRD5A2, and \SRD5A3, which vary in different tissues with age. 

The enzyme is produced in many tissues in both males and females, in the reproductive tract, testes and ovaries, skin, seminal vesicles, [[prostate|Prostate]], epididymis and many organs, including the [[nervous system|Nervous System]].

5α-reductase is most known for converting testosterone, the male sex hormone, into the more potent [[dihydrotestosterone|Dihydrotestosterone]]. 

{{center{[img(577px+, )[images5/5-alpha-reductase.jpg]]}}}
See also:
* [[5α-reductase inhibitor|5α-reductase Inhibitor]]

* [[WIKIPEDIA - 5α-Reductase|]]
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In recent decades we have seen theories in physics like [[superstring theory|Superstring Theory]] or [[AdS/CFT|AdS/CFT Correspondence]] which have gained a lot of attention and popularity even though there is no experimental backing so far. A very nasty hallmark of several such theories is that they are quite removed from the world we perceive around us. String theory predicts many more spatial dimension, it is incompatible with a positive [[cosmological constant|Cosmological Constant]], it requires [[supersymmetry|Supersymmetry]], etc. \AdS/CFT works in [[Anti-de Sitter space|Anti-de Sitter Space]] but after two decades a similar duality in [[de Sitter space|De Sitter Space]] is not available. One sometimes hears arguments like "these are just toy models but nevertheless they may allow us to learn something about the real world", or "these models allow us to compute something whereas in the other case we don't know what to do". Concerning the first argument, it is just plain wrong. Suppose a toy model gives you the answer "yes" to a certain question. Then you would conclude that the answer should also be "yes" for the real world situation. But if this is not the case, then you are not just wrong but even worse, you are convinced of the wrong answer because it seems to be backed by your toy model. Concerning the second argument, it is clear why it is given, researchers want to get funding, want to keep their Ph.D. students busy, all in all, it's about the red tape.
By the way, all this is reminiscent of medical research. There, most experiments are done with simple [[model organisms|Model Organism]] because it's cheaper, quicker and of course, one can learn a lot about humans, or doesn't one? No, one doesn't because why would one do costly clinical trials in humans when it comes to the introduction of new medications when it would be enough to test it in [[C. elegans|Caenorhabditis Elegans]], for instance.

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Coming back to physics, what is going wrong here? Suppose you develop a physical theory from scratch, the way mathematicians do. Then, what you need is a consistent set of axioms. Yet the subtle difference between physics and mathematics is that in the latter case consistency is necessary and sufficient whereas in the former case it is merely necessary.
Now let's assume your physical axiom system is consistent and makes predictions, how could you figure out if it is also sufficient? The answer is simple, just do the experiments and see if they confirm the theory (the [[Popper|Karl Raimund Popper]] kind of thing). But if your predictions are such that they are not accessible to current experiments and there is little hope that this will change in the foreseeable future, what can you do? There are at least two things one could do. The first thing is to rely on blind faith, i.e. one makes science to kind of a religion, but this is just what science shouldn't be. The second thing is to ask yourself the following question: "How would a flawed physical theory look like, how would it manifest itself?". The claim made here is that the consistent failure in realistic situations which is a hallmark of superstring theory or \AdS/CFT, for instance, is exactly what one would expect for a flawed physical theory. Why is that? It is so because one has taken the venerable axioms of physics and just pushed them too far. There is nothing wrong with doing that, and it should be the default when trying to construct a new theory ([[Occam's razor|Occam's Razor]]) but when one realizes that one has pushed too far with the axioms one should ask oneself if it wouldn't be better to reconsider them. There are two famous examples in the history of physics where this was done, the transitions from classical physics to the theory of relativity and to [[quantum mechanics|Quantum Mechanics]] which required to give up some sacred assumptions like absolute time or [[determinism|Determinism]]. But there is a difference between then and the current situation. Then it was more or less forced upon us because we had very concrete experimental anomalies which cried out for an explanation like the constancy of the [[speed of light|Speed of Light]] measured in the \Michelson-Morley experiment. Today it seems to require a bit more courage.
So how do we go from here? Maybe a quote from Thomas Stearns Eliot can help, who said "Only those who will risk going too far can possibly find out how far one can go." So then, let'sstep back and rethink things.

© 2019 Markus Maute

See also:
* [[Beyond quantum mechanics|Beyond Quantum Mechanics]]
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The maintenance of [[genomic|Genome]] integrity is crucial for the survival of all organisms. In humans, compromised genomic integrity contributes to genetic disorders, [[aging|Biological Aging]] and [[cancers|Cancer]]. The task of safeguarding the genome is accomplished by the concerted action of a number of cellular processes, including [[DNA replication|DNA Replication]], [[DNA repair|DNA Repair]], [[senescence|Cellular Senescence]] and [[apoptosis|Apoptosis]]. 
Many, if not all, of these processes are regulated and coordinated by the [[DNA-damage|DNA Damage]] [[checkpoint|Cell Cycle Checkpoint]], which is a complex signaling network that is triggered by [[DNA damage|DNA Damage]] or genomic instability. Two [[phospho-inositide 3-kinase-like protein kinases (\PIKKs)|PIKK]] – ''Ataxia Telangiectasia Mutated (ATM)'', and ''Ataxia Telangiectasia and Rad3-related (ATR)'' – are master regulators of two major checkpoint pathways. ATM is primarily activated by [[DNA double-strand breaks (DSBs)|Double Strand Break Repair]], whereas ATR responds to a much broader spectrum of DNA damage, including \DSBs and many types of DNA damage that interfere with DNA replication. In contrast to ATM, ATR has a crucial role in stabilizing the genome during DNA replication and is essential for cell survival.

* [[ATM and ATR Signaling at a Glance (2015) - P. Awasthi, M. Foiani, A. Kumar|]] [[local|papers3/4255.full.pdf]] {{t100Cite{[[pct. 265|,5&hl=en]]}}}
* [[ATM and ATR: Sensing DNA Damage (2004) - J. Yang, Z.-P. Xu, Y. Huang, H. E. Hamrick, P. J. Duerksen-Hughes, Y.-N. Yu|]] [[local|papers3/WJG-10-155.pdf]] {{t100Cite{[[pct. 165|,5&hl=en]]}}}
* [[ATR Signaling at a Glance (2009) - B. Shiotani, L. Zou|]] [[local|papers3/301.full.pdf]] {{t100Cite{[[pct. 100|,5&hl=en]]}}}

* [[WIKIPEDIA - ATM_serine/threonine Kinase|]]
* [[WIKIPEDIA - Ataxia Telangiectasia and Rad3 Related|]]
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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 [[tiddlers|What is a Tiddler]] tagged with "polyvector" and the WIKI may still be of some value to you).

As this notebook is based on a [[TiddlyWiki Classic|]] the information is nicely chunked. So for each "Tiddler" I can decide if I dare to put it on the web or not.

Disclaimer: A lot in this Wiki is second hand information and it is prone to be erroneous. I take no guarantee whatsoever. (I encourage you to do kind of a "peer reviewing" in that you write [[comments|Comments]] - this would be very helpful). Furthermore I am quite lazy painting pictures or reformulating every adequate text found somewhere, so I often just copy the information (occasionally mutatis mutandis) into this Wiki. Therefore if you think there is a problem with violations of copyrights, please let me know and I'll remove the respective content immediately.
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An ''Active Transformation'' transforms the basis elements of an algebra. See also [[passive transformations|Passive Transformation]].
> Polchinski admits that the condensed-matter sceptics have a point. "I don't think that string theorists have yet come up with anything that condensed-matter theorists don't already know," he says. The quantitative results tend to be re-derivations of answers that condensed-matter theorists had already calculated using more mundane methods.
> - [1] -

> 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|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]]

* [[Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories (1998) - E. Witten|]] [[local|papers/9803131.pdf]] {{t1000Cite{[[pct. 3094|,5&hl=en]]}}}
* [[Josephson Junctions and AdS/CFT Networks (2011) - E. Kiritsis, V. Niarchos|]] [[local|papers/1105.6100.pdf]] [[pct. 26|,5&hl=en]]

* [[A Black Hole Full of Answers (2007) - J. Zaanen|]] [[local|magazines/zaanen-nature.pdf]] [[mct. 15|,5&hl=en]]
* [[Strange and Stringy (2013) - S. Sachdev|]] [[local|magazines/c63.pdf]] [[mct. 11|,5&hl=de]]
* [[[1] String Theory Finds a Bench Mate (2011) - Z. Merali|]] [[local|magazines/478302a.pdf]] [[mct. 7|,5&hl=en]]

* [[WIKIPEDIA - AdS/CFT Correspondence|]]

* [[Conformal Gravity and AdS/CFT (2012) - K. Skenderis|]]
* [[KITP Program: Holographic Duality and Condensed Matter Physics 2011|]]
''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 $15$.
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''Albert'' is a computer algebra system for doing calculus with [[nonassociative algebras|Nonassociative Algebra]].

* [[Albert website|]]
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* [[WIKIPEDIA - Apophis|]]
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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. ?

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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
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<<tiddler [[include_tiddlers/Artificial Neural Network.html#"Artificial Neural Network"]]>>
<<tiddler [[include_tiddlers/Artificial Neuron.html#"Artificial Neuron"]]>>
<<tiddler [[include_tiddlers/Ashtekar Formalism.html#"Ashtekar Formalism"]]>>
<<tiddler [[include_tiddlers/Aspect Ratio.html#"Aspect Ratio"]]>>
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]] $\mathcal K(5)$.

* [[Root Systems and Generalized Associahedra - S. Fomin, N. Reading|]] [[pct. 45|]]
* [[The Diagonal of the Stasheff Polytope - J.-L. Loday|]]
* [[Cluster Algebras: Notes for the CDM-03 Conference - S. Fomin, A. Zelevinsky|]]
* [[The Multiple Facets of the Associahedron - J.-L. Loday|]] pct. 0

* [[Strange Associations|]]
<<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]].

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/Astaxanthin.html#"Astaxanthin"]]>>
<<tiddler [[include_tiddlers/Asthma.html#"Asthma"]]>>
<<tiddler [[include_tiddlers/Astro Camera.html#"Astro Camera"]]>>
<<tiddler [[include_tiddlers/AstroToaster.html#"AstroToaster"]]>>
<<tiddler [[include_tiddlers/Astrobiology.html#"Astrobiology"]]>>
<<tiddler [[include_tiddlers/Astrocyte.html#"Astrocyte"]]>>
<<tiddler [[include_tiddlers/Astronomical Interferometer.html#"Astronomical Interferometer"]]>>
<<tiddler [[include_tiddlers/Astronomical Objects.html#"Astronomical Objects"]]>>
<<tiddler [[include_tiddlers/Astronomical Seeing.html#"Astronomical Seeing"]]>>
<<tiddler [[include_tiddlers/Astronomical Weather Forecast.html#"Astronomical Weather Forecast"]]>>
<<tiddler [[include_tiddlers/Astronomy Software.html#"Astronomy Software"]]>>
<<tiddler [[include_tiddlers/Astrophotography.html#"Astrophotography"]]>>
<<tiddler [[include_tiddlers/Astrophotography Automation Software.html#"Astrophotography Automation Software"]]>>
<<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/Atherosclerosis.html#"Atherosclerosis"]]>>
<<tiddler [[include_tiddlers/Atiyah-Singer Index Theorem.html#"Atiyah-Singer Index Theorem"]]>>
<<tiddler [[include_tiddlers/Atmospheric Refraction.html#"Atmospheric Refraction"]]>>
<<tiddler [[include_tiddlers/Atom Laser.html#"Atom Laser"]]>>
<<tiddler [[include_tiddlers/Attention Scheme Theory.html#"Attention Scheme Theory"]]>>
<<tiddler [[include_tiddlers/Autocorrelation.html#"Autocorrelation"]]>>
<<tiddler [[include_tiddlers/Autoimmunity.html#"Autoimmunity"]]>> 
<<tiddler [[include_tiddlers/Automated Reasoning.html#"Automated Reasoning"]]>>
<<tiddler [[include_tiddlers/Automorphism.html#"Automorphism"]]>>
<<tiddler [[include_tiddlers/Automorphism Group.html#"Automorphism Group"]]>>
<<tiddler [[include_tiddlers/Autonomic Nervous System.html#"Autonomic Nervous System"]]>> 
<<tiddler [[include_tiddlers/Autoparallelity.html#"Autoparallelity"]]>>
<<tiddler [[include_tiddlers/Autophagy.html#"Autophagy"]]>>
<<tiddler [[include_tiddlers/Autophagy Induction.html#"Autophagy Induction"]]>>
<<tiddler [[include_tiddlers/Autophagy and Aging.html#"Autophagy and Aging"]]>>
<<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/Axiomatic Quantum Field Theory.html#"Axiomatic Quantum Field Theory"]]>>
<<tiddler [[include_tiddlers/Axion.html#"Axion"]]>>
<<tiddler [[include_tiddlers/Axion-like Particle.html#"Axion-like Particle"]]>>
<<tiddler [[include_tiddlers/Azimuth.html#"Azimuth"]]>>
<<tiddler [[include_tiddlers/B Cell.html#"B Cell"]]>>
<<tiddler [[include_tiddlers/B-L.html#"B-L"]]>>
''BCH Code'' = ''Bose\-Chaudhuri\-Hocquenghem Code'' belong to a large class of cyclic [[blockcodes|Blockcode]].

* [[Notes on Coding Theory, The Definition of BCH and RS Codes - J. Beachy|]]
<<tiddler [[include_tiddlers/BF Theory.html#"BF Theory"]]>>
<<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/Babesia.html#"Babesia"]]>>
<<tiddler [[include_tiddlers/Baby Monster Group.html#"Baby Monster Group"]]>>
<<tiddler [[include_tiddlers/Bach Brackets.html#"Bach Brackets"]]>>
<<tiddler [[include_tiddlers/Back-illuminated Sensor.html#"Back-illuminated Sensor"]]>>
<<tiddler [[include_tiddlers/Background Extraction.html#"Background Extraction"]]>>
<<tiddler [[include_tiddlers/Background Independence.html#"Background Independence"]]>>
<<tiddler [[include_tiddlers/Bacteria Toxin.html#"Bacteria Toxin"]]>>
<<tiddler [[include_tiddlers/Bacterial Generation Time.html#"Bacterial Generation Time"]]>>
<<tiddler [[include_tiddlers/Bacterial Stress Response.html#"Bacterial Stress Response"]]>>
<<tiddler [[include_tiddlers/Bactericide.html#"Bactericide"]]>>
<<tiddler [[include_tiddlers/Bacteriophage.html#"Bacteriophage"]]>>
<<tiddler [[include_tiddlers/Bacteriostatic Agent.html#"Bacteriostatic Agent"]]>>
<<tiddler [[include_tiddlers/Bacterium.html#"Bacterium"]]>>
<<tiddler [[include_tiddlers/Bahtinov Mask.html#"Bahtinov Mask"]]>>
<<tiddler [[include_tiddlers/Baicalein.html#"Baicalein"]]>>
<<tiddler [[include_tiddlers/Baker-Campbell-Hausdorff Formula.html#"Baker-Campbell-Hausdorff Formula"]]>>
<<tiddler [[include_tiddlers/Banach Algebra.html#"Banach Algebra"]]>>
<<tiddler [[include_tiddlers/Barabási-Albert Model.html#"Barabási-Albert Model"]]>>
<<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 Acoustic Oscillations.html#"Baryon Acoustic Oscillations"]]>>
<<tiddler [[include_tiddlers/Baryon Asymmetry.html#"Baryon Asymmetry"]]>>
<<tiddler [[include_tiddlers/Baryonic Tully-Fisher Relation.html#"Baryonic Tully-Fisher Relation"]]>>
<<tiddler [[include_tiddlers/Basal-cell Carcinoma.html#"Basal-cell Carcinoma"]]>>
<<tiddler [[include_tiddlers/Base Excision Repair.html#"Base Excision Repair"]]>>
<<tiddler [[include_tiddlers/Basic Reproduction Number.html#"Basic Reproduction Number"]]>>  
This tiddler contains some basic $\LaTeX$ macros.  Documentation can be found [[here|BasicMacros Documentation]].

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.


% 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\cunit{\mathbf i_{\small \mathbb C}}
\def\Chr#1#2{\left\{ {\begin{array}{*{20}c}  #1\\ {#2} \\ \end{array} } \right\}}

% Symbols




% Sets and set operations

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

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



\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\map#1->#2{\parentheses{{#1} \mapsTo {#2}}}

\def\restriction#1#2{{{\left. {#1} \right\lvert}_{#2}}}



% Topology related notation




\def\interval#1#2,#3#4{{\left#1 {#2}, {#3} \right#4}}
\def\unitInterval{\interval[0, 1]}

% General notation

\def\modulus#1{{\left\lvert {#1} \right\rvert}}
\def\norm#1{{\left\Vert {#1} \right\Vert}}

% Aliases










% 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.

% Because 'blackboard' is one word the second 'b' should not be capitalised in the command name.

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

<<tiddler [[include_tiddlers/Bat.html#"Bat"]]>>
<<tiddler [[include_tiddlers/Bayesian Probability.html#"Bayesian Probability"]]>>
<<tiddler [[include_tiddlers/Beam Splitter.html#"Beam Splitter"]]>>
<<tiddler [[include_tiddlers/Bee Venom.html#"Bee Venom"]]>>
<<tiddler [[include_tiddlers/Bekenstein-Hawking Entropy.html#"Bekenstein-Hawking Entropy"]]>>
<<tiddler [[include_tiddlers/Bel-Robinson Tensor.html#"Bel-Robinson Tensor"]]>>
[[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 Spaceship Paradox.html#"Bell's Spaceship Paradox"]]>>
<<tiddler [[include_tiddlers/Bell's Theorem.html#"Bell's Theorem"]]>>
<<tiddler [[include_tiddlers/Belousov-Zhabotinsky Reaction.html#"Belousov-Zhabotinsky Reaction"]]>>
<<tiddler [[include_tiddlers/Benign Prostatic Hyperplasia.html#"Benign Prostatic Hyperplasia"]]>>
<<tiddler [[include_tiddlers/Berberine.html#"Berberine"]]>>
<<tiddler [[include_tiddlers/Berezin Calculus.html#"Berezin Calculus"]]>>
<<tiddler [[include_tiddlers/Berry Phase.html#"Berry Phase"]]>>
* [[WIKIPEDIA - Bertrand Russell|]]

* [[Bertrand Russell - To our Descendants|]]
* [[Bertrand Russell on God (1959)|]]
* [[Bertrand Russell on Clarity and Exact Thinking|]]
<<tiddler [[include_tiddlers/Bertrand's Theorem.html#"Bertrand's Theorem"]]>>
* [[WIKIPEDIA - Bessel Function|]]
<<tiddler [[include_tiddlers/Bethe-Salpeter Equation.html#"Bethe-Salpeter Equation"]]>>
<<tiddler [[include_tiddlers/Betti Number.html#"Betti Number"]]>>
<<tiddler [[include_tiddlers/Beyond Quantum Mechanics.html#"Beyond Quantum Mechanics"]]>>
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.

* [[WIKIPEDIA - Bialgebra|]]
<<tiddler [[include_tiddlers/Bianchi Classification.html#"Bianchi Classification"]]>>
<<tiddler [[include_tiddlers/Big Bang.html#"Big Bang"]]>>
<<tiddler [[include_tiddlers/Big Bang Models.html#"Big Bang Models"]]>>
<<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/Bilateria.html#"Bilateria"]]>>
<<tiddler [[include_tiddlers/Bilinear Covariants.html#"Bilinear Covariants"]]>>
<<tiddler [[include_tiddlers/Billionaire.html#"Billionaire"]]>>
<<tiddler [[include_tiddlers/Binary Black Hole.html#"Binary Black Hole"]]>>
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/Biohacking.html#"Biohacking"]]>>
<<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 Membrane.html#"Biological Membrane"]]>>
<<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/Biomass.html#"Biomass"]]>>
<<tiddler [[include_tiddlers/Biophoton.html#"Biophoton"]]>>
<<tiddler [[include_tiddlers/Biquaternion.html#"Biquaternion"]]>>
<<tiddler [[include_tiddlers/Bird Longevity.html#"Bird Longevity"]]>>
''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 Bomb.html#"Black Hole Bomb"]]>>
<<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 Observation.html#"Black Hole Observation"]]>>
<<tiddler [[include_tiddlers/Black Hole Remnant.html#"Black Hole Remnant"]]>>
<<tiddler [[include_tiddlers/Black Hole Shadow.html#"Black Hole Shadow"]]>>
<<tiddler [[include_tiddlers/Black Hole Starship.html#"Black Hole Starship"]]>>
<<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/Blackjack.html#"Blackjack"]]>>
<<tiddler [[include_tiddlers/Blaschke Conjecture.html#"Blaschke Conjecture"]]>>
<<tiddler [[include_tiddlers/Blind Man and an Elephant.html#"Blind Man and an Elephant"]]>>
<<tiddler [[include_tiddlers/Block Universe.html#"Block Universe"]]>>
<<tiddler [[include_tiddlers/Blockcode.html#"Blockcode"]]>>
<<tiddler [[include_tiddlers/Blood Cleansing.html#"Blood Cleansing"]]>>
<<tiddler [[include_tiddlers/Blueberry.html#"Blueberry"]]>>
<<tiddler [[include_tiddlers/Body Mass Index.html#"Body Mass Index"]]>>
<<tiddler [[include_tiddlers/Body Size and Longevity.html#"Body Size and Longevity"]]>>
<<tiddler [[include_tiddlers/Body Temperature.html#"Body Temperature"]]>>
<<tiddler [[include_tiddlers/Body Temperature and Lifespan.html#"Body Temperature and Lifespan"]]>>
<<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/Borrelia.html#"Borrelia"]]>>
<<tiddler [[include_tiddlers/Borrelia Complement Evasion.html#"Borrelia Complement Evasion"]]>>
<<tiddler [[include_tiddlers/Borrelia Enzootic Cycle.html#"Borrelia Enzootic Cycle"]]>>
<<tiddler [[include_tiddlers/Borrelia Garinii.html#"Borrelia Garinii"]]>>
<<tiddler [[include_tiddlers/Borrelia Genetics.html#"Borrelia Genetics"]]>>
<<tiddler [[include_tiddlers/Borrelia Immune Escape.html#"Borrelia Immune Escape"]]>>
<<tiddler [[include_tiddlers/Borrelia Persister.html#"Borrelia Persister"]]>>
<<tiddler [[include_tiddlers/Borrelia Pleomorphic Forms.html#"Borrelia Pleomorphic Forms"]]>>
<<tiddler [[include_tiddlers/Borrelia Round Body.html#"Borrelia Round Body"]]>>
<<tiddler [[include_tiddlers/Bortle Scale.html#"Bortle Scale"]]>>
<<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/Boson Star.html#"Boson Star"]]>>
<<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/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/Brandt's Bat.html#"Brandt's Bat"]]>>
<<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"]]>>
|Author|Eric Shulman|
|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).
see [[BreadcrumbsPluginInfo]]
<<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

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
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,
		chkBreadcrumbsHideHomeLink:	false,
		chkBreadcrumbsSave:		false,
		txtBreadcrumbsHomeSeparator:	'|',
		txtBreadcrumbsCrumbSeparator:	''
	init: function() {
	handler: function(place,macroName,params,wikifier,paramString,tiddler) {
		var area=createTiddlyElement(place,"span",null,"breadCrumbs",null);
	add: function (title) {
		var thisCrumb = title;
		var ind = this.crumbs.indexOf(thisCrumb);
		if(ind === -1)
		else if (config.options.chkReorderBreadcrumbs)
			this.crumbs.push(this.crumbs.splice(ind,1)[0]); // reorder crumbs
			this.crumbs=this.crumbs.slice(0,ind+1); // trim crumbs
		// if (config.options.chkBreadcrumbsLimitOpenTiddlers)
		//	this.limitOpenTiddlers();
		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); "none";
			var targetArea= document.getElementById("tiddlerDisplay");
		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";
	render: function(here) {
		var co=config.options; var out=""
		if (!co.chkBreadcrumbsHideHomeLink) {
			createTiddlyButton(here,"Home",null,this.home,"tiddlyLink tiddlyLinkExisting");
		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]))
		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();
	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:{};
		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)) {
					else story.closeTiddler(this.crumbs[c]);
// // 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;
			return config.commands.previousTiddler.handler(ev,this)
// update crumbs when a tiddler is displayed
if (Story.prototype.breadCrumbs_coreDisplayTiddler==undefined)
Story.prototype.displayTiddler = function(srcElement,tiddler) {
	var title=(tiddler instanceof Tiddler)?tiddler.title:tiddler;
	if (!startingUp || config.options.chkShowStartupBreadcrumbs)

// update crumbs when a tiddler is deleted
if (TiddlyWiki.prototype.breadCrumbs_coreRemoveTiddler==undefined)
TiddlyWiki.prototype.removeTiddler= function() {
<<tiddler [[include_tiddlers/Breakthrough Initiatives.html#"Breakthrough Initiatives"]]>>
<<tiddler [[include_tiddlers/Breakthrough Starshot Initiative.html#"Breakthrough Starshot Initiative"]]>>
<<tiddler [[include_tiddlers/Bremsstrahlung.html#"Bremsstrahlung"]]>>
<<tiddler [[include_tiddlers/Brian Johnson.html#"Brian Johnson"]]>>
<<tiddler [[include_tiddlers/Broadman Area.html#"Broadman Area"]]>>
<<tiddler [[include_tiddlers/Broca's Area.html#"Broca's Area"]]>>
<<tiddler [[include_tiddlers/Bronstein Cube.html#"Bronstein Cube"]]>>
<<tiddler [[include_tiddlers/Brosche-Wesson-Tassie Relation.html#"Brosche-Wesson-Tassie Relation"]]>>
<<tiddler [[include_tiddlers/Browser IDE.html#"Browser IDE"]]>>
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/Buffet Indicator.html#"Buffet Indicator"]]>>
<<tiddler [[include_tiddlers/Buhner Protocol.html#"Buhner Protocol"]]>>
<<tiddler [[include_tiddlers/Built Tool.html#"Built Tool"]]>>
<<tiddler [[include_tiddlers/Bullet Cluster.html#"Bullet Cluster"]]>>
<<tiddler [[include_tiddlers/Bunch-Davies Vacuum.html#"Bunch-Davies Vacuum"]]>>
A ''Burgers Vector'' characterizes a [[dislocation|Dislocation]].

{{center{[img(407px+, )[images/BurgersVector.jpg]]}}}
* [[Dislocations - Institut für Angewandte Physik der Technischen Universität Wien|]]
<<tiddler [[include_tiddlers/C (Programming Language).html#"C (Programming Language)"]]>>
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]].

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.

See [1].

* [[Jordan C*-Algebras (1977) - J. D. Maitland Wright|]] [[local|papers/JordanCStar.pdf]] [[pct. 93|,5&hl=de]]
* [[State Spaces of C*-Algebras (1980) - E. M. Alfsen, H. Hanche-Olsen, F. W. Shultz|]] [[local|papers/144_8.pdf]] [[pct. 84|,5&hl=de]]
* [[Ring C*-Algebras (2009) - X. Li|]] [[local|papers/0905.4861v1.pdf]] [[pct. 8|]]
* [[[1] C*-algebras in Tensor Categories (2007) - P. Bouwknegt, K. Hannabuss, V. Mathai|]] [[local|papers/0702802v2.pdf]] [[pct. 5|]]
* [[Algebraic Quantum Mechanics (2009) - N. P. Landsman|]] [[local|papers/algebraicQM.pdf]] [[pct. 2|]]
* [[On the Definition of C*-algebra (2007) - J.M. Egger|]] [[local|papers/egger.pdf]] pct. 0

* [[WIKIPEDIA - C*-Algebra|*-algebra#Some_history:_B.2A-algebras_and_C.2A-algebras]]
* [[WIKIPEDIA - Von Neumann Algebra|]]
<<tiddler [[include_tiddlers/C-Theorem.html#"C-Theorem"]]>>
<<tiddler [[include_tiddlers/C-reactive Protein.html#"C-reactive Protein"]]>>
<<tiddler [[include_tiddlers/C-value.html#"C-value"]]>>
<<tiddler [[include_tiddlers/C60 Olive Oil.html#"C60 Olive Oil"]]>>
<<tiddler [[include_tiddlers/CCDciel.html#"CCDciel"]]>>
<<tiddler [[include_tiddlers/CMB Anisotropy.html#"CMB Anisotropy"]]>>
<<tiddler [[include_tiddlers/CMB Cold Spot.html#"CMB Cold Spot"]]>>
<<tiddler [[include_tiddlers/CMB Dipole.html#"CMB Dipole"]]>>
<<tiddler [[include_tiddlers/CMB Polarization.html#"CMB Polarization"]]>>
<<tiddler [[include_tiddlers/CML.html#"CML"]]>>
<<tiddler [[include_tiddlers/CP Violation.html#"CP Violation"]]>>
<<tiddler [[include_tiddlers/CPT Theorem.html#"CPT Theorem"]]>>
<<tiddler [[include_tiddlers/CPT Transformations.html#"CPT Transformations"]]>>
<<tiddler [[include_tiddlers/CR2.html#"CR2"]]>>
<<tiddler [[include_tiddlers/CRAM File Format.html#"CRAM File Format"]]>>
<<tiddler [[include_tiddlers/CRISPER.html#"CRISPER"]]>>
<<tiddler [[include_tiddlers/CRISPR-Cas3.html#"CRISPR-Cas3"]]>>
<<tiddler [[include_tiddlers/CW-Complex.html#"CW-Complex"]]>>
<<tiddler [[include_tiddlers/Caenorhabditis Elegans.html#"Caenorhabditis Elegans"]]>>
* [[Cages - A. E. Brouwer|]]
<<tiddler [[include_tiddlers/Calabi-Yau Manifold.html#"Calabi-Yau Manifold"]]>>
<<tiddler [[include_tiddlers/Calibrated Geometry.html#"Calibrated Geometry"]]>>
<<tiddler [[include_tiddlers/Calibration Frame.html#"Calibration Frame"]]>>
<<tiddler [[include_tiddlers/Callan-Symanzik Equation.html#"Callan-Symanzik Equation"]]>>
<<tiddler [[include_tiddlers/Callias Index Theorem.html#"Callias Index Theorem"]]>>
<<tiddler [[include_tiddlers/Calorie.html#"Calorie"]]>>
<<tiddler [[include_tiddlers/Calorie Restriction.html#"Calorie Restriction"]]>>
<<tiddler [[include_tiddlers/Calorie Restriction Mimetic.html#"Calorie Restriction Mimetic"]]>>
<<tiddler [[include_tiddlers/Calorie Restriction Primate Studies.html#"Calorie Restriction Primate Studies"]]>>
<<tiddler [[include_tiddlers/Calorie Restriction and Body Temperature.html#"Calorie Restriction and Body Temperature"]]>>
<<tiddler [[include_tiddlers/Calorie Restriction in Humans.html#"Calorie Restriction in Humans"]]>>
<<tiddler [[include_tiddlers/Calorie Restriction versus Fasting.html#"Calorie Restriction versus Fasting"]]>>
<<tiddler [[include_tiddlers/Cambrian Explosion.html#"Cambrian Explosion"]]>>
<<tiddler [[include_tiddlers/Camera Gain.html#"Camera Gain"]]>>
<<tiddler [[include_tiddlers/Cancer Early Detection Blood Test.html#"Cancer Early Detection Blood Test"]]>>
<<tiddler [[include_tiddlers/Cancer Epigenetics.html#"Cancer Epigenetics"]]>>  
<<tiddler [[include_tiddlers/Cancer Genetics.html#"Cancer Genetics"]]>>  
<<tiddler [[include_tiddlers/Cancer Screening.html#"Cancer Screening"]]>>
<<tiddler [[include_tiddlers/Cancer Statistics.html#"Cancer Statistics"]]>>
<<tiddler [[include_tiddlers/Cancer and Environment.html#"Cancer and Environment"]]>>
<<tiddler [[include_tiddlers/Canon EOS 600D.html#"Canon EOS 600D"]]>>
<<tiddler [[include_tiddlers/Canonical Commutation Relation.html#"Canonical Commutation Relation"]]>>
<<tiddler [[include_tiddlers/Canonical Coordinates.html#"Canonical Coordinates"]]>>
<<tiddler [[include_tiddlers/Canonical Expansion.html#"Canonical Expansion"]]>>
<<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.

* 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]]
<<tiddler [[include_tiddlers/Canonical Quantization.html#"Canonical Quantization"]]>>
<<tiddler [[include_tiddlers/Canonical Quantum Gravity.html#"Canonical Quantum Gravity"]]>>
<<tiddler [[include_tiddlers/Cantor Set.html#"Cantor Set"]]>>
<<tiddler [[include_tiddlers/Cantor's Diagonal Argument.html#"Cantor's Diagonal Argument"]]>>
<<tiddler [[include_tiddlers/Cap.html#"Cap"]]>>
<<tiddler [[include_tiddlers/Capric Acid.html#"Capric Acid"]]>>
<<tiddler [[include_tiddlers/Carbohydrate Metabolism.html#"Carbohydrate Metabolism"]]>>
<<tiddler [[include_tiddlers/Carbon Capture.html#"Carbon Capture"]]>>
<<tiddler [[include_tiddlers/Carbonyl Group.html#"Carbonyl Group"]]>>
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"]]>>
<<tiddler [[include_tiddlers/Carl Friedrich von Weizsäcker.html#"Carl Friedrich von Weizsäcker"]]>>
<<tiddler [[include_tiddlers/Carl Gustav Jung.html#"Carl Gustav Jung"]]>>
<<tiddler [[include_tiddlers/Carlos Castro.html#"Carlos Castro"]]>>
<<tiddler [[include_tiddlers/Carnosine.html#"Carnosine"]]>>
<<tiddler [[include_tiddlers/Carotenoid.html#"Carotenoid"]]>>
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]].

* [[Strings on Orbifolds: An Introduction - H.-P. Nilles|]] [[pct. 2|]]
<<tiddler [[include_tiddlers/Cartan Subalgebra.html#"Cartan Subalgebra"]]>>
The ''Cartan Tensor'' $C^\rho_{\mu\nu}$ (a.k.a. ''Modified Torsion Tensor'') is defined by
C^\rho_{\mu\nu} = T^\rho_{\mu\nu} + T_\mu \delta_\nu^\rho + T_\nu \delta_\mu^\rho
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.

* [[On a Completely Antisymmetric Cartan Torsion Tensor - L. Fabbri|]] [[pct. 2|]] - There exists yet another different version.
<<tiddler [[include_tiddlers/Cartan-Laptev Method.html#"Cartan-Laptev Method"]]>>
<<tiddler [[include_tiddlers/Cartes du Ciel.html#"Cartes du Ciel"]]>>
<<tiddler [[include_tiddlers/Cascading Style Sheets.html#"Cascading Style Sheets"]]>>
<<tiddler [[include_tiddlers/Casimir Effect.html#"Casimir Effect"]]>>
<<tiddler [[include_tiddlers/Casimir Effect Dark Energy.html#"Casimir Effect Dark Energy"]]>>
<<tiddler [[include_tiddlers/Casson Handle.html#"Casson Handle"]]>>
<<tiddler [[include_tiddlers/Cat State.html#"Cat State"]]>>
<<tiddler [[include_tiddlers/Catalan Number.html#"Catalan Number"]]>>
<<tiddler [[include_tiddlers/Catalase.html#"Catalase"]]>>
<<tiddler [[include_tiddlers/Catastrophe Theory.html#"Catastrophe Theory"]]>>
<<tiddler [[include_tiddlers/Catechin.html#"Catechin"]]>>
<<tiddler [[include_tiddlers/Category Theory.html#"Category Theory"]]>>
<<tiddler [[include_tiddlers/Cauchy-Riemann Operator.html#"Cauchy-Riemann Operator"]]>>
<<tiddler [[include_tiddlers/Causal Dynamical Triangulation.html#"Causal Dynamical Triangulation"]]>>
<<tiddler [[include_tiddlers/Causal Sets.html#"Causal Sets"]]>>
<<tiddler [[include_tiddlers/Causal Structure.html#"Causal Structure"]]>>
<<tiddler [[include_tiddlers/Causality.html#"Causality"]]>>
<<tiddler [[include_tiddlers/Causality Condition.html#"Causality Condition"]]>>
<<tiddler [[include_tiddlers/Cavity Optomechanics.html#"Cavity Optomechanics"]]>>
<<tiddler [[include_tiddlers/Cayley Ternary Algebra.html#"Cayley Ternary Algebra"]]>>
<<tiddler [[include_tiddlers/Cayley's Theorem.html#"Cayley's Theorem"]]>>
<<tiddler [[include_tiddlers/Cayley-Dickson Algebra.html#"Cayley-Dickson Algebra"]]>>
<<tiddler [[include_tiddlers/Cayley-Dickson Doubling.html#"Cayley-Dickson Doubling"]]>>
<<tiddler [[include_tiddlers/Cayley-Dickson Quantum Mechanics.html#"Cayley-Dickson Quantum Mechanics"]]>>
<<tiddler [[include_tiddlers/Celestial Coordinate System.html#"Celestial Coordinate System"]]>>
<<tiddler [[include_tiddlers/Celestron 8.html#"Celestron 8"]]>>
<<tiddler [[include_tiddlers/Cell Consciousness.html#"Cell Consciousness"]]>>
<<tiddler [[include_tiddlers/Cell Division.html#"Cell Division"]]>>
<<tiddler [[include_tiddlers/Cell Intelligence.html#"Cell Intelligence"]]>>
<<tiddler [[include_tiddlers/Cell Memory.html#"Cell Memory"]]>>
<<tiddler [[include_tiddlers/Cell Nucleus.html#"Cell Nucleus"]]>>
<<tiddler [[include_tiddlers/Cell Surface Receptor.html#"Cell Surface Receptor"]]>>
<<tiddler [[include_tiddlers/Cellular Automaton.html#"Cellular Automaton"]]>>
<<tiddler [[include_tiddlers/Cellular Noise.html#"Cellular Noise"]]>>
<<tiddler [[include_tiddlers/Cellular Senescence.html#"Cellular Senescence"]]>>
<<tiddler [[include_tiddlers/Cellular Stress Response.html#"Cellular Stress Response"]]>>
<<tiddler [[include_tiddlers/Center.html#"Center"]]>>
<<tiddler [[include_tiddlers/Central Charge.html#"Central Charge"]]>>
<<tiddler [[include_tiddlers/Central Dogma of Molecular Biology.html#"Central Dogma of Molecular Biology"]]>>
<<tiddler [[include_tiddlers/Central Nervous System.html#"Central Nervous System"]]>>
<<tiddler [[include_tiddlers/Centralizer.html#"Centralizer"]]>>
<<tiddler [[include_tiddlers/Centriole.html#"Centriole"]]>>
<<tiddler [[include_tiddlers/Cerebellum.html#"Cerebellum"]]>>
<<tiddler [[include_tiddlers/Chameleon Particle.html#"Chameleon Particle"]]>>
<<tiddler [[include_tiddlers/Chamomile.html#"Chamomile"]]>>
<<tiddler [[include_tiddlers/Chaos.html#"Chaos"]]>>
<<tiddler [[include_tiddlers/Chaotic Inflation.html#"Chaotic Inflation"]]>>
<<tiddler [[include_tiddlers/Chaotic Map.html#"Chaotic Map"]]>>
<<tiddler [[include_tiddlers/Chaotic Quantization.html#"Chaotic Quantization"]]>>
<<tiddler [[include_tiddlers/Chaperone.html#"Chaperone"]]>>
<<tiddler [[include_tiddlers/Character.html#"Character"]]>>
<<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
p_{\mb M} (\lambda) \equiv \det(\lambda \mb I_n- \mb M)
It is the solution to the [[eigenvalue problem|Eigenvalue Theory]] $ \mb M \mb A = \lambda \mb A$.
The equation
p_{\mb M} (\lambda)=0
is called the ''Characteristic Equation''.

$1 \times 1$''-matrix''
p_{m} (\lambda)= \lambda-m
$2 \times 2$''-matrix''
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)
<<tiddler [[include_tiddlers/Charles Musès.html#"Charles Musès"]]>>
<<tiddler [[include_tiddlers/Chart.js.html#"Chart.js"]]>>
<<tiddler [[include_tiddlers/ChatGPT.html#"ChatGPT"]]>>
<<tiddler [[include_tiddlers/Chatbot.html#"Chatbot"]]>>
<<tiddler [[include_tiddlers/Chebyshev Polynomial.html#"Chebyshev Polynomial"]]>>
<<tiddler [[include_tiddlers/Checkerboard Lattice.html#"Checkerboard Lattice"]]>>
<<tiddler [[include_tiddlers/Chelation Therapy.html#"Chelation Therapy"]]>>
<<tiddler [[include_tiddlers/Chemistry of Biological Aging.html#"Chemistry of Biological Aging"]]>>
<<tiddler [[include_tiddlers/Chemotherapy.html#"Chemotherapy"]]>>
The ''Chevalley Groups'' are the [[automorphism groups|Automorphism]] of the [[Lie algebras|Lie Algebra]] defined over the [[finite fields|Galois Field]].
<<tiddler [[include_tiddlers/Chicxulub Crater.html#"Chicxulub Crater"]]>>
<<tiddler [[include_tiddlers/Chiral Anomaly.html#"Chiral Anomaly"]]>>
<<tiddler [[include_tiddlers/Chiral Anomaly in Materials.html#"Chiral Anomaly in Materials"]]>>
<<tiddler [[include_tiddlers/Chiral Dirac Equation.html#"Chiral Dirac Equation"]]>>
<<tiddler [[include_tiddlers/Chiral Symmetry.html#"Chiral Symmetry"]]>>
<<tiddler [[include_tiddlers/Chiral Symmetry Breaking.html#"Chiral Symmetry Breaking"]]>>
<<tiddler [[include_tiddlers/Chirality.html#"Chirality"]]>>
<<tiddler [[include_tiddlers/Chocolate.html#"Chocolate"]]>>
<<tiddler [[include_tiddlers/Cholesterol.html#"Cholesterol"]]>>
<<tiddler [[include_tiddlers/Choline.html#"Choline"]]>>
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''
''Christoffel Symbols of second kind''
\Chr{\lambda}{\mu\nu} \equiv g^{\lambda\rho} \{\rho,\mu,\nu\}
A definition of the latter is required as the Christoffel connection is not tensorial which would imply the possibility of raising and lowering indices.
<<tiddler [[include_tiddlers/Chromatic Aberration.html#"Chromatic Aberration"]]>>
<<tiddler [[include_tiddlers/Chromatin Remodelling.html#"Chromatin Remodelling"]]>>
<<tiddler [[include_tiddlers/Chronic Fatigue Syndrome.html#"Chronic Fatigue Syndrome"]]>>
<<tiddler [[include_tiddlers/Chronology Protection Conjecture.html#"Chronology Protection Conjecture"]]>>
<<tiddler [[include_tiddlers/Church-Turing Hypothesis.html#"Church-Turing Hypothesis"]]>>
<<tiddler [[include_tiddlers/Citric Acid Cycle.html#"Citric Acid Cycle"]]>>
<<tiddler [[include_tiddlers/Cl(8).html#"Cl(8)"]]>>
<<tiddler [[include_tiddlers/Classical Electron Radius.html#"Classical Electron Radius"]]>>
<<tiddler [[include_tiddlers/Classicality.html#"Classicality"]]>>
<<tiddler [[include_tiddlers/Claustrum.html#"Claustrum"]]>>
<<tiddler [[include_tiddlers/Clientside Python.html#"Clientside Python"]]>>
<<tiddler [[include_tiddlers/Clifford Algebra.html#"Clifford Algebra"]]>>
<<tiddler [[include_tiddlers/Clifford Algebra and Electroweak Interactions.html#"Clifford Algebra and Electroweak Interactions"]]>>
<<tiddler [[include_tiddlers/Clifford Analysis.html#"Clifford Analysis"]]>>
<<tiddler [[include_tiddlers/Clifford Geometric Algebra.html#"Clifford Geometric Algebra"]]>>
<<tiddler [[include_tiddlers/Climate Change.html#"Climate Change"]]>>
<<tiddler [[include_tiddlers/Climate Change Criticism.html#"Climate Change Criticism"]]>>
<<tiddler [[include_tiddlers/Cloning Paradox.html#"Cloning Paradox"]]>>
<<tiddler [[include_tiddlers/Closed Manifold.html#"Closed Manifold"]]>>
<<tiddler [[include_tiddlers/Closed Timelike Curve.html#"Closed Timelike Curve"]]>>
<<tiddler [[include_tiddlers/Cloud Computing.html#"Cloud Computing"]]>>
<<tiddler [[include_tiddlers/Cloud IDE.html#"Cloud IDE"]]>>
<<tiddler [[include_tiddlers/Cloud Storage.html#"Cloud Storage"]]>>
<<tiddler [[include_tiddlers/Cloud-based Quantum Computing.html#"Cloud-based Quantum Computing"]]>>
<<tiddler [[include_tiddlers/Cluster Decomposition Theorem.html#"Cluster Decomposition Theorem"]]>>
<<tiddler [[include_tiddlers/Clustering Coefficient.html#"Clustering Coefficient"]]>>
<<tiddler [[include_tiddlers/Cnidaria.html#"Cnidaria"]]>>
<<tiddler [[include_tiddlers/Code Coverage.html#"Code Coverage"]]>>
<<tiddler [[include_tiddlers/Code Loop.html#"Code Loop"]]>>
<<tiddler [[include_tiddlers/CodePen.html#"CodePen"]]>>
<<tiddler [[include_tiddlers/Coding Theory.html#"Coding Theory"]]>>
<<tiddler [[include_tiddlers/Codon.html#"Codon"]]>>
<<tiddler [[include_tiddlers/Coenzyme Q10.html#"Coenzyme Q10"]]>>
<<tiddler [[include_tiddlers/Coffee.html#"Coffee"]]>>
<<tiddler [[include_tiddlers/Coherence Law.html#"Coherence Law"]]>>
<<tiddler [[include_tiddlers/Coherent State.html#"Coherent State"]]>>
<<tiddler [[include_tiddlers/Cold Dark Matter.html#"Cold Dark Matter"]]>>
<<tiddler [[include_tiddlers/Cold Fusion.html#"Cold Fusion"]]>>
''Hot'' and ''Cold Mirrors'' use the properties of dielectric coatings to selectively reflect and transmit light at prescribed wavelengths:
* Hot mirrors reflect near-infrared (NIR) and infrared (IR) and allow the transmission of ultraviolet (UV) and visible light. 
* Cold mirrors reflect UV and visible light and allow the transmission of NIR and IR.
Typically, off-the-shelf hot mirrors are designed for either 0° or 45° angle of incidence. These are the angles where the dielectric coatings are optimized for maximum NIR and IR reflection. Custom hot mirrors can be made for any angle of incidence between 0° and 45°.

* [[WIKIPEDIA - Cold Mirror|]]
* [[WIKIPEDIA - Hot Mirror|]]
* [[Hot Mirror vs Cold Mirror|]]
* [[Thorlabs - Longpass Dichroic Mirrors/Beamsplitters|]]
* [[Thorlabs - Hot and Cold Mirrors: UV Fused Silica Substrate|]]
* [[Edmund Optics - High Performance Cold Mirrors|]]
* [[Where to Buy Cold Mirrors|]]
<<tiddler [[include_tiddlers/Coleman-Mandula Theorem.html#"Coleman-Mandula Theorem"]]>>
<<tiddler [[include_tiddlers/Coleman-Weinberg Mechanism.html#"Coleman-Weinberg Mechanism"]]>>
<<tiddler [[include_tiddlers/Collagen.html#"Collagen"]]>>
<<tiddler [[include_tiddlers/Collapse of the Wavefunction.html#"Collapse of the Wavefunction"]]>>
<<tiddler [[include_tiddlers/Collective Unconscious.html#"Collective Unconscious"]]>>
A ''Collineation'' (or ''Projective Transformation'', ''Projectivity'') is a bijection between [[projective planes or spaces|Projective Space]], that maps straight lines to straight lines. Hence rectangles are mapped to rectangles (in particular squares are mapped to rectangles). Collineations do not preserve sizes or angles but do preserve coincidences and cross-ratios: two properties which are important in projective geometry. Collineations form a [[group|Group]].

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

<html><center> <br><iframe name="content" src="" width=65% height=450></iframe></center>

<iframe name="poll" src="polls.html" width=16% height=260></iframe>
<<tiddler [[include_tiddlers/Commission E Monographs.html#"Commission E Monographs"]]>>
<<tiddler [[include_tiddlers/Commodore C64.html#"Commodore C64"]]>>
<<tiddler [[include_tiddlers/Commutant.html#"Commutant"]]>>
<<tiddler [[include_tiddlers/Commutator.html#"Commutator"]]>>
<<tiddler [[include_tiddlers/Commutators of Degree 4.html#"Commutators of Degree 4"]]>>
<<tiddler [[include_tiddlers/Commutators of Degree 5.html#"Commutators of Degree 5"]]>>
<<tiddler [[include_tiddlers/Comoving Coordinates.html#"Comoving Coordinates"]]>>
<<tiddler [[include_tiddlers/Comoving Distance.html#"Comoving Distance"]]>>
<<tiddler [[include_tiddlers/Compact Space.html#"Compact Space"]]>>
<<tiddler [[include_tiddlers/Compartmental Models in Epidemiology.html#"Compartmental Models in Epidemiology"]]>>
<<tiddler [[include_tiddlers/Complementarity.html#"Complementarity"]]>>
<<tiddler [[include_tiddlers/Complemented Lattice.html#"Complemented Lattice"]]>>
<<tiddler [[include_tiddlers/Complex Action.html#"Complex Action"]]>>
<<tiddler [[include_tiddlers/Complex General Relativity.html#"Complex General Relativity"]]>>
<<tiddler [[include_tiddlers/Complex Hamiltonian.html#"Complex Hamiltonian"]]>>
<<tiddler [[include_tiddlers/Complex Number.html#"Complex Number"]]>>
<<tiddler [[include_tiddlers/Complex Spacetime.html#"Complex Spacetime"]]>>
<<tiddler [[include_tiddlers/Complex Time.html#"Complex Time"]]>>
<<tiddler [[include_tiddlers/Complexity.html#"Complexity"]]>>
<<tiddler [[include_tiddlers/Complexity Class.html#"Complexity Class"]]>>
A ''Composition Algebra'' (or [[normed algebra|Normed Algebra]]) is an algebra with a [[multiplicative norm|Norm]].

* 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.

* [[Composition Algebras and their Automorphisms - N. Jacobson|]]  [[local|papers/CompositionAlgebrasAndTheirAutomorphisms.pdf]] {{t100Cite{[[pct. 136|]]}}}

Google books:
* [[Octonions, Jordan Algebras, and Exceptional Groups - T. A. Springer, F. D. Veldkamp|,M1]] {{t100Cite{[[bct. 120|]]}}}
<<tiddler [[include_tiddlers/Composition of the Human Body.html#"Composition of the Human Body"]]>>
<<tiddler [[include_tiddlers/Compressibility.html#"Compressibility"]]>>
<<tiddler [[include_tiddlers/Compton Wavelength.html#"Compton Wavelength"]]>>
<<tiddler [[include_tiddlers/Computer Algebra System.html#"Computer Algebra System"]]>>
<<tiddler [[include_tiddlers/Comtrans Algebra.html#"Comtrans Algebra"]]>>
<<tiddler [[include_tiddlers/Condensate.html#"Condensate"]]>>
<<tiddler [[include_tiddlers/Condensed Matter.html#"Condensed Matter"]]>>
config.macros.breadcrumbs.homeSeparator=" ";
config.macros.breadcrumbs.crumbSeparator=" ";
<<tiddler [[include_tiddlers/Confinement.html#"Confinement"]]>>
<<tiddler [[include_tiddlers/Conformal Anomaly.html#"Conformal Anomaly"]]>>
<<tiddler [[include_tiddlers/Conformal Cosmology.html#"Conformal Cosmology"]]>>
<<tiddler [[include_tiddlers/Conformal Cyclic Cosmology.html#"Conformal Cyclic Cosmology"]]>>
<<tiddler [[include_tiddlers/Conformal Field Theory.html#"Conformal Field Theory"]]>>
<<tiddler [[include_tiddlers/Conformal Gravity.html#"Conformal Gravity"]]>>
<<tiddler [[include_tiddlers/Conformal Gravity and Missing Mass.html#"Conformal Gravity and Missing Mass"]]>>
<<tiddler [[include_tiddlers/Conformal Group.html#"Conformal Group"]]>>
<<tiddler [[include_tiddlers/Conformal Invariance Hypothesis.html#"Conformal Invariance Hypothesis"]]>>
<<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/Conformally Flat Manifold.html#"Conformally Flat Manifold"]]>>
<<tiddler [[include_tiddlers/Conic Sedenion.html#"Conic Sedenion"]]>>
* [[Website Kevin Carmody - Hypernumbers|]]
* [[Tony Smith's Homepage - Zero Divisor Algebras|]]

* [[Circular and Hyperbolic Quaternions, Octonions, and Sedenions - K. Carmody|]] [[local|papers/sedenions1.pdf]] [[pct. 18|]]
* [[Circular and Hyperbolic Quaternions, Octonions, and Sedenions - Further Results - K. Carmody|]] [[local|papers/sedenions2.pdf]] [[pct. 18|]]
<<tiddler [[include_tiddlers/Conjugation.html#"Conjugation"]]>>
<<tiddler [[include_tiddlers/Connectome.html#"Connectome"]]>>
<<tiddler [[include_tiddlers/Consciousness.html#"Consciousness"]]>>
<<tiddler [[include_tiddlers/Consciousness and Experiment.html#"Consciousness and Experiment"]]>>
<<tiddler [[include_tiddlers/Constructive Quantum Field Theory.html#"Constructive Quantum Field Theory"]]>>
<<tiddler [[include_tiddlers/Content Management System.html#"Content Management System"]]>>
<<tiddler [[include_tiddlers/Continued Fraction.html#"Continued Fraction"]]>>
<<tiddler [[include_tiddlers/Continuum Hypothesis.html#"Continuum Hypothesis"]]>>
<<tiddler [[include_tiddlers/Contorsion.html#"Contorsion"]]>>
<<tiddler [[include_tiddlers/Convergent Evolution.html#"Convergent Evolution"]]>>
<<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)) {
// The following settings are applied as defaults for all users:           //
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	// Make sure FAQ panels are *always* closed on startup (so load-on-demand can be used later on):
// 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") {
|Author|Eric Shulman|
|Description|view/add/delete browser-based cookies and "bake" cookies to CookieJar tiddler for 'sticky' settings|
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.
{{smallform small center{
<<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//
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
version.extensions.CookieManagerPlugin= {major: 2, minor: 4, revision: 1, date: new Date(2011,1,16)};
config.macros.cookieManager = {
	allowBrowserCookie: function(name) {
		return true;
	displayStatus: function(msg) {
		if (config.options.chkMonitorBrowserCookies && !startingUp)
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			+"!!![[Browser cookies:|CookieManagerPlugin]] "
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		if (config.options.chkCookieManagerAddToAdvancedOptions===undefined)
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		--><input type="button" style="width:50%;margin:0;" value="bake all options" \
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		+"!!![[Baked cookies:|CookieManagerPlugin]]\n"
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		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+='if (config.options.txtUserName=="'+who+'") {\n';
		for (i=0; i<opts.length; i++) t+=config.macros.cookieManager.format(opts[i])+"\n";
		var msg=opts.length+(all?' options':' cookies')+' have been saved in ''.  ';
		msg+='Please review all stored settings.';
		return false;
// Hijack saveOptionCookie() to add cookie blocking and monitoring messages
	var cmc=config.macros.cookieManager; // abbrev
	if (force || ((config.options.chkAllowBrowserCookies || name=="chkAllowBrowserCookies")
		&& !cmc.blockedCookies.contains(name) && cmc.allowBrowserCookie(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
	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.displayStatus("deleted "+name);
	} else cmc.displayStatus("deletion of '"+name+"' is blocked");
|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

// // 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"]]>>
|Version|M.M. - 831 removed due to collision with MathJax.|
|Author|Eric Shulman|
|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{
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 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")//
 <<tiddler ScrollBox with: CoreTweaks##1151test 12em>>}}}/%
<<tiddler About>>
<<showPopup tiddler:About label:"test popup" tip:About popupClass:sticky>>
window.findScrollOffsetX=function(obj) {
	var x=0;
	while(obj) {
		if (obj.scrollLeft && obj.nodeName!='HTML')
	return -x;

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

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) +');
// // }}}}}}// // {{block{
!!!1147 tiddler macro with params does not refresh
// // {{groupbox small{
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) {
//			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
// 	} // 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;
	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 {
// // }}}}}}// // {{block{
!!!1134 allow leading whitespace in section headings / TBD handle shadow tiddler sections
// // {{groupbox small{
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;
		text = this.getShadowTiddlerText(title);
		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
			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{
/*** - 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 (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{
/*** - 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()
	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{
/*** - CLOSED:FIXED - TW2.4.3 -
When invoking a macro, this tweak makes the current containing tiddler object and DOM rendering location available as global variables (window.tiddler and, 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);
// // }}}}}}
// // fixed in ~TW2.4.2:
// // {{block{
!!!823 apply option values via paramifiers (e.g. #chk...and #txt...)
// // {{groupbox small{
/*** - CLOSED:FIXED - TW2.4.2
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)
	for(var t=1; t<params.length; t++) {
		var p = config.paramifiers[params[t].name];
		if(p && p[handler] instanceof Function)
		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)
// // }}}}}}
// // closed: won't fix //(leave as core tweaks)//
// // {{block{
!!!637 TiddlyLink tooltip - custom formatting
// // {{groupbox small{
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';


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{
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);
config.macros.permaview.setHREF = function(event){
	var links = [];
	story.forEachTiddler(function(title,element) {
	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{
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.
	// create the core button, then add the HREF (to internal links only)
	var link=window.coreTweaks_createTiddlyLink.apply(this,arguments);
	if (!isStatic)
	return link;
// // }}}}}}
// // open tickets:
// // {{block{
!!!608/609/610 toolbars - toggles, separators and transclusion
// // {{groupbox small{
/*** - OPEN (more/less toggle) - OPEN (separators) - 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 |
	moreLabel: 'more\u25BC',
	morePrompt: 'Show additional commands',
	lessLabel: '\u25C4less',
	lessPrompt: 'Hide additional commands',
	separator: '|'
config.macros.toolbar.onClickMore = function(ev) {
	var e = this.nextSibling; = 'inline'; // show menu = 'none'; // hide button
	return false;
config.macros.toolbar.onClickLess = function(ev) {
	var e = this.parentNode;
	var m = e.previousSibling; = 'none'; // hide menu = '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)
			case '*':  // ELS - LINEBREAK (added)
			case '<': // ELS - LESS COMMAND (added)
				var btn = createTiddlyButton(place,
			case '>':
				var btn = createTiddlyButton(place,
				var e = createTiddlyElement(place,'span',null,'moreCommand'); = 'none';
				place = e;
				var theClass = '';
				switch(c.substr(0,1)) {
					case '+':
						theClass = 'defaultCommand';
						c = c.substr(1);
					case '-':
						theClass = 'cancelCommand';
						c = c.substr(1);
				if(c in config.commands)

				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
						// trim PRE format wrapper if any
						// render content into toolbar
				} // ELS - end WIKIFY CONTENT
// // }}}}}}// // {{block{
!!!529 IE fixup - case-sensitive element lookup of tiddler elements
// // {{groupbox small{
/*** - 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.getElementById=function(id) {
	var e=document.coreTweaks_coreGetElementById(id);
	if (!e || !e.parentNode ||!='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{
/*** - 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;
// // }}}}}}// // {{block{
!!!831 backslash-quoting for embedding newlines in 'line-mode' formats ---- REMOVED due to collision with MathJax
// // {{groupbox small{
/*** - 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{
/*** - 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 - 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{
/*** - 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 {'UniversalXPConnect');
		var nsIFilePicker = window.Components.interfaces.nsIFilePicker;
		var picker = Components.classes[';1'].createInstance(nsIFilePicker);
		picker.init(window, msg, mustExist?nsIFilePicker.modeOpen:nsIFilePicker.modeSave);
		var thispath = Components.classes[';1'].createInstance(Components.interfaces.nsILocalFile);
		if (!=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;
		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{
/*** - 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');
	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.title = prompt;
		if(config.options[cookie] == label)
			validTab = true;
		config.options[cookie] = params[1];
// // }}}}}}// // {{block{
!!!628 hide 'no such macro' errors
// // {{groupbox small{
/*** - OPEN
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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]].

* [[The Genus of the Coxeter-Todd Lattice - R. Scharlau, B. B. Venkov|]] [[pct. 4|]]
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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

\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

* [[WIKIPEDIA - D'Alembert Operator|'Alembert_operator]]
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* [[Mechanism for Vanishing Zero Point Energy (2003) - R. D. Klauber|]] [[local|papers/0309679v3.pdf]] [[pct. 1|,5&hl=de]]

* [[Field Theory I - Lecture 6 (2011) - K. Zarembo|]]
* [[Lecture Notes and Videos (2011) - P. Siegel|]]
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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!)!$.
$10!! = 10\cdot 8\cdot 6\cdot 4\cdot 2 = 3.840 $
$7!! = 7 \cdot 5 \cdot 3 \cdot 1 = 105 $

* $(2n)!! = 2^n n!$
* $ (2n+1)!! = \frac{(2n+1)!}{2^n n!}$
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''$E_7$'' 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.

* [[The Chevalley group G2(2) of order 12096 and the octonionic root system of E7 - M. Koca, R. Koc, N. O. Koca|]]
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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.

* [[The Bianchi Identity of Differential Geometry M. W. Evans, H. Eckardt|]] pct.0

* [[Evans on Torsion|]]
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* [[Special-Relativistic Resolution of Ehrenfest's Paradox: Comments on Some Recent Statements by T. E. Phipps, Jr. -O. Gron|]]
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<<tiddler [[include_tiddlers/Einstein Field Equations.html#"Einstein Field Equations"]]>>
<<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 and Jordan Frame.html#"Einstein and Jordan Frame"]]>>
<<tiddler [[include_tiddlers/Einstein's Constant.html#"Einstein's Constant"]]>>
<<tiddler [[include_tiddlers/Einstein-Cartan Theory.html#"Einstein-Cartan Theory"]]>>
<<tiddler [[include_tiddlers/Einstein-Hilbert Action.html#"Einstein-Hilbert Action"]]>>
<<tiddler [[include_tiddlers/Einstein-Klein Gordon Equations.html#"Einstein-Klein Gordon Equations"]]>>
<<tiddler [[include_tiddlers/Einstein-Podolsky-Rosen Paradox.html#"Einstein-Podolsky-Rosen Paradox"]]>>
<<tiddler [[include_tiddlers/Einstein-de Sitter Universe.html#"Einstein-de Sitter Universe"]]>>
{{center{[img(513px+, )[images/CyclicModel.jpg]]}}}

* [[The Ekpyrotic Universe: Colliding Branes and the Origin of the Hot Big Bang (2001) - J. Khoury, B. A. Ovrut, P. J. Steinhardt, N. Turok|]] [[local|papers/0103239v3.pdf]] {{t500Cite{[[pct. 961|,5&hl=de]]}}}
* [[From Big Crunch to Big Bang (2002) - J. Khoury., B. A. Ovrut, N. Seiberg, P. J. Steinhardt, N. Turok|]] [[local|papers/0108187v4.pdf]] {{t500Cite{[[pct. 473|,5&hl=de]]}}}

* [[WIKIPEDIA - Ekpyrotic Universe|]]

* [[Inflationary Cosmology on Trial (2011) - P. J. Steinhardt|]]
* [[Interview with Neil Turok (2008)|]]
* [[Authors@Google (2007) - Paul Steinhardt|]]
<<tiddler [[include_tiddlers/Elasticity.html#"Elasticity"]]>>
<<tiddler [[include_tiddlers/Electron.html#"Electron"]]>>
<<tiddler [[include_tiddlers/Electron Anomalous Magnetic Moment.html#"Electron Anomalous Magnetic Moment"]]>>
<<tiddler [[include_tiddlers/Electron Magnetic Moment.html#"Electron Magnetic Moment"]]>>
<<tiddler [[include_tiddlers/Electron Volt.html#"Electron Volt"]]>>
<<tiddler [[include_tiddlers/Electronically Assisted Astronomy.html#"Electronically Assisted Astronomy"]]>>
<<tiddler [[include_tiddlers/Electrosmog.html#"Electrosmog"]]>>
<<tiddler [[include_tiddlers/Electroweak Interactions.html#"Electroweak Interactions"]]>>
<<tiddler [[include_tiddlers/Elliptic Function.html#"Elliptic Function"]]>>
<<tiddler [[include_tiddlers/Embedding.html#"Embedding"]]>>
<<tiddler [[include_tiddlers/Emergence.html#"Emergence"]]>>
<<tiddler [[include_tiddlers/Emergent Gravity.html#"Emergent Gravity"]]>>
<<tiddler [[include_tiddlers/Emergent Quantum Mechanics.html#"Emergent Quantum Mechanics"]]>>
<<tiddler [[include_tiddlers/Emergent Relativity.html#"Emergent Relativity"]]>>
<<tiddler [[include_tiddlers/Emergent Spacetime.html#"Emergent Spacetime"]]>>
<<tiddler [[include_tiddlers/Encephalitis.html#"Encephalitis"]]>>
<<tiddler [[include_tiddlers/End of Science.html#"End of Science"]]>>
<<tiddler [[include_tiddlers/Endoplasmatic Reticulum.html#"Endoplasmatic Reticulum"]]>>
<<tiddler [[include_tiddlers/Endothelial Cell.html#"Endothelial Cell"]]>>
<<tiddler [[include_tiddlers/Energy Condition.html#"Energy Condition"]]>>
<<tiddler [[include_tiddlers/Energy Condition Violation.html#"Energy Condition Violation"]]>>
<<tiddler [[include_tiddlers/Energy Rate Density.html#"Energy Rate Density"]]>>
<<tiddler [[include_tiddlers/Energy of the Universe.html#"Energy of the Universe"]]>>
<<tiddler [[include_tiddlers/Enormous Theorem.html#"Enormous Theorem"]]>>
<<tiddler [[include_tiddlers/Enteric Nervous System.html#"Enteric Nervous System"]]>>
<<tiddler [[include_tiddlers/Entire Function.html#"Entire Function"]]>>
<<tiddler [[include_tiddlers/Entropic Force.html#"Entropic Force"]]>>
<<tiddler [[include_tiddlers/Entropic Gravity.html#"Entropic Gravity"]]>>
<<tiddler [[include_tiddlers/Entropy.html#"Entropy"]]>>
<<tiddler [[include_tiddlers/Entropy Production of a Living Cell.html#"Entropy Production of a Living Cell"]]>>
<<tiddler [[include_tiddlers/Entropy of the Universe.html#"Entropy of the Universe"]]>>
<<tiddler [[include_tiddlers/Environment Induced Superselection.html#"Environment Induced Superselection"]]>>
<<tiddler [[include_tiddlers/Epidemic.html#"Epidemic"]]>>
<<tiddler [[include_tiddlers/Epigallocatechin Gallate.html#"Epigallocatechin Gallate"]]>>
<<tiddler [[include_tiddlers/Epigenetic Clock.html#"Epigenetic Clock"]]>>
<<tiddler [[include_tiddlers/Epigenetic Drift.html#"Epigenetic Drift"]]>>
<<tiddler [[include_tiddlers/Epigenetic Reprogramming.html#"Epigenetic Reprogramming"]]>>
<<tiddler [[include_tiddlers/Epigenetics and Aging.html#"Epigenetics and Aging"]]>>
<<tiddler [[include_tiddlers/Epigenome.html#"Epigenome"]]>>
<<tiddler [[include_tiddlers/Epilobium.html#"Epilobium"]]>>
<<tiddler [[include_tiddlers/Epimutation.html#"Epimutation"]]>>
<<tiddler [[include_tiddlers/Epiphenomenalism.html#"Epiphenomenalism"]]>>
<<tiddler [[include_tiddlers/Epithelium.html#"Epithelium"]]>>
<<tiddler [[include_tiddlers/Epitope.html#"Epitope"]]>>
<<tiddler [[include_tiddlers/Epstein-Barr Virus.html#"Epstein-Barr Virus"]]>>
''Equations of Motion'' are [[differential equations|Differential Equation]] which remain valid if transformed differentially to new coordinates, even if the transformation is not integrable in the Schwarz sense.
<<tiddler [[include_tiddlers/Equatorial Coordinate System.html#"Equatorial Coordinate System"]]>>
<<tiddler [[include_tiddlers/Equipartition Theorem.html#"Equipartition Theorem"]]>>
<<tiddler [[include_tiddlers/Equivalence Principle and Quantum Mechanics.html#"Equivalence Principle and Quantum Mechanics"]]>>
<<tiddler [[include_tiddlers/Equivalence Principles.html#"Equivalence Principles"]]>>
<<tiddler [[include_tiddlers/Equivalence of Heisenberg- and Spin Algebra.html#"Equivalence of Heisenberg- and Spin Algebra"]]>>
<<tiddler [[include_tiddlers/Equivalent Dose.html#"Equivalent Dose"]]>>
<<tiddler [[include_tiddlers/Error Catastrophe Theory of Aging.html#"Error Catastrophe Theory of Aging"]]>>
<<tiddler [[include_tiddlers/Error Threshold.html#"Error Threshold"]]>>
<<tiddler [[include_tiddlers/Erwin Schrödinger.html#"Erwin Schrödinger"]]>>
<<tiddler [[include_tiddlers/Essential Amino Acid.html#"Essential Amino Acid"]]>>
<<tiddler [[include_tiddlers/Essential Fatty Acid.html#"Essential Fatty Acid"]]>>
<<tiddler [[include_tiddlers/Eternal Inflation.html#"Eternal Inflation"]]>>
<<tiddler [[include_tiddlers/Eternal Nothingness.html#"Eternal Nothingness"]]>>
<<tiddler [[include_tiddlers/Euchromatin.html#"Euchromatin"]]>>
<<tiddler [[include_tiddlers/Euclidean Path Integral.html#"Euclidean Path Integral"]]>> 
<<tiddler [[include_tiddlers/Eukaryote.html#"Eukaryote"]]>>
<<tiddler [[include_tiddlers/Euler Characteristic.html#"Euler Characteristic"]]>>
<<tiddler [[include_tiddlers/Euler-Lagrange Equations.html#"Euler-Lagrange Equations"]]>>
<<tiddler [[include_tiddlers/Eutely.html#"Eutely"]]>>
<<tiddler [[include_tiddlers/Even Spectral Triple.html#"Even Spectral Triple"]]>>
<<tiddler [[include_tiddlers/Event Horizon.html#"Event Horizon"]]>>
<<tiddler [[include_tiddlers/Event Horizon Telescope.html#"Event Horizon Telescope"]]>>
<<tiddler [[include_tiddlers/Evolutionary Computing.html#"Evolutionary Computing"]]>>
<<tiddler [[include_tiddlers/Evolutionary Theory of Aging.html#"Evolutionary Theory of Aging"]]>>
<<tiddler [[include_tiddlers/Exact Renormalization Group Equation.html#"Exact Renormalization Group Equation"]]>>
<<tiddler [[include_tiddlers/Exact Sequence.html#"Exact Sequence"]]>>
<<tiddler [[include_tiddlers/Exceptional Lie Group.html#"Exceptional Lie Group"]]>>
<<tiddler [[include_tiddlers/Excision Repair.html#"Excision Repair"]]>>
// // Excludes any tiddlers from timeline that have been tagged with ''excludeTimeline''
config.macros.timeline.handler = function(place,macroName,params)
	var field = params[0] ? params[0] : "modified";
	var tiddlers = store.reverseLookup("tags","excludeLists",false,field);
	var lastDay = "";
	var last = params[1] ? tiddlers.length-Math.min(tiddlers.length,parseInt(params[1])) : 0;
	for(var t=tiddlers.length-1; t>=last; t--)
		var tiddler = tiddlers[t];
		var theDay = tiddler[field].convertToLocalYYYYMMDDHHMM().substr(0,8);

                if(theDay > 20130101)
		if(theDay != lastDay)
			var theDateList = document.createElement("ul");
			lastDay = theDay;
		var theDateListItem = createTiddlyElement(theDateList,"li",null,"listLink");
<<tiddler [[include_tiddlers/Exercise and Autophagy.html#"Exercise and Autophagy"]]>>
<<tiddler [[include_tiddlers/Exocytosis.html#"Exocytosis"]]>>
<<tiddler [[include_tiddlers/Exon.html#"Exon"]]>>
<<tiddler [[include_tiddlers/Exoplanet.html#"Exoplanet"]]>>
<<tiddler [[include_tiddlers/Exoplanet Atmosphere.html#"Exoplanet Atmosphere"]]>>
<<tiddler [[include_tiddlers/Exosome.html#"Exosome"]]>>
<<tiddler [[include_tiddlers/Exotic 4-Geometry and Cosmology.html#"Exotic 4-Geometry and Cosmology"]]>>
<<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"]]>>
<<tiddler [[include_tiddlers/Exotic Differential Structure.html#"Exotic Differential Structure"]]>>
<<tiddler [[include_tiddlers/Exotic Geometry.html#"Exotic Geometry"]]>>
<<tiddler [[include_tiddlers/Exotic Pair.html#"Exotic Pair"]]>>
<<tiddler [[include_tiddlers/Exotic R4.html#"Exotic R4"]]>>
<<tiddler [[include_tiddlers/Exotic Sphere.html#"Exotic Sphere"]]>>
<<tiddler [[include_tiddlers/Experimental Quantum Gravity.html#"Experimental Quantum Gravity"]]>>
<<tiddler [[include_tiddlers/Explicit Symmetry Breaking.html#"Explicit Symmetry Breaking"]]>>
* [[The Exponential Map on the Cayley-Dickson Algebras - G. Moreno|]] [[pct. 1|]]
<<tiddler [[include_tiddlers/Exponential Integral.html#"Exponential Integral"]]>>
<<tiddler [[include_tiddlers/Exterior Derivative.html#"Exterior Derivative"]]>>
<<tiddler [[include_tiddlers/External Field Effect.html#"External Field Effect"]]>>
|Author|Eric Shulman|
|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.
>see [[ExternalTiddlersPluginInfo]]
<<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]]''
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
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"];

	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
		else // TW document with "permaview-like" suffix... copy tiddler(s) from remote store
			for (var t=0;t<tids.length;t++)
	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)
		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)
		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
			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
		// 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(; if (!wrapper) return; 
				wikify(substitute(extract(text,params.tids),params.args),wrapper); // ASYNC RENDER
			var callbackparams={ url:src,, 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)
				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/Extracellular Bacterium.html#"Extracellular Bacterium"]]>>
<<tiddler [[include_tiddlers/Extracellular Matrix.html#"Extracellular Matrix"]]>>
<<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"]]>>
* [[Extremal Lattices - R. Scharlau, R. Schulze-Pillot|]] [[pct. 30|]]
<<tiddler [[include_tiddlers/Extremely Large Telescope.html#"Extremely Large Telescope"]]>>
<<tiddler [[include_tiddlers/Extremophile.html#"Extremophile"]]>>
<<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/FOXO.html#"FOXO"]]>>
<<tiddler [[include_tiddlers/FOXO3.html#"FOXO3"]]>>
<<tiddler [[include_tiddlers/FWHM.html#"FWHM"]]>>
<<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/Faithful Representation.html#"Faithful Representation"]]>>
<<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/Fasting.html#"Fasting"]]>>
<<tiddler [[include_tiddlers/Fasting and Cancer.html#"Fasting and Cancer"]]>>
<<tiddler [[include_tiddlers/Fate of the Universe.html#"Fate of the Universe"]]>>
<<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|]] [[local|papers/ijms-10-02084.pdf]] [[pct. 1|]]

* [[Bose-Einstein Condensation in a Dilute Gas; the First 70 Years and some Recent Experiments - E. A. Cornell, E. Wieman|]] [[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/Fibroblast.html#"Fibroblast"]]>>
<<tiddler [[include_tiddlers/Field.html#"Field"]]>>
<<tiddler [[include_tiddlers/Field Rotation.html#"Field Rotation"]]>>
<<tiddler [[include_tiddlers/Field Rotator.html#"Field Rotator"]]>>
<<tiddler [[include_tiddlers/Field Strength Tensor.html#"Field Strength Tensor"]]>>
<<tiddler [[include_tiddlers/Field Theory.html#"Field Theory"]]>>
* [[Les Médailles Fields|]]
<<tiddler [[include_tiddlers/Fifth Force.html#"Fifth Force"]]>>
<<tiddler [[include_tiddlers/Finasteride.html#"Finasteride"]]>>
<<tiddler [[include_tiddlers/Fine Structure Constant.html#"Fine Structure Constant"]]>>
<<tiddler [[include_tiddlers/Fine Tuning.html#"Fine Tuning"]]>>
<<tiddler [[include_tiddlers/Fine Tuning of Inflation.html#"Fine Tuning of Inflation"]]>>
<<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"]]>>
<<tiddler [[include_tiddlers/Fish.html#"Fish"]]>>
<<tiddler [[include_tiddlers/Fixed Point.html#"Fixed Point"]]>>
<<tiddler [[include_tiddlers/Flagellin.html#"Flagellin"]]>>
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/Flat Frame.html#"Flat Frame"]]>>
<<tiddler [[include_tiddlers/Flatness Problem.html#"Flatness Problem"]]>>
<<tiddler [[include_tiddlers/Flavan.html#"Flavan"]]>>
<<tiddler [[include_tiddlers/Flavone.html#"Flavone"]]>>
<<tiddler [[include_tiddlers/Flavonoid.html#"Flavonoid"]]>>
<<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\}
* [[WIKIPEDIA - Floor Function|]]
<<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/Folate.html#"Folate"]]>>
<<tiddler [[include_tiddlers/Foliation.html#"Foliation"]]>>
|Created by|SaqImtiaz|
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.

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

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.

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

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.

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.

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



//configuration settings
config.fontSize.settings =
            defaultSize : 120,  // 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;
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");

config.macros.fontSize.handler = function (place,macroName,params,wikifier,paramString,tiddler)

               var sp = createTiddlyElement(place,"span",null,"fontResizer");
               if (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 ()
               setStylesheet(".tiddler .viewer {font-size:"+config.options.txtFontSize+"%;}\n","fontResizerStyles");

               if (config.options.txtFontSize < fontSettings.maxSize)
                  config.options.txtFontSize = (config.options.txtFontSize*1)+fontSettings.stepSize;


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



config.paramifiers.font =
               onstart: function(v)
                   config.options.txtFontSize = v;
<<tiddler [[include_tiddlers/Food Additive.html#"Food Additive"]]>>
<<tiddler [[include_tiddlers/Forcing.html#"Forcing"]]>>
<<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/Frailty Index.html#"Frailty Index"]]>>
<<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.html#"Free Radical"]]>>
<<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/Fried Parameter.html#"Fried Parameter"]]>>
<<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/Functionalism.html#"Functionalism"]]>>
<<tiddler [[include_tiddlers/Full Stack Web Developer.html#"Full Stack Web Developer"]]>>
<<tiddler [[include_tiddlers/Full Well Capacity.html#"Full Well Capacity"]]>>
<<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/Functionalism.html#"Functionalism"]]>>
<<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/Future of Computing.html#"Future of Computing"]]>>
<<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/GABA.html#"GABA"]]>>
<<tiddler [[include_tiddlers/GAP.html#"GAP"]]>>
<<tiddler [[include_tiddlers/GIMP.html#"GIMP"]]>>
<<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/GUT Inflation.html#"GUT Inflation"]]>>
<<tiddler [[include_tiddlers/Gaia Hypothesis.html#"Gaia Hypothesis"]]>>
<<tiddler [[include_tiddlers/Gain Function Research.html#"Gain Function Research"]]>>
<<tiddler [[include_tiddlers/Galaxy.html#"Galaxy"]]>>
<<tiddler [[include_tiddlers/Galaxy Filament.html#"Galaxy Filament"]]>>
<<tiddler [[include_tiddlers/Galaxy Group.html#"Galaxy Group"]]>>
<<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/Ganzfeld Experiment.html#"Ganzfeld Experiment"]]>>
<<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.

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

* [[WIKIPEDIA - Gaussian Binomial|]]
* [[WolframMathWorld - q-Binomial Coefficient|]]
<<tiddler [[include_tiddlers/Gaussian Curvature.html#"Gaussian Curvature"]]>>
<<tiddler [[include_tiddlers/Gaussian Integer.html#"Gaussian Integer"]]>>
<<tiddler [[include_tiddlers/Gaussian Integral.html#"Gaussian Integral"]]>>
<<tiddler [[include_tiddlers/Gelatinase.html#"Gelatinase"]]>>
<<tiddler [[include_tiddlers/Gelatine.html#"Gelatine"]]>>
<<tiddler [[include_tiddlers/Gelfand-Naimark Theorem.html#"Gelfand-Naimark Theorem"]]>>
* [[WIKIPEDIA - Geomerical Frustration|]]
* [[Geomerical Frustration - Physics Today 02/2006|]]
<<tiddler [[include_tiddlers/Gender and Lifespan.html#"Gender and Lifespan"]]>>
<<tiddler [[include_tiddlers/Gene Drive.html#"Gene Drive"]]>>
<<tiddler [[include_tiddlers/Gene Transcription.html#"Gene Transcription"]]>>
<<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"]]>>
<<tiddler [[include_tiddlers/Generative Adversarial Network.html#"Generative Adversarial Network"]]>>
!!!!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.

* $SO(3)$: 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)]]: $n = 120$
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.
\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}
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 Code.html#"Genetic Code"]]>>
<<tiddler [[include_tiddlers/Genetic Engineering.html#"Genetic Engineering"]]>>
<<tiddler [[include_tiddlers/Genome.html#"Genome"]]>>
<<tiddler [[include_tiddlers/Genome-wide Association Study.html#"Genome-wide Association Study"]]>>
<<tiddler [[include_tiddlers/Geodesic Equation.html#"Geodesic Equation"]]>>
<<tiddler [[include_tiddlers/Geodesic Loop.html#"Geodesic Loop"]]>>
<<tiddler [[include_tiddlers/Geologic Timescale.html#"Geologic Timescale"]]>>
<<tiddler [[include_tiddlers/Geomagnetic Reversal.html#"Geomagnetic Reversal"]]>>
<<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]]

* [[Unity of all Elementary-particle Forces (1974) - H. Georgi, S. L. Glashow|]] [[local|papers/georgi_prl_32_438_74.pdf]] {{t1000Cite{[[pct. 4394|,5&hl=de]]}}}

* [[Introduction to Group Theory for Physicists (2011) - M. von Steinkirch|]] [[local|documents/group.pdf]]

* [[WIKIPEDIA - Georgi-Glashow Model|]]
<<tiddler [[include_tiddlers/Gerard 't Hooft.html#"Gerard 't Hooft"]]>>
<<tiddler [[include_tiddlers/Germ Theory.html#"Germ Theory"]]>>
<<tiddler [[include_tiddlers/Germline Mutation.html#"Germline Mutation"]]>>
<<tiddler [[include_tiddlers/Germline Mutation Rate.html#"Germline Mutation Rate"]]>>
<<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/Giant Arc.html#"Giant Arc"]]>>
<<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]].

* [[Weight Polynomials of Self-dual Codes and the MacWilliams Identities (1970) - A. M. Gleason|]] [[local|papers/icm1970.3.0211.0216.ocr.pdf]] {{t100Cite{[[pct. 108|,5&hl=de]]}}}
* [[Gleason's Theorem on Self-Dual Codes and its Generalizations (2006) - N. J. A. Sloane|]] [[local|papers/0612535v1.pdf]] [[pct. 2|]]

* [[The Type of a Code (2008) - G. Nebe|]] [[local|presentations/nato1.pdf]]
<<tiddler [[include_tiddlers/Gleason's Theorem (Quantum Mechanics).html#"Gleason's Theorem (Quantum Mechanics)"]]>>
<<tiddler [[include_tiddlers/Glial Cell.html#"Glial Cell"]]>>
<<tiddler [[include_tiddlers/Global Anomaly.html#"Global Anomaly"]]>>
<<tiddler [[include_tiddlers/Global Consciousness Project.html#"Global Consciousness Project"]]>>
<<tiddler [[include_tiddlers/Global Symmetry.html#"Global Symmetry"]]>>
<<tiddler [[include_tiddlers/Globally Hyperbolic Manifold.html#"Globally Hyperbolic Manifold"]]>>
<<tiddler [[include_tiddlers/Glucosepane.html#"Glucosepane"]]>>
<<tiddler [[include_tiddlers/Glueball.html#"Glueball"]]>>
<<tiddler [[include_tiddlers/Gluon.html#"Gluon"]]>>
<<tiddler [[include_tiddlers/Glutamate.html#"Glutamate"]]>>
<<tiddler [[include_tiddlers/Glutathione.html#"Glutathione"]]>>
<<tiddler [[include_tiddlers/Glutathione Peroxidases.html#"Glutathione Peroxidases"]]>>
<<tiddler [[include_tiddlers/Glutathione Reductase.html#"Glutathione Reductase"]]>>
<<tiddler [[include_tiddlers/Glutathione-S-transferase.html#"Glutathione-S-transferase"]]>>
<<tiddler [[include_tiddlers/Glycated Hemoglobin.html#"Glycated Hemoglobin"]]>>
<<tiddler [[include_tiddlers/Glycation.html#"Glycation"]]>>
<<tiddler [[include_tiddlers/Glycerol.html#"Glycerol"]]>>
<<tiddler [[include_tiddlers/Glycerophospholipid.html#"Glycerophospholipid"]]>>
<<tiddler [[include_tiddlers/Glycogen.html#"Glycogen"]]>>
<<tiddler [[include_tiddlers/Glycolysis.html#"Glycolysis"]]>>
''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|]] is available.

* [[Guplot Homepage|]]
* [[Gnuplot Manual (Version 4.6)|]] [[local|documents/gnuplot.pdf]]
* [[WIKIPEDIA - Gnuplot|]]
<<tiddler [[include_tiddlers/GoTo Dobsonian.html#"GoTo Dobsonian"]]>>
<<tiddler [[include_tiddlers/God and Physics.html#"God and Physics"]]>>
<<tiddler [[include_tiddlers/God and the Laws of Nature.html#"God and the Laws of Nature"]]>>
<<tiddler [[include_tiddlers/Godbillon-Vey Invariant.html#"Godbillon-Vey Invariant"]]>>
<<tiddler [[include_tiddlers/Golay Code.html#"Golay Code"]]>>
<<tiddler [[include_tiddlers/Goldberg-Sachs Theorem.html#"Goldberg-Sachs Theorem"]]>>
<<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"]]>>
|Author|Eric Shulman - ELS Design Studios|
|License| <br>and [[Creative Commons Attribution-ShareAlike 2.5 License|]]|
|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.
>see [[GotoPluginInfo]]
*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
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
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'...",
	function(place,macroName,params,wikifier,paramString,tiddler) {
		var quiet	=params.contains("quiet");
		var showlist	=params.contains("showlist");
		var search	=params.contains("search");
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	"<table style='width:100%;display:inline;padding:0;margin:0;border:0;'>\
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	function(t) {
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		try{event.keyCode=0;}catch(e){}; // IE5
		if (window.event) ev.returnValue=false; // IE6
		if (ev.preventDefault) ev.preventDefault(); // moz/opera/konqueror
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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.

* [[Graded Lie Algebras and q-commutative and r-associative Parameters - L. A. Wills-Toro, J. D. Vaelez, T. Craven|]] [[pct. 1|]]
<<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
* 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"]]>>
<<tiddler [[include_tiddlers/Grandmother Cell.html#"Grandmother Cell"]]>>
<<tiddler [[include_tiddlers/Granulocyte.html#"Granulocyte"]]>>
<<tiddler [[include_tiddlers/Granulocyte-macrophage Colony-stimulating Factor.html#"Granulocyte-macrophage Colony-stimulating Factor"]]>>
<<tiddler [[include_tiddlers/Grapefruit Seed Extract.html#"Grapefruit Seed Extract"]]>>
* [[WolframMathWorld - Distance-Transitive Graph|]]
<<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"]]>>
<<tiddler [[include_tiddlers/Gravitational Lensing.html#"Gravitational Lensing"]]>>
<<tiddler [[include_tiddlers/Gravitational Memory.html#"Gravitational Memory"]]>>
<<tiddler [[include_tiddlers/Gravitational Wave.html#"Gravitational Wave"]]>>
<<tiddler [[include_tiddlers/Gravitationally Induced State Reduction.html#"Gravitationally Induced State Reduction"]]>>
<<tiddler [[include_tiddlers/Gravitino.html#"Gravitino"]]>>
<<tiddler [[include_tiddlers/Gravitoelectromagnetism.html#"Gravitoelectromagnetism"]]>>
<<tiddler [[include_tiddlers/Graviton.html#"Graviton"]]>>
<<tiddler [[include_tiddlers/Graviton Mass.html#"Graviton Mass"]]>>
<<tiddler [[include_tiddlers/Graviton as Goldstone Boson.html#"Graviton as Goldstone Boson"]]>>
<<tiddler [[include_tiddlers/Graviweak Theory.html#"Graviweak Theory"]]>>
* [[Prefixes & Words Based On Greek Number Names|]]
<<tiddler [[include_tiddlers/Green's Function.html#"Green's Function"]]>>
<<tiddler [[include_tiddlers/Greenwood Frequency.html#"Greenwood Frequency"]]>>
<<tiddler [[include_tiddlers/Gregory Chaitin.html#"Gregory Chaitin"]]>>
<<tiddler [[include_tiddlers/Grey Hair.html#"Grey Hair"]]>>
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) -

* [[WIKIPEDIA - Grigori Perelman|]]

* [[SWR2 Wissen 31.03.2008: Die Perelman-Vermutung|]]
<<tiddler [[include_tiddlers/Gronwall Conjecture.html#"Gronwall Conjecture"]]>>
<<tiddler [[include_tiddlers/Groq.html#"Groq"]]>>
<<tiddler [[include_tiddlers/Group.html#"Group"]]>>
<<tiddler [[include_tiddlers/Group Algebra.html#"Group Algebra"]]>>
<<tiddler [[include_tiddlers/Group Determinant.html#"Group Determinant"]]>>
<<tiddler [[include_tiddlers/Group Ring.html#"Group Ring"]]>>
<<tiddler [[include_tiddlers/Groupoid.html#"Groupoid"]]>>
<<tiddler [[include_tiddlers/Guiding Software.html#"Guiding Software"]]>>
<<tiddler [[include_tiddlers/Guitar Tuning.html#"Guitar Tuning"]]>>
<<tiddler [[include_tiddlers/Gut-brain Axis.html#"Gut-brain Axis"]]>>
<<tiddler [[include_tiddlers/Gyrogroup.html#"Gyrogroup"]]>>
<<tiddler [[include_tiddlers/Gyromagnetic Ratio.html#"Gyromagnetic Ratio"]]>>
<<tiddler [[include_tiddlers/Gâteaux Derivative.html#"Gâteaux Derivative"]]>>
<<tiddler [[include_tiddlers/Gödel Cosmos.html#"Gödel Cosmos"]]>>
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).

* [[Nonassociative Real H*-algebras - M. Cabrera, J. Martínez, A. Rodríguez|]] [[pct. 7|]]
* [[Malcev H*-algebras - M. Cabrera, J. Martínez, A. Rodríguez|]] [[pct. 5|]]
* [[Moufang H*-algebras - J. A. C. Mira|]] pct. 0
* [[On Lie Derivations of 3-Graded Algebras A. J. C. Martín, C. M. González|]] pct. 0

* [[WIKIPEDIA - H*-Algebra|*-Algebra]]
<<tiddler [[include_tiddlers/H-Space.html#"H-Space"]]>>
<<tiddler [[include_tiddlers/HLA Typing.html#"HLA Typing"]]>>
<<tiddler [[include_tiddlers/HP Prime.html#"HP Prime"]]>>
<<tiddler [[include_tiddlers/HTML Obfuscator.html#"HTML Obfuscator"]]>>
<<tiddler [[include_tiddlers/HTML5.html#"HTML5"]]>>
<<tiddler [[include_tiddlers/Haag's Theorem.html#"Haag's Theorem"]]>>
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]].

* All Possible Generators of Supersymmetries of the S Matrix (1975) - R. Haag, J. T. AopuszaDski, M. Sohnius {{t500Cite{[[jct. 959|]]}}}
<<tiddler [[include_tiddlers/Habitable Planet.html#"Habitable Planet"]]>>
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|]]^^[[Help|Sage]]^^ examples
Ham = gap.HadamardCode(16)
N = gap.Elements(Ham)
Aut = Ham.AutomorphismGroup()

* [[Z4-linear Hadamard and Extended Perfect Codes -D. S.Krotov|]] [[pct. 51|]]
<<tiddler [[include_tiddlers/Hadamard Design.html#"Hadamard Design"]]>>
<<tiddler [[include_tiddlers/Hadamard Matrices of Order 24.html#"Hadamard Matrices of Order 24"]]>>
<<tiddler [[include_tiddlers/Hadamard Matrices of Orders 1 to 16.html#"Hadamard Matrices of Orders 1 to 16"]]>>
<<tiddler [[include_tiddlers/Hadamard Matrix.html#"Hadamard Matrix"]]>>
<<tiddler [[include_tiddlers/Hagedorn Temperature.html#"Hagedorn Temperature"]]>>
<<tiddler [[include_tiddlers/Half Flux Diameter.html#"Half Flux Diameter"]]>> 
<<tiddler [[include_tiddlers/Hallmarks of Aging.html#"Hallmarks of Aging"]]>>
<<tiddler [[include_tiddlers/Hallmarks of Cancer.html#"Hallmarks of Cancer"]]>>
<<tiddler [[include_tiddlers/Hallucinogen.html#"Hallucinogen"]]>>
<<tiddler [[include_tiddlers/Halting Problem.html#"Halting Problem"]]>>
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.
\mathcal H = \mathcal Q_8 \times \oplus_i\ \mathcal C_2 \times \mathcal A
!!!! Generalizations:
* See [[Hamilton Loop]]. 
<<tiddler [[include_tiddlers/Hamilton Loop.html#"Hamilton Loop"]]>>
<<tiddler [[include_tiddlers/Hamiltonian Mechanics.html#"Hamiltonian Mechanics"]]>>
<<tiddler [[include_tiddlers/Hamming Code.html#"Hamming Code"]]>>
<<tiddler [[include_tiddlers/Hamming Distance.html#"Hamming Distance"]]>>
The ''Hamming\-Weight'' (or ''Hamming\-Norm'') ''$w(x)$'' of a word of a [[code|Code]] $C$ is equal to the number of its letters not equal to "zero".

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

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

For an [[orthogonal code|Dual Code]] one has: $w(x) \in 2\mathbb Z$.
For a [[selfdual code|Dual Code]]: $w(x) \in 4\mathbb Z$. 
<<tiddler [[include_tiddlers/Handlebody Decomposition.html#"Handlebody Decomposition"]]>>
<<tiddler [[include_tiddlers/Hard Problem of Consciousness.html#"Hard Problem of Consciousness"]]>>
<<tiddler [[include_tiddlers/Hardy Space.html#"Hardy Space"]]>>
<<tiddler [[include_tiddlers/Harmonic Oscillator.html#"Harmonic Oscillator"]]>>
<<tiddler [[include_tiddlers/Hartle-Hawking Wavefunction.html#"Hartle-Hawking Wavefunction"]]>>
<<tiddler [[include_tiddlers/Hartman-Grobman Theorem.html#"Hartman-Grobman Theorem"]]>>
<<tiddler [[include_tiddlers/Hawking Radiation.html#"Hawking Radiation"]]>>
<<tiddler [[include_tiddlers/Hawking Radiation Laboratory Analogue.html#"Hawking Radiation Laboratory Analogue"]]>>
<<tiddler [[include_tiddlers/Hawking Temperature.html#"Hawking Temperature"]]>>
<<tiddler [[include_tiddlers/Hayflick Limit.html#"Hayflick Limit"]]>>
<<tiddler [[include_tiddlers/Head Transplant.html#"Head Transplant"]]>>
<<tiddler [[include_tiddlers/Heart Neuron.html#"Heart Neuron"]]>>
<<tiddler [[include_tiddlers/Heart Rate.html#"Heart Rate"]]>>
<<tiddler [[include_tiddlers/Heat Equation.html#"Heat Equation"]]>>
<<tiddler [[include_tiddlers/Heat Kernel.html#"Heat Kernel"]]>>
<<tiddler [[include_tiddlers/Heat Kernel Expansion.html#"Heat Kernel Expansion"]]>>
<<tiddler [[include_tiddlers/Heat Shock Factor.html#"Heat Shock Factor"]]>>
<<tiddler [[include_tiddlers/Heim Theory.html#"Heim Theory"]]>>
<<tiddler [[include_tiddlers/Heisenberg Algebra.html#"Heisenberg Algebra"]]>>
<<tiddler [[include_tiddlers/Heisenberg Cut.html#"Heisenberg Cut"]]>>
<<tiddler [[include_tiddlers/Heisenberg Picture.html#"Heisenberg Picture"]]>>
<<tiddler [[include_tiddlers/Helicobacter Pylori Natural Treatment.html#"Helicobacter Pylori Natural Treatment"]]>>
<<tiddler [[include_tiddlers/Helmholtz Conditions.html#"Helmholtz Conditions"]]>>
<<tiddler [[include_tiddlers/Heme.html#"Heme"]]>>
<<tiddler [[include_tiddlers/Hentzel-Peresi Identity.html#"Hentzel-Peresi Identity"]]>>
<<tiddler [[include_tiddlers/Heraclitus.html#"Heraclitus"]]>>
<<tiddler [[include_tiddlers/Herbal Tea.html#"Herbal Tea"]]>>
>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] -

* [[[1] An Interview with Michael Atiyah|]]
* [[WIKIPEDIA - Hermann Weyl|]]
* [[|]]
<<tiddler [[include_tiddlers/Hermitian Conjugate.html#"Hermitian Conjugate"]]>>
<<tiddler [[include_tiddlers/Hermitian Inner Product.html#"Hermitian Inner Product"]]>>
<<tiddler [[include_tiddlers/Hermitian Manifold.html#"Hermitian Manifold"]]>>
<<tiddler [[include_tiddlers/Hermitian Product.html#"Hermitian Product"]]>>
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 $\mb x$.

* [[The Curvature of a Hessian Metric - B. Totaro|]] [[pct. 7|]]
<<tiddler [[include_tiddlers/Heterochromatin.html#"Heterochromatin"]]>>
<<tiddler [[include_tiddlers/Heterochronic Parabiosis.html#"Heterochronic Parabiosis"]]>>
<<tiddler [[include_tiddlers/Heteroplasmy.html#"Heteroplasmy"]]>>
<<tiddler [[include_tiddlers/Heterosis.html#"Heterosis"]]>>
<<tiddler [[include_tiddlers/Heterotherm.html#"Heterotherm"]]>>
<<tiddler [[include_tiddlers/Heterotic String.html#"Heterotic String"]]>>
<<tiddler [[include_tiddlers/Heterotroph.html#"Heterotroph"]]>>
<<tiddler [[include_tiddlers/Heun Function.html#"Heun Function"]]>>
<<tiddler [[include_tiddlers/Hexagonal 3-Web.html#"Hexagonal 3-Web"]]>>
<<tiddler [[include_tiddlers/Hickson Compact Group.html#"Hickson Compact Group"]]>>
<<tiddler [[include_tiddlers/Hidden Variable Theory.html#"Hidden Variable Theory"]]>>
<<tiddler [[include_tiddlers/Hierarchy Problem.html#"Hierarchy Problem"]]>>
<<tiddler [[include_tiddlers/Higgs Boson.html#"Higgs Boson"]]>>
<<tiddler [[include_tiddlers/Higgs Inflation.html#"Higgs Inflation"]]>>
<<tiddler [[include_tiddlers/Higgs Mass.html#"Higgs Mass"]]>>
<<tiddler [[include_tiddlers/Higgs Mechanism.html#"Higgs Mechanism"]]>>
<<tiddler [[include_tiddlers/Higgs Portal.html#"Higgs Portal"]]>>
<<tiddler [[include_tiddlers/Higgs Self-couplings Lagrangian.html#"Higgs Self-couplings Lagrangian"]]>>
<<tiddler [[include_tiddlers/High Temperature Superconductivity.html#"High Temperature Superconductivity"]]>>
<<tiddler [[include_tiddlers/High-intensity Interval Training.html#"High-intensity Interval Training"]]>>
<<tiddler [[include_tiddlers/High-resolution Astronomical Imaging.html#"High-resolution Astronomical Imaging"]]>>
<<tiddler [[include_tiddlers/Higher Order Derivative Gravity.html#"Higher Order Derivative Gravity"]]>>
<<tiddler [[include_tiddlers/Higher Spin Theory.html#"Higher Spin Theory"]]>>
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.

* [[On the Graphs of Hoffman-Singleton and Higman-Sims (2004) - P. R. Hafner|]] [[local|papers/v11i1r77.pdf]] [[pct. 5|]]

* [[Projections of the Higman-Sims Graph from the Leech Lattice|]] [[local|videos/Projections of the Higman Sims graph from the Leech lattice.wmv]]
<<tiddler [[include_tiddlers/Hilbert Space.html#"Hilbert Space"]]>>
<<tiddler [[include_tiddlers/Hilbert's Hotel.html#"Hilbert's Hotel"]]>>
<<tiddler [[include_tiddlers/Hilbert's Problems.html#"Hilbert's Problems"]]>>
<<tiddler [[include_tiddlers/Hippocampus.html#"Hippocampus"]]>>
<<tiddler [[include_tiddlers/Histone.html#"Histone"]]>>
<<tiddler [[include_tiddlers/Histone Deacetylase.html#"Histone Deacetylase"]]>>
<<tiddler [[include_tiddlers/Histone Modification.html#"Histone Modification"]]>>
<<tiddler [[include_tiddlers/Hoffman-Singleton Graph.html#"Hoffman-Singleton Graph"]]>>
* [[Stanford Encyclopedia of Philosophy|]]
<<tiddler [[include_tiddlers/Holographic Principle.html#"Holographic Principle"]]>>
<<tiddler [[include_tiddlers/Holomorphic Function.html#"Holomorphic Function"]]>>
<<tiddler [[include_tiddlers/Home Observatory.html#"Home Observatory"]]>>
<<tiddler [[include_tiddlers/Homeomorphism.html#"Homeomorphism"]]>>
<<tiddler [[include_tiddlers/Homogeneous Space.html#"Homogeneous Space"]]>>
<<tiddler [[include_tiddlers/Homology Sphere.html#"Homology Sphere"]]>>
<<tiddler [[include_tiddlers/Homomorphism.html#"Homomorphism"]]>>
<<tiddler [[include_tiddlers/Homothety.html#"Homothety"]]>>
<<tiddler [[include_tiddlers/Homotopy.html#"Homotopy"]]>>
<<tiddler [[include_tiddlers/Hoop Conjecture.html#"Hoop Conjecture"]]>>
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].

* [[Hopf Algebras, Renormalization and Noncommutative Geometry (1998) - A. Connes, D. Kreimer|]] [[local|papers/9808042v1.pdf]] {{t100Cite{[[pct. 482|,5&hl=de]]}}}
* [[A Primer of Hopf Algebras (2006) - P. Cartier|]] [[local|papers/M-06-40.pdf]] [[pct. 44|,5&hl=de]]
* [[Quantum Groups and Noncommutative Geometry (2000) - S. Majid|]] [[local|papers/0006167v1.pdf]] [[pct. 39|,5&hl=de]]
* [[Hopf Algebras in Noncommutative Geometry (2001) - J. C. Várilly|]] [[local|papers/0109077v2.pdf]] [[pct. 18|,5&hl=de]]
* [[[1] On Hopf Algebras and their Generalizations (2008) - G. Karaali|]] [[local|papers/0703441.pdf]] [[pct. 10|,5&hl=de]]
* [[The Hopf Algebra Structure of GL(1,Hq) and the Isomorphism between SPq(1) and SUq(2) - S. Celik|]] [[local|papers/0112118v1.pdf]] [[pct. 1|,5&hl=de]]

* [[Lecture 1 - Hopf Algebras and Combinatorics (Federico Ardila)|]] - Excellent lectures (Math olymia... )
* [[Compact Quantum Groups & their Representation Categories|]] - 10 lectures by Sergey Neshveyev.

* [[WIKIPEDIA - Hopf Algebra|]]@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/HeinzHopf.jpg][Comments]]}}}&nbsp;@@
<<tiddler [[include_tiddlers/Hopf Fibration.html#"Hopf Fibration"]]>>
<<tiddler [[include_tiddlers/Hopf Map.html#"Hopf Map"]]>>
<<tiddler [[include_tiddlers/Horizontal Coordinate System.html#"Horizontal Coordinate System"]]>>
<<tiddler [[include_tiddlers/Hormesis.html#"Hormesis"]]>>
<<tiddler [[include_tiddlers/Hour Angle.html#"Hour Angle"]]>>
''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 $d_S \approx 2$.

* [[Quantum Gravity at a Lifshitz Point (2009) - P. Hořava|]] [[local|papers/0901.3775v2.pdf]] {{t1000Cite{[[pct. 1030|]]}}}
* [[Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point (2009) - P. Hořava|]] [[local|papers/0902.3657.pdf]] {{t100Cite{[[pct. 339|,5&hl=de]]}}}
* [[Quantum Gravity without Lorentz Invariance (2009) - T. P. Sotiriou, M. Visser, S. Weinfurtner|]] [[local|papers/0905.2798v3.pdf]] {{t100Cite{[[pct. 182|]]}}}

* [[Quantum Gravity with Anisotropic Scaling (2010) - P. Hořava|]] [[local|presentations/horava.pdf]]

* [[Quantum Gravity with Anisotropic Scaling and the Multicritical Universe (2011) - P. Hořava|]]
<<tiddler [[include_tiddlers/Hubble Constant.html#"Hubble Constant"]]>>
<<tiddler [[include_tiddlers/Hubble Space Telescope.html#"Hubble Space Telescope"]]>>
<<tiddler [[include_tiddlers/Hubble Tension.html#"Hubble Tension"]]>>
<<tiddler [[include_tiddlers/Human Leukocyte Antigen.html#"Human Leukocyte Antigen"]]>>
<<tiddler [[include_tiddlers/Humanism.html#"Humanism"]]>>
<<tiddler [[include_tiddlers/Humanoid Robot.html#"Humanoid Robot"]]>>
<<tiddler [[include_tiddlers/Humoral Immunity.html#"Humoral Immunity"]]>>
<<tiddler [[include_tiddlers/Hurwitz Integer.html#"Hurwitz Integer"]]>>
<<tiddler [[include_tiddlers/Hurwitz Theorem.html#"Hurwitz Theorem"]]>>
<<tiddler [[include_tiddlers/Hurwitz-Radon Theorem.html#"Hurwitz-Radon Theorem"]]>>
<<tiddler [[include_tiddlers/Hyaluronic Acid.html#"Hyaluronic Acid"]]>>
<<tiddler [[include_tiddlers/Hydrino.html#"Hydrino"]]>>
<<tiddler [[include_tiddlers/Hydrogen Peroxide.html#"Hydrogen Peroxide"]]>>
<<tiddler [[include_tiddlers/Hydrolysis.html#"Hydrolysis"]]>>
<<tiddler [[include_tiddlers/Hydroxyl Radical.html#"Hydroxyl Radical"]]>>
<<tiddler [[include_tiddlers/Hype Cycle.html#"Hype Cycle"]]>>
<<tiddler [[include_tiddlers/Hyperbolic Quaternion.html#"Hyperbolic Quaternion"]]>>
<<tiddler [[include_tiddlers/Hyperbolic Space.html#"Hyperbolic Space"]]>>
<<tiddler [[include_tiddlers/Hyperboloid.html#"Hyperboloid"]]>>
<<tiddler [[include_tiddlers/Hypercomplex Analysis.html#"Hypercomplex Analysis"]]>>
<<tiddler [[include_tiddlers/Hypergraph.html#"Hypergraph"]]>>
<<tiddler [[include_tiddlers/Hyperthermia Therapy.html#"Hyperthermia Therapy"]]>>
<<tiddler [[include_tiddlers/Hypo-sensitization.html#"Hypo-sensitization"]]>>
<<tiddler [[include_tiddlers/Hypoxia-inducible Factor.html#"Hypoxia-inducible Factor"]]>>
<<tiddler [[include_tiddlers/IGF-1.html#"IGF-1"]]>>
<<tiddler [[include_tiddlers/IKKT Model.html#"IKKT Model"]]>>
<<tiddler [[include_tiddlers/INDI.html#"INDI"]]>>
<<tiddler [[include_tiddlers/IQ.html#"IQ"]]>>
<<tiddler [[include_tiddlers/IRIS.html#"IRIS"]]>>
<<tiddler [[include_tiddlers/Icosahedral Group.html#"Icosahedral Group"]]>>
<<tiddler [[include_tiddlers/Icosian.html#"Icosian"]]>>
<<tiddler [[include_tiddlers/Ideal.html#"Ideal"]]>>
<<tiddler [[include_tiddlers/Idealizations in Physics.html#"Idealizations in Physics"]]>>
<<tiddler [[include_tiddlers/Idempotency.html#"Idempotency"]]>>
<<tiddler [[include_tiddlers/Ideomotor Effect.html#"Ideomotor Effect"]]>>
<<tiddler [[include_tiddlers/IgM.html#"IgM"]]>>
<<tiddler [[include_tiddlers/Ilya Prigogine.html#"Ilya Prigogine"]]>>
<<tiddler [[include_tiddlers/Image File Format.html#"Image File Format"]]>>
|Author|Eric Shulman|
|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.
The extended image syntax is:
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).
[<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(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]]
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
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.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.title = config.macros.attach.linkTooltip + link;
				} else
					e = createExternalLink(w.output,link);
			} else
				e = createTiddlyLink(w.output,link,false,null,w.isStatic);

		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())} };;
			if (stretchW||stretchH) config.formatterHelpers.addStretchHandlers(img,stretchW,stretchH);
		if(tooltip) img.title = tooltip;

		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) {
					return false;
			} else
		w.nextMatch = this.lookaheadRegExp.lastIndex;

	tip: 'SHIFT-CLICK=show full size, CTRL-CLICK=restore initial size',
	dragtip: 'DRAG=stretch/shrink, '

config.formatterHelpers.addStretchHandlers=function(e,stretchW,stretchH) {
	e.statusMsg='width=%0, height=%1';'move';;;
	e.onmousedown=function(ev) { var ev=ev||window.event;
		return false;
	e.onmousemove=function(ev) { var ev=ev||window.event;
		if (this.sizing) {
			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) {''; }
		if (ev.ctrlKey)  {;; }
		return false;
	e.onmouseout=function(ev) { var ev=ev||window.event;
		return false;
<<tiddler [[include_tiddlers/Immortal Jellyfish.html#"Immortal Jellyfish"]]>>
<<tiddler [[include_tiddlers/Immortality.html#"Immortality"]]>>
<<tiddler [[include_tiddlers/Immune System.html#"Immune System"]]>>
<<tiddler [[include_tiddlers/Immunoglobulin.html#"Immunoglobulin"]]>>
<<tiddler [[include_tiddlers/Immunosenescence.html#"Immunosenescence"]]>>
<<tiddler [[include_tiddlers/Incidence Matrix.html#"Incidence Matrix"]]>>
<<tiddler [[include_tiddlers/Incidence Structure.html#"Incidence Structure"]]>>
<<tiddler [[include_tiddlers/Indel.html#"Indel"]]>>
''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]] $G/H$.
The special [[orthogonal group|Orthogonal Group]] $SO(n)$ has index 2 in respect to the orthogonal group $O(n)$.
<<tiddler [[include_tiddlers/Index Catalogue.html#"Index Catalogue"]]>>
<<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:
\langle \; , \; \rangle: T\mathcal M \times T\mathcal M \rightarrow \mathbb R
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/Inevitability of Biological Aging.html#"Inevitability of Biological Aging"]]>>
<<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/Inflammation.html#"Inflammation"]]>>
<<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"]]>>
|Author|Eric Shulman|
|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.
>see [[InlineJavascriptPluginInfo]]
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
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
				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);"
						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((typeof(r)==="string")&&r.length) {
								return false;
							} else return r!==undefined?r:false;
						} catch(e){alert(e.description||e.toString());return false;}
					var URIcode='javascript:void(eval(decodeURIComponent(%22(function(){try{';
					URIcode+=encodeURIComponent(encodeURIComponent(code.replace(/\n/g,' ')));
					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 $(id) { return document.getElementById(id.replace(/^#/,'')); } }
<<tiddler [[include_tiddlers/Innate Immune System.html#"Innate Immune System"]]>>
<<tiddler [[include_tiddlers/Insomnia.html#"Insomnia"]]>>
<<tiddler [[include_tiddlers/Instanton.html#"Instanton"]]>>
An ''$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]]).
* $N(2) = 8$
* $N(4) = 384$
* $N(8) = 10.321.920$
* $N(16) = 1.371.195.958.099.968.000$
See also: [[Sloane's A000165|]].
<<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/Interactome.html#"Interactome"]]>>
<<tiddler [[include_tiddlers/Interest Rate.html#"Interest Rate"]]>>
<<tiddler [[include_tiddlers/Interface Theory of Perception.html#"Interface Theory of Perception"]]>>
<<tiddler [[include_tiddlers/Interferon.html#"Interferon"]]>>
<<tiddler [[include_tiddlers/Interferon Type I.html#"Interferon Type I"]]>>
<<tiddler [[include_tiddlers/Interleukin 1 Family.html#"Interleukin 1 Family"]]>>
<<tiddler [[include_tiddlers/Interleukin 17.html#"Interleukin 17"]]>>
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* [[WIKIPEDIA - Indra's Net|'s_net]]
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> 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]]

* [[Conscious Realism and the Mind-Body Problem (2008) - D. D. Hoffman|]] [[local|papers/ConsciousRealism2.pdf]] [[pct. 14|,5&hl=en]]
* [[A Theory of Consciousness (2009) - C. Koch|]] [[local|papers/CR-Complexity-09.pdf]] [[pct. 7|,5&hl=en]]
* [[Objects of Consciousness (2014) - D. D.Hoffman, C. Prakash|]] [[local|papers/fpsyg-05-00577.pdf]] pct. 0

* [[Consciousness and the Interface Theory of Perception - D. Hoffman|]]
* [[Physics From Consciousness - D. Hoffman|]]
* [[Donald Hoffman- Sages and Scientists 2013|]]
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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

* [[BBC Horizon - Infinity|]]
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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 inequalities|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.

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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 safe 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 comes 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 by William James in psychology.

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

* [[[1] Complementarity in Classical Dynamical Systems (2004) - P. beim Graben, H. Atmanspacher|]] [[local|papers/0407046v5.pdf]] [[pct. 39|,5&hl=en]] - "We consider pairs of arbitrary observables of a classical dynamical system and show that those pairs of observables are in general complementary with respect to the coarsegrainings, or partitions, of the phase space ..."
* [[Generalized Quantum Theory: Overview and Latest Developments (2012) - T. Filk, H. Römer|]] [[local|papers2/1202.1659.pdf]] [[pct. 37|,5&hl=de]]
> 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 -

With respect to the existence of a theory of [[quantum gravity|Quantum Gravity]] there are at least four possibilities:
# [[Gravity|Gravitation]] needs to be [[quantized|Quantization]]. This is the approach taken by the majority of people ([[string theory|Superstring Theory]], [[loop quantum gravity|Loop Quantum Gravity]], etc.)
# [[Gravity is emergent|Emergent Gravity]] and it is rather the underlying constituents that need to be quantized.
# Gravity is through and through a [[classical|Classicality]] theory which coexists with [[quantum mechanics|Quantum Mechanics]], each theory being the appropriate description in its respective domain of applicability. A prominent adherent of this point of view is [[Freeman Dyson]].
# "Everything is geometry", hence quantum mechanics in some way must follow from [[General Relativity]] or a variant thereof (e.g. [[Einstein-Cartan theory|Einstein-Cartan Theory]], [[teleparallel gravity|Teleparallel Gravity]], [[Brans-Dicke theory|Brans-Dicke Theory]]). The idea of geometrizing gravity and matter is actually quite old, reaching at least back to people like [[Clifford|William Kingdon Clifford]] and [[Einstein|Albert Einstein]]. However, when quantum mechanics came up, it largely fell out of favour and it became common lore that quantum mechanics must be more fundamental than is General Relativity. For more on this point, see [[quantum mechanics from General Relativity|Quantum Mechanics from General Relativity]]).
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An argument often heard in favour of the existence of a theory of quantum gravity is that General Relativity and [[quantum mechanics|Quantum Mechanics]] are inconsistent at very small scales and that the quantization of gravity would be the cure. Although the argument seems reasonable, the conclusion is doubtful. The reason is the following: The argument presumes that classical spacetime breaks down around the [[Planck scale|Planck Length]] as quantum fluctuations are so big that [[black holes|Black Hole]] form (see also [[Compton wavelength|Compton Wavelength]]). The implicit assumption is that quantum mechanics remains valid whereas (classical) GR breaks down. But here one faces a situation comparable to someone sitting on a branch of a tree and cutting it off. The problem is that quantum mechanics requires observables for it to make sense. But with the breakdown of classical variables such as spacial position and time, associated observables like momentum and energy also break down (see also [[Noether's theorem|Noether Theorem]]). Hence, there is no more [[uncertainty relation|Uncertainty Relations]] between momentum and location and also energy and time. But these are essential in the argument for the formation of black holes. A way out of this circularity of argumentation could by [[purporting conformal invariance|Conformal Invariance Hypothesis]] at very small scales.

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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]] (GR) 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 all 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, it does not make sense to quantize temperature [1].<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.

See also:
* [[Graviton]]

* [[Is Quantum Gravity Necessary? (2008) - S. Carlip|]] [[local|papers/0803.3456.pdf]] {{t100Cite{[[pct. 131|,5&hl=en]]}}}
* [[Are the Spectra of Geometrical Operators in Loop Quantum Gravity Really Discrete? (2007) - B. Dittrich, T. Thiemann|]] [[local|papers/0708.1721v2.pdf]] [[pct. 71|,5&hl=de]]
* [[Using Cosmology to Establish the Quantization of Gravity - L. M. Krauss, F. Wilczek|]] [[local|papers2/89003.pdf]] [[pct. 60|,5&hl=en]]
* [[Measurement Analysis and Quantum Gravity (2008) - M. Albers, C. Kiefer, M. Reginatto|]] [[local|papers2/0802.1978.pdf]] [[pct. 44|,5&hl=en]]
* [[Is Quantum Gravity Necessary? (1999) - J. Mattingly|]] [[local|papers/IsGravityNecessarilyQuantized.proofs.pdf]] [[pct. 27|,5&hl=en]]
* [[The Philosophy behind Quantum Gravity (2006) - H. Zinkernagel|]] [[local|papers2/PhilBehindQGFinalWref.pdf]] [[pct. 9|,5&hl=en]]

* [[The World on a String (2004) - F. Dyson|]]

* [[Is a Graviton Detectable? (2013) - F. Dyson|]]
* [[Why is the Quantum So Mysterious? - F. Dyson|]]
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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. $q(v) < 0, \; \forall \,v$.
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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}
* [[The Jacobian Determinant - Jeff Knisley|]] - Doesn't work well with firefox, better use other browser.
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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.
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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 $v=15$.

* [[[1] WIKIPEDIA - Magic Square|]]
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* [[History and Physics of the Klein Paradox (1999) - A. Calogeracos, N. Dombey|]] [[local|papers/9905076v1.pdf]] [[pct. 43|]]
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The ''Kretschmann Scalar K'' for an n-dimensional Riemann manifold is given by
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
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:
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

<<tiddler [[include_tiddlers/Kynurenine Pathway.html#"Kynurenine Pathway"]]>>
<<tiddler [[include_tiddlers/Kähler Manifold.html#"Kähler Manifold"]]>>
<<tiddler [[include_tiddlers/Kähler-Dirac Equation.html#"Kähler-Dirac Equation"]]>>
<<tiddler [[include_tiddlers/LMMS.html#"LMMS"]]>>
<<tiddler [[include_tiddlers/LOFAR.html#"LOFAR"]]>>
<<tiddler [[include_tiddlers/LSZ Reduction Formula.html#"LSZ Reduction Formula"]]>>
<<tiddler [[include_tiddlers/Laboratory Mouse.html#"Laboratory Mouse"]]>>
<<tiddler [[include_tiddlers/Lactate Dehydrogenase.html#"Lactate Dehydrogenase"]]>>
<<tiddler [[include_tiddlers/Lactic Acid.html#"Lactic Acid"]]>>
<<tiddler [[include_tiddlers/Lactic Acidosis.html#"Lactic Acidosis"]]>>
<<tiddler [[include_tiddlers/Lactoferrin.html#"Lactoferrin"]]>>
<<tiddler [[include_tiddlers/Lactose Intolerance.html#"Lactose Intolerance"]]>>
<<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"]]>>
<<tiddler [[include_tiddlers/Landscape Astrophotography.html#"Landscape Astrophotography"]]>>
<<tiddler [[include_tiddlers/Langlands Program.html#"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 Gauge Transformation.html#"Large Gauge Transformation"]]>>
<<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 Scale Structure of the Universe.html#"Large Scale Structure of the Universe"]]>>
<<tiddler [[include_tiddlers/Large Telescopes.html#"Large Telescopes"]]>>
<<tiddler [[include_tiddlers/Laser.html#"Laser"]]>>
<<tiddler [[include_tiddlers/Late Life Cancer Incidence.html#"Late Life Cancer Incidence"]]>>
<<tiddler [[include_tiddlers/Late-life Mortality Convergence.html#"Late-life Mortality Convergence"]]>>
<<tiddler [[include_tiddlers/Late-life Mortality Plateau.html#"Late-life Mortality Plateau"]]>>
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.

%We hook the usual macro-creating Latex commands up to Richard's javascript routines.

%Here we make copies of the standard Latex commands which we may displace.

%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.

%We preload the macrosets which are considered global.

%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.
<<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:
|$\mb A \wedge \mb A = \mb A$; $\;\mb A \vee \mb A = \mb A\quad$|[[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"]]>>
<<tiddler [[include_tiddlers/Law of Excluded Middle.html#"Law of Excluded Middle"]]>>
<<tiddler [[include_tiddlers/Le Sage Gravity.html#"Le Sage Gravity"]]>>
<<tiddler [[include_tiddlers/Lean Body Mass.html#"Lean Body Mass"]]>>
<<tiddler [[include_tiddlers/Leech Lattice.html#"Leech Lattice"]]>>
<<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"]]>>
<<tiddler [[include_tiddlers/Lense-Thirring Effect.html#"Lense-Thirring Effect"]]>>
<<tiddler [[include_tiddlers/Leptogenesis.html#"Leptogenesis"]]>>
<<tiddler [[include_tiddlers/Lepton.html#"Lepton"]]>>
<<tiddler [[include_tiddlers/Lepton Flavour Universality.html#"Lepton Flavour Universality"]]>>
<<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.

<html><center><iframe name="content" src="" width=51% height=86></iframe></center></html>
<<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.

* [[Right Angle Lever Paradox - J. C. Nickerson, R. T. McAdory|]] [[pct. 2|]]
* [[The Lewis-Tolman Lever Paradox - J. W. Butler|]] [[pct. 1|]]
* [[Covariant Formulation of Hooke's Law - O. Gron|]] pct. 0
* [[The Lack of Rotation in a Moving Right Angle Lever - J. Franklin|]] pct. 0
* [[Relativistic Angular Momentum - N. Menicucci|]] 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/Life History Theory.html#"Life History Theory"]]>>
<<tiddler [[include_tiddlers/Lifespan Clock.html#"Lifespan Clock"]]>>
<<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 Pollution.html#"Light Pollution"]]>>
<<tiddler [[include_tiddlers/Light Scattering.html#"Light Scattering"]]>>
<<tiddler [[include_tiddlers/Limits to Science.html#"Limits to Science"]]>>
<<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/Lipoprotein.html#"Lipoprotein"]]>>
<<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/Live Stacking.html#"Live Stacking"]]>>
<<tiddler [[include_tiddlers/Live Streaming.html#"Live Streaming"]]>>
<<tiddler [[include_tiddlers/Live View.html#"Live View"]]>>
<<tiddler [[include_tiddlers/Liver Disease.html#"Liver Disease"]]>>
<<tiddler [[include_tiddlers/Logic.html#"Logic"]]>>
<<tiddler [[include_tiddlers/Logic Gate.html#"Logic Gate"]]>>
<<tiddler [[include_tiddlers/Long Short-term Memory.html#"Long Short-term Memory"]]>>
<<tiddler [[include_tiddlers/Longevity.html#"Longevity"]]>>
<<tiddler [[include_tiddlers/Longevity Gene.html#"Longevity Gene"]]>>
<<tiddler [[include_tiddlers/Longevity Pathway.html#"Longevity Pathway"]]>>
<<tiddler [[include_tiddlers/Longevity Quotient.html#"Longevity Quotient"]]>>
<<tiddler [[include_tiddlers/Longitudinal Study.html#"Longitudinal Study"]]>>
<<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"]]>>
<<tiddler [[include_tiddlers/Loschmidt's Paradox.html#"Loschmidt's Paradox"]]>>
''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.

* [[WIKIPEDIA - Lovelock Theory of Gravity|]]
<<tiddler [[include_tiddlers/Low Energy Quantum Gravity.html#"Low Energy Quantum Gravity"]]>>
<<tiddler [[include_tiddlers/Low Fat Diet.html#"Low Fat Diet"]]>>
<<tiddler [[include_tiddlers/Low Protein Diet.html#"Low Protein Diet"]]>>
<<tiddler [[include_tiddlers/Low-carbohydrate Diet.html#"Low-carbohydrate 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.

* [[Low Density Parity Check Codes Based on Finite Geometries: A Rediscovery and New Results - Y. Kou, S. Lin, M. P.C. Fossorier|]] [[local|papers/ucd1.pdf]] {{t500Cite{[[pct. 598|]]}}}
* [[Structured Low-Density Parity-Check Codes - J. M. F. Moura, J. Lu, H. Zhang|]] [[local|papers/spm-jan04-moura-lu-zhang-ieeeexplore.pdf]] [[pct. 35|]]
<<tiddler [[include_tiddlers/Low-protein High-carbohydrate Diet.html#"Low-protein High-carbohydrate Diet"]]>>
<<tiddler [[include_tiddlers/Lucas' Problem.html#"Lucas' Problem"]]>>
<<tiddler [[include_tiddlers/Ludwig Wittgenstein.html#"Ludwig Wittgenstein"]]>>
<<tiddler [[include_tiddlers/Lumbar Spinal Stenosis.html#"Lumbar Spinal Stenosis"]]>>
<<tiddler [[include_tiddlers/Lumbar Spine.html#"Lumbar Spine"]]>>
<<tiddler [[include_tiddlers/Lyman-alpha Forest.html#"Lyman-alpha Forest"]]>>
<<tiddler [[include_tiddlers/Lyme Disease Herbal Treatment.html#"Lyme Disease Herbal Treatment"]]>>
<<tiddler [[include_tiddlers/Lyme Disease Natural Treatment.html#"Lyme Disease Natural Treatment"]]>>
<<tiddler [[include_tiddlers/Lyme Disease Symptoms Etiology.html#"Lyme Disease Symptoms Etiology"]]>>
<<tiddler [[include_tiddlers/Lyme Disease Treatment.html#"Lyme Disease Treatment"]]>>
<<tiddler [[include_tiddlers/Lymphocyte.html#"Lymphocyte"]]>>
<<tiddler [[include_tiddlers/Lysosome.html#"Lysosome"]]>>
<<tiddler [[include_tiddlers/Lyth Bound.html#"Lyth Bound"]]>>
<<tiddler [[include_tiddlers/M-Algebra.html#"M-Algebra"]]>>
<<tiddler [[include_tiddlers/M-theory.html#"M-theory"]]>>
<<tiddler [[include_tiddlers/MACHO.html#"MACHO"]]>>
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.

* [[MAGMA Computational Algebra System Home Page|]]
* [[MAGMA Online Calculator|]]
* [[WIKIPEDIA - MAGMA Computer Algebra System|]]
* [[Solving Problems with MAGMA - W. Bosma, J. Cannon, C. Playoust, A. Steel|]]  [[local|lectures/SolvingProblemsWithMAGMA.pdf]] [[lct. 10|]]
* [[Handbook of MAGMA Functions|]] [[local|documents/MAGMA]]
** [[Lattices|]] [[local|documents/MAGMA/html/text826.htm]]
** [[Coding Theory|]] [[local|documents/MAGMA/html/part16.htm]]
** [[Hadamard Matrices|]] [[local|file:///E:/Trajectory/documents/MAGMA/html/text1517.htm]]
** [[Incidence Structures and Designs|]] [[local|file:///E:/Trajectory/documents/MAGMA/html/text1502.htm]]

* [[Applied Abstract Algebra - D. Joyner, R. Kreminski, J. Turisco|]]
<<tiddler [[include_tiddlers/MAXIMA.html#"MAXIMA"]]>>
<<tiddler [[include_tiddlers/MHC Class II.html#"MHC Class II"]]>>
<<tiddler [[include_tiddlers/MOND Derivations.html#"MOND Derivations"]]>>
<<tiddler [[include_tiddlers/MOND and Cosmology.html#"MOND and Cosmology"]]>>
<<tiddler [[include_tiddlers/MSM.html#"MSM"]]>>
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/Macrophage.html#"Macrophage"]]>>
<<tiddler [[include_tiddlers/Macroscopic Quantum State.html#"Macroscopic Quantum State"]]>>
<<tiddler [[include_tiddlers/Magic Lantern.html#"Magic Lantern"]]>>
<<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/Magnitude.html#"Magnitude"]]>>
<<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/Majumdar-Papapetrou Solution.html#"Majumdar-Papapetrou Solution"]]>>
<<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/Manifold.html#"Manifold"]]>>
<<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/Mars Exploration Rover.html#"Mars Exploration Rover"]]>>
<<tiddler [[include_tiddlers/Mars Rover.html#"Mars Rover"]]>>
<<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/Mass of the Universe.html#"Mass of the Universe"]]>>
<<tiddler [[include_tiddlers/Massive Photon.html#"Massive Photon"]]>>
<<tiddler [[include_tiddlers/Master Equation.html#"Master Equation"]]>>
|Author|Eric Shulman|
|Description|'tag matching' with full boolean expressions (AND, OR, NOT, and nested parentheses)|
> see [[MatchTagsPluginInfo]]
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
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
	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
			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")
		if (mode=="report" || mode=="panel") {
			var target=params.shift()||this.defTiddler;
		if (mode=="popup") {
			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");
			var tags=t.tags.length?'[['+t.tags.join(']] [[')+']]':'';
		return out.join(sep);
	createInline: function(place,expr,fmt,sep,sortBy) {
	createPopup: function(place,label,expr,prompt,sortBy) {
		var btn=createTiddlyButton(place,
			function(ev){ return config.macros.matchTags.showPopup(this,ev||window.event); });
	showPopup: function(here,ev) {
		var p=Popup.create(here); if (!p) return false;
		var tids=store.getMatchingTiddlers(here.getAttribute("expr"));
		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"),
				function() {
					var list=this.getAttribute("list").readBracketedList();
			b.setAttribute("list","[["+list.join("]] [[")+"]]");
		var out=this.formatList(tids," &nbsp;[[%0]]&nbsp; ","\n"); wikify(out,p);;
		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])
		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;
<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;
	return false;">
// 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;
// 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+"]]");
<<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
\dot{x} & = & \sigma(y-x) \\
\dot{y} & = & \rho x - y - xz \\
\dot{z} & = & -\beta z + xy
!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
\[  1 +  \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =
\quad\quad \textrm{for}\quad |q|<1. \]
!Maxwell's Equations
\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
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 Constant.html#"Mathematical Constant"]]>>
<<tiddler [[include_tiddlers/Mathematical Structure.html#"Mathematical Structure"]]>>
* [[FamousWhy|]] - Dto.
<<tiddler [[include_tiddlers/Mathieu Group.html#"Mathieu Group"]]>>
<<tiddler [[include_tiddlers/Mathisson-Papapetrou Equations.html#"Mathisson-Papapetrou Equations"]]>>
* [[The Matrix Reference Manual|]]
<<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/Maximum Tension Principle.html#"Maximum Tension Principle"]]>>
<<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/Medical Error.html#"Medical Error"]]>>
<<tiddler [[include_tiddlers/Mediocrity Principle.html#"Mediocrity Principle"]]>>
<<tiddler [[include_tiddlers/Meditation.html#"Meditation"]]>>
<<tiddler [[include_tiddlers/Melanoma.html#"Melanoma"]]>>
<<tiddler [[include_tiddlers/Melatonin.html#"Melatonin"]]>>
<<tiddler [[include_tiddlers/Melittin.html#"Melittin"]]>>
<<tiddler [[include_tiddlers/Membrane Theory of Aging.html#"Membrane Theory of Aging"]]>>
<<tiddler [[include_tiddlers/Meme.html#"Meme"]]>>
<<tiddler [[include_tiddlers/Memory.html#"Memory"]]>>
<<tiddler [[include_tiddlers/Meninges.html#"Meninges"]]>>
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.

* [[WIKIPEDIA - Mersenne Primes|]]
<<tiddler [[include_tiddlers/Messier Object.html#"Messier Object"]]>>
<<tiddler [[include_tiddlers/Metabolic Network.html#"Metabolic Network"]]>>
<<tiddler [[include_tiddlers/Metabolic Theory of Cancer.html#"Metabolic Theory of Cancer"]]>>
<<tiddler [[include_tiddlers/Metabolism.html#"Metabolism"]]>>
<<tiddler [[include_tiddlers/Metal-catalized Protein Oxidation.html#"Metal-catalized Protein Oxidation"]]>>
<<tiddler [[include_tiddlers/Metamaterial.html#"Metamaterial"]]>>
<<tiddler [[include_tiddlers/Metamath.html#"Metamath"]]>>
<<tiddler [[include_tiddlers/Metazoan.html#"Metazoan"]]>>
<<tiddler [[include_tiddlers/Methionine Restriction.html#"Methionine Restriction"]]>>
<<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/Microaerophile.html#"Microaerophile"]]>>
<<tiddler [[include_tiddlers/Microbiome.html#"Microbiome"]]>>
<<tiddler [[include_tiddlers/Microcausality.html#"Microcausality"]]>>
<<tiddler [[include_tiddlers/Microsatellite.html#"Microsatellite"]]>>
<<tiddler [[include_tiddlers/Microtubule.html#"Microtubule"]]>>
<<tiddler [[include_tiddlers/Mie Scattering.html#"Mie Scattering"]]>>
<<tiddler [[include_tiddlers/Milky Way.html#"Milky Way"]]>>
<<tiddler [[include_tiddlers/Millimetre of Mercury.html#"Millimetre of Mercury"]]>>
<<tiddler [[include_tiddlers/Minimal Coupling.html#"Minimal Coupling"]]>>
<<tiddler [[include_tiddlers/Minimal Mass.html#"Minimal Mass"]]>>
<<tiddler [[include_tiddlers/Minimum Bactericidal Concentration.html#"Minimum Bactericidal Concentration"]]>>
<<tiddler [[include_tiddlers/Minimum Inhibitory Concentration.html#"Minimum Inhibitory Concentration"]]>>
<<tiddler [[include_tiddlers/Minisuperspace Model.html#"Minisuperspace Model"]]>>
<<tiddler [[include_tiddlers/Minkowski Catalog of Planetary Nebulae.html#"Minkowski Catalog of Planetary Nebulae"]]>>
<<tiddler [[include_tiddlers/Miracle Ocatad Generator (MOG).html#"Miracle Ocatad Generator (MOG)"]]>>
<<tiddler [[include_tiddlers/Missing Satellites Problem.html#"Missing Satellites Problem"]]>>
<<tiddler [[include_tiddlers/Mitochondria-targeted Antioxidant.html#"Mitochondria-targeted Antioxidant"]]>>
<<tiddler [[include_tiddlers/Mitochondrial Biogenesis.html#"Mitochondrial Biogenesis"]]>>
<<tiddler [[include_tiddlers/Mitochondrial DNA.html#"Mitochondrial DNA"]]>>
<<tiddler [[include_tiddlers/Mitochondrial DNA Repair.html#"Mitochondrial DNA Repair"]]>>
<<tiddler [[include_tiddlers/Mitochondrial DNA Replication.html#"Mitochondrial DNA Replication"]]>>
<<tiddler [[include_tiddlers/Mitochondrial Dynamics.html#"Mitochondrial Dynamics"]]>>
<<tiddler [[include_tiddlers/Mitochondrial Eve.html#"Mitochondrial Eve"]]>>
<<tiddler [[include_tiddlers/Mitochondrial Theory of Aging.html#"Mitochondrial Theory of Aging"]]>>
<<tiddler [[include_tiddlers/Mitochondrial Transfer.html#"Mitochondrial Transfer"]]>>
<<tiddler [[include_tiddlers/Mitochondrion.html#"Mitochondrion"]]>>
<<tiddler [[include_tiddlers/Mitophagy.html#"Mitophagy"]]>>
<<tiddler [[include_tiddlers/Mixed Anomaly.html#"Mixed Anomaly"]]>>
<<tiddler [[include_tiddlers/Mobile Element.html#"Mobile Element"]]>>
<<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/Moment of Inertia.html#"Moment of Inertia"]]>>
<<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
[\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
\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.

Any [[flexible algebra|Flexible Algebra]] is monoassociative, as it satisfies $[\mb A,\mb B,\mb A] = 0$.
<<tiddler [[include_tiddlers/Monocyte.html#"Monocyte"]]>>
<<tiddler [[include_tiddlers/Monogamy of Entanglement.html#"Monogamy of Entanglement"]]>>
<<tiddler [[include_tiddlers/Monopole Problem.html#"Monopole Problem"]]>>
<<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.html#"Moon"]]>>
<<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"]]>>
* [[Morita Equivalence in Geometry and Algebra - R. Mayer|]]
<<tiddler [[include_tiddlers/Mortal Computation.html#"Mortal Computation"]]>>
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/Multi-point Alignment.html#"Multi-point Alignment"]]>>
<<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/Muon Anomalous Magnetic Moment.html#"Muon Anomalous Magnetic Moment"]]>>
<<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>

* [[My Favorite Integer Sequences N. J. A. Sloane|]] [[pct. 9|]]

* [[Cognitive Theoretic Model of the Universe (CMTU) - C. Langan|]] - How a man with an alledged I.Q. of around 200 "sees" the universe.
* [[Ben Goertzel Essays|]]
* [[University of Toronto Mathematics Network - Question Corner and Discussion Area|]]
* [[Articles by S. M. Phillips|]] - 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|]]
* [[Strings and Loops in Event Symmetric Space-Time - P. Gibbs|]]
* [[Rafiki Inc.|]]
* [[The Cellular Universe website - C. Ranzan|]]
* [[Tony Smith's Homepage, 240 Thoughts|]]
* [[Verman University Mathematical Quotations Server|]]
* [[God does not play Dice|]]
* [[Gennady I. Shipov|]] - Torsion, warp drives and all that ...
* [[The Orientation Congruent Algebra and the Native Exterior Calculus of Twisted Differential Forms|]]
* [[|]] - The alternative arXiv.
<<tiddler [[include_tiddlers/Mutation Accumulation Theory of Aging.html#"Mutation Accumulation Theory of Aging"]]>>
<<tiddler [[include_tiddlers/Mutation Rate.html#"Mutation Rate"]]>>
<<tiddler [[include_tiddlers/Myelin.html#"Myelin"]]>>
<<tiddler [[include_tiddlers/Mysterious Analogies.html#"Mysterious Analogies"]]>>
<<tiddler [[include_tiddlers/Möbius Function.html#"Möbius Function"]]>>
<<tiddler [[include_tiddlers/Möbius Strip.html#"Möbius Strip"]]>>
<<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]]

* [[N-ary Algebras: A Review with Applications (2010) - J. A. de Azcárraga, J. M. Izquierdo|]] [[local|papers/1005.1028v2.pdf]] [[pct. 17|]]
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The [[NAD⁺|Nicotinamide Adenine Dinucleotide]]/[[NADH]] redox pair is coupled to the energy consumption of a cell and therefore the ratio of the two molecules reflects the metabolic state of a cell.
A shift from [[cytosolic|Cytosol]] fermentation to [[aerobic mitochondrial metabolism|Oxidative Phosphorylation]] leads to decreasing intracellular levels of NADH and an increased NAD⁺/NADH ratio. Thus, the NAD⁺/NADH ratio can be affected by [[hypoxia|Hypoxia]] or by substrate metabolism, where low glucose leads to an increased NAD⁺/NADH ratio, and high glucose, lactate, or ethanol leads to a decrease in the NAD⁺/NADH ratio. Similarly, changes in the dinucleotide ratio can be accomplished in vivo by [[exercise|Physical Exercise]] (decreased ratio), [[calorie restriction|Calorie Restriction]] (increase of the NAD⁺/NADH ratio by decreasing NADH levels), or through a [[high fat|High Fat Diet]] intake.

The NAD⁺/NADH ratio is __quite complicated__; it actually varies wildly between different locations within the cell. While cytoplasmic NAD⁺/NADH ratios range between 60 and 700 in a typical [[eukaryotic cell|Eukaryote]], [[mitochondrial|Mitochondrion]] NAD⁺/NADH ratios are maintained at 7 to 8.
Even when the cytoplasmic pool of NAD⁺ is severely depleted, the mitochondrial NAD⁺ levels remain steadfast for up to three days. This is believed to be due to the inability of NAD⁺ to cross the mitochondrial membrane (the "gateway" to the mitochondria), so changes in the cytoplasmic NAD⁺/NADH ratio have absolutely no effect on the mitochondria. Only when the cell is truly in distress, and the mitochondrial membrane is compromised, is the mitochondrial NAD⁺/NADH ratio lost.

In cells, the two dinucleotides may be either "free" or bound to protein with two major pools compartmentalized in the cytosol and the mitochondria, respectively. Studies have estimated that 40–90% of the total NADH may be bound to proteins. However, the redox state depends on the unbound concentrations of dinucleotides in the different [[organelles|Organelle]], concentrations that are inherently difficult to measure.

NAD⁺ consuming enzymes, like [[sirtuins|Sirtuin]], also require NAD⁺ in order to function properly. Unlike alcohol dehydrogenase and reactions in cellular respiration, these are not redox reactions and do not generate NADH when they use NAD⁺. Rather, they cleave NAD⁺ to form [[nicotinamide (a form of vitamin B3)|Niacin]] as a byproduct.

Endogenous [[NAD⁺|Nicotinamide Adenine Dinucleotide]] and [[NADH]] coexist in the body and are converted from one to the other in many metabolic reactions. However, little is known about whether the metabolic fates of NAD⁺ and NADH are similar or whether administration of one compound results in alterations in the systemic levels of the other.

Electron acceptor availability can restrict cell proliferation, and the NAD⁺/NADH ratio has been found to correlate with tumor growth in vivo [1].

* [[The Redox State of Free Nicotinamide-adenine Dinucleotide in the Cytoplasm and Mitochondria of Rat Liver (1967) - D. H. Williamson, P. Lund , H. A. Krebs|]] [[local|papers4/biochemj00743-0232.pdf]] {{t1000Cite{[[pct. 1921|,5&hl=en]]}}}
* [[The Dynamic Regulation of NAD Metabolism in Mitochondria (2012) - L. R. Stein, S. Imai|]] [[local|papers4/nihms396349.pdf]] {{t100Cite{[[pct. 321|,5&hl=en]]}}}
* [[Investigating Mitochondrial Redox State Using NADH and NADPH Autofluorescence (2016) - T. S.Blacker, M. R. Duchen|]] [[local|papers2/1-s2.0-S0891584916303926-main.pdf]] {{t100Cite{[[pct. 196|,5&hl=en]]}}}
* [[[1] Environment Dictates Dependence on Mitochondrial Complex I for NAD and Aspartate Production and Determines Cancer Cell Sensitivity to Metformin (2016) - D. Y. Gui, L. B. Sullivan, A. Luengo, A. M.Hosios, L. N. Bush, N. Gitego, S. M.Davidson, E. Freinkman, C. J. Thomas, M. G. Vander Heiden|]] [[local|papers4/1-s2.0-S1550413116304843-main.pdf]] {{t100Cite{[[pct. 169|,5&hl=en]]}}}
* [[Nicotinamide-induced Mitophagy: Event Mediated by High NAD⁺/NADH Ratio and SIRT1 Protein Activation (2012) - S. Y. Jang, H. T. Kang, E. S. Hwang|]] [[local|papers2/zbc19304.pdf]] {{t100Cite{[[pct. 159|,5&hl=de]]}}}
* [[Metabolic Control by Sirtuins and Other Enzymes that Sense NAD⁺, NADH, or their Ratio (2017) - K. A. Anderson, A. S. Madsen, C. A. Olsen, M. D. Hirschey|]] [[local|papers2/1-s2.0-S0005272817301457-main.pdf]] [[pct. 63|,5&hl=de]]
* [[Nutritional Ketosis Increases NAD⁺/NADH Ratio in Healthy Human Brain: An in Vivo Study by ³¹P-MRS (2018) - L. Xin, Ö. Ipek, M. Beaumont, M. Shevlyakova, N. Christinat, M. Masoodi, N. Greenberg, R. Gruetter, B. Cuenoud|]] [[local|papers2/fnut-05-00062.pdf]] [[pct. 24|,5&hl=de]]
* [[Tolerance to NADH/NAD⁺ Imbalance Anticipates Aging and Anti-aging Interventions (3333) - A. J. Alonso-Lavin, D. Bajić, J. F. Poyatos|]] [[local|papers4/719401v1.full.pdf]] pct. 0

* [[What's the Difference Between NAD+ and NADH?|]]
''\NF-κB'' (''Nuclear Factor Kappa-light-chain-enhancer of Activated [[B Cells|B Cell]]'') is a ubiquitously expressed proinflammatory [[transcription factor|Transcription Factor]] that regulates the expression of over 500 [[genes|Gene]] involved in cellular transformation, survival, proliferation, invasion, angiogenesis, [[metastasis|Metastasis]], and the expression of certain [[viral|Virus]] genes. It is a centerpiece of the immune response. 

See also:
* [[NF-kB inhibitor|NF-kB Inhibitor]]

* [[WIKIPEDIA - NF-κB|]]
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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]]).
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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
[[\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
[\mb A, [\mb B, \mb C], \mb D]  &= & A^i B^j C^k D^l  A_{iml} T_{jk}^m
[\mb A, \mb B, [\mb C, \mb D]]  &= & A^i B^j C^k D^l  A_{ijm} T_{kl}^m
Hence the position of the commutator equals the position of the summation index of the associator.

!!!!Inner associators
[[\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}
and accordingly
[\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}
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.
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* [[WIKIPEDIA - Nilradical of a Ring|]]
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* [[On the Notion of Lower Central Series for Loops - J. Mostovoy|]]
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''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.

* [[WIKIPEDIA - Nonsymmetric Gravitational Theory|]]
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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]].
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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.

* [[The Octonions - J. C. Baez|]] [[local|papers/S0273-0979-01-00934-X.pdf]]  {{t100Cite{[[pct. 318|]]}}}
* [[Submanifolds of the Cayley Projective Plane with bounded Second Fundamental Form - P. Coulton, J. Glazenbrook|]] [[pct. 4|]]
* [[OP2 Bundles in M-theory - H. Sati|]] [[pct. 2|]]
* [[Plurisubharmonic Functions on the Octonionic Plane and Spin(9)-Invariant Valuations on Convex Sets - S. Alesker|]] [[pct. 1|]]
* [[On the Geometry of the Supermultiplet in M-theory - H. Sati|]] pct. 0
* [[On the Cell Structure of the Octonion Projective Plane - I. Yokota|]] pct. 0
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* [[WIKIPEDIA - Olber's Paradox|'_paradox]]

* [[Physics Lecture:- Olbers Paradox and the Distant Universe - R. Nemiroff|]]
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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''.
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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.

* [[Algebraic Methods in the Design of Space-time Codes - J- Hiltunen|]] - pct. 0
* [[Delay-optimal Rectangular Real Orthogonal Designs from Cayley-Dickson Algebra - S. Das, B. S. Rajan|]] 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:
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An ''Orthogonal Tranfsormation'' $\mb O$ is a linear transformation that preserves the scalar product
\langle \mb O(\mb A)|\mb O (\mb B)\rangle = \langle \mb A| \mb B \rangle
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<font color="lightgrey"><b> Song of the week: </b></font>

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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.

* [[The Universality of Einstein Equations (1993) - M. Ferraris, M. Francaviglia, I. Volovich|]] [[pct. 72|]]
*[[On the so-called "Palatini Method" of Variation in Covariant Gravitational Theories (1973) - A. A. El-Kholy, R. U. Sexl, H. K. Urbantke|]] pct. 0
*[[Palatini Variational Principle for an Extended Einstein-Hilbert Action (1997) - H. Burton, R. B. Mann|]] [[pct. 5|]]

* [[On the Palatini Variation and Connection Theories of Gravity - H. S. Burton (1998)|]] [[local|theses/NQ38225.pdf]] tct. 0
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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 ...
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<html><center><img src="images/paralleltransport.jpg" style="width: 320px; "/></center></html>
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).

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

See also: [[autoparallelity|Autoparallelity]].

* [[On the Parallel Transport of Tetrad in a Riemann-Cartan Spacetime - D.-C. Chern|]] [[pct. 1|]]
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:
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
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.
R_{\mu\nu} =\tilde{R}_{\mu\nu}  + D_\lambda {T^{\lambda}}_{\mu\nu} + T_{\mu\sigma\lambda} {T^{\sigma\lambda}}_{\nu} = 0
This implies that the symmtric and the antisymmetric parts of the Ricci tensor vanish, i.e.
\tilde{R}_{\mu\nu}  + T_{\mu\sigma\lambda} {T^{\sigma\lambda}}_{\nu} = 0
D_\lambda {T^{\lambda}}_{\mu\nu} = 0
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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.

[[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.

Given a [[polyvector|Polyvector Space]] $\mb A = A_\alpha \mb E_\alpha$, a passive transformation is required to not change it. Therefore
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 \\
Hence the [[covariant polyvector derivative|Polyvector Covariant Derivative]] must be zero.
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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]].
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

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

* [[WIKIPEDIA - Perfect Number|]]
* [[The MacTutor History of Mathematics Archive - Perfect Numbers|]]
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A ''Pfaffian Equation'' is defined by relationship between [[forms|Form]]. The theory of Pfaffian equations is a rather difficult subject.
$xdy ? ydx = 0$
''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]].

$n = 1$: $\langle\langle a_1\rangle\rangle = \langle 1, -a_1 \rangle = x_1^2 - a_1x_2^2$.
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 $(8,0)$).

* [[Topics on Quadratic Forms - A. Vishik|]]
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> 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.

* [[Complex Coordinates and Quantum Mechanics (1996) - F. Strocchi|]] [[local|papers/FStrocchi-RevModPhys-38-36-1966.pdf]] {{t100Cite{[[pct. 105|,5&hl=de]]}}}

* [[WIKIPEDIA - Phase Space|]]
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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
\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\}
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{{{<<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.
Default player size:
width: <<option txtPlayerDefaultWidth>> height: <<option txtPlayerDefaultHeight>>
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
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")
	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')
	var show=true;
	if (params[0]=='showcontrols'||params[0]=='true'||params[0]=='false')
	var extras="";
	while (params[0]!=undefined)
		extras+="<param name='"+params.shift()+"' value='"+params.shift()+"'> ";

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 :-(

	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
		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
		case "realone":
			height-=show?60:0; // leave room for controls
		case "quicktime":
		case "image":
		case "iframe":

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

	var html=config.macros.player.html[type];

// // 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
	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;
	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
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; \

// JPG/GIF/PNG still images
	<a href="%u%" target="_blank"><img width="%w%" height="%h%" style="display:%s%;" src="%u%"></a>';

// IFRAME web page viewer
	<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' \
	<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=",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="" \
			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> \

// 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="AUTOSTART" Value="%a%"> \
		<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="NOJAVA" 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> \

// 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=""> \
		<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"> \
		%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" \
		</EMBED> \

// Flash Player
	<object id="%i%" width="%w%" height="%h%" style="margin:0;padding:0;" \
		classid="clsid:D27CDB6E-AE6D-11cf-96B8-444553540000" \
		codebase=",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="" \
			type="application/x-shockwave-flash" scale="exactfit"> \
		</embed> \
<<tiddler [[include_tiddlers/Plotkin Construction.html#"Plotkin Construction"]]>>
|''Description''|Displays TeX math using MathJax|
|''Author''|[[Canada East|]]|
|''Requires''|[[MathJax v1.01|]]|
|''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.|
This plugin uses [[MathJax|]] to typeset ([[AMS|]]) [[LaTeX|]]  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."
Right click any math display for a MathJax menu. The user can select the renderer and zoom settings. It performs best in [[Webkit|]] 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|]]
#''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).
|{{{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|]] |>|
[[Additional MathJax Examples|MathJax Examples]]
MathJax can be manually configured if desired by editing the code below (advanced). See the [[MathJax documentation|]] 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

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
             * I've had to point to the 'MathJax
             * Content Delivery Network' instead.
            // 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", ""],' +
            '"HTML-CSS": {' +
                'scale: 85' +
                '}' +
            '});' +


            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;


            // 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,
                    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.nextMatch = endRegExp.lastIndex;

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

              name: "inlineMath1",
              match: "\\\$",
              terminator: "\\\$",
              termRegExp: "\\\$",
              element: "script",
              type: "math/tex",
              handler: config.formatterHelpers.mathFormatHelper

            var backslashformatters = new Array(0);

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

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

              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} ...
              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.
                name: "escape",
                match: "\\\\.",
                handler: function(w) {
                    w.nextMatch = w.matchStart+2;


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




<<tiddler [[include_tiddlers/Podkletnov Experiments.html#"Podkletnov Experiments"]]>>
<<tiddler [[include_tiddlers/Poikilotherm.html#"Poikilotherm"]]>>
<<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/Point Mutation.html#"Point Mutation"]]>>
<<tiddler [[include_tiddlers/Point Spread Function.html#"Point Spread Function"]]>>
<<tiddler [[include_tiddlers/Poisson Bracket.html#"Poisson Bracket"]]>>
<<tiddler [[include_tiddlers/Poisson Distribution.html#"Poisson Distribution"]]>>
<<tiddler [[include_tiddlers/Poisson Manifold.html#"Poisson Manifold"]]>>
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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.
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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).

* [[Spinning Relativistic Particles in External Fields - I.B. Khriplovich|]]
* [[Classical and Quantum Spins in Curved Spacetimes - A. J. Silenko|]]   [[Transparencies|]] 
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The ''Poynting Theorem'' in electrodynamics states:
\vec E \cdot \vec  j + \vec \nabla \cdot \vec  S  -\frac{\partial \rho}{\partial t} = 0
With $\rho$ the energy density of the electromagnetic field:
\rho = \frac{1}{2} ( \vec E  \cdot \vec D + \vec B \cdot \vec H)
and $\vec S$ the Poynting vector
\vec S = \vec E \times \vec H
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* [[This Time - What a Strange Turn of Events! (2010)- P. E. Gibbs|]] [[local|papers/9.pdf]] pct. 0

* [[WIKIPEDIA - Event Symmetry|]]
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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]]. 
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A ''Pyrochlore Lattice'' is built up of tetrahedra.
<html><center><img src="images/pyrochlore.jpg" style="width: 265px; "/></center></html>
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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 $b(\mb A, \mb B) \equiv \mathcal Q(\mb A + \mb B) - \mathcal Q(\mb A) - \mathcal Q( \mb B)$.
* [[Arizona Winter School 2009: Quadratic Forms|]]
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<br> {{center{[img(462px+, )[images/QFC.jpg]]}}}
It's hard to have an idea and somebody didn't have it before. This is what happened to me with the ''Quantum Field Computer'' (''QFC''), although it seems I have a bit of a different take on the subject. More details can be found under [[unitary inequivalence|Unitary Inequivalence]].
Quantum field computing, which is computing with the "continuum" rather than digital computing, I suspect to be the ultimate computing paradigm, way superior to any form of classical or [[quantum computing|Quantum Computer]]. I think it should allow to proof the [[Church-Turing Hypothesis|Church-Turing Hypothesis]] to be wrong. In its full fledged form it must involve all the forces of nature, in particular [[gravity|Gravitation]]. (An implementation though requires a better understanding of [[quantum gravity|Quantum Gravity]]). Moreover it should allow for answering the question as to how to construct real [[AI|Artificial Intelligence]] and solve the [["hard problem" of consciousness|Hard Problem of Consciousness]] in philosophy, paving the way to building "conscious machines". If the conscious mind involves computations based on [[quantum field theory|Quantum Field Theory]], there had to be elements of [[non-computability|Non-Computability]], something which has already been suspected by some people (e.g. [[Roger Penrose]]).
For a good understanding of the subject, it would be crucial to know how to build a quantum field computer "from scratch". At the moment I do not have a good idea how to do this.

If the human [[brain|Brain]] is a quantum field computer, then it must have an uncountably infinite number of states. If one divides this number by the number of atoms in the brain (which is finite), one is left with an uncountably infinite number of states. The consequence is that elementary particles must already be conscious and are the building blocks of "higher" forms of consciousness. Again, not really a new idea. Thus an elementary particle is a small cosmos all by its own, having an incredible computational capacity, by far exceeding that of any classical computer. Therefore, emulating such a system on a classical computer will never be possible. The best one can do is to approximate it to a certain degree.

!!!! Some postulates
A QFC ...
* is non-[[deterministic|Determinism]]. Although it may be constrained to a certain degree, leading to [[superselection rules|Superselection]]. Thus in a way any quantum field computer has generic [[free will|Free Will]]. <br><br>
* can emulate biological systems ([[quantum field biology|Quantum Field Biology]]). <br><br>
* can simulate a [[Hilbert hotel|Hilbert's Hotel]]. A rearrangement in the hotel can be interpreted as an elementary operation of a QFC. (An example would be a "global shift operation", letting all people go to rooms with even numbers). The point is that any operation is global, i.e. it does not involve the propagation of a signal limited by the speed of light. An idea is that this "update" of Hilbert's hotel is a [[quantum tunneling|Tunneling in Quantum Field Theory]] process between inequivalent quantum vacua. I.e. the paradigm of a QFC would be very much that of an [[infinite cellular automaton|Quantum Field Cellular Automaton]] with Hilbert's hotel being a nice illustrative example. The crucial difference between a Turing machine and a QFC is that the former only can do local changes (on the Turing strip), i.e. a finite number of bits are flipped at a time, whereas the latter is bound to do an infinite number of changes of states per time step otherwise there is no transition to a new vacuum. <br><br>
* never "crashes" - it just can't do so by definition. Concerning nature, what has been very intriguing to me is that if it is doing a huge computation (some even believe it's a [[simulation|The Simulation Hypothesis]]) based on a "digital" algorithm, why does it "never" crash ("never", for all practical purposes) ? But if the fundamental building blocks of nature are quantum fields - which is state of the art of our understanding - then an explanation is at hand. (See also [1] for more details).

* What is the smallest QFC possible ? The answer could come from biology and "living" systems. <br><br>
* Can we find a generic QFT effect we "cannot put on a conventional machine". (There are some hints of such effects, e.g. [[chiral fermions|Chiral Fermion]] in [[lattice field theory|Lattice QCD]]).
&nbsp;&nbsp; ... etc. pp. -  an awful lot remains to be understood !

See also:
* [[Quantum field cellular automaton|Quantum Field Cellular Automaton]]
* [[Digital physics|Digital Physics]]
* [[Is nature infinite ?|Is Nature Infinite ?]]

* [[P/NP, and the Quantum Field Computer (1998) - M. H. Freedman|]] [[local|papers/98.full.pdf]] {{t100Cite{[[pct. 176|,5&hl=de]]}}}
* [[Quantum Algorithms for Quantum Field Theories (2011) - S. P. Jordan, K. S. M. Lee, J. Preskill|]] [[local|papers/1111.3633v2.pdf]] {{t100Cite{[[pct. 133|,5&hl=de]]}}}
* [[[1] The Dissipative Brain (2004) - G. Vitiello|]] [[local|papers/0409037v1.pdf]] [[pct. 58|,5&hl=de]]
* [[The Unity between Quantum Field Computation, Real Computation, and Quantum Computation (2001) - A. C. Manoharan|]] [[local|papers/0109015v1.pdf]] [[pct. 5|]]
* [[Beyond Quantum Computation and Towards Quantum Field Computation (2003) - A. C. Manoharan|]] [[local|papers/0304136v1.pdf]] pct. 0
* [[QFT + NP = P Quantum Field Theory (QFT): A Possible Way of Solving NP-Complete Problems in Polynomial Time (1996) - A. Beltran, V. Kreinovich, L. Longpré|]] [[local|papers/search.htm]] pct. 0
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See [[Hopf Algebra]]. 
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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]].

* [[Quasialgebra Structure of the Octonions - H. Albuquerque, S. Majid|]] [[pct. 49|]]
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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 -
<<tiddler [[include_tiddlers/Quotient Group.html#"Quotient Group"]]>>
<<tiddler [[include_tiddlers/R (Programming Language).html#"R (Programming Language)"]]>>
<<tiddler [[include_tiddlers/REDUCE.html#"REDUCE"]]>>
<<tiddler [[include_tiddlers/REM Sleep.html#"REM Sleep"]]>>
<<tiddler [[include_tiddlers/REST API.html#"REST API"]]>>
<<tiddler [[include_tiddlers/RNA.html#"RNA"]]>>
<<tiddler [[include_tiddlers/Radial Acceleration Relation.html#"Radial Acceleration Relation"]]>>
<<tiddler [[include_tiddlers/Radiation Dose Quantities.html#"Radiation Dose Quantities"]]>>
<<tiddler [[include_tiddlers/Radiation Reaction.html#"Radiation Reaction"]]>>
<<tiddler [[include_tiddlers/Radiation Related Units.html#"Radiation Related Units"]]>>
<<tiddler [[include_tiddlers/Radical Life Extension.html#"Radical Life Extension"]]>>
<<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/Rare Earth Hypothesis.html#"Rare Earth Hypothesis"]]>>
<<tiddler [[include_tiddlers/Rate of Living Theory.html#"Rate of Living Theory"]]>>
<<tiddler [[include_tiddlers/Raw Foodism.html#"Raw Foodism"]]>>
<<tiddler [[include_tiddlers/Raw Image Format.html#"Raw Image Format"]]>>
<<tiddler [[include_tiddlers/Ray.html#"Ray"]]>>
* [[WIKIPEDIA - Raychaudhuri Equation|]]
<<tiddler [[include_tiddlers/Rayleigh Scattering.html#"Rayleigh Scattering"]]>>
<<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/Read Noise.html#"Read Noise"]]>>
<<tiddler [[include_tiddlers/Real Projective 3-space.html#"Real Projective 3-space"]]>>
<<tiddler [[include_tiddlers/Real Projective Plane.html#"Real Projective Plane"]]>>
<<tiddler [[include_tiddlers/Real Spectral Triple.html#"Real Spectral Triple"]]>>
<<tiddler [[include_tiddlers/Real Structure.html#"Real Structure"]]>>
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<<tiddler [[include_tiddlers/Reconstruction Theorem.html#"Reconstruction Theorem"]]>>
<<tiddler [[include_tiddlers/Red Cell Distribution Width.html#"Red Cell Distribution Width"]]>>
<<tiddler [[include_tiddlers/Redox State.html#"Redox State"]]>>
<<tiddler [[include_tiddlers/Redshift Quantization.html#"Redshift Quantization"]]>>
<<tiddler [[include_tiddlers/Reductio Ad Absurbum.html#"Reductio Ad Absurbum"]]>>
<<tiddler [[include_tiddlers/Reductionism.html#"Reductionism"]]>>
A (locally) ''Red