Cognitive Physics, What [i]could[/i] physics rationally be?

Some basic stastistical concepts...


GENERALIZED MEAN
http://en.wikipedia.org/wiki/Generalized_mean
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A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric and harmonic means.

Signal processing
A power mean serves a non-linear moving average which is shifted towards small signal values for small p and emphasizes big signal values for big p. Given an efficient implementation of a moving arithmetic mean called smooth you can implement a moving power mean according to the following Haskell code.

powerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a]
powerSmooth smooth p =
map (** recip p) . smooth . map (**p)
For big p it can serve an envelope detector on a rectified signal.
For small p it can serve an baseline detector on a mass spectrum
.
http://en.wikipedia.org/wiki/Generalized_mean


PYTHAGOREAN MEANS
http://en.wikipedia.org/wiki/Pythagorean_means
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The three classical Pythagorean means are the arithmetic mean (A), the geometric mean (G), and the harmonic mean (H). They are defined by:




MOVING AVERAGE
http://en.wikipedia.org/wiki/Moving_average
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In statistics, a moving average or rolling average is one of a family of similar techniques used to analyze time series data. It is applied in finance and especially in technical analysis. It can also be used as a generic smoothing operation, in which case the raw data need not be a time series.

A moving average series can be calculated for any time series. In finance it is most often applied to stock prices, returns or trading volumes. Moving averages are used to smooth out short-term fluctuations, thus highlighting longer-term trends or cycles. The threshold between short-term and long-term depends on the application, and the parameters of the moving average will be set accordingly.

Mathematically, each of these moving averages is an example of a convolution. These averages are also similar to the low-pass filters used in signal processing.



ROOT MEAN SQUARE
http://en.wikipedia.org/wiki/Root_mean_square
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In mathematics, root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g. WAVES.

It can be calculated for a series of discrete values or for a continuously varying function. The name comes from the fact that it is the square root of the mean of the squares of the values. It is a power mean with the power p = 2.

sqrt S (x1^2 + x2^2 + x3^3.... + xn^2)/n)

This post has been edited by Raphie Frank on Dec 3 2007, 12:05 AM

ABOUT SYNCRETISM
( a necessary concept to master for any aspiring "Cognitive Physicist"

The coinage of "Syncreticism" in its modern form, as I recall, has been attributed to Carl Jung who also coined the term "synchronicity."

MORE:
http://profile.myspace.com/index.cfm?fusea...endID=196999694.

MORE:
(via Wikipedia aka "The Syncretic Global Mind" http://en.wikipedia.org/wiki/Syncretism)

Syncretism consists of the attempt to reconcile disparate or contradictory beliefs, often while melding practices of various schools of thought. The term may refer to attempts to merge and analogize several originally discrete traditions, especially in the theology and mythology of religion, and thus assert an underlying unity allowing for an inclusive approach to other faiths.
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The Oxford English Dictionary first attests the word syncretism in English in 1618. It derives from modern Latin syncretismus, drawing on Greek... (synkretismos), meaning "a union of communities".
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The Greek word occurs in Plutarch's (1st century AD) essay on "Fraternal Love" in his Moralia (2.490b). He cites the example of the Cretans, who reconciled their differences and came together in alliance when faced with external dangers. "And that is their so-called Syncretism".

In linguistic syncretism, one word-form serves two or more morphosyntactic functions. Some inflected words or word forms in some natural languages indicate (morphologically) a distinction in syntax, while some other words in the same language do not. For example in Russian, some nouns have different word forms (inflections) in nominative and accusative (kniga and knigu respectively) while some other nouns (pismo, pismo) inflect without a distinction. The former indicate a distinction in the Russian syntax while the latter hide that distinction.
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Religious syncretism exhibits the blending of two or more religious belief systems into a new system, or the incorporation into a religious tradition of beliefs from unrelated traditions. This can occur for many reasons, and the latter scenario happens quite commonly in areas where multiple religious traditions exist in proximity and function actively in the culture.
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Other forms of syncretism not directly related to religion appear in the modern world as well: thus one can sometimes speak of cultural and/or social syncretism. Japanese culture after World War II and the moderate tendencies within Neo-Tribalism may serve as examples.


RELATED POST ON "SYNCHRONICITY"
http://forum.physorg.com/index.php?showtop...60&#entry286799


photo by Mattijn Frannsen of The Netherlands
http://www.flickr.com/photos/mattijn/321728633/

Yes, Majkl. Synchronicity, coined by Carl Jung, is certainly one of the implications of Garrett Lisi's theory and it is a step towards unification of mind and matter, animate and inanimate, continuum and bit string. A bit more dramatically stated with an eastern overtone:

The elephant of many parts is coming into the light.

A very good book I suggest is "Synchronicity: The Bridge Between Matter and Mind" by F. David Peat

"For the sceptic, coincidences are the jokers in the deck of life. For the searcher, they are the key to synchronicity. As physicists search for a unified field theory, so Carl Jung and others searched for synchronicity -- the unifying principle behind meaningful coincidence, individual consciousness and the totality of space and time. Now, F. David Peat joins these two quests into one intriguing journey to show the connection between quantum theory and synchronicity, and to open the way to an exciting new understanding about the bridge between matter and mind."

This post has been edited by Raphie Frank on Dec 3 2007, 04:13 AM

ABOUT TWIN PRIMES
http://en.wikipedia.org/wiki/Twin_prime

A twin prime is a prime number that differs from another prime number by two. Except for the pair (2, 3), this is the smallest possible difference between two primes. Some examples of twin prime pairs are (5, 7), (11, 13),(41, 43), and (821, 823). Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin.

The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture. A strong form of the twin prime conjecture, the Hardy-Littlewood conjecture, postulates a distribution law for twin primes akin to the prime number theorem.

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199 and 201 also comprise a twin prime pair, a point I make in relation to "KOLO MATH" as this pair "bounds" D == 200 == "Diameter" , conceived as 2R, where R == Rho == 100 == "Radius", used in mathematics to represent the radius in a system of spherical polar coordinates...
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FROM SLOANE'S Search: id:A001359,

Lesser of twin primes...
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3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607
http://www.research.att.com/~njas/sequences/A001359

Greater of twin primes...
---------------------------
5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609
http://www.research.att.com/~njas/sequences/A006512

This post has been edited by Raphie Frank on Dec 3 2007, 04:21 AM

EVOLUTION OF NUMBERS, LEONARDO DA PISA & THE DISUNITY OF MATHEMATICS

an interesting blog post I came across...
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The Evolution of Numbers
Posted on Tuesday 13 February 2007

Counting in groups of ten, a practice evidently suggested by the number of fingers on two hands has been practiced by many cultures for millennia . But the act of writing them down using a system of place notation system which can expand without limit is only comparatively recent innovation.

Here then is the story of 1 to 9 and of 0 and 10...

From LaputanLogic.com
http://www.laputanlogic.com/articles/2003/...1-95210802.html

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Also of interest is to note that the Arabic Number System was introduced to Europe by none other than "crackpot" extraordinairre :-) Leonardo of Pisa, otherwise know as Finonacci"

Fibonacci, an Italian mathematician who had studied in Bejaia (Bougie), Algeria, promoted the Arabic numeral system in Europe with his book Liber Abaci, which was written in 1202, still describing the numerals as Indian rather than Arabic.

"When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it.."

MORE
http://en.wikipedia.org/wiki/Arabic_numerals

ABOUT LEONARDO OF PISA
http://en.wikipedia.org/wiki/Leonardo_of_Pisa

Back in the day, or so they say, Fibonacci used to take on the abacists in the then contemporary version of Kasparov against Deep Blue and would "smoke" them in a runaway. The Arabic Number system won out in the end and replaced all those D's and X's and M's because the system he introduced simply made more sense. In the post-post modern age, however, the issue is nor one of "replacement," but far more one of "extension" and "syncretic blending" in what I fondly refer to as our "Babelized" or "Humpty-Dumptified" world culture and what Sir Michael Atiyah might term as mathematical "disunity"

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"If Newton, Gauss or even Archimedes were to return, I believe that after a short course to learn the new jargon, they would understand and even approve of the progress that has been made. . . . Can mathematics continue at this ever-increasing rate and still remain the subject we love?"

from a review of...

Mathematics: Frontiers and Perspectives. By V. Arnold, M. Atiyah, P. Lax, and B. Mazur (editors) American Mathematical Society, Providence, RI, 1999, xi + 459 pages, $49.00.
http://www.siam.org/news/news.php?id=658
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ABOUT SYNCRETISM

excerpt from previous post...
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Syncretism consists of the attempt to reconcile disparate or contradictory beliefs, often while melding practices of various schools of thought. The term may refer to attempts to merge and analogize several originally discrete traditions, especially in the theology and mythology of religion, and thus assert an underlying unity allowing for an inclusive approach to other faiths.

http://forum.physorg.com/index.php?showtop...ndpost&p=287889

This post has been edited by Raphie Frank on Dec 3 2007, 04:58 AM

Dear Isee,

I replied to you in two places, three, including this post regarding observer POV and E8 as it relates to Quasicrystals...

THE QUANTUM GOSPEL
http://forum.physorg.com/index.php?act=ST&...160#entry287904

excerpt...
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Each Gospel is one writer's "truth," but the "Truth" is in the totality and requires the observer POV, the "fifth eye" so to speak, to interpret and recognize that totality. That said, the "Truth" one discerns will always be RELATIVE to one's frame of reference. Think about that in relation to why Jesus the man, spoke in parables and allegories. He was giving shape to what one might term the "crystal," or perhaps better stated the "quasicrystal" of Man and Woman and all manner of lifeform.

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E8 AND QUASICRYSTALS
http://forum.physorg.com/index.php?showtop...60&#entry287906

The E8 lattice and quasicrystals: geometry, number theory and quasicrystals

J F Sadoc and R Mosseri
Lab. de Phys. des Solides, Orsay, France
J F Sadoc et al 1993 J. Phys. A: Math. Gen. 26 1789-1809 doi:10.1088/0305-4470/26/8/009
Print publication: Issue 8 (21 April 1993)

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I might add that the mere fact of having to post the reply in two different places is itself evidence of the "fractured" nature of knowledge...

Also Related (a response in advance so to speak...)

THE DOWNSIDE OF KNOWING TOO MUCH
Each academic discipline or industry carries with it it's own argot (aka specific vocabulary). Talk to an economist and you'll throw out concepts such as Laffer Curve, Keynsian Intervention or Utility Theory without a pause. Talk to a social anthropologist and expressions such as the other, normative or the rules of the game are no brainers. A poet will nod knowingly at your mention of metonymy or mellissonance, while a physicist will laugh at your mention of Schrodinger's Date. Especially beware the film lighting technician! He will walkie you with jabberwockish (aka nonsensical) ramblings along the lines of: Hey, after you get done twisting the rag and stopping by crafty, 86 that pigeon on a pancake and the blond for the Abbie Singer and give me a senior on a turtle instead..

from Snipes, Logomancy & So So Psychosis
A potpourri of the creative, political, hopeful, amusing and eclectic
Wednesday, March 22, 2006
MORE: http:/boojummy.blogs.friendster.com/snipe...ownside_of.html

Best,
Raphie

This post has been edited by Raphie Frank on Dec 3 2007, 06:29 AM

@Raphie
I still think your simple manipulations of numbers is missing the bigger picture. I identify many of the patterns you find as simple reflections of deeper principles. And, others as arbitrary associations with no further significance. I say this with sincerity (and I'm sure in your mind, ignorance).

I recommended once before that you read "Chaos" by James Gleick. I'm not sure you read that post, so I'll recommend it again. Seriously. If you are searching for clues about nature, about patterns, about the mathematics of reality, then that specific book will kick you in the head.

MORE ON QUASICRYSTALS: SOMEWHERE BETWEEN ORDER AND DISORDER


FROM QUASICRYSTALS TO KLEENEX
by Alison Boyle
http://plus.maths.org/issue16/features/penrose/index.html
NOTE: a well-written quite accessible read related to Penrose Tiling and it's connection to Quasicrystals


excerpts from Wikipedia
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Quasicrystals are structural forms that are both ordered and nonperiodic. The term and the concept were introduced originally to denote a specific arrangement observed in solids which can be said to be in a state intermediary between crystal and glass. Producing Bragg diffraction, they share a defining property with crystals, but differ from them by lacking a simply repeating structure. Mathematical artefacts, known as aperiodic tiling, were invented in the early 1960s, but some twenty years later physical experiments gave evidence of their material existence. Within the field of crystallography and solid state physics the discovery has produced a paradigm shift which is indeed a minor scientific revolution.[1]

An ordering is nonperiodic if it lacks translational symmetry, which means that a shifted copy will never match exactly with its original.

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RELATED DISCIPLINE
Noncommutative geometry, or NCG, is concerned with the possible spatial interpretations of algebraic structures for which the commutative law fails; that is, for which xy does not always equal yx. For example; 3 steps of 4 units and 4 steps of 3 units length might be different in noncommutative spaces.
MORE: http://en.wikipedia.org/wiki/Noncommutative_geometry]
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Mathematical description

The intuitive considerations obtained from simple model aperiodic tilings are formally expressed in the concepts of Meyer and Delone sets. The mathematical counterpart of physical diffraction is the Fourier transform and the qualitative description of a diffraction picture as 'clear cut' or 'sharp' means that singularities are present in the Fourier spectrum. There are different methods to construct model quasicrystals. These are the same methods that produce aperiodic tilings with the additional constraint for the diffractive property. Thus for a Substitution tiling the eigenvalues of the substitution matrix should be Pisot numbers. The aperiodic structures obtained by the cut and project method are made diffractive by chosing a suitable orientation for the construction. This is indeed a geometric approach which has also a great appeal for physicists.
http://en.wikipedia.org/wiki/Quasicrystal


Quasicrystals: Somewhere Between Order And Disorder
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ScienceDaily (May 29, 2007) — Professionally speaking, things in David Damanik's world don't line up -- and he can prove it.

In new research that's available online and slated for publication in July's issue of the Journal of the American Mathematical Society, Damanik and colleague Serguei Tcheremchantsev offer a key proof in the study of quasicrystals, crystal-like materials whose atoms don't line up in neat, unbroken rows like the atoms found in crystals. Damanik's latest work focused on a popular model mathematicians use to study quasicrystals. The research, which was 10 years in the making, proves that quasicrystals in the model are not electrical conductors and sheds light on a little-understood corner of materials science.

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Until 1982, quasicrystals weren't just undiscovered, they were believed to be physically impossible. To understand why, it helps to understand how atoms line up in a crystal.

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"Mathematically speaking, quasicrystals fall into a middle ground between order and disorder," Damanik said. "Over the past decade, it's become increasingly clear that the mathematical tools that people have used for decades to predict the electronic properties of materials will not work in this middle ground."

Best,
Raphie

P.S. @MeBigGuy: Thank you for the book recommendation. I hope to have a moment to check it out in the near future. Best, RF

This post has been edited by Raphie Frank on Dec 3 2007, 06:40 AM

IONIAN ENCHANTMENT

I have mentioned numerous times within the confines of this forum, the notion of Consilience aka "The Unity of Knowledge." At heart of the concept is the notion of "Ionian Enchantment," a concept that captured my imagination a couple years ago and only gained force with Garrett Lisi's Recent Paper "An Exceptionally Simple Theory of Everything..."

See...

Toward an EXCEPTIONAL Simple Theory Of Everything :: Celestial Chiaroscuro
http://raphie.wordpress.com/2007/11/24/tow...olemys-revenge/

Below is a bit more about the concept as put forth by Evolutionary Biologist Edward O. Wilson. It is highly worth a read...

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Consilience: Unity of Knowledge by Edward O. Wilson by Alfred A. Knopf
http://www.cnn.com/books/beginnings/9805/consilience/

(CNN) -- In "Consilience", Edward Wilson aims to establish that all knowledge and understanding is bound together by some as yet unknown common theory. He argues that there is one grand scheme to explain and unite all that we know and can know.

=======================================================
From Chapter One

By inwardly manipulating time and space, I found I could climb the steps in biological organization from microscopic particles in cells to the forests that clothe mountain slopes. A new enthusiasm surged through me. The animals and plants I loved so dearly reentered the stage as lead players in a grand drama. Natural history was validated as a real science.

I had experienced the Ionian Enchantment. That recently coined expression I borrow from the physicist and historian Gerald Holton. It means a belief in the unity of the sciences -- a conviction, far deeper than a mere working proposition, that the world is orderly and can be explained by a small number of natural laws. Its roots go back to Thales of Miletus, in Ionia, in the sixth century B.C. The legendary philosopher was considered by Aristotle two centuries later to be the founder of the physical sciences. He is of course remembered more concretely for his belief that all matter consists ultimately of water. Although the notion is often cited as an example of how far astray early Greek speculation could wander, its real significance is the metaphysics it expressed about the material basis of the world and the unity of nature.

The Enchantment, growing steadily more sophisticated, has dominated scientific thought ever since. In modern physics its focus has been the unification of all the forces of nature -- electroweak, strong, and gravitation -- the hoped-for consolidation of theory so tight as to turn the science into a "perfect" system of thought, which by sheer weight of evidence and logic is made resistant to revision. But the spell of the Enchantment extends to other fields of science as well, and in the minds of a few it reaches beyond into the social sciences, and still further, as I will explain later, to touch the humanities. The idea of the unity of science is not idle. It has been tested in acid baths of experiment and logic and enjoyed repeated vindication. It has suffered no decisive defeats. At least not yet, even though at its center, by the very nature of the scientific method, it must be thought always vulnerable. On this weakness I will also expand in due course.

NEXT POST: THE C-IONIAN MODE (C-MAJOR) AND & SOME BASIC INFO ON OCTAVES & RATIOS IN MUSIC
http://forum.physorg.com/index.php?showtop...ndpost&p=288251

This post has been edited by Raphie Frank on Dec 4 2007, 10:20 AM

THE C-IONIAN MODE (C-MAJOR) AND & SOME BASIC INFO ON OCTAVES & RATIOS IN MUSIC

Related Post:
IONIAN ENCHANTMENT
http://forum.physorg.com/index.php?showtop...ndpost&p=288248

Basically, in a nutshell, from Guitar Secrets, here is the run down...

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The first mode you will learn out of the Key of C Major will be the C Ionian mode. Each mode in the key of C has a different name. The names are from the ancient Greek language. The Ionian mode is a medieval church mode corresponding to the modern major diatonic scale, or the first mode of the major key. The first mode in the major key is always the Ionian mode. This means if you are in the key of G major, G would be the Ionian mode.
http://guitarsecrets.com/c_scales.htm
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In other words...

[DO RE MI FA SO LA TI...] DO

7 notes and then back around to the beginning... FULL CIRCLE

7 + 1 = 8

An octave, of course as Wikipedia consicely notes is "sometimes abbreviated 8ve or P8" and it is "the interval between one musical note and another with half or double its frequency."

From a mathematical perspective, here are the numbers underlying the concept of the octave, whch can be conceived of as a "circle" with orders of exponential progression. You go around full circle in spiral manner and end up back where you started *2 or /2...

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For example, if one note has a frequency of 400 Hz, the note an octave above it is at 800 Hz, and the note an octave below is at 200 Hz. The ratio of frequencies of two notes an octave apart is therefore 2:1. Further octaves of a note occur at 2n times the frequency of that note (where n is an integer), such as 2, 4, 8, 16, etc. and the reciprocal of that series. For example, 50 Hz and 400 Hz are one and two octaves away from 100 Hz because they are (or 2 ? 1) and 4 (or 22) times the frequency, respectively. However, 300 Hz is not a whole number octave above 100 Hz, despite being a harmonic of 100 Hz.
MORE: http://en.wikipedia.org/wiki/Octave
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Interestingly enough, set theory, abstract algebra, and number theory. Wikipedia notes that "the notes in an equal temperament octave form an abelian group with 12 elements. It is in fact possible to describe just intonation in terms of free abelian group."

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Free abelian group
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In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Unlike vector spaces, not all abelian groups have a basis, hence the special name for those that do. A typical example of a free abelian group is the direct sum Z ? Z of two copies of the infinite cyclic group Z; a basis is {(1,0),(0,1)}. The trivial abelian group {0} is also considered to be free abelian, with basis the empty set.
http://en.wikipedia.org/wiki/Free_abelian_group
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And here are the ratios...

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Note Ratio Interval
0 1:1 unison
1 135:128 major chroma or minor second
2 9:8 major second
3 6:5 minor third
4 5:4 major third
5 4:3 perfect fourth
6 45:32 diatonic tritone
7 3:2 perfect fifth
8 8:5 minor sixth
9 27:16 Pythagorean major sixth
10 9:5 minor seventh
11 15:8 major seventh
12 2:1 octave

To calculate the frequency of a note in a scale given in terms of ratios, the frequency ratio is multiplied by the frequency we associate to the unison, which will often be the tonic frequency. For instance, with a tonic of A4 (A natural above middle C), the frequency is 440 Hz, and a justly tuned fifth above it (E5) is simply 440*(3/2) = 660 Hz.
MORE: http://en.wikipedia.org/wiki/Music_and_mathematics
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Of particular interest, to me at least, is the notion of the "syntonic comma" based on the ratio 81 : 80 equal to 1.0125

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In music theory, the syntonic comma, also known as the comma of Didymus or Ptolemaic comma, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents. Two notes that differ by this interval would sound different from each other even to untrained ears[1], but would be close enough that they would be more likely interpreted as out-of-tune versions of the same note than as different notes.

Another way of describing the syntonic comma, as a combination of more commonly encountered intervals, is the difference between four justly tuned perfect fifths, and two octaves plus a justly tuned major third. A just perfect fifth has its notes in the frequency ratio 3:2, which is equal to 701.955 cents, and four of them are equal to 2807.82 cents. A just major third has its notes in the frequency ratio 5:4, which is equal to 386.31 cents, and one of them plus two octaves is equal to 2786.31 cents. The difference between these is 21.51, a syntonic comma.
MORE: http://en.wikipedia.org/wiki/Syntonic_comma
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And here is a bit more on the evolution of the Ionian Scale...

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IONIAN SCALE AS ELEVENTH MODE

In 1547, Heinrich Glarean published his Dodecachordon. Central to its premise was the idea that there were twelve diatonic modes rather than eight. It seems that the additional modes were used in popular folk music, but were not part of the official church repertory. Glarean borrowed the Greek term Ionian for a quite different mode.

He added Ionian as the name of the new eleventh mode: the relative natural mode in C with the perfect fifth as its dominant, reciting note or tenor. The twelfth mode was the plagal version of the Ionian mode, called Hypoionian (under Ionian), based on the same relative scale, but with the major third as its tenor, and having a melodic range from a perfect fourth below the tonic, to a perfect fifth above it.
MORE: http://en.wikipedia.org/wiki/Ionian_mode
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TANTEXTUAL LINK (Qualitative Reference Point)

Introducing “Trickle Up Economics” (aka “The One Song”)
http://raphie.wordpress.com/2006/07/02/int...a-the-one-song/

This post has been edited by Raphie Frank on Dec 4 2007, 10:24 AM

Evolving Points of Reference - Cognitive Physics (Part IV: Conceptual Influences)

CONCEPTUAL INFLUENCES
Interdisciplinary Bricolage and Distillations
=======================

Also see…
Part I - MAINSTREAM PRESS
Part II - BOOKS
Part III - SELECTED PAPERS & ESSAYS

http://raphie.wordpress.com/2007/12/14/evo...ual-influences/

Best,
Raphie

P.S. StevenA, Here is a fun "Looking Glass Transform" equation for you to play with:

x^2 + 2xy + y^2

I derived it by deconstructing Base 10 into component digits. Thus, for instance, 81 --> x = 8, y = 1. As such, given that the x component is a multiple of 10, you can also play with the following formula which is essentially related to Taylor Expansions, specifically the MacLaurin Series

(10*x)^2 + 2(10*x)y + y^2

Try plugging in different values such as x = y or x = y+1 etc. You'll see the pattern emerge quite quickly I am sure. Also play with exponents of phi 1/phi, etc. and remember that phi^n + phi^-n = L_n where L_n is a Lucas Number.

========================
The Maclaurin series for any polynomial is the polynomial itself.

The Maclaurin series for (1 ? x) ? 1 is the geometric series

http://en.wikipedia.org/wiki/Taylor_series

This post has been edited by Raphie Frank on Dec 14 2007, 06:42 PM

How can anyone go on and on about musical scales without ever mentioning the twelfth root of 2?

http://en.wikipedia.org/wiki/Twelfth_root_of_two

History of the Twelfth root of two...



The twelfth root of two was first calculated accurately by the Chinese mathematician Prince Chu Tsai-Yu of the Ming Dynasty. In 1596, he published a work, Lu lu ching i ("A clear explanation of that which concerns the lu (musical pipes)"), which gave theoretical pipe lengths for 12-tone equal temperament correct to nine places. Prince Chu made note of the difference between his ideal mathematically-tuned lu and traditional pipes, which used a form of Pythagorean tuning.

This would be calculated again later in 1636 by the French mathematician Marin Mersenne, and as the techniques for calculating logarithms became widely known, this calculation would eventually become trivial.

========================================

Thank you to MeBigGuy who also posted a very informative post on three phase electricity and Tesla coils, etc...
SEE: http://forum.physorg.com/index.php?showtop...75&#entry291823

Mathematics of the Twelfth root of two...
http://en.wikipedia.org/wiki/Twelfth_root_of_two

This post has been edited by Raphie Frank on Dec 15 2007, 09:07 PM

Raphie,

POV is the source of the confusion. And how we eliminate this confusion is through the use of the scientific method. If you look and listen to all the things people talking about are mostly things that they think they know or seen. How do we know what the truth is? This is a matter of problem solving.

POV, opinions, and questions are the things we do in social club. Lots of idea, lots of question, lots of gray area, and lots of loose end. If we look at today sports such as the NFL, tennis, and basket ball we see that the use of cameras in combination with officials eliminates all doubts and suspicion. When the leading official made his final judgement call it is said that he has the TRUTH or the correct definition and resolution for the situation at hand. The definition standard here is the rule of the game and the reolution standard is how clear the officials see the actions. The resolution is gain with the help of the cameras. This is one way to eliminate the gray area in sports and this is one problem solving methology. Now, if this gray area is not resolved things can get out of hand. Instead of one problem you may have multiple problems over time. And before long everything is a suspect.
In physic the problem solving methology is very much the same. Any way more of this later on. I have to go.

A few mathematical concepts below that I personally find of interest and would like to learn more about...

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Game Theory
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In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a solution concept of a game involving two or more players, in which no player has anything to gain by changing only his or her own strategy unilaterally. If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.
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One can use the Kakutani fixed point theorem to prove that b has a fixed point. That is, there is a ? * such that . Since b(? * ) represents the best response for all players to ? * , the existence of the fixed point proves that there is some strategy set which is a best response to itself. No player could do any better by deviating, and it is therefore a Nash equilibrium.

When Nash made this point to John von Neumann in 1949, von Neumann famously dismissed it with the words, "That's trivial, you know. That's just a fixed point theorem." (See Nasar, 1998, p. 94.)
http://en.wikipedia.org/wiki/Nash_equilibrium


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Kakutani fixed point theorem
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In mathematical analysis, the Kakutani fixed point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to itself. The Kakutani fixed point theorem is a generalization of Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions.

The theorem was developed by Shizuo Kakutani in 1941[1] and was famously used by John Nash[2] in his description of Nash equilibrium. It has subsequently found widespread application in game theory and economics.[3]
http://en.wikipedia.org/wiki/Kakutani_fixed_point_theorem


Topological space
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Topological spaces are mathematical structures that allow the formalization of concepts such as convergence, connectedness and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology.

Definition

The most common way to define a topological space is as a set X together with a collection T of subsets of X satisfying the following axioms:

The empty set and X are in T.
The union of any collection of sets in T is also in T.
The intersection of any finite collection of sets in T is also in T.

The collection T is called a topology on X, and the elements of X are called points. Under this definition, the sets in T are the open sets, and their complements in X are the closed sets. By convention, the union of the empty collection is the empty set, and the intersection of the empty collection is X. A convention is used as these statements are true only vacuously, so the converse could equally be considered to be true.
http://en.wikipedia.org/wiki/Topological_space

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Continuous function (topology)
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In topology and related areas of mathematics a continuous function is a morphism between topological spaces. Intuitively, this is a function f where a set of points near f(x) always contain the image of a set of points near x. For a general topological space, this means a neighbourhood of f(x) always contains the image of a neighbourhood of x.

In a metric space (for example, the real numbers) this means that the points within a given distance of f(x) always contain the images of all the points within some other distance of x, giving the ?-? definition.
http://en.wikipedia.org/wiki/Continuous_fu..._%28topology%29

GAME THEORY & DISCRIMINATION


Noah and his Mom by tmcdaily...
http://www.flickr.com/photos/tmcdaily/227458629/

From the CogPhy and Sociological perspective, the implications of Game Theory are rather obvious with regards to issues of discrimination.

(RELATED POST: http://forum.physorg.com/index.php?showtop...dpost&p=292762)

If "no player can benefit by changing his or her strategy while the other players keep theirs unchanged," and if one is excluded from meaningful participation within the "game system" that we call "society," then one must go elsewhere if one wishes not to be an outcast within a community or else one must work to change the system from within, not an easy task for the excluded, whether they be a victim of discrimination by race, color, creed, economic class or, for that matter, a phenomeneon I have noticed on these boards, "intellectual orientation."

In a more easily accessible if perchance unpleasant vein, one need only think of those who drowned in their attics down in New Orleans...

"Why didn't they just leave town!" people said.

And go WHERE and by what means of transport? Should they have WALKED? Well, some of them tried to...

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No Charges in Katrina Bridge Blockade
October 31, 2007 - 5:46pm

By MICHAEL KUNZELMAN
Associated Press Writer

GRETNA, La. (AP) - A grand jury will not charge anyone in a police blockade that kept hundreds of evacuees from crossing a Mississippi River bridge on foot after Hurricane Katrina flooded New Orleans, authorities said Wednesday.

Several hundred people claimed police from suburban Gretna blocked them as they tried to flee New Orleans on Sept. 1, three days after the storm hit.

Many evacuees, who had been stranded at the New Orleans convention center without food and water, said they were told to cross the bridge to be evacuated from the city, only to be forced to turn around upon reaching the other side.

Police later said they blocked the evacuees because there were no supplies or services for them on the other side.

Gretna Police Chief Arthur Lawson also acknowledged that one of his officers fired a shot into the air during the blockade in an attempt to quell what he described as unrest among the evacuees.
http://www.wtopnews.com/index.php?nid=660&sid=1283693


RELATED POST
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Pressure is Aggression to the Monkey in the Man
http://raphie.wordpress.com/2007/04/06/pre...key-in-the-man/
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photo by Greg Whiting aka "egg"
http://www.flickr.com/photos/eggie/

Barack Obama, I suspect, knows a thing or two about what I’m talking about here. Part black, part Muslim in a country not known once upon a time for it’s hospitality to the former or these days for it’s hospitality to the latter, I dare say that “the audacity to hope” was, for him not a choice, but a necessity. The truth is, it’s not much fun to never be accepted as part of a group and told to “stick to your own kind” when you’re one of a kind, and the truth is that that kind of ostracization might just feel a bit like aggression to you, something Scarlett with the “A” could probably tell you about.

And that’s where the contradiction kind of comes in because if pressure is aggression , then how does one get in from the outside if the door is locked and no one will open it?

The answer?

You’ve got to apply a little pressure. You don’t really have a choice in the matter when you’ve no place in the manner, unless, that is, you’re okay being locked out, which most people aren’t, actually, because being locked out also feels a lot like being locked up and that’s liable to make you feel a bit like a monkey. Maybe how Martin Luther King felt when he wrote “Letter From Birmingham Jail” way back in 1963…
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A Letter From 1963 to Today c/o Martin Luther King ATTN: America
http://boojummy.blogs.friendster.com/snipe...ter_from_1.html

This post has been edited by Raphie Frank on Dec 18 2007, 01:26 AM