'contextual' Mathematics/number Theory, Enhancing their 'Reality Correlation'

And the difference between magic and mysticism?

As for saying that about the nacreous clouds and exposure times, I sure hope them up there clouds weren't listening. I kind of imagine them as even more beautiful than the photos and one of my lifelong dreams, actually, is to see the Northern Lights.

Best,
Raphie

P.S. I used to give magic shows as a kid. The real "magic" was the reactions of others to the tricks. That kind of wonder and amazement, THAT is the "real" magic in its purest and most beautiful form (IMO). :-)

This post has been edited by Raphie Frank on Feb 27 2008, 05:02 AM

WHAT ARE OPTICAL VORTICES?
(in relation to "The Dark Side of Light")

from the website of:
KEVIN O'HOLLERAN
http://www.physics.gla.ac.uk/~kevin/research2.html
(There are pictures on the page above referred to below, so it's worth a click...)

All of my research is concerned with optical vortices. Optical vortices are essentially the parts of light field that are perfectly dark. Energy swirls around these points like water round a plug hole (hence the term 'vortex'). It also turns out that the phase of the light is 'twisted' around the vortices. In 3 dimensions the vortices are lines or loops. My research is concerned with the nature of these vortices in random fields. Shown on the right is the phase from a cross-section of a light beam. The vortices are points where the phase is not defined i.e. where all the colours (which represent phase, not real colour) meet. The two shown here differ in the direction that energy is flowing around them.

Through experiment and computer simulation I intend to find general properties of vortex lines in 3 dimensions. It is interesting to note that vortices may be linked or even knotted.


AND FROM WIKIPEDIA...
==================================================
An optical vortex (also known as a screw dislocation or phase singularity) is a zero of an optical field, a point of zero intensity. Research into the properties of vortices has thrived since a comprehensive paper by Nye and Berry, in 1974,[1] described the basic properties of 'dislocations in wave trains'. The research that followed became the core of what is now known as 'singular optics'.

Layman's explanation
---------------------------

Light can be twisted like a corkscrew around its axis of travel. Because of the twisting, the light waves at the axis itself cancel each other out. On a flat surface, an optical vortex looks like a ring of light, with a dark hole in the center. This corkscrew of light, with darkness at the center, is called an optical vortex.

The vortex is given a number, called the topological charge, according to how many twists the light does in one wavelength. The number is always an integer, and can be positive or negative, depending on the direction of the twist. The higher the number of the twist, the faster the light is spinning around the axis. This spinning carries orbital angular momentum with the wave train, and will induce torque on an electric dipole.

This orbital angular momentum of light can be observed in the orbiting motion of trapped particles. Interfering an optical vortex with a plane wave of light reveals the spiral phase as concentric spirals. The number of arms in the spiral equals the topological charge.

Optical vortices are studied by creating them in the lab in various ways. A laser beam can be twisted into vortex using a "fork" computer generated hologram. The "fork" hologram can be used in a spatial light modulator, a specialized type of liquid crystal display controlled by a computer; or in a diffraction grating on a film or piece of glass.

Several different groups around the world have worked with optical vortices. Their communications are usually found in specialized physics and optics journals.

Properties
-----------------
An optical singularity is a zero of an optical field. The phase in the field circulates around these points of zero intensity (giving rise to the name 'vortex'). Vortices are points in 2D fields and lines in 3D fields (as they have codimension two). Integrating the phase of the field around a path enclosing a vortex yields an integer multiple of 2*pi. This integer is known as the topological charge, or strength, of the vortex.

MORE: http://en.wikipedia.org/wiki/Optical_Vortex
==================================================

In relation to "contextual" number theory, note in particular the following lines from above:

#1
"The vortex is given a number, called the topological charge, according to how many twists the light does in one wavelength. The number is always an integer, and can be positive or negative, depending on the direction of the twist. The higher the number of the twist, the faster the light is spinning around the axis."

#2
Interfering an optical vortex with a plane wave of light reveals the spiral phase as concentric spirals. The number of arms in the spiral equals the topological charge.

#3
"Vortices are points in 2D fields and lines in 3D fields (as they have codimension two). Integrating the phase of the field around a path enclosing a vortex yields an integer multiple of 2*pi."

Best,
Raphie

This post has been edited by Raphie Frank on Feb 27 2008, 06:35 AM

>And the difference between magic and mysticism?

It depends on how one interprets the meaning of these words as there are various.

>nacreous clouds

I found the images surprising but I'm still not sure I believe them.

>northern lights

They're truly a delight to behold.

>symbolic mysticism

I made that up. Not quite sure what it means.

>ethnomath

An interesting word indeed...

>optical vortices

FASCINATING!



This post has been edited by PJParent001 on Feb 27 2008, 09:47 PM

Dear PJParent,

As a small contextual aside...

SYMBOLIC LOGIC IN ACTION
Modified from "Imagination Into Action"
(posted November 15, 2007 on Wordpress, and December 2006 on Friendster)
http://raphie.wordpress.com/2007/11/15/ima...on-into-action/

I said please, she said Yes, I said Thank you.

Please
http://www.flickr.com/photos/wakalani/325429540

Yes (Do it)
http://www.flickr.com/photos/artivist/122182862/favorites/

Thank You (Glue it)
http://www.flickr.com/photos/wakalani/277562544/

That's a clue to how the artists knew it.

Best,
Raphie

This post has been edited by Raphie Frank on Feb 28 2008, 05:03 AM

I'm keeping things here, since this is the thread in which the subject arose, and I believe it important for people to see what a cretin you are.

Well yes, and it certainly seems to have stumped you! I'll ask again: you claim to be reading a certain book, the prerequisites* of which lie (by your own admittance) well beyond your level of knowledge and understanding. If you aren't lying, then you wont mind proving your claim is true. Perhaps, by means of an example, you'd like to display your understanding of the material in the book by answering the most basic of questions involving string theory? Better yet - how about you display the knowledge of a child by answering a high school question on mathematics!?

Now why do I get the feeling you're going to shy away from this challenge...

*That is, the necessary requirements for the reader to understand the material in the book.

Except that those are two typical pages from Polchinski. I can post more 'sample pages'. Someone give me two numbers between 1 and about 300. The first few pages of the book is a wordy introduction and outline of what the book goes through. After that, it's that level of maths.

And in answer to my own question, the picture you corrected the link of is the justification for IIB string theory being SL(2,Z) dual to itself. It's a concept I had to read about when writing my 4th year essay and which now relates to my PhD.

The other picture is the derivation of renormalisation group flow equations in string theory, a concept which got the 2004 Nobel Prize when applied to QCD, because it implies that the coupling 'constants' of QFT are not constants but functions of energy. At high energy QCD's coupling gets weaker.

The language of the book is maths, not English.

Thank you Alphanumeric for the explanations. Feel free to tell me and others more about your dissertation. I know it has to do with Calabi-Yau manifolds and I am sure many on these boards are interested. Alternatively, of course, feel free to direct us to prior posts in which you discuss it.

Best,
Raphie

P.S. Oh, and as for page numbers...

...how about posting pages 32 and 55, which is the chapter on "Conformal Field Theory" and offering some insights, not just conceptually, but also in relation to the mathematical notations on those pages which involve the Virasoro Algebra, "Virasoro" being a new term for me. TIA. Best, RF

This post has been edited by Raphie Frank on Feb 29 2008, 04:55 AM

KEY STRING THEORY CONCEPT:
------------------------------------------
THE PATH INTEGRAL FORMULATION
User posted image: http://en.wikipedia.org/wiki/Image:Three_paths_from_A_to_B.png

The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique history for a system with a sum, or functional integral, over an infinity of possible histories to compute a quantum amplitude.

The path integral formulation was developed in 1948 by Richard Feynman. Some preliminaries were worked out earlier, in the course of his doctoral thesis work with John Archibald Wheeler.
This formulation has proved crucial to the subsequent development of theoretical physics, since it provided the basis for the grand synthesis of the 1970s called the renormalization group which unified quantum field theory with statistical mechanics. If we realize that the Schrödinger equation is essentially a diffusion equation with an imaginary diffusion constant, then the path integral is a method for the enumeration of random walks. For this reason path integrals had also been used in the study of Brownian motion and diffusion before they were introduced in quantum mechanics.
http://en.wikipedia.org/wiki/Path_integral_formulation

==================================================
A COUPLE RELATED CONCEPTS FROM DIFFERENT FIELDS

The Random Walk (in Economics)
----------------------------------------------------------
Talking the Walk With Burton Malkiel
http://www.thestreet.com/_yahoo/funds/fina...n/10079582.html
(related to Brownian Motion)

Spanning Tree Protocol (in Information Technology)
------------------------------------------------------------
http://en.wikipedia.org/wiki/Spanning_tree_protocol
==================================================

Or, in plain English, as the Eastern philosophers might say "One destination, many paths." The same principle, I might add, applies in principle, to knowledge and understanding, much less life itself from cradle to grave.

Best,
Raphie

P.S. For anyone desirous of understanding Polchinski specifically, and string theory in general from a conceptual standpoint, myself included, the notion of the path integral would seem a not bad place to start, as the Polyakov path integral (again a new term) is central to Joseph Polchinski's "Volume I: An Introduction to the Bosonic String." (pub. 1998)

This post has been edited by Raphie Frank on Feb 29 2008, 05:30 AM

EXAMPLE OF "ARBITRARY" DEFINITION #1 - Prime Number

In a prior post to RPenner, in relation to infinity (see original post below), I noted that I believed some of our definitions could be extended without "harm" coming to the fundamental structural integrity of contemporary mathematics. The definition of "prime number" is one such example.

It is considered neither perfect nor prime and I simply see no real good reason for it, given the ease with which one can "split" numbers into two different parts, SELF and UNITY. Do this and then ONE (1) becomes both Perfect and Prime (with regards to Perfect #'s, the definition would be to add all divisors excluding the self and unity PLUS self and unity and then divide by 2. If the result is equal to the self, the number would then be Perfect.)

Prime Number Definition
==========================
In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid in about 300 BC. The first thirty prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113 (sequence A000040 in OEIS)

See the list of prime numbers for a longer list. The number one is by definition not a prime number; see the discussion below under Primality of one.
MORE: http://en.wikipedia.org/wiki/Prime_number

Primality of one
==========================
Until the 19th century, most mathematicians considered the number 1 a prime, and there is still a large body of mathematical work that is valid despite labeling 1 a prime, such as the work of Stern and Zeisel. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. The change in label occurred so that the fundamental theorem of arithmetic, as stated, is valid, i.e., “each number has a unique factorization into primes”[2][3][4]
MORE: http://en.wikipedia.org/wiki/Prime_number#Primality_of_one

About Henri Lebesque
Born: 28 June 1875 in Beauvais, Oise, Picardie, France
Died: 26 July 1941 in Paris, France
http://www-groups.dcs.st-and.ac.uk/~histor...s/Lebesgue.html

Fundamental Theorem of Arithmetic
=========================
In number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every natural number greater than 1 can be written as a unique product of prime numbers. For instance,

There are no other possible factorizations of 6936 or 1200 into prime numbers. The above representation collapses repeated prime factors into powers for easier identification. Because multiplication is commutative, the order of factors is irrelevant and usually written from smallest to largest.
MORE: http://en.wikipedia.org/wiki/Fundamental_t...m_of_arithmetic

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

Now think about it. 3*3*2 = 18. Uniqueness? We get to count 3 twice, so "uniquety" is clearly not based on redundancy. And as for a natural number being a product of primes...

1*1 = ?

a zillion?

Hmmm... let me think... 5?

Wait, maybe 22?

No. "Uniquety" is based on the concept of two DIFFERENT prime numbers being required? Hmmm... I'm baffled. 3*3 = 9, no?

In any case, last I checked 1 * 1 (i.e. unity * self) = precisely and exactly ONE and near as I can figure, the only reason 1 is not considered prime is because it is excluded from the definition by definition. Talk about circular logic that one can not break out of!

--------------------------------------------------------------------------
QUOTE FROM A PREVIOUS POST:
Modular thinking that one can not break out of is one of the symptoms [of autism]...

========================================================
Pioneering theatre therapy helps children with autism and Asperger's syndrome 'play to their strengths'
http://www.autismconnect.org/news.asp?sect...pe=news&id=5771
========================================================

Many people without autism, I might add, are modular thinkers. It only becomes a problem when the "loop" becomes closed
--------------------------------------------------------------------------

And, I might add, to slightly twist terminology, what could be more "unique" than one and only one natural number that when multiplied by itself equals itself?

In a manner of speaking, to my view, ONE is little more than a victim of the Law of Unintended Consequences. Excluded by definition, it is simply not counted and represents what might be termed a "hidden population," a hidden population of ONE.

CONTEXTUAL QUOTE
(in anthropomorphized terms)
=====================================
Pressure is Aggression to the Monkey in the Man
posted on 4/13/2007
=====================================
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.
FULL ESSAY: http://raphie.wordpress.com/2007/04/06/pre...key-in-the-man/
=====================================

Best,
Raphie

Read RPenner's Response here:
http://forum.physorg.com/index.php?showtop...ndpost&p=315782

Incidentally, regarding the "death" of mathematics, Lebesque wrote:

"Reduced to general theories, mathematics would be a beautiful form without content. It would quickly die."

This post has been edited by Raphie Frank on Feb 29 2008, 06:41 AM

THE DISCONTINUOUS FUNCTION - BASIC DEFINITION

In (contextual) relation to some heated debate within this forum upon the .9r thread...

SEE: http://forum.physorg.com/index.php?showtopic=20002&st=300

...a visual representation of a discontinuous function:

http://www.mathwords.com/d/discontinuous_function.htm

And a basic explanation of the concept:

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College
=====================================================
This is not so simple a question as it first appears. A discontinuous function is one that has a "break" in the curve, that is easy to "say" but as your question implies how does one "prove" that assertion, and that is not quite so straightforward. In the first place there are different "kinds" of discontinuities.

The "simplest" one is a curve that has a "break". For example, Y= 2*X for X<1 and Y=4*X for X = 1 or X>1. Here the function is defined for all values of X. There is another kind of discontinuity where this is not the case. For example, Y=1 for X < 0 and Y= 0 for X = 0 or X > 0. Here the two pieces are not connected. Another type of similar discontinuity is the same except leave unspecified what happens to Y when X=0. Then there is a missing point in the discontinuity.

Another type of discontinuity is one where the value of Y approaches + infinity for some value of X greater than a specified value and - infinity for X less than a certain value. An example is 1/sin(X) for X >0 and X<0. There can even be some pathological discontinuities such as Y= 1 if X is an even integer and Y = -1 if X is an odd integer. and Y= 0 for all other values of X.

Lastly, there are some functions that "look like" they should be discontinuous, but are not. An example is:

Y(X)= [sin(X)]/X as X approaches 0. That limit is Y(X)=1! Surprise.

Usually, the subject of calculus develops a number of theorems for determining continuity because that is an important concept in that branch of mathematics, but calculus does not address all possible cases of discontinuous mathematical functions.
http://www.newton.dep.anl.gov/askasci/math99/math99194.htm

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

In relation to that last sentence, I (and perhaps others who visit this thread) would be curious to know the instances wherein calculus does NOT address the exceptional case(s) of discontinuous mathematical functions.

Best,
Raphie

This post has been edited by Raphie Frank on Mar 1 2008, 04:55 AM

Well of course you wont! Doing so would force you to leave your land of make believe, and be confronted with the fact that you're just another not-very-bright individual who spends an awful lot of time on internet forums trying to talk about advanced topics! I find it hard to convey quite how pathetic your situation is.

So now you've moved onto path integrals. You claim that for someone to understand Polchinski's book, you'll need to understand these things. You have two options:

• Claim you do understand them (in which case, you wont mind computing the most trivial of examples).
• Or you don't understand them, and hence don't understand Polchinski's book.

Given that you refuse to answer mathematics questions accessible to children, I somehow find it difficult to believe you'll be jumping on the first option. But hey, I'm more than happy for you to prove me wrong.

You won't learn much conceptually because path quantisation is staggeringly complicated, especially for strings.

I just looked up the section on the Polyakov action in Polchinski. I see nothing that you could get concepts-wise in terms a non-mathematician could grasp. Liberal use of Wikipedia and Google to find the meaning of things like 'Euler number' might help a little but not much. And within 2 pages it's into gauge fixing (do you know what a gauge is?) due to a large 'gauge volume' of diffeomorphisms and Weyl symmetries. And then 2 pages later, it's the Faddev-Poppov gauge fixing method using gauge slices. When I first saw that, even with half a dozen quantum courses under my belt, it was a struggle to understand. To someone who doesn't know what a gauge is or why you'd want to 'fix it', it's meaningless.

I'm half tempted to post the pages in question and ask you to explain them. The same will happen as last time I did that, you'll be unable to. And I wasn't unfair posting those two pages, they related to important physical concepts in string theory and they were no more mathematical than the majority of the book. That's my point, even when the result leads to important conceptual understanding, it's expressed mathematically and you wouldn't, and don't, understand it.

Do us a favour and stop pretending. And stop bouncing from topic to topic. You aren't generating discussion, you just post reams and reams of unrelated crap on topics you don't understand and which you copied and pasted from elsewhere.

I asked YOU to explain them Alphanumeric. Please do post those pages and see if you can manage not to turn a question and an invitation to contribute in constructive manner into an exam.

Let me say it again: CONCEPT MINING.

Shall we write it 100 times together upon the chalkboard?

I might add, to borrow a line from Daniel Dennett in "Consciousness Explained": PLEASE do not confuse a lack of imagination for an insight into necessity.

Best,
Raphie

This post has been edited by Raphie Frank on Mar 1 2008, 10:12 AM

Dear PJParent, RC, et al,

Here is a book I think you would very much like, very accessible and approaching the "mystical" with a keen and critical eye:

The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (2002)

============================================================
Mario Livio
From Wikipedia, the free encyclopedia
----------------------------------------------
Mario Livio... is an astrophysicist and an author of works that popularize science and mathematics. He is currently Senior Astrophysicist at the Hubble Space Telescope Science Institute. He is perhaps best known for his book on the irrational number phi: The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (2002). The book won the Peano Prize and the International Pythagoras Prize for popular books on mathematics.
============================================================

I may not be too "bright" as at least one poster on this board has repeatedly posited, but Dr. Cliff Pickover, whom I have briefly corresponded with over email, recommended it to me, and I think we can safely suggest that he ain't no dummy.

"Dr. Cliff Pickover has published nearly a book a year in which he stretches the limit of computers, art, and thought." - Los Angeles Times
http://sprott.physics.wisc.edu/pickover/home.htm

Best,
Raphie

Incidentally, my friend Mariana Tomas is an occasional contributor to his new blog. Definitely check it out. It's a truly interesting mix of the serious and the bizarre.

Reality Carnival Unleashed
"Six-word news headlines that shatter the ice of our unconscious!" - Cliff Pickover
http://realitycarnival.blogspot.com/

Best,
Raphie

Ahem...