The Fourfold Square and Eightfold Cube
Related material: A Google image search for “field dream” + log24.
Monday, January 20, 2020
Tuesday, August 7, 2018
The Search for Harmonic Analysis
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Tuesday, January 12, 2016
Harmonic Analysis and Galois Spaces
The above sketch indicates, in a vague, hand-waving, fashion,
a connection between Galois spaces and harmonic analysis.
For more details of the connection, see (for instance) yesterday
afternoon's post Space Oddity.
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Sunday, November 24, 2013
Tuesday, October 24, 2023
Two Views of Mathieu Geometry*
For related remarks, see a reference from OEIS, A001438 —
David Joyner and Jon-Lark Kim,
<a href="http://dx.doi.org/10.1007/978-0-8176-8256-9_3">
Kittens, Mathematical Blackjack, and Combinatorial Codes</a>,
Chapter 3 in Selected Unsolved Problems in Coding Theory,
Applied and Numerical Harmonic Analysis, Springer, 2011,
pp. 47-70, DOI: 10.1007/978-0-8176-8256-9_3.
Today happens to be a related online-publication anniversary —
* A part of what might be called, more generally,. "figurate geometry."
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Sunday, December 18, 2022
“Square Round” — Ulysses, end of Ch. 17
Circle and Square at the Court of King Minos —
Harmonic analysis based on the circle involves the
circular functions. Dyadic harmonic analysis involves …
For some related history, see (for instance) . . .
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Wednesday, April 7, 2021
Timeless Capsules
Drilling down . . .
My own, more abstract, academic interests are indicated by
a post from this journal on January 20, 2020 —
Dyadic Harmonic Analysis: The Fourfold Square and Eightfold Cube.
Those poetically inclined may regard that post as an instance of the
“intersection of the timeless with time.”
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Sunday, January 19, 2020
For 6 Prescott Street*
"Freshman Seminar Program Department Administrator Corinna S. Rohse
described the program’s courses, which allow students to study subjects
that vary from Sanskrit to the mathematical basis for chess, as
'jewel-like: small and incredibly well-cut.' "
— The Harvard Crimson , Dec. 10, 2008
For remarks related to Sanskrit, chessboard structure, and "jewel-like"
mathematics, see A Prince of Darkness (Log24, March 28, 2006).
See also Walsh Functions in this journal and …
Lecture notes on dyadic harmonic analysis
(Cuernavaca, 2000)
Compare and contrast these remarks of Pereyra with the following
remarks, apparently by the same Corinna S. Rohse quoted above.
* Location of the Harvard Freshman Seminar program in the 2008
article above. The building at 6 Prescott was moved there from
5 Divinity Avenue in 1978. When the seminar program was started
in the fall of 1959, it was located in a house at 8 Prescott St. (In
1958-1959 this was a freshman dorm, the home of Ted Kaczynski.)
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Monday, April 2, 2018
Three Mother Cubes
From a Toronto Star video pictured here on April 1 three years ago:
The three connected cubes are labeled "Harmonic Analysis," 'Number Theory,"
and "Geometry."
Related cultural commentary from a review of the recent film "Justice League" —
"Now all they need is to resurrect Superman (Henry Cavill),
stop Steppenwolf from reuniting his three Mother Cubes
(sure, whatever) and wrap things up in under two cinematic
hours (God bless)."
The nineteenth-century German mathematician Felix Christian Klein
as Steppenwolf —
Volume I of a treatise by Klein is subtitled
"Arithmetic, Algebra, Analysis." This covers
two of the above three Toronto Star cubes.
Klein's Volume II is subtitled "Geometry."
An excerpt from that volume —
Further cultural commentary: "Glitch" in this journal.
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Wednesday, June 28, 2017
You Say Goodbye, I Say …
The title is from a Beatles song. See a link to 2008 in the previous post.
Harmonic Analysis for Illiterates —
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Friday, March 17, 2017
To Coin a Phrase
(A sequel to the previous post, Narrative for Westworld)
"That corpse you planted last year . . . ." — T. S. Eliot
Circle and Square at the Court of King Minos —
Harmonic analysis based on the circle involves the
circular functions. Dyadic harmonic analysis involves …
For some related history, see (for instance) E. M. Stein
on square functions in a 1982 AMS Bulletin article.
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Friday, December 23, 2016
Requiem for a Mathematician
From a Dec. 21 obituary posted by the
University of Tennessee at Knoxville —
"Wade was ordained as a pastor and served
at Oakwood Baptist Church in Knoxville."
Other information —
In a Log24 post, "Seeing the Finite Structure,"
of August 16, 2008, Wade appeared as a co-author
of the Walsh series book mentioned above —
Walsh Series: An Introduction
to Dyadic Harmonic Analysis,
by F. Schipp et al.,
Taylor & Francis, 1990
From the 2008 post —
The patterns on the faces of the cube on the cover
of Walsh Series above illustrate both the
Walsh functions of order 3 and the same structure
in a different guise, subspaces of the affine 3-space
over the binary field. For a note on the relationship
of Walsh functions to finite geometry, see
Symmetry of Walsh Functions.
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Friday, January 29, 2016
Excellent Adventure*
(Continued from Dec. 9, 2013)
|
"…it would be quite a long walk
Swiftly Mrs. Who brought her hands… together.
"Now, you see," Mrs. Whatsit said,
– A Wrinkle in Time , |
From a media weblog yesterday, a quote from the video below —
"At 12:03 PM Eastern Standard Time, January 12th, 2016…."
This weblog on the previous day (January 11th, 2016) —
"There is such a thing as harmonic analysis of switching functions."
— Saying adapted from a young-adult novel
* For some backstory, see a Caltech page.
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Tuesday, January 12, 2016
Lechner’s Beginning
… Professionally, at least …
Click image to enlarge.
See also the previous post, Lechner's End.
For a more up-to-date look at harmonic analysis
and switching functions (i.e., Boolean functions),
see Ryan O'Donnell, Analysis of Boolean Functions ,
Cambridge U. Press, 2014. Page 40 gives an
informative overview of the history of this field.
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Monday, January 11, 2016
Space Oddity
It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone
unremarked by mathematicians.
A Google search today for “Galois spaces” + “Boolean spaces”
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself.
Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …
“Harmonic Analysis of Switching Functions” ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor,
Recent Developments in Switching Theory , Academic Press, 1971.
“Galois Switching Functions and Their Applications,”
by B. Benjauthrit and I. S. Reed,
JPL Deep Space Network Progress Report 42-27 , 1975
D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239-249, Mar. 1978
“Switching functions constructed by Galois extension fields,”
by Iwaro Takahashi, Information and Control ,
Volume 48, Issue 2, pp. 95–108, February 1981
An illustration from the Lechner paper above —
“There is such a thing as harmonic analysis of switching functions.”
— Saying adapted from a young-adult novel
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Wednesday, April 1, 2015
Math’s Big Lies
Two mathematicians, Barry Mazur and Edward Frenkel,
have, for rhetorical effect, badly misrepresented the
history of some basic fields of mathematics. Mazur and
Frenkel like to emphasize the importance of new
research by claiming that it connects fields that previously
had no known connection— when, in fact, the fields were
known to be connected since at least the nineteenth century.
For Mazur, see The Proof and the Lie; for Frenkel, see posts
tagged Frenkel-Metaphors.
See also a story and video on Robert Langlands from the
Toronto Star on March 27, 2015:
"His conjectures are called functoriality and
reciprocity. They made it possible to link up
three branches of math: harmonic analysis,
number theory, and geometry.
To mathematicians, this is mind-blowing stuff
because these branches have nothing to do
with each other."
For a much earlier link between these three fields, see the essay
"Why Pi Matters" published in The New Yorker last month.
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Thursday, December 5, 2013
Blackboard Jungle
Continued from Field of Dreams, Jan. 20, 2013.
That post mentioned the March 2011 AMS Notices ,
an issue on mathematics education.
In that issue was an interview with Abel Prize winner
John Tate done in Oslo on May 25, 2010, the day
he was awarded the prize. From the interview—
|
Research Contributions
Raussen and Skau: This brings us to the next
Tate: Well, first of all, it was not a new result, except
[For a different perspective on the highlighted areas of |
"So often there is a time in mathematics for something to be done."
— John Tate in Oslo on May 25, 2010.
See also this journal on May 25, 2010, as well as
Galois Groups and Harmonic Analysis on Nov. 24, 2013.
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Fields
Edward Frenkel recently claimed for Robert Langlands
the discovery of a link between two "totally different"
fields of mathematics— number theory and harmonic analysis.
He implied that before Langlands, no relationship between
these fields was known.
See his recent book, and his lecture at the Fields Institute
in Toronto on October 24, 2013.
Meanwhile, in this journal on that date, two math-related
quotations for Stephen King, author of Doctor Sleep—
|
"Danvers is a town in Essex County, Massachusetts, |
For those who prefer their mathematics presented as fact, not fiction—
(Click for a larger image.)
The arrows in the figure at the right are an attempt to say visually that
the diamond theorem is related to various fields of mathematics.
There is no claim that prior to the theorem, these fields were not related.
See also Scott Carnahan on arrow diagrams, and Mathematical Imagery.
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Tuesday, November 26, 2013
Edward Frenkel, Your Order Is Ready.
Backstory: Frenkel's Metaphors and Waitressing for Godot.
In a recent vulgarized presentation of the Langlands program,
Edward Frenkel implied that number theory and harmonic
analysis were, before Langlands came along, quite unrelated.
This is false.
"If we think of different fields of mathematics as continents,
then number theory would be like North America and
harmonic analysis like Europe."
— Edward Frenkel, Love and Math , 2013
For a discussion of pre-Langlands connections between
these "continents," see…
"Fourier Analysis in Number Theory, my senior thesis, under the advisory of Patrick Gallagher.
This thesis contains no original research, but is instead a compilation of results from analytic
number theory that involve Fourier analysis. These include quadratic reciprocity (one of 200+
published proofs), Dirichlet's theorem on primes in arithmetic progression, and Weyl's criterion.
There is also a function field analogue of Fermat's Last Theorem. The presentation of the
material is completely self-contained."
— Shanshan Ding, University of Pennsylvania graduate student
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Saturday, November 23, 2013
Saturday, January 12, 2013
Abstract*
|
Saturday January 12, 2013, MAA Invited Paper Session on Room 2, Upper Level, San Diego 9:30 a.m. Mathematicians develop habits of thought and employ |
Remarks for a dead mathematician—
Click on the above image for the original post.
Then click on the Harmonic Analysis link for
some exposition by Folland.
* As opposed to concrete —
See yesterday morning's Grapevine Hill and…
| SFGate 1/12/13 — Californians bring out gloves, hats for cold spell http://www.sfgate.com/news/us/article/ Zookeepers-growers-prepare-for- California-freeze-4185448.php A 40-mile stretch of a major highway north of The California Highway Patrol shut the Grapevine "There must have been 1,000 Mack trucks lined up," |
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Saturday, December 1, 2012
Star Wars
Source: Rudolf Koch, The Book of Signs
The American Mathematical Society
(AMS) yesterday:
|
Lars Hörmander (1931-2012) Hörmander, who received a Fields Medal in 1962, |
Some related material:
- A 1990 AMS review of a book by another mathematician,
Harmonic Analysis in Phase Space , that is closely
related to Hörmander's research (and perhaps more
accessible to the nonspecialist than his books), - Mathematical Imagery (Log24, Jan. 19, 2012),
- The Eight-Pointed Star, and yesterday's
- Point.
See also posts on Damnation Morning and, from the
date of Hörmander's death,
- Will and Representation.
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Sunday, September 9, 2012
Decomposition Sermon
(Continued from Walpurgisnacht 2012)
Wikipedia article on functional decomposition—
"Outside of purely mathematical considerations,
perhaps the greatest value of functional decomposition
is the insight it provides into the structure of the world."
Certainly this is true for the sort of decomposition
known as harmonic analysis .
It is not, however, true of my own decomposition theorem,
which deals only with structures made up of at most four
different sorts of elementary parts.
But my own approach has at least some poetic value.
See the four elements of the Greeks in (for instance)
Eliot's Four Quartets and in Auden's For the Time Being .
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Monday, February 6, 2012
Beach Boy
| Cached from artnet.com |
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|
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In memory of a dealer in artists' ephemera,
Steven Leiber, who died on January 28, 2012—
a link to a post from the date of Leiber's death—
See also Me and My Shadow, a post from
the date the above photo was offered for sale.
Related ephemeral art— a post titled, with irony,
Introduction to Harmonic Analysis.
Related non -ephemeral art—
Mathematical Imagery.
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Friday, January 20, 2012
Brightness at Noon
See "harmonic analysis" in Mathematical Imagery and elsewhere in this journal.
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Saturday, December 26, 2009
Annals of Philosophy
Towards a Philosophy of Real Mathematics, by David Corfield, Cambridge U. Press, 2003, p. 206:
"Now, it is no easy business defining what one means by the term conceptual…. I think we can say that the conceptual is usually expressible in terms of broad principles. A nice example of this comes in the form of harmonic analysis, which is based on the idea, whose scope has been shown by George Mackey (1992) to be immense, that many kinds of entity become easier to handle by decomposing them into components belonging to spaces invariant under specified symmetries."
For a simpler example of this idea, see the entities in The Diamond Theorem, the decomposition in A Four-Color Theorem, and the space in Geometry of the 4×4 Square. The decomposition differs from that of harmonic analysis, although the subspaces involved in the diamond theorem are isomorphic to Walsh functions— well-known as discrete analogues of the trigonometric functions of traditional harmonic analysis.
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Thursday, November 6, 2008
Thursday November 6, 2008
Death of a Classmate
Michael Crichton,
Harvard College, 1964
Authors Michael Crichton and
David Foster Wallace in today’s
New York Times obituaries
The Times’s remarks above
on the prose styles of
Crichton and Wallace–
“compelling formula” vs.
“intricate complexity”–
suggest the following works
of visual art in memory
of Crichton.
“Crystal”—
“Dragon”
(from Crichton’s
Jurassic Park)–
For the mathematics
(dyadic harmonic analysis)
relating these two figures,
see Crystal and Dragon.
Some philosophical
remarks related to
the Harvard background
that Crichton and I share–
Hitler’s Still Point
and
The Crimson Passion.
Comments Off on Thursday November 6, 2008
Saturday, August 16, 2008
Saturday August 16, 2008
Seeing the Finite Structure
The following supplies some context for remarks of Halmos on combinatorics.
From Paul Halmos: Celebrating 50 years of Mathematics, by John H. Ewing, Paul Richard Halmos, Frederick W. Gehring, published by Springer, 1991–
Interviews with Halmos, “Paul Halmos by Parts,” by Donald J. Albers–
“Part II: In Touch with God*“– on pp. 27-28:
The Root of All Deep Mathematics
“Albers. In the conclusion of ‘Fifty Years of Linear Algebra,’ you wrote: ‘I am inclined to believe that at the root of all deep mathematics there is a combinatorial insight… I think that in this subject (in every subject?) the really original, really deep insights are always combinatorial, and I think for the new discoveries that we need– the pendulum needs– to swing back, and will swing back in the combinatorial direction.’ I always thought of you as an analyst.
Halmos: People call me an analyst, but I think I’m a born algebraist, and I mean the same thing, analytic versus combinatorial-algebraic. I think the finite case illustrates and guides and simplifies the infinite.
Some people called me full of baloney when I asserted that the deep problems of operator theory could all be solved if we knew the answer to every finite dimensional matrix question. I still have this religion that if you knew the answer to every matrix question, somehow you could answer every operator question. But the ‘somehow’ would require genius. The problem is not, given an operator question, to ask the same question in finite dimensions– that’s silly. The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question.
Combinatorics, the finite case, is where the genuine, deep insight is. Generalizing, making it infinite, is sometimes intricate and sometimes difficult, and I might even be willing to say that it’s sometimes deep, but it is nowhere near as fundamental as seeing the finite structure.”
Finite Structure
on a Book Cover:
on a Book Cover:
Walsh Series: An Introduction
to Dyadic Harmonic Analysis,
by F. Schipp et al.,
Taylor & Francis, 1990
Halmos’s above remarks on combinatorics as a source of “deep mathematics” were in the context of operator theory. For connections between operator theory and harmonic analysis, see (for instance) H.S. Shapiro, “Operator Theory and Harmonic Analysis,” pp. 31-56 in Twentieth Century Harmonic Analysis– A Celebration, ed. by J.S. Byrnes, published by Springer, 2001.
Walsh Series states that Walsh functions provide “the simplest non-trivial model for harmonic analysis.”
Walsh Series states that Walsh functions provide “the simplest non-trivial model for harmonic analysis.”
The patterns on the faces of the cube on the cover of Walsh Series above illustrate both the Walsh functions of order 3 and the same structure in a different guise, subspaces of the affine 3-space over the binary field. For a note on the relationship of Walsh functions to finite geometry, see Symmetry of Walsh Functions.
Whether the above sketch of the passage from operator theory to harmonic analysis to Walsh functions to finite geometry can ever help find “the right finite question to ask,” I do not know. It at least suggests that finite geometry (and my own work on models in finite geometry) may not be completely irrelevant to mathematics generally regarded as more deep.
* See the Log24 entries following Halmos’s death.
Comments Off on Saturday August 16, 2008
Wednesday, October 24, 2007
Wednesday October 24, 2007
Descartes’s Twelfth Step
An earlier entry today (“Hollywood Midrash continued“) on a father and son suggests we might look for an appropriate holy ghost. In that context…
Descartes
A search for further background on Emmanuel Levinas, a favorite philosopher of the late R. B. Kitaj (previous two entries), led (somewhat indirectly) to the following figures of Descartes:
This trinity of figures is taken from Descartes’ Rule Twelve in Rules for the Direction of the Mind. It seems to be meant to suggest an analogy between superposition of colors and superposition of shapes.Note that the first figure is made up of vertical lines, the second of vertical and horizontal lines, and the third of vertical, horizontal, and diagonal lines. Leon R. Kass recently suggested that the Descartes figures might be replaced by a more modern concept– colors as wavelengths. (Commentary, April 2007). This in turn suggests an analogy to Fourier series decomposition of a waveform in harmonic analysis. See the Kass essay for a discussion of the Descartes figures in the context of (pdf) Science, Religion, and the Human Future (not to be confused with Life, the Universe, and Everything).
Compare and contrast:
The harmonic-analysis analogy suggests a review of an earlier entry’s
link today to 4/30– Structure and Logic— as well as
re-examination of Symmetry and a Trinity
(Dec. 4, 2002).
See also —
A Four-Color Theorem,
The Diamond Theorem, and
The Most Violent Poem,
Comments Off on Wednesday October 24, 2007
Tuesday, March 28, 2006
Tuesday March 28, 2006
A Prince of Darkness
“What did he fear? It was not a fear or dread, It was a nothing that he knew too well. It was all a nothing and a man was a nothing too. It was only that and light was all it needed and a certain cleanness and order. Some lived in it and never felt it but he knew it all was nada y pues nada y nada y pues nada. Our nada who art in nada, nada be thy name thy kingdom nada thy will be nada in nada as it is in nada. Give us this nada our daily nada and nada us our nada as we nada our nadas and nada us not into nada but deliver us from nada; pues nada. Hail nothing full of nothing, nothing is with thee.”
— From Ernest Hemingway,
“A Clean, Well-Lighted Place“
“A Clean, Well-Lighted Place“
| “By groping toward the light we are made to realize how deep the darkness is around us.” — Arthur Koestler, The Call Girls: A Tragi-Comedy, Random House, 1973, page 118 |
From a review of
Teilhard de Chardin’s
The Phenomenon of Man:
Teilhard de Chardin’s
The Phenomenon of Man:
|
“It would have been |
“He’s good.”
“Good? He’s the fucking
Prince of Darkness!”
— Paul Newman
and Jack Warden
in “The Verdict“
| Sanskrit (transliterated) —
nada: “So Nada Brahma means not only: — Joachim-Ernst Berendt, |
“This book is the outcome of
a course given at Harvard
first by G. W. Mackey….”
— Lynn H. Loomis, 1953, preface to
An Introduction to
Abstract Harmonic Analysis
For more on Mackey and Harvard, see
the Log24 entries of March 14-17.
Comments Off on Tuesday March 28, 2006
Friday, March 17, 2006
Friday March 17, 2006
Mackey was born, according to Wikipedia, on Feb. 1, 1916. He died, according to Harvard University, on the night of March 14-15, 2006. He was the author of, notably, “Harmonic Analysis as the Exploitation of Symmetry — A Historical Survey,” pp. 543-698 in Bulletin of the American Mathematical Society (New Series), Vol. 3, No. 1, July 1980. This is available in a hardcover book published in 1992 by the A.M.S., The Scope and History of Commutative and Noncommutative Harmonic Analysis. (370 pages, ISBN 0-8218-9903-1). A paperback edition of this book will apparently be published this month by Oxford University Press (ISBN 978-0-8218-3790-7).
- Introduction
- Harmonic analysis as the exploitation of symmetry: A historical survey
- Herman Weyl and the application of group theory to quantum mechanics
- The significance of invariant measures for harmonic analysis
- Weyl’s program and modern physics
- Induced representations and the applications of harmonic analysis
- Von Neumann and the early days of ergodic theory
- Final remarks
Related material:
Log24, Oct. 22, 2002.
Women’s history month continues.
Comments Off on Friday March 17, 2006
Sunday, July 31, 2005
Sunday July 31, 2005
Looney Tunes
|
LOS ANGELES, July 30 (AP) – Kayo Hatta, an independent filmmaker… died on July 20. She was 47. She accidentally drowned at a friend’s home in the San Diego area, her sister Julie Hatta said….
Ms. Hatta graduated from Stanford University with a degree in English and received a master’s degree in film from the University of California, Los Angeles. She recently completed a 30-minute coming-of-age film called “Fishbowl,” based on the writings of Hawaiian author Lois-Ann Yamanaka. |
From Log24 on Moon Day, July 20,
the date of Hatta’s death:
the date of Hatta’s death:
Quote from an earlier entry:
Quote from July 20:
|
Quote from Lois-Ann Yamanaka:
| Blu’s Hanging
… Poppy still plays “Moon River” in the background. |
There will be a public memorial service in Honolulu
open to friends and the general public:
Date: Sunday, July 31st
Time: 1:00 pm
Location: Moiliili Hongwanji Buddhist Church,
902 University Avenue
In lieu of flowers, donations may be sent to:
Asian Improv aRts / Kayo Hatta Fund
201 Spear St., Ste 1650
San Francisco, CA 94105
open to friends and the general public:
Date: Sunday, July 31st
Time: 1:00 pm
Location: Moiliili Hongwanji Buddhist Church,
902 University Avenue
In lieu of flowers, donations may be sent to:
Asian Improv aRts / Kayo Hatta Fund
201 Spear St., Ste 1650
San Francisco, CA 94105
Comments Off on Sunday July 31, 2005
Wednesday, July 20, 2005
Wednesday July 20, 2005
| Words that may or may not have been said on July 20, 1969:
“That’s one small step for |
Another rhetorical contrast,
from a different date —
from a different date —
One small step for me:
Sunday, November 03, 2002
|
Music to Read By In honor of Roger Cooke’s review of Helson’s Harmonic Analysis, 2nd Edition, today’s site music is “Moonlight in Vermont.” |
|
One giant leap for mankind:
“The ‘Diamond Theory’ website of Steven Cullinane shows a man who is incapable of telling the truth: a pathological liar who hates and despises the mathematical community; a sociopath caught between the conflicting desires to earn the admiration of mathematicians, and his desire to insult those who ignore him and refuse him his self-perceived due measure of honor and reverie. As such, Steven Cullinane is constantly trying to purchase recognition when he has the funds to advertise on google.com, or steal that recognition by lying and deceiving dmoz.org when money isn’t enough. As you can see from the correspondence below, Jed Pack has clearly pointed out serious errors in Steven Cullinane’s calculations. Now, instead of admitting that he has been caught with his pants down, Steven Cullinane is questioning Jed Pack’s education! Surely, Jed Pack is a more competent mathematician than Steven Cullinane.”
For further details, see Crankbuster.
Comments Off on Wednesday July 20, 2005
Wednesday, September 15, 2004
Wednesday September 15, 2004
11:59 PM: The Last Minute
For the benefit of Grace (Paley, Enormous Changes at the Last Minute), here are the September 15 lottery numbers for Pennsylvania, the State of Grace (Kelly):
Midday: 053 Evening: 373.
For the significance of the evening number, 373, see Directions Out and Outside the World (both of 4/26/04). In both of these entries, and others to which they are linked, the number 373 signifies eternity.
The two most obvious interpretations of the midday number, 53, are as follows:
-
As a famous number of tones in musical harmonic analysis (i.e., tuning theory), as opposed to mathematical harmonic analysis ( The Square Wheel, 9/14/04), and
-
as a reference to the year 1953– a good year for Grace Kelly and the year of the classic film From Here to Eternity (the latter being signified, as noted above, by yesterday’s evening lottery number in the State of Grace).
“Time and chance
happeneth to them all.”
Ecclesiastes 9-11
Comments Off on Wednesday September 15, 2004
Tuesday, September 14, 2004
Tuesday September 14, 2004
The Square Wheel
Harmonic analysis may be based either on the circular (i.e., trigonometric) functions or on the square (i. e., Walsh) functions. George Mackey's masterly historical survey showed that the discovery of Fourier analysis, based on the circle, was of comparable importance (within mathematics) to the discovery (within general human history) of the wheel. Harmonic analysis based on square
For some observations of Stephen Wolfram on square-wheel analysis, see pp. 573 ff. in Wolfram's magnum opus, A New Kind of Science (Wolfram Media, May 14, 2002). Wolfram's illustration of this topic is closely related, as it happens, to a note on the symmetry of finite-geometry hyperplanes that I wrote in 1986. A web page pointing out this same symmetry in Walsh functions was archived on Oct. 30, 2001.
That web page is significant (as later versions point out) partly because it shows that just as the phrase "the circular functions" is applied to the trigonometric functions, the phrase "the square functions" might well be applied to Walsh
"While the reader may draw many a moral from our tale, I hope that the story is of interest for its own sake. Moreover, I hope that it may inspire others, participants or observers, to preserve the true and complete record of our mathematical times."
— From Error-Correcting Codes
Through Sphere Packings
To Simple Groups,
by Thomas M. Thompson,
Mathematical Association of America, 1983
Comments Off on Tuesday September 14, 2004
Wednesday, May 19, 2004
Wednesday May 19, 2004
Style
In memory of Lynn H. Loomis:
The above diagram is from a
(paper) journal note of October 21, 1999.
It pictures the relationship of my own discovery, diamond theory (at center), to the field, harmonic analysis, of Professor Loomis, a writer whose style I have long admired.
A quotation from the 1999 note:
"…it is not impossible to draw a fairly sharp dividing line between our mental disposition in the case of esthetic response and that of the responses of ordinary life. A far more difficult question arises if we try to distinguish it from the responses made by us to certain abstract mental constructions such as those of pure mathematics…. Perhaps the distinction lies in this, that in the case of works of art the whole end and purpose is found in the exact quality of the emotional state, whereas in the case of mathematics the purpose is the constatation of the universal validity of the relations without regard to the quality of the emotion accompanying apprehension. Still, it would be impossible to deny the close similarity of the orientation of faculties and attention in the two cases."
— Roger Fry, Transformations (1926), Doubleday Anchor paperback, 1956, p. 8
In other words, appreciating mathematics is much like appreciating art.
(Digitized diagram courtesy of Violet.)
Comments Off on Wednesday May 19, 2004
Thursday, November 6, 2003
Thursday November 6, 2003
Legacy Codes:
The Most Violent Poem
Lore of the Manhattan Project:
From The Trinity Site —
“I imagined Oppenheimer saying aloud,
‘Batter my heart, three person’d God,”
unexpectedly recalling John Donne’s ‘Holy Sonnet [14],’
and then he knew, ‘ “Trinity” will do.’
Memory has its reasons.
‘Batter my heart’ — I remember these words.
I first heard them on a fall day at Duke University in 1963.
Inside a classroom twelve of us were
seated around a long seminar table
listening to Reynolds Price recite this holy sonnet….
I remember Reynolds saying, slowly, carefully,
‘This is the most violent poem in the English language.’ ”
Related Entertainment
Today’s birthday:
director Mike Nichols
From a dead Righteous Brother:
“If you believe in forever
Then life is just a one-night stand.”
— Bobby Hatfield, found dead
in his hotel room at
7 PM EST Wednesday, Nov. 5, 2003,
before a concert scheduled at
Western Michigan University, Kalamazoo.
From a review of The Matrix Revolutions:
“You’d have to be totally blind at the end
to miss the Christian symbolism….
Trinity gets a glimpse of heaven…. And in the end…
God Put A Rainbow In The Clouds.”
Moral of the
Entertainment:
According to Chu Hsi [Zhu Xi],
“Li” is
“the principle or coherence
or order or pattern
underlying the cosmos.”
— Smith, Bol, Adler, and Wyatt,
Sung Dynasty Uses of the I Ching,
Princeton University Press, 1990
Related Non-Entertainment
Symmetry and a Trinity
(for the dotting-the-eye symbol above)
Introduction to Harmonic Analysis
(for musical and historical background)
Mathematical Proofs
(for the spirit of Western Michigan
University, Kalamazoo)
Moral of the
Non-Entertainment:
“Many kinds of entity
become easier to handle
by decomposing them into
components belonging to spaces
invariant under specified symmetries.”
— The importance of
mathematical conceptualisation
by David Corfield,
Department of History and
Philosophy of Science,
University of Cambridge
See, too,
Symmetry of Walsh Functions and
Geometry of the I Ching.
Comments Off on Thursday November 6, 2003
Friday, April 25, 2003
Friday April 25, 2003
Mark
Today is the feast of Saint Mark. It seems an appropriate day to thank Dr. Gerald McDaniel for his online cultural calendar, which is invaluable for suggesting blog topics.
Yesterday's entry "Cross-Referenced" referred to a bizarre meditation of mine titled "The Matthias Defense," which combines some thoughts of Nabokov on lunacy with some of my own thoughts on the Judeo-Christian tradition (i.e., also on lunacy). In this connection, the following is of interest:
From a site titled Meaning of the Twentieth Century —
"Freeman Dyson has expressed some thoughts on craziness. In a Scientific American article called 'Innovation in Physics,' he began by quoting Niels Bohr. Bohr had been in attendance at a lecture in which Wolfgang Pauli proposed a new theory of elementary particles. Pauli came under heavy criticism, which Bohr summed up for him: 'We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that is not crazy enough.' To that Freeman added: 'When a great innovation appears, it will almost certainly be in a muddled, incomplete and confusing form. To the discoverer, himself, it will be only half understood; to everyone else, it will be a mystery. For any speculation which does not at first glance look crazy, there is no hope!' "
— Kenneth Brower, The Starship and the Canoe, 1979, pp. 146, 147
It is my hope that the speculation, implied in The Matthias Defense, that the number 162 has astonishing mystical properties (as a page number, article number, etc.) is sufficiently crazy to satisfy Pauli and his friend Jung as well as the more conventional thinkers Bohr and Dyson. It is no less crazy than Christianity, and has a certain mad simplicity that perhaps improves on some of that religion's lunatic doctrines.
Some fruits of the "162 theory" —
Searching on Google for muses 162, we find the following Orphic Hymn to Apollo and a footnote of interest:
27 Tis thine all Nature's music to inspire,
28 With various-sounding, harmonising lyre;
29 Now the last string thou tun'ft to sweet accord,
30 Divinely warbling now the highest chord….
"Page 162 Verse 29…. Now the last string…. Gesner well observes, in his notes to this Hymn, that the comparison and conjunction of the musical and astronomical elements are most ancient; being derived from Orpheus and Pythagoras, to Plato. Now, according to the Orphic and Pythagoric doctrine, the lyre of Apollo is an image of the celestial harmony…."
For the "highest chord" in a metaphorical sense, see selection 162 of the 1919 edition of The Oxford Book of English Verse (whose editor apparently had a strong religious belief in the Muses (led by Apollo)). This selection contains the phrase "an ever-fixèd mark" — appropriately enough for this saint's day. The word "mark," in turn, suggests a Google search for the phrase "runes to grave" Hardy, after a poem quoted in G. H. Hardy's A Mathematician's Apology.
Such a search yields a website that quotes Housman as the source of the "runes" phrase, and a further search yields what is apparently the entire poem:
Smooth Between Sea and Land
by A. E. Housman
Smooth between sea and land
Is laid the yellow sand,
And here through summer days
The seed of Adam plays.Here the child comes to found
His unremaining mound,
And the grown lad to score
Two names upon the shore.Here, on the level sand,
Between the sea and land,
What shall I build or write
Against the fall of night?Tell me of runes to grave
That hold the bursting wave,
Or bastions to design
For longer date than mine.Shall it be Troy or Rome
I fence against the foam
Or my own name, to stay
When I depart for aye?Nothing: too near at hand
Planing the figured sand,
Effacing clean and fast
Cities not built to last
And charms devised in vain,
Pours the confounding main.(Said to be from More Poems (Knopf, 1936), p. 64)
Housman asks the reader to tell him of runes to grave or bastions to design. Here, as examples, are one rune and one bastion.
|
Represents |
Dagaz: (Pronounced thaw-gauze, but with the "th" voiced as in "the," not unvoiced as in "thick") (Day or dawn.)
From Rune Meanings:
Dagaz means "breakthrough, awakening, awareness. Daylight clarity as opposed to nighttime uncertainty. A time to plan or embark upon an enterprise. The power of change directed by your own will, transformation. Hope/happiness, the ideal. Security and certainty. Growth and release. Balance point, the place where opposites meet."
Also known as "the rune of transformation."
For the Dagaz rune in another context, see Geometry of the I Ching. The geometry discussed there does, in a sense, "hold the bursting wave," through its connection with Walsh functions, hence with harmonic analysis.
Temple of Athena Nike on the Nike Bastion, the Acropolis, Athens. Here is a relevant passage from Paul Valéry's Eupalinos ou L'Architecte about another temple of four columns:
Et puis… Écoute, Phèdre (me disait-il encore), ce petit temple que j'ai bâti pour Hermès, à quelques pas d'ici, si tu savais ce qu'il est pour moi ! — Où le passant ne voit qu'une élégante chapelle, — c'est peu de chose: quatre colonnes, un style très simple, — j'ai mis le souvenir d'un clair jour de ma vie. Ô douce métamorphose ! Ce temple délicat, nul ne le sait, est l'image mathématique d'une fille de Corinthe que j'ai heureusement aimée. Il en reproduit fidèlement les proportions particulières. Il vit pour moi !
Four columns, in a sense more suited to Hardy's interests, are also a recurrent theme in The Diamond 16 Puzzle and Diamond Theory.
Apart from the word "mark" in The Oxford Book of English Verse, as noted above, neither the rune nor the bastion discussed has any apparent connection with the number 162… but seek and ye shall find.
Comments Off on Friday April 25, 2003
Wednesday, February 12, 2003
Wednesday February 12, 2003
Diamond Life
(Von Neumann’s Song, Part II)
A reader of yesterday’s entry “St. John von Neumann’s Song” suggested the relevance of little Dougie Hofstadter‘s book Gödel, Escher, Bach: An Eternal Golden Braid. While the title of this work does continue the “golden” theme of my last three entries, Dougie is not playing in von Neumann’s league. The nature of this league is suggested by yesterday’s citation of
For work that is more in von Neumann’s league than in Hofstadter’s, see the following
VECTOR-VALUED EXTENSIONS
OF SOME CLASSICAL THEOREMS
IN HARMONIC ANALYSIS
Abstract:
…. The approach used combines methods from Fourier analysis and the geometry of Banach spaces, such as R-boundedness.
A related paper by the same authors:
CRITERIA FOR R-BOUNDEDNESS
OF OPERATOR FAMILIES
Abstract:
…smooth operator-valued functions have a R-bounded range, where the degree of smoothness depends on the geometry of the Banach space.
Those who would like to make a connection to music in the charmingly childlike manner of Hofstadter are invited to sing a few choruses of “How do you solve a problem like Maria?“
Personally, I prefer the following lyrics:
Diamond life, lover boy;
We move in space with minimum waste and maximum joy.
City lights and business nights
When you require streetcar desire for higher heights.
No place for beginners or sensitive hearts
When sentiment is left to chance.
No place to be ending but somewhere to start.
No need to ask.
He’s a smooth operator….
Words and Music: Sade Adu and Ray St. John
Some may wish to alter the last five syllables of these lyrics in accordance with yesterday’s entry on another St. John.
Comments Off on Wednesday February 12, 2003
Tuesday, February 11, 2003
Tuesday February 11, 2003
St. John von Neumann's Song
The mathematician John von Neumann, a heavy drinker and party animal, advocated a nuclear first strike on Moscow.* Confined to a wheelchair before his death, he was, some say, the inspiration for Kubrick's Dr. Strangelove. He was a Jew converted to Catholicism. His saint's day was February 8. Here is an excerpt from a book titled Abstract Harmonic Analysis**, just one of the fields illuminated by von Neumann's brilliance:
"…von Neumann showed that an intrinsic definition can be given for the mean M(f) of an almost periodic function…. Von Neumann proved the existence and properties of M(f) by completely elementary methods…."
Should W. B. Yeats wander into the Catholic Anticommunists' section of Paradise, he might encounter, as in "Sailing to Byzantium," an unexpected set of "singing-masters" there: the Platonic archetypes of the Hollywood Argyles.
The Argyles' attire is in keeping with Yeats's desire for gold in his "artifice of eternity"… In this case, gold lamé, but hey, it's Hollywood. The Argyles' lyrics will no doubt be somewhat more explicit in heaven. For instance, in "Alley Oop," the line
"He's a mean motor scooter and a bad go-getter"
will in its purer heavenly version be rendered
"He's a mean M(f)er and…"
in keeping with von Neumann's artifice of eternity described above.
This theological meditation was suggested by previous entries on Yeats, music and Catholicism (see Feb. 8, von Neumann's saint's day) and by the following recent weblog entries of a Harvard senior majoring in mathematics:
"I changed my profile picture to Oedipus last night because I felt cursed by fate…."
"It's not rational for me to believe that I am cursed, that the gods are set against me. Because I don't even believe in any gods!"
The spiritual benefits of a Harvard education are summarized by this student's new profile picture:
M(f)
*Source: Von Neumann and the Development of Game Theory
**by Harvard professor Lynn H. Loomis, Van Nostrand, 1953, p. 169.
Wednesday, November 6, 2002
Wednesday November 6, 2002
Today's birthdays: Mike Nichols and Sally Field.
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Who is Sylvia? What is she? |
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From A Beautiful Mind, by Sylvia Nasar:
Prologue
Where the statue stood
Of Newton with his prism and silent face,
The marble index of a mind for ever
Voyaging through strange seas of Thought, alone.
— WILLIAM WORDSWORTH
John Forbes Nash, Jr. — mathematical genius, inventor of a theory of rational behavior, visionary of the thinking machine — had been sitting with his visitor, also a mathematician, for nearly half an hour. It was late on a weekday afternoon in the spring of 1959, and, though it was only May, uncomfortably warm. Nash was slumped in an armchair in one corner of the hospital lounge, carelessly dressed in a nylon shirt that hung limply over his unbelted trousers. His powerful frame was slack as a rag doll's, his finely molded features expressionless. He had been staring dully at a spot immediately in front of the left foot of Harvard professor George Mackey, hardly moving except to brush his long dark hair away from his forehead in a fitful, repetitive motion. His visitor sat upright, oppressed by the silence, acutely conscious that the doors to the room were locked. Mackey finally could contain himself no longer. His voice was slightly querulous, but he strained to be gentle. "How could you," began Mackey, "how could you, a mathematician, a man devoted to reason and logical proof…how could you believe that extraterrestrials are sending you messages? How could you believe that you are being recruited by aliens from outer space to save the world? How could you…?"
Nash looked up at last and fixed Mackey with an unblinking stare as cool and dispassionate as that of any bird or snake. "Because," Nash said slowly in his soft, reasonable southern drawl, as if talking to himself, "the ideas I had about supernatural beings came to me the same way that my mathematical ideas did. So I took them seriously."
What I take seriously:
Introduction to Topology and Modern Analysis, by George F. Simmons, McGraw-Hill, New York, 1963
An Introduction to Abstract Harmonic Analysis, by Lynn H. Loomis, Van Nostrand, Princeton, 1953
"Harmonic Analysis as the Exploitation of Symmetry — A Historical Survey," by George W. Mackey, pp. 543-698, Bulletin of the American Mathematical Society, July 1980
Walsh Functions and Their Applications, by K. G. Beauchamp, Academic Press, New York, 1975
Walsh Series: An Introduction to Dyadic Harmonic Analysis, by F. Schipp, P. Simon, W. R. Wade, and J. Pal, Adam Hilger Ltd., 1990
The review, by W. R. Wade, of Walsh Series and Transforms (Golubov, Efimov, and Skvortsov, publ. by Kluwer, Netherlands, 1991) in the Bulletin of the American Mathematical Society, April 1992, pp. 348-359
Sunday, November 3, 2002
Sunday November 3, 2002
Music to Read By
In honor of Roger Cooke’s review of Helson’s Harmonic Analysis, 2nd Edition, today’s site music is “Moonlight in Vermont.”
Tuesday, October 22, 2002
Tuesday October 22, 2002
Introduction to
Harmonic Analysis
From Dr. Mac’s Cultural Calendar for Oct. 22:
- The French actress Catherine Deneuve was born on this day in Paris in 1943….
- The Beach Boys released the single “Good Vibrations” on this day in 1966.
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“I hear the sound of a On the wind that lifts — The Beach Boys |
In honor of Deneuve and of George W. Mackey, author of the classic 156-page essay, “Harmonic analysis* as the exploitation of symmetry† — A historical survey” (Bulletin of the American Mathematical Society (New Series), Vol. 3, No. 1, Part 1 (July 1980), pp. 543-698), this site’s music is, for the time being, “Good Vibrations.”
For more on harmonic analysis, see “Group Representations and Harmonic Analysis from Euler to Langlands,” by Anthony W. Knapp, Part I and Part II.
* For “the simplest non-trivial model for harmonic analysis,” the Walsh functions, see F. Schipp et. al., Walsh Series: An Introduction to Dyadic Harmonic Analysis, Hilger, 1990. For Mackey’s “exploitation of symmetry” in this context, see my note Symmetry of Walsh Functions, and also the footnote below.
† “Now, it is no easy business defining what one means by the term conceptual…. I think we can say that the conceptual is usually expressible in terms of broad principles. A nice example of this comes in form of harmonic analysis, which is based on the idea, whose scope has been shown by George Mackey… to be immense, that many kinds of entity become easier to handle by decomposing them into components belonging to spaces invariant under specified symmetries.”
— The importance of mathematical conceptualisation,
by David Corfield, Department of History and Philosophy of Science, University of Cambridge
by David Corfield, Department of History and Philosophy of Science, University of Cambridge
Saturday, October 19, 2002
Saturday October 19, 2002
What is Truth?
My state of mind |
My state of mind |
In light of the entry below (“Mass Confusion,” Oct. 19, 2002), some further literary reflections seem called for. Since this is, after all, a personal journal, allow me some personal details…
Yesterday I picked up some packages, delivered earlier, that included four books I had ordered. I opened these packages this morning before writing the entry below; their contents may indicate my frame of mind when I later read this morning’s New York Times story that prompted my remarks. The books are, in the order I encountered them as I opened packages,
- Prince Ombra, by Roderick MacLeish (1982, reprinted in August 2002 as a Tom Doherty Associates Starscape paperback)
- Truth, edited by Simon Blackburn and Keith Simmons, from the Oxford Readings in Philosophy series (Oxford University Press, 1999, reprinted as a paperback, 2000)
- The Savage and Beautiful Country, by Alan McGlashan (1967, reprinted in a revised and expanded edition in 1988 as a Daimon Verlag paperback)
- Abstract Harmonic Analysis, by Lynn H. Loomis (Van Nostrand, 1953… a used copy)
Taken as a whole, this quartet of books supplies a rather powerful answer to the catechism question of Pontius Pilate, “What is truth?”…
The answer, which I pray will some day be delivered at heaven’s gate to all who have lied in the name of religion, is, in Jack Nicholson’s classic words,
You can’t handle the truth!