(Continued from May 2, 2023 and December 18, 2022)
Harmonic analysis based on the circle involves the
circular functions. Dyadic harmonic analysis involves …
Summary, as an illustration of a title by George Mackey —
Friday, September 27, 2024
To Phrase a Coin
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Saturday, November 23, 2013
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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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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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.
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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.
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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
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Wednesday, November 6, 2002
Wednesday November 6, 2002
Today's birthdays: Mike Nichols and Sally Field.
|
Who is Sylvia? What is she? |
|
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
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.
 |
“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