The Fourfold Square and Eightfold Cube
Related material: A Google image search for “field dream” + log24.
Monday, January 20, 2020
Wednesday, November 8, 2017
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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Thursday, July 14, 2022
For Dark Jokester*
Also on April 1, 2015 —
From the essay cited above —
For the formula of building blocks , see Walsh series.
* See Google.
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Wednesday, April 7, 2021
Drilling Down . . .
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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 26, 2020
Harmonic-Analysis Building Blocks
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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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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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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.
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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.
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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
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