Log24

Tuesday, August 7, 2018

The Search for Harmonic Analysis

Filed under: General — m759 @ 11:45 AM

See Harmonic Analysis in this journal.

See also Loop.

'Loop De Loop,' Johnny Thunder, Diamond Records, 1962

"Here we go loop de lie."

Tuesday, January 12, 2016

Harmonic Analysis and Galois Spaces

Filed under: General,Geometry — Tags: — m759 @ 7:59 AM

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.

Sunday, November 24, 2013

Galois Groups and Harmonic Analysis

Filed under: General,Geometry — Tags: — m759 @ 9:29 AM

"In 1967, he [Langlands] came up with revolutionary
insights tying together the theory of Galois groups 
and another area of mathematics called harmonic
analysis. These two areas, which seem light years
apart
, turned out to be closely related."

— Edward Frenkel, Love and Math, 2013

"Class field theory expresses Galois groups of
abelian extensions of a number field F
in terms of harmonic analysis on the
multiplicative group of [a] locally compact
topological ring, the adèle ring, attached to F."

— Michael Harris in a description of a Princeton
    mathematics department talk of October 2012

Related material: a Saturday evening post.

See also Wikipedia on the history of class field theory.
For greater depth, see Tate's [1950] thesis and the book
Fourier Analysis on Number Fields .

Monday, April 2, 2018

Three Mother Cubes

Filed under: General,Geometry — Tags: — m759 @ 1:44 PM

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.

Wednesday, June 28, 2017

You Say Goodbye, I Say …

Filed under: General — m759 @ 9:38 PM

The title is from a Beatles song. See a link to 2008 in the previous post.

Harmonic Analysis for Illiterates

The image “http://www.log24.com/music/images/Keys-Values.gif” cannot be displayed, because it contains errors.

Notes and Frequency Ratios

Friday, March 17, 2017

To Coin a Phrase

Filed under: General,Geometry — Tags: — m759 @ 9:26 PM

(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.

Friday, December 23, 2016

Requiem for a Mathematician

Filed under: General,Geometry — m759 @ 2:10 PM

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.

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.

Friday, January 29, 2016

Excellent Adventure*

Filed under: General — Tags: — m759 @ 9:29 PM

(Continued from Dec. 9, 2013)

"…it would be quite a long walk
for him if he had to walk straight across."

The image “http://www.log24.com/log/pix07A/070831-Ant1.gif” cannot be displayed, because it contains errors.

Swiftly Mrs. Who brought her hands… together.

"Now, you see," Mrs. Whatsit said,
"he would be  there, without that long trip.
That is how we travel."

The image “http://www.log24.com/log/pix07A/070831-Ant2.gif” cannot be displayed, because it contains errors.

– A Wrinkle in Time 
Chapter 5, "The Tesseract"

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.

Tuesday, January 12, 2016

Lechner’s Beginning

Filed under: General — m759 @ 12:00 AM

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.

Monday, January 11, 2016

Space Oddity

Filed under: General,Geometry — Tags: — m759 @ 3:15 PM

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

Wednesday, April 1, 2015

Math’s Big Lies

Filed under: General,Geometry — Tags: , — m759 @ 11:00 AM

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.

Thursday, December 5, 2013

Blackboard Jungle

Filed under: General,Geometry — Tags: , — m759 @ 11:07 AM

Continued from Field of Dreams, Jan. 20, 2013.

IMAGE- Richard Kiley in 'Blackboard Jungle,' with grids and broken records

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
topic: Your Ph.D. thesis from 1950, when you were
twenty-five years old. It has been extensively cited
in the literature under the sobriquet “Tate’s thesis”.
Several mathematicians have described your thesis
as unsurpassable in conciseness and lucidity and as
representing a watershed in the study of number
fields. Could you tell us what was so novel and fruitful
in your thesis?

Tate: Well, first of all, it was not a new result, except
perhaps for some local aspects. The big global
theorem had been proved around 1920 by the
great German mathematician Erich Hecke, namely
​the fact that all L -functions of number fields,
abelian -functions, generalizations of Dirichlet’s
L -functions, have an analytic continuation
throughout the plane with a functional equation
of the expected type. In the course of proving
it Hecke saw that his proof even applied to a new
kind of L -function, the so-called L -functions with
Grössencharacter. Artin suggested to me that one
might prove Hecke’s theorem using abstract
harmonic analysis on what is now called the adele
ring, treating all places of the field equally, instead
of using classical Fourier analysis at the archimedian 
places and finite Fourier analysis with congruences 
at the p -adic places as Hecke had done. I think I did
a good job —it might even have been lucid and
concise!—but in a way it was just a wonderful 
exercise to carry out this idea. And it was also in the
air. So often there is a time in mathematics for 
something to be done. My thesis is an example. 
Iwasawa would have done it had I not.

[For a different perspective on the highlighted areas of
mathematics, see recent remarks by Edward Frenkel.]

"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.

Fields

Filed under: General,Geometry — Tags: , , — m759 @ 1:20 AM

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, 
United States, located on the Danvers River near the
northeastern coast of Massachusetts. Originally known
as Salem Village, the town is most widely known for its
association with the 1692 Salem witch trials. It is also
known for the Danvers State Hospital, one of the state's
19th-century psychiatric hospitals, which was located here." 

"The summer's gone and all the roses fallin' "

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.

Tuesday, November 26, 2013

Edward Frenkel, Your Order Is Ready.

Filed under: General — Tags: — m759 @ 11:00 AM

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

Ding!

"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

Saturday, November 23, 2013

Light Years Apart?

Filed under: General,Geometry — Tags: — m759 @ 9:00 PM

From a recent attempt to vulgarize the Langlands program:

"Galois’ work is a great example of the power of a mathematical insight…. 

And then, 150 years later, Langlands took these ideas much farther. In 1967, he came up with revolutionary insights tying together the theory of Galois groups and another area of mathematics called harmonic analysis. These two areas, which seem light years apart, turned out to be closely related."

— Frenkel, Edward (2013-10-01).
     Love and Math: The Heart of Hidden Reality
     (p. 78, Basic Books, Kindle Edition) 

(Links to related Wikipedia articles have been added.)

 

Wikipedia on the Langlands program

The starting point of the program may be seen as Emil Artin's reciprocity law [1924-1930], which generalizes quadratic reciprocity. The Artin reciprocity law applies to a Galois extension of algebraic number fields whose Galois group is abelian, assigns L-functions to the one-dimensional representations of this Galois group; and states that these L-functions are identical to certain Dirichlet L-series or more general series (that is, certain analogues of the Riemann zeta function) constructed from Hecke characters. The precise correspondence between these different kinds of L-functions constitutes Artin's reciprocity law.

For non-abelian Galois groups and higher-dimensional representations of them, one can still define L-functions in a natural way: Artin L-functions.

The insight of Langlands was to find the proper generalization of Dirichlet L-functions, which would allow the formulation of Artin's statement in this more general setting.

 

From "An Elementary Introduction to the Langlands Program," by Stephen Gelbart (Bulletin of the American Mathematical Society, New Series , Vol. 10, No. 2, April 1984, pp. 177-219)

On page 194:

"The use of group representations in systematizing and resolving diverse mathematical problems is certainly not new, and the subject has been ably surveyed in several recent articles, notably [ Gross and Mackey ]. The reader is strongly urged to consult these articles, especially for their reformulation of harmonic analysis as a chapter in the theory of group representations.

In harmonic analysis, as well as in the theory of automorphic forms, the fundamental example of a (unitary) representation is the so-called 'right regular' representation of G….

Our interest here is in the role representation theory has played in the theory of automorphic forms.* We focus on two separate developments, both of which are eventually synthesized in the Langlands program, and both of which derive from the original contributions of Hecke already described."

Gross ]  K. I. Gross, On the evolution of non-commutative harmonic analysis . Amer. Math. Monthly 85 (1978), 525-548.

Mackey ]  G. Mackey, Harmonic analysis as the exploitation of symmetry—a historical survey . Bull. Amer. Math. Soc. (N.S.) 3 (1980), 543-698.

* A link to a related Math Overflow article has been added.

In 2011, Frenkel published a commentary in the A.M.S. Bulletin  
on Gelbart's Langlands article. The commentary, written for
a mathematically sophisticated audience, lacks the bold
(and misleading) "light years apart" rhetoric from his new book 
quoted above.

In the year the Gelbart article was published, Frenkel was
a senior in high school. The year was 1984.

For some remarks of my own that mention
that year, see a search for 1984 in this journal.

Saturday, January 12, 2013

Abstract*

Filed under: General — m759 @ 7:59 AM
 

Saturday January 12, 2013,
8:00 a.m.-10:50 a.m. Pacific Standard Time

MAA Invited Paper Session on
Writing, Talking, and Sharing Mathematics

Room 2, Upper Level, San Diego
Convention Center

9:30 a.m.
Mathematics, Meaning, and Misunderstanding.
Gerald B. Folland, University of Washington
(1086-AH-1058)

Abstract:

Mathematicians develop habits of thought and employ
ways of expressing their ideas that are not always
shared by others who wish to learn mathematics or
use mathematics in their own disciplines. We shall
comment on various aspects of this phenomenon
and the (often amusing) pitfalls it creates for e ffective
communication. (Received September 18, 2012)

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
Los Angeles reopened some 17 hours after snow
shut the route and forced hundreds of truckers 
to spend the cold night in their rigs.

The California Highway Patrol shut the Grapevine
segment of Interstate 5 on Thursday afternoon,
severing a key link between the Central Valley 
and Los Angeles.

"There must have been 1,000 Mack trucks lined up,"
said traveler Heidi Blood, 40.

Saturday, December 1, 2012

Star Wars

Filed under: General — m759 @ 2:01 AM

IMAGE- Rudolf Koch's version of the 'double cross' symbol

  Source: Rudolf KochThe Book of Signs

The American Mathematical Society
(AMS) yesterday:

Lars Hörmander (1931-2012)
Friday November 30th 2012

Hörmander, who received a Fields Medal in 1962,
died November 25 at the age of 81. …

more »

Some related material:

See also posts on Damnation Morning and, from the
date of Hörmander's death,

Sunday, September 9, 2012

Decomposition Sermon

Filed under: General,Geometry — m759 @ 11:00 AM

(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 .

Monday, February 6, 2012

Beach Boy

Filed under: General — m759 @ 8:18 PM
 
Cached from artnet.com
     
Artist   Daniel Simon
Title   Yves Montand et Catherine Deneuve
dans Le Sauvage, Bahamas
Medium   gelatin silver print
Size   11.8 x 15.7 in. / 30 x 40 cm.
Year   1975 –
Edition   1/10
Misc.   Signed, Stamped
Sale Of   Ader: Monday, October 20, 2008
[Lot 140]
Paris – Célébrités II
Estimate   300 – 400 EUR (USD 403 – 537)
Sold For   *
* Complete data is available for subscribers.

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—

The Sweet Smell of Avon.

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.

Friday, January 20, 2012

Brightness at Noon

Filed under: General — m759 @ 12:00 PM

(Continued)

See "harmonic analysis" in Mathematical Imagery and elsewhere in this journal.

Saturday, December 26, 2009

Annals of Philosophy

Filed under: General,Geometry — m759 @ 12:00 PM

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.

Thursday, November 6, 2008

Thursday November 6, 2008

Filed under: General,Geometry — m759 @ 10:07 AM
Death of a Classmate

Michael Crichton,
Harvard College, 1964

Authors Michael Crichton and David Foster Wallace in NY Times obituaries, Thursday, Nov.  6, 2008

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”

Crystal from 'Diamond Theory'

“Dragon”

(from Crichton’s
Jurassic Park)–


Dragon Curve from '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.

Saturday, August 16, 2008

Saturday August 16, 2008

Filed under: General,Geometry — m759 @ 8:00 AM

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:

Walsh Series: An Introduction to Dyadic Harmonic Analysis, by F. Schipp et. al.

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.”

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.

Wednesday, October 24, 2007

Wednesday October 24, 2007

Filed under: General,Geometry — Tags: , — m759 @ 11:11 PM
 
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:

The color-analogy 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,

Emma Thompson in 'Wit'

from Mike Nichols's birthday, 2003.

Tuesday, March 28, 2006

Tuesday March 28, 2006

Filed under: General — Tags: — m759 @ 4:00 PM
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

“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:

“It would have been
 a great disappointment
 to me if Vibration did not
 somewhere make itself felt,
 for all scientific mystics
 either vibrate in person
 or find themselves
 resonant with cosmic
 vibrations….”

Sir Peter Brian Medawar

“He’s good.”
“Good? He’s the fucking
Prince of Darkness!”

— Paul Newman
and Jack Warden
in “The Verdict

Sanskrit (transliterated) —

    nada:
 
 
  the universal sound, vibration.

“So Nada Brahma means not only:
 God the Creator is sound; but also
 (and above all), Creation,
 the cosmos, the world, is sound.
 And: Sound is the world.”

Joachim-Ernst Berendt,  
   author of Nada Brahma

 
“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.

Friday, March 17, 2006

Friday March 17, 2006

Filed under: General,Geometry — m759 @ 2:28 AM
George W. Mackey,
Harvard mathematician,
is dead at 90.

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). 

From Oxford U.P.–

Contents

  • 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.

Sunday, July 31, 2005

Sunday July 31, 2005

Filed under: General — m759 @ 5:24 AM
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:

Quote from an earlier entry:

“In honor of Roger Cooke’s review of Helson’s Harmonic Analysis, 2nd Edition, today’s site music is “Moonlight in Vermont.”

Quote from July 20: 

“And if the band you’re in
   starts playing different tunes
 I’ll see you on
   the dark side of the moon.”

Quote from Lois-Ann Yamanaka:

Blu’s Hanging

   … Poppy still plays “Moon River” in the background.
   He sings aloud:
   “Old dreammaker, you heartbreaker, wherever you’re going, I’m going your way.”
   He makes me afraid.
   I know where he wants to go.
   And who the dreammaker is.

The image “http://www.log24.com/log/pix05A/050731-Hatta.jpg” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix05A/050731-Moon.jpg” cannot be displayed, because it contains errors.

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

Wednesday, July 20, 2005

Wednesday July 20, 2005

Filed under: General — m759 @ 7:20 PM
Moon Day
Words that may or may not have been said on July 20, 1969:

“That’s one small step for a man; one giant leap for mankind.”

Another rhetorical contrast,
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:

Date Posted: 11/03/02 Sun


“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.

Wednesday, September 15, 2004

Wednesday September 15, 2004

Filed under: General — m759 @ 11:59 PM

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

Tuesday, September 14, 2004

Tuesday September 14, 2004

Filed under: General,Geometry — Tags: , — m759 @ 3:00 PM

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 functions– the "square wheel," as it were– is also not without its importance.

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 functions– which have, in fact, properties very like those of the trig functions.  For details, see Symmetry of Walsh Functions, updated today.

"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

Wednesday, May 19, 2004

Wednesday May 19, 2004

Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

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.)

Thursday, November 6, 2003

Thursday November 6, 2003

Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

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.

Friday, April 25, 2003

Friday April 25, 2003

Filed under: General,Geometry — Tags: , , — m759 @ 7:59 PM

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.

 


The rune known as
"Dagaz"

Represents
the balance point or "still point."


The Nike Bastion

 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.
 

Wednesday, February 12, 2003

Wednesday February 12, 2003

Filed under: General,Geometry — m759 @ 3:00 AM

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

Abstract Harmonic Analysis. 

For work that is more in von Neumann’s league than in Hofstadter’s, see the following

harmonic analysis abstract:

VECTOR-VALUED EXTENSIONS
OF SOME CLASSICAL THEOREMS
IN HARMONIC ANALYSIS

Maria Girardi and Lutz Weis

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.

Tuesday, February 11, 2003

Tuesday February 11, 2003

Filed under: General — m759 @ 5:10 PM

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:

The image “http://log24.com/log/pix03/030211-oedipus.gif” cannot be displayed, because it contains errors.

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

Filed under: General — m759 @ 2:22 PM

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

Music courtesy of Franz Schubert.

Sunday, November 3, 2002

Sunday November 3, 2002

Filed under: General — m759 @ 12:00 AM

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

Filed under: General,Geometry — m759 @ 1:16 AM

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
   gentle word

On the wind that lifts
   her perfume
   through the air.”

— 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

Saturday, October 19, 2002

Saturday October 19, 2002

Filed under: General — m759 @ 9:47 AM

What is Truth?

My state of mind
before reading the
New York Times

My state of mind
after reading the
New York Times

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!  

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