Log24

Monday, January 23, 2023

“You’re so vain…” — Song lyric

Filed under: General — m759 @ 3:22 pm

A remark by Esther Dyson on Jan. 19:

See as well the above group-theory author here  on Jan. 19.

Related material:  "Same Staircase, DIfferent Day."

Thursday, January 19, 2023

Two Approaches to Local-Global Symmetry

Filed under: General — Tags: , — m759 @ 2:34 am

Last revised: January 20, 2023 @ 11:39:05

The First Approach — Via Substructure Isomorphisms —

From "Symmetry in Mathematics and Mathematics of Symmetry"
by Peter J. Cameron, a Jan. 16, 2007, talk at the International
Symmetry Conference, Edinburgh, Jan. 14-17, 2007

Local or global?

"Among other (mostly more vague) definitions of symmetry, the dictionary will typically list two, something like this:

• exact correspondence of parts;
• remaining unchanged by transformation.

Mathematicians typically consider the second, global, notion, but what about the first, local, notion, and what is the relationship between them?  A structure M  is homogeneous * if every isomorphism between finite substructures of M  can be extended to an automorphism of ; in other words, 'any local symmetry is global.' "

A related discussion of the same approach — 

"The aim of this thesis is to classify certain structures
which are, from a certain point of view,
as homogeneous as possible, that is
which have as many symmetries as possible.
the basic idea is the following: a structure S  is
said to be homogeneous  if, whenever two (finite)
substructures Sand S2 of S  are isomorphic,
there is an automorphism of S  mapping S1 onto S2.”

— Alice Devillers,
Classification of Some Homogeneous
and Ultrahomogeneous Structures
,”
Ph.D. thesis, Université Libre de Bruxelles,
academic year 2001-2002

The Wikipedia article Homogeneous graph discusses the local-global approach
used by Cameron and by Devillers.

For some historical background on this approach
via substructure isomorphisms, see a former student of Cameron:

Dugald Macpherson, "A survey of homogeneous structures,"
Discrete Mathematics , Volume 311, Issue 15, 2011,
Pages 1599-1634.

Related material:

Cherlin, G. (2000). "Sporadic Homogeneous Structures."
In: Gelfand, I.M., Retakh, V.S. (eds)
The Gelfand Mathematical Seminars, 1996–1999.
Gelfand Mathematical Seminars. Birkhäuser, Boston, MA.
https://doi.org/10.1007/978-1-4612-1340-6_2

and, more recently, 

Gill et al., "Cherlin's conjecture on finite primitive binary
permutation groups," https://arxiv.org/abs/2106.05154v2
(Submitted on 9 Jun 2021, last revised 9 Jul 2021)

This approach seems to be a rather deep rabbit hole.

The Second Approach — Via Induced Group Actions —

My own interest in local-global symmetry is of a quite different sort.

See properties of the two patterns illustrated in a note of 24 December 1981 —

Pattern A above actually has as few  symmetries as possible
(under the actions described in the diamond theorem ), but it
does  enjoy, as does patttern B, the local-global property that
a group acting in the same way locally on each part  induces
a global group action on the whole .

* For some historical background on the term "homogeneous,"
    see the Wikipedia article Homogeneous space.

Friday, October 14, 2016

A Little Solitaire

Filed under: General — m759 @ 10:00 pm

Detail of a photograph from 1954

A related tale

Monday, May 16, 2016

Fake Religion

Filed under: General — Tags: — m759 @ 1:00 am

(A companion-piece to Fake Eliot.)

The President at Rutgers on Sunday

Point number one

When you hear someone longing for the ‘good old days,’
take it with a grain of salt.  (Laughter and applause.)
Take it with a grain of salt.”

Boris Karloff as a modernist architect in a 1934 horror film —

Cum grano salis.

Friday, April 3, 2015

Math Humor for Holy Week

Filed under: General — m759 @ 12:00 am

See also the home page of Cornell mathematician Steven Strogatz:

Strogatz is the author of "Why Pi Matters."

Backstory —

Tuesday, September 23, 2014

Both Hands and an Ass Map

Filed under: General — Tags: , — m759 @ 9:48 am

Remarks by University Diaries  this morning suggested a search for
Rutgers in this journal. That search yields a post from Dec. 30, 2005,
that is closely related to both this morning's previous post and to recent
Log24 remarks on entities and concrete universals .

Monday, March 4, 2013

Occupy Galois Space

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

Continued from February 27, the day Joseph Frank died

"Throughout the 1940s, he published essays
and criticism in literary journals, and one,
'Spatial Form in Modern Literature'—
a discussion of experimental treatments
of space and time by Eliot, Joyce, Proust,
Pound and others— published in
The Sewanee Review  in 1945, propelled him
to prominence as a theoretician."

— Bruce Weber in this morning's print copy
of The New York Times  (p. A15, NY edition)

That essay is reprinted in a 1991 collection
of Frank's work from Rutgers University Press:

See also Galois Space and Occupy Space in this journal.

Frank was best known as a biographer of Dostoevsky.
A very loosely related reference… in a recent Log24 post,
Freeman Dyson's praise of a book on the history of
mathematics and religion in Russia:

"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."

Frank is survived by, among others, his wife, a mathematician.

Sunday, November 25, 2012

Will*

Filed under: General — m759 @ 11:00 am

Today's sermon is for Martha B. Helfer, author of
the treatise on Darstellung  in today's previous post
and of the following—

(Click for clearer image.)

Helfer's The Word Unheard  was published by Northwestern University Press
on St. Andrew's Day, 2011. Log24 posts on that day—

Lines, Grids, and Fatuity for St. Andrew's Day.

The last of these warned of an upcoming Jewish Book Week event
on February 22, 2012.

That date turned out to be Ash Wednesday. See a Log24 post on that topic
that quotes a poet, T.S. Eliot, with anti-Semitic proclivities—

"And the light shone in darkness and
Against the Word the unstilled world still whirled
About the centre of the silent Word."

— T. S. Eliot, "Ash Wednesday"

This is perhaps not entirely irrelevant to Helfer's title, The Word Unheard .

* A concept of Schopenhauer and Hitler, and the first name of
   a fictional Boston mathematician.

Representation

Filed under: General — m759 @ 10:31 am

The title of yesterday's post Will and Representation is of course
a reference to Schopenhauer's philosophical work of that name.

As the post itself indicates, the title is also a punning reference to
mathematical representation theory . 

To avoid confusion, it should be noted that Schopenhauer's
representation , in the original German, was Vorstellung .

The German for mathematical  representation theory is,
on the other hand, Darstellungstheorie . (The mathematical
use of Vorstellung  is non-technical, referring to concepts
of pedagogy. (A group presentation  is a Präsentation .))

For a discussion of the Vorstellung-Darstellung  distinction
in philosophy, not mathematics, see… 

The Retreat of Representation: The Concept of  Darstellung
in German Critical Discourse , by Martha B. Helfer,
State University of New York Press, 1996, esp. pp. 24-26.

Saturday, July 14, 2012

Lemma

Filed under: General,Geometry — Tags: , — m759 @ 2:25 pm

IMAGE- 'Lemma (mathematics)' in Wikipedia

For example—

A letter to the editor of the American Mathematical Monthly
from the June-July 1985 issue has—

… a "square-triangle" lemma:

   ( t ∈ T , t  is an  -replica )
    if and only if  
n  is a square.

  [I.e., "Every triangle is an -replica"
   is true if and only if n  is a square.]

For definitions, see the 1985 letter in Triangles Are Square.

(The 1984 lemma discussed there has now, in response to an article
in Wolfram MathWorld, been renamed the square-triangle theorem .)

A search today for related material yielded the following—

"Suppose that one side of a triangle
has length . Then it can be cut
into n  2 congruent triangles which
are similar to the original one and
whose corresponding sides to the
side of length  have lengths 1."

This was supplied, without attribution, as part of the official solution
to Problem 3 in the 17th Asian Pacific Mathematics Olympiad
from March 2005. Apparently it seemed obvious to the composer
of the problem. As the 1985 letter notes, it may be not quite  obvious.

At any rate, it served in Problem 3 as a lemma , in the sense
described above by Wikipedia. See related remarks by Doron Zeilberger.

Saturday, March 3, 2012

Decomposition

Filed under: General,Geometry — m759 @ 3:33 am

A search tonight for material related to the four-color
decomposition theorem yielded the Wikipedia article
Functional decomposition.

The article, of more philosophical than mathematical
interest, is largely due to one David Fass at Rutgers.

(See the article's revision history for mid-August 2007.)

Fass's interest in function decomposition may or may not
be related to the above-mentioned theorem, which 
originated in the investigation of functions into the
four-element Galois field from a 4×4 square domain.

Some related material involving Fass and 4×4 squares—

A 2003 paper he wrote with Jacob Feldman—

(Click to enlarge.)

"Design is how it works." — Steve Jobs

An assignment for Jobs in the afterlife—

Discuss the Fass-Feldman approach to "categorization under
complexity" in the context of the Wikipedia article's
philosophical remarks on "reductionist tradition."

The Fass-Feldman paper was assigned in an MIT course
for a class on Walpurgisnacht 2003.

Friday, January 20, 2012

The Nothing That Is

Filed under: General — m759 @ 9:00 pm

"The 'one' with whom the reader has identified himself
has now become 'the listener, who listens in the snow';
he has become the snow man, and he knows winter
with a mind of winter, knows it in its strictest reality,
stripped of all imagination and human feeling.
But at that point when he sees the winter scene
reduced to absolute fact, as the object not of the mind,
but of the perfect perceptual eye that sees
'nothing that is not there,' then the scene,
devoid of  its imaginative correspondences,
has become 'the nothing that is.'"

Robert Pack, Wallace Stevens:
An Approach to His Poetry and Thought
.
New Brunswick: Rutgers UP, 1958.

 

IMAGE- The Ninefold Square at Ninevine.net

Wednesday, April 11, 2007

Wednesday April 11, 2007

Filed under: General — m759 @ 12:00 pm
It’s Not Easy
Being Green

The image “http://www.log24.com/log/pix07/070411-Imus.jpg” cannot be displayed, because it contains errors.

Don Imus, 1974 album


Rutgers women’s basketball
coach C. Vivian Stringer:

“It’s about us as people– black, white, purple or green. And as much as I speak about that, it’s not even black and white– the color is green.”

Imus flap about
black, white, and green

David Lieberman, Laura Petrecca and Gary Strauss in USA Today:

“So amid all the uproar over Imus’ remarks and the national discussion over race relations that they ignited, why wasn’t he fired?

Stringer and others think that has less to do with relations between blacks and whites than it does with another color.

‘The color is green– if we can tolerate as a society what’s just taken place,’ she said. ‘I don’t know how anyone could have heard this and not been offended.’

As one of the country’s most popular radio talk show hosts, Imus is the centerpiece of a multimillion-dollar business that would collapse without him.

To get a sense of its size: Advertisers spent $11.3 million last year on his show at just one station, New York’s WFAN, according to Nielsen. That accounted for nearly 24% of all the station’s ad sales.

Sponsors paid MSNBC an additional $8.4 million last year for spots on Imus’ show, according to TNS Media Intelligence.”

Mike Lupica in the
New York Daily News,
April 11, 2007:

“Essence Carson talked about what Imus had said about her and her teammates, and about everything that has happened since.

‘It has stolen a moment of pure grace from us,’ she said.

The moment of pure grace was Essence Carson….”

From ESSENCE.com:

“Essence Communications Inc. (ECI) was founded in 1968. In October 2000, ECI signed an agreement with Time Inc., a subsidiary of Time Warner Inc., to form a joint venture known as Essence Communications Partners. ESSENCE was the majority owner of the venture. In March 2005, Time Inc. acquired the portion it did not already own. The company’s name changed back to Essence Communications Inc. The ECI corporate headquarters are in New York City, with offices in Chicago, Los Angeles, Atlanta and Detroit.

ESSENCE magazine

ESSENCE magazine

During the past 36 years, the company has grown into a vital business of diverse media properties and communications systems that include ESSENCE, its flagship magazine launched in 1970. Its success is linked to its unique relationship with the readers of ESSENCE magazine and the strong alliances it has forged with America’s leading corporations and financial institutions.”

Monday, January 9, 2006

Monday January 9, 2006

Filed under: General,Geometry — m759 @ 5:01 am
Cornerstone

“In 1782, the Swiss mathematician Leonhard Euler posed a problem whose mathematical content at the time seemed about as much as that of a parlor puzzle. 178 years passed before a complete solution was found; not only did it inspire a wealth of mathematics, it is now a cornerstone of modern design theory.”

— Dean G. Hoffman, Auburn U.,
    July 2001 Rutgers talk

Diagrams from Dieter Betten’s 1983 proof
of the nonexistence of two orthogonal
6×6 Latin squares (i.e., a proof
of Tarry’s 1900 theorem solving
Euler’s 1782 problem of the 36 officers):

The image “http://www.log24.com/log/pix06/060109-TarryProof.gif” cannot be displayed, because it contains errors.

Compare with the partitions into
two 8-sets of the 4×4 Latin squares
discussed in my 1978 note (pdf).

Friday, December 30, 2005

Friday December 30, 2005

Filed under: General — Tags: — m759 @ 1:00 am

Expression

Expression is an impossible word.
If you want to use it
I think you have to
explain it further…
— Ad Reinhardt,
Art as Art

“… an equation is a very
abstract expression
of knowledge
about something.”
The Hidden Side
of Visualization

Well, yes.
For example:

The image “http://www.log24.com/log/pix05B/051230-Shema.jpg” cannot be displayed, because it contains errors.
Abstract Expression
,
pixels on screen, 12/30/05

The Scope and Limits
of Quotation
(pdf), in
The Philosophy of
Donald Davidson,
ed. L. E. Hahn,
Open Court Publishers,
1999, pp. 691-714,
by Dr. Ernest Lepore,
Rutgers University:

“… an assumption I wish to reject, namely, that in quotation an abstract expression (shape) is denoted….

Since Davidson, however, only says ‘we may  take [an expression] to be an abstract shape’ [1979, p.85, my emphasis], his theory is compatible with expressions being something else. We need only find something that can be instantiated by radically differently shaped objects. Whatever it is must be such that written tokens, spoken tokens, signed tokens, Braille tokens, Semaphore tokens, finger language tokens, and any other way in which words can be produced, can be instantiated by it….

Such entities might  exist; if they do, they might ultimately play some role in the metaphysics of language.”

My emphasis.

Reference:

Davidson, D., 1979, “Quotation,” in Inquiries into Truth and Interpretation , Oxford, Oxford University Press, 1984, pp.79-92.

For more on zero and other entities,
see Is Nothing Sacred?

For more on Davidson,
see Shema .

Monday, November 7, 2005

Monday November 7, 2005

Filed under: General — m759 @ 9:00 pm
Butterfly and Wheel

The illustration for the previous entry,

The image “http://www.log24.com/log/pix05B/051107-Aristos2.jpg” cannot be displayed, because it contains errors.

taken from fowlesbooks.com,
suggests more links:

1. To Butterflies and Wheels* (banner below),

The image “http://www.log24.com/log/pix05B/051107-Butt.jpg” cannot be displayed, because it contains errors.

a site that attacks Mary Midgley, and

2. To an AAAS site offering Midgley’s

Evolution as a Religion:
A Comparison of Prophecies

I personally prefer Midgley,
an Oxford-trained philosopher
also known as
Mary, Mary, Quite Contrary.

Related material:
a meditation on the phrase
crucified on the wheel of time.”

* “Who breaks a Butterfly upon a Wheel?”
Alexander Pope

Friday, October 1, 2004

Friday October 1, 2004

Filed under: General — m759 @ 4:07 pm

Ig Nobel
Geography Prize

Background from David Brooks (NYT Magazine, Aug. 29, 2004):

“The war on Islamic extremism….
This has been miscast as a war on terror, but terror is just the means our enemies use. In reality, we’re fighting a war against a specific brand of Islamic extremism, a loose federation of ideologues who seek to dominate the Middle East and return it to the days of the caliphate.

We are in the beginning of this war, where we were against Bolshevism around 1905 or Fascism in the early 1930’s, with enemies that will continue to gain strength, thanks to the demographic bulge in the Middle East….”

On last night’s Ig Nobel ceremony at Harvard:

“Coming a week before their more noble cousins are announced in Stockholm, the awards have become a keenly contested, globally reported event.”

The Guardian, Oct. 1 

From the Bush-Kerry debate last night:

Kerry: The president just talked about Iraq as a center of the war on terror. Iraq was not even close to the center of the war on terror before the president invaded it.”

As David Brooks has pointed out, the so-called war on terror is actually a war on an Islamic extremism that seeks to dominate the Middle East.  The United States government may deny, and Kerry may decry, the strategy of building permanent bases at the very center of the Middle East, Iraq (shown above), but such a strategy seems not without merit if the long-term strategic interests of the United States are placed ahead of political considerations.

Kerry’s apparent ignorance of such a strategy’s merits entitles him to this year’s Ig Nobel prize in geography– the above map.

Thursday, May 20, 2004

Thursday May 20, 2004

Filed under: General,Geometry — Tags: — m759 @ 7:00 am

Parable

"A comparison or analogy. The word is simply a transliteration of the Greek word: parabolé (literally: 'what is thrown beside' or 'juxtaposed'), a term used to designate the geometric application we call a 'parabola.'….  The basic parables are extended similes or metaphors."

http://religion.rutgers.edu/nt/
    primer/parable.html

"If one style of thought stands out as the most potent explanation of genius, it is the ability to make juxtapositions that elude mere mortals.  Call it a facility with metaphor, the ability to connect the unconnected, to see relationships to which others are blind."

Sharon Begley, "The Puzzle of Genius," Newsweek magazine, June 28, 1993, p. 50

"The poet sets one metaphor against another and hopes that the sparks set off by the juxtaposition will ignite something in the mind as well. Hopkins’ poem 'Pied Beauty' has to do with 'creation.' "

Speaking in Parables, Ch. 2, by Sallie McFague

"The Act of Creation is, I believe, a more truly creative work than any of Koestler's novels….  According to him, the creative faculty in whatever form is owing to a circumstance which he calls 'bisociation.' And we recognize this intuitively whenever we laugh at a joke, are dazzled by a fine metaphor, are astonished and excited by a unification of styles, or 'see,' for the first time, the possibility of a significant theoretical breakthrough in a scientific inquiry. In short, one touch of genius—or bisociation—makes the whole world kin. Or so Koestler believes."

— Henry David Aiken, The Metaphysics of Arthur Koestler, New York Review of Books, Dec. 17, 1964

For further details, see

Speaking in Parables:
A Study in Metaphor and Theology

by Sallie McFague

Fortress Press, Philadelphia, 1975

Introduction
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7

"Perhaps every science must start with metaphor and end with algebra; and perhaps without metaphor there would never have been any algebra."

— attributed, in varying forms (1, 2, 3), to Max Black, Models and Metaphors, 1962

For metaphor and algebra combined, see

"Symmetry invariance in a diamond ring," A.M.S. abstract 79T-A37, Notices of the Amer. Math. Soc., February 1979, pages A-193, 194 — the original version of the 4×4 case of the diamond theorem.

Wednesday, March 17, 2004

Wednesday March 17, 2004

Filed under: General — Tags: — m759 @ 4:31 pm

Readings for
St. Patrick's Day

Books:

Finnegans Wake (1939)

Gravity's Rainbow (1978)

Masks of the Illuminati (1981)

Quotations:

"Nature does not know extinction;
all it knows is transformation.
Everything science has taught me,
and continues to teach me,
strengthens my belief in
the continuity of our
spiritual existence
after death."

Wernher von Braun

"I faced myself that day
with the nonplused apprehension
of someone who has
come across a vampire
and has no crucifix in hand."

— Joan Didion, "On Self-Respect,"
in Slouching Towards Bethlehem

 

"For every kind of vampire,
there is a kind of cross.
"

— Thomas Pynchon,
Gravity's Rainbow

Inscribed
Carpenter's Square:

In Latin, NORMA

Multa renascentur quae iam cecidere, cadentque
quae nunc sunt in honore uocabula, si uolet usus,
quem penes arbitrium est et ius et norma loquendi.

Horace, Ars Poetica

Many terms will be born again
that by now have sunk into oblivion,
and many that are now held in respect
will die out if that is what use should dictate
in whose power is the judgment and the law
and the rule of speech.

All, all must perish — but, surviving last,
The love of Letters half preserves the past;
True — some decay, yet not a few revive,
Though those shall sink, which now
     appear to thrive,
As Custom arbitrates, whose shifting sway
Our life and language must alike obey.

Hints from Horace

"Norma was the latin word for what we now call a carpenter's square. It was used to construct lines which were at right angles to another line, so the created line was said to be 'normal.'  The norma was also used as a standard to compare if objects, like a wall, might be erect (perpendicular to the ground) and so those that met the standard were called 'normal' and this use extended to the 'typical' element of any type of set. Eventually normal came to mean anything that 'met the standard.' "

Pat Ballew on mathematical usage

 

"317 is a prime,
not because we think so,
or because our minds are shaped
in one way rather than another,
but
because it is so,
because mathematical reality
is built that way."

— G. H. Hardy,
A Mathematician's Apology

Tuesday, September 2, 2003

Tuesday September 2, 2003

Filed under: General,Geometry — Tags: , — m759 @ 1:11 pm

One Ring to Rule Them All

In memory of J. R. R. Tolkien, who died on this date, and in honor of Israel Gelfand, who was born on this date.

Leonard Gillman on his collaboration with Meyer Jerison and Melvin Henriksen in studying rings of continuous functions:

“The triple papers that Mel and I wrote deserve comment. Jerry had conjectured a characterization of beta X (the Stone-Cech compactification of X) and the three of us had proved that it was true. Then he dug up a 1939 paper by Gelfand and Kolmogoroff that Hewitt, in his big paper, had referred to but apparently not appreciated, and there we found Jerry’s characterization. The three of us sat around to decide what to do; we called it the ‘wake.’  Since the authors had not furnished a proof, we decided to publish ours. When the referee expressed himself strongly that a title should be informative, we came up with On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions. (This proved to be my second-longest title, and a nuisance to refer to.) Kolmogoroff died many years ago, but Gelfand is still living, a vigorous octogenarian now at Rutgers. A year or so ago, I met him at a dinner party in Austin and mentioned the 1939 paper. He remembered it very well and proceeded to complain that the only contribution Kolmogoroff had made was to point out that a certain result was valid for the complex case as well. I was intrigued to see how the giants grouse about each other just as we do.”

Leonard Gillman: An Interview

This clears up a question I asked earlier in this journal….

Wednesday, May 14, 2003

Common Sense

On the mathematician Kolmogorov:

“It turns out that he DID prove one basic theorem that I take for granted, that a compact hausdorff space is determined by its ring of continuous functions (this ring being considered without any topology) — basic discoveries like this are the ones most likely to have their origins obscured, for they eventually come to be seen as mere common sense, and not even a theorem.”

Richard Cudney, Harvard ’03, writing at Xanga.com as rcudney on May 14, 2003

That this theorem is Kolmogorov’s is news to me.

See

The above references establish that Gelfand is usually cited as the source of the theorem Cudney discusses.  Gelfand was a student of Kolmogorov’s in the 1930’s, so who discovered what when may be a touchy question in this case.  A reference that seems relevant: I. M. Gelfand and A. Kolmogoroff, “On rings of continuous functions on topological spaces,” Doklady Akad. Nauk SSSR 22 (1939), 11-15.  This is cited by Gillman and Jerison in the classic Rings of Continuous Functions.

There ARE some references that indicate Kolmogorov may have done some work of his own in this area.  See here (“quite a few duality theorems… including those of Banaschewski, Morita, Gel’fand-Kolmogorov and Gel’fand-Naimark”) and here  (“the classical theorems of M. H. Stone, Gelfand & Kolmogorov”).

Any other references to Kolmogorov’s work in this area would be of interest.

Naturally, any discussion of this area should include a reference to the pioneering work of M. H. Stone.  I recommend the autobiographical article on Stone in McGraw-Hill Modern Men of Science, Volume II, 1968.

A response by Richard Cudney:

“In regard to your entry, it is largely correct.  The paper by Kolmogorov and Gelfand that you refer to is the one that I just read in his collected works.  So, I suppose my entry was unfair to Gelfand.  You’re right, the issue of credit is a bit touchy since Gelfand was his student.  In a somewhat recent essay, Arnol’d makes the claim that this whole thread of early work by Gelfand may have been properly due to Kolmogorov, however he has no concrete proof, having been but a child at the time, and makes this inference based only on his own later experience as Kolmogorov’s student.  At any rate, I had known about Gelfand’s representation theorem, but had not known that Kolmogorov had done any work of this sort, or that this theorem in particular was due to either of them. 

And to clarify-where I speak of the credit for this theorem being obscured, I speak of my own experience as an algebraic geometer and not a functional analyst.  In the textbooks on algebraic geometry, one sees no explanation of why we use Spec A to denote the scheme corresponding to a ring A.  That question was answered when I took functional analysis and learned about Gelfand’s theorem, but even there, Kolmogorov’s name did not come up.

This result is different from the Gelfand representation theorem that you mention-this result concerns algebras considered without any topology(or norm)-whereas his representation theorem is a result on Banach algebras.  In historical terms, this result precedes Gelfand’s theorem and is the foundation for it-he starts with a general commutative Banach algebra and reconstructs a space from it-thus establishing in what sense that the space to algebra correspondence is surjective, and hence by the aforementioned theorem, bi-unique.  That is to say, this whole vein of Gelfand’s work started in this joint paper.

Of course, to be even more fair, I should say that Stone was the very first to prove a theorem like this, a debt which Kolmogorov and Gelfand acknowledge.  Stone’s paper is the true starting point of these ideas, but this paper of Kolmogorov and Gelfand is the second landmark on the path that led to Grothendieck’s concept of a scheme(with Gelfand’s representation theorem probably as the third).

As an aside, this paper was not Kolmogorov’s first foray into topological algebra-earlier he conjectured the possibility of a classification of locally compact fields, a problem which was solved by Pontryagin.  The point of all this is that I had been making use of ideas due to Kolmogorov for many years without having had any inkling of it.”

Posted 5/14/2003 at 8:44 PM by rcudney

Sunday, August 10, 2003

Sunday August 10, 2003

Filed under: General — m759 @ 8:35 am

Death of a Holy Man

Part I:  An American Religion

Hiroshima Mayor Says
US Worships Nukes

“HIROSHIMA — Hiroshima Mayor Tadatoshi Akiba warned that the world is moving toward war and accused Washington of ‘worshipping’ nuclear weapons during Wednesday’s ceremony marking the 58th anniversary of the atomic bombing of the city….

… the Hiroshima mayor blamed the United States for making the world a more uncertain place through its policy of undermining the Treaty on the Non-Proliferation of Nuclear Weapons.

‘A world without nuclear weapons and war that the victims of the atomic bomb have long sought for is slipping into the shadows of growing black clouds that could turn into mushroom clouds at any moment,’ Akiba said. ‘The chief cause of this is the United States’ nuclear policy which, by openly declaring the possibility of a pre-emptive nuclear strike and by starting research into small ‘useable’ nuclear weapons, appears to worship nuclear weapons as God.’ “

Mainichi Shimbun, Aug. 6, 2003

Part II: Holy Men and
             Sons of Bitches

“I am become Death, the Destroyer of Worlds.”

Dr. J. Robert Oppenheimer,
    Director of Los Alamos

John Steinbeck describing Cannery Row in Monterey:

“Its inhabitants are, as the man once said, ‘whores, pimps, gamblers, and sons of bitches,’ by which he meant Everybody. Had the man looked through another peephole he might have said, ‘Saints and angels and martyrs and holy men,’ and he would have meant the same thing.”  

“Now we are all sons of bitches.”

Dr. Kenneth Bainbridge,
    Director of Trinity Test

Part III: Death of a Holy Man

The New York Times, Aug. 10, 2003:

Atom-Bomb Physicist Dies at 98

“Henry A. Boorse, a physicist who was one of the original scientists who worked on the Manhattan Project in the development of the atomic bomb, died on July 28 in Houston, where he lived….

Dr. Boorse was a consultant to the United States Atomic Energy Commission from 1946 to 1958 and to the Brookhaven National Laboratory from 1951 to 1955.

He and Lloyd Motz wrote a two-volume work, The World of the Atom (1966), and — with Jefferson Hane Weaver — a one-volume book, The Atomic Scientists (1989).”

From a review of The Atomic Scientists:

“… the authors try to add a personal element that can excite the reader about science.”

For more excitement, see Timequake, by Kurt Vonnegut, Jr.

Thursday, April 17, 2003

Thursday April 17, 2003

Filed under: General — m759 @ 2:14 pm

Holiday Affair

From a site recommended by oOMisfitOo:

In The Star of Bethlehem: The Legacy of the Magi (Rutgers University Press, 1999), Michael R. Molnar explains how the purchase of a $50 Roman coin led him to discover the real date of Jesus’s birth.

The coin that provided the clue portrayed Aries the Ram looking back at a star.

From Molnar’s own site, Star of Bethlehem:

“On April 17, 6 BC, two years before King Herod died, Jupiter emerged in the east as a morning star in the sign of the Jews, Aries the Ram.”

Therefore, according to Molnar, today is Christmas.  Accordingly, let us sing a (slightly improved) carol in memory of the late Murray L. Bob (see April 15 entries):

God rest ye, merry gentleman.

Let us also voice a rousing chorus of one of my personal all-time favorites, in memory of a film director (see previous entry), who gave us a vision of Robert Mitchum (Ram) and Sarah Miles (“Lady Caroline Lamb“) united in marriage (Ding-Dong):

Who put the Ram in the
Ram-a-Lamb-a Ding-Dong?

Why, David Lean, of course.

Update of April 21, 2003:

When You Care Enough
to Send the Very Best

“Jan Scott, 88, a television art director and production designer who had won 11 Emmy Awards, died April 17 at her home in Hollywood Hills, Calif. The cause of death was not reported.

She started working in television in the 1950s and earned her first Emmy nomination in 1956 for a “Hallmark Hall of Fame” production. Her first Emmy Award came in 1968 for her work as an art director for “Kismet,” which appeared on ABC. Her last Emmy was awarded in 1989 for “I’ll Be Home for Christmas,” on NBC.”

The Washington Post, April 21, 2003

Wednesday, January 29, 2003

Wednesday January 29, 2003

Filed under: General — m759 @ 6:09 pm

Inaugural Address
for Cullinane College

(undelivered):

The Prisoner

Cullinane College was scheduled to open its doors officially on January 29, 2003.  The following might have been an appropriate inaugural address.

From The Prisoner: Comments
 on the Final Episode, “Fall Out”
:

“When the President asks for a vote, he says: ‘All in favor.’ But he never asks for those opposed. (Though it appears that none will be opposed — and though he says its a democratic assembly, it is hardly that. The President even says that the society is in a ‘democratic crisis,’ though without democracy present, it’s just a sham.)

#48/Young Man sings ‘Dry Bones,’, which is his rebellion (notice its chaotic effect on ‘society’). But then the song gets taken over, ‘polished,’ and sung by a voice-over (presumably set up by #1). Does this mean that society is stealing the thunder (i.e. the creative energy) of youth, and cheapening it, or does it mean that youth is just rebelling in the same way that their fathers did (with equal ineffectiveness)? Perhaps it is simply a comment on the ease with which society can deal with the real rebellion of the 1960’s, which purported to be led by musicians; one that even the Beatles said was impossible in ‘Revolution.'”

President: Guilty! Read the Charge!

#48 is guilty, of something, and then the society pins something on him.”

The Other Side of the Coin

The Weinman Dime

From the CoinCentric website:

In 1916, sculptor Adolph A. Weinman produced a new design for the dime called the Liberty Head type. The motif features Miss Liberty facing left, wearing a Phrygian cap with wings, symbolizing “liberty of thought”. The word “LIBERTY” encircles her head, with “IN GOD WE TRUST” and the date below her head.

The reverse depicts Roman fasces, a bundle of rods with the center rod being an ax, against a branch in the background. It is a symbol of state authority, which offers a choice: “by the rod or by the ax”. The condemned was either beaten to death with the rods or allowed the mercy of the ax. The words “UNITED STATES OF AMERICA” and “ONE DIME” surround the border. “E PLURIBUS UNUM” appears at the lower right.

Excerpt from the poem that Robert Frost (who died on this date in 1963) meant to read at the 1961 inauguration of John F. Kennedy:

It makes the prophet in us all presage
The glory of a next Augustan age
Of a power leading from its strength and pride,
Of young ambition eager to be tried,
Firm in our free beliefs without dismay,
In any game the nations want to play.
A golden age of poetry and power
Of which this noonday’s the beginning hour.

I greatly prefer Robinson Jeffers’s “Shine, Perishing Republic“:

While this America settles in the mould of its vulgarity,
    heavily thickening to empire,
And protest, only a bubble in the molten mass, pops and sighs out, 
    and the mass hardens,
I sadly smiling remember….

See also the thoughts on Republic vs. Empire in the work of Alec Guinness (as Marcus Aurelius and as Obi-Wan Kenobi).

Friday, January 10, 2003

Friday January 10, 2003

Filed under: General,Geometry — Tags: — m759 @ 8:15 pm

Story

"How much story do you want?" 
— George Balanchine

While researching yesterday's entry on Balanchine, Apollo, and the nine Muses, I came across this architect's remarks, partially quoted yesterday and continued here:

"The icon that I use for this element is the nine-fold square…. This is the garden of Apollo, the field of Reason….  This is the Temple of Solomon, as inscribed, for example, by a nine-fold compartmentation to provide the ground plan of Yale, as described to me by Professor Hersey."

Duncanology Part 3

Checking this out yesterday, I came across the following at a Yale University Art Gallery site:

"This exhibition of nine boldly colored, asymmetrically designed quilts selected from a private collection will be displayed in the Matrix Gallery….

With the guidance of Professor Maude Southwell Wahlman, author of 'Signs and Symbols: African Images in African American Quilts,' the collector has explored and gathered examples…."

Exploring and gathering examples myself today, I received a book in the mail — W. M. Spackman's On the Decay of Humanism (Rutgers University Press, 1967) — and picked up a second-hand book at a sale — Barbara Michaels's Stitches in Time (Harper Collins Publishers, 1995).

The Spackman book includes the following poem at the end:

In sandarac etui for sepulchre
  lies the cered body of a poisoned queen;
     and in her mouth and hair, and at her feet,
     and in the grey folds of her winding-sheet,
  there sifts a dreamy powder, smooth and green,
the magic of an idle sorcerer,
  an ancient spell, cast when the shroud was spun.
     In death her hands clasp amourously a bowl
     that still contains the fragments of her soul,
  a tale of Beauty sought, and Beauty won,
his false lips kissed, and Beauty dead for her.

— Alexander B. Griswold, Princeton '28, in the
    Nassau Literary Magazine of December 1925

From a synopsis of Michaels's Stitches in Time:

"Michaels follows Rachel, a graduate student studying women's crafts–weaving, spinning, quilting, embroidery–and the superstitions connected with them. Linking all important rites of passage to the garments created as markers of these occasions leads Rachel to her theory: in societies in which magic was practiced, the garment was meant to protect its wearer. She gains evidence that her theory is valid when an evil antique bridal quilt enters her life."

Although Stitches in Time is about a quilt — stitched, not spun — Griswold's line

"an ancient spell, cast when the shroud was spun" 

is very closely related to the evil spell in Michaels's book. 

The above events display a certain synchronicity that Wallace Stevens might appreciate, especially in light of the following remark in a review of Stitches in Time:

"…the premise is too outlandish for even the suspension of disbelief…." (Publishers Weekly, 4/24/95)

Stevens might reply,

The very man despising honest quilts
Lies quilted to his poll in his despite.

— "The Comedian as the Letter C," Part V

Finally, those who prefer stories to the more formal qualities of pure dance (ballet) pure mathematics (see previous entry), pure (instrumental) music, and pure (abstract, as in quilt designs) art, can consult the oeuvre of Jodie Foster — as in my 

Pearl Harbor Day entry on Buddhism.

An art historian named Griswold — perhaps that very same Griswold quoted above — might have a thing or two to say to Jodie on her recent film "Anna and the King."  In the April, 1957, issue of The Journal of the Siam Society, Alexander B. Griswold takes issue with Broadway's and Hollywood's "grotesque caricature" of Siamese society, and ultimately with Anna herself:

"The real fault lies in the two books they ultimately spring from — The English Governess at the Court of Siam and The Romance of the Harem — both written by Mrs. Anna Leonowens.''

Is a puzzlement.

See also The Diamond 16 Puzzle for some quilt designs.

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