Log24

Wednesday, October 2, 2019

Stevens at 140

Filed under: General — m759 @ 12:38 AM

Poet Wallace Stevens was born 140 years ago today.

For another 140, see Diamond Theory in 1937.

For some notes related to a Stevens poem from 1937,
see "arrowy, still strings" in this journal.

Friday, July 12, 2019

Holloway Today

Filed under: General — Tags: — m759 @ 8:23 AM

"The area is home to many artists and people who work in
 the media, including many journalists, writers and professionals 
working in film and television." — Wikipedia

Tusen takk to My Square Lady —

Tuesday, December 18, 2018

CV

Filed under: General — m759 @ 8:01 PM

The title abbreviates* that of a collection of Wittgenstein's remarks:

Ludwig Wittgenstein — Culture and Value 
Revised Edition, Wiley-Blackwell (1998)

Showing 20 results for spirit

page 18, rubble & finally a heap of ashes; but spirits will hover over the ashes. MS 107 229:

page 18, Page 5 Only something supernatural can expre

page 20, contemplating it from above in its†c flight.†

page 21, spirit in which it is written.†f This spirit is, I believe, different from that of t

page 21, and American civilization. The spirit of this civilization the expression of

page 21, day†h fascism & socialism, is a spirit that is alien & uncongenial†i to the au

page 21, he Page Break 9 can work in the spirit of the whole, and his strength can with

page 21, straight for what is concrete. Which is chara

page 22, danger in a long foreword is that the spirit of a book has to be evident in the book

page 22, It is all one to me whether the typical weste

page 23, a great temptation to want to make the spirit explicit. MS 109 204: 6-7.11.1930 Page

page 23, readers that will be clear just from the fact

page 28, Foggy day. Grey autumn haunts us. Laughter se

page 42, If one wanted to characterize the essence of

page 51, attention from what matters.) The Spirit puts what is essential, essential for y

page 51, how far all this is exactly in the spirit of Kierkegaard.) MS 119 151: 22.10.1937

page 51, something feminine about this outlook?) MS 11

page 100, comfortable, clearer expression, but cannot b

page 106, act otherwise."–Perhaps, though, one might s

page 210, Page 7 †b function Page 7 †c from its Page

****************************************************************

The above "spirit guide" was suggested by yesterday's post
on Knuth as Yoda and by the paper in today's previous post, 
"Shadowhunter Tales."

This  post's title, "CV," is from . . .

Wednesday, January 24, 2018

The Pentagram Papers

Filed under: General,Geometry — Tags: — m759 @ 12:40 PM

(Continued)

From a Log24 post of March 4, 2008 —

SINGER, ISAAC:
"Are Children the Ultimate Literary Critics?"
— Top of the News 29 (Nov. 1972): 32-36.

"Sets forth his own aims in writing for children and laments
'slice of life' and chaos in children's literature. Maintains that
children like good plots, logic, and clarity, and that they
have a concern for 'so-called eternal questions.'"

— An Annotated Listing of Criticism
by Linnea Hendrickson

"She returned the smile, then looked across the room to
her youngest brother, Charles Wallace, and to their father,
who were deep in concentration, bent over the model
they were building of a tesseract: the square squared,
and squared again: a construction of the dimension of time."

— A Swiftly Tilting Planet,
by Madeleine L'Engle

Cover of 'A Swiftly Tilting Planet' and picture of tesseract

For "the dimension of time," see A Fold in TimeTime Fold,
and Diamond Theory in 1937

A Swiftly Tilting Planet  is a fantasy for children 
set partly in Vespugia, a fictional country bordered by
Chile and Argentina.

Ibid.

The pen's point:

Wm. F. Buckley as Archimedes, moving the world with a giant pen as lever. The pen's point is applied to southern South America.
John Trever, Albuquerque Journal, 2/29/08

Note the figure on the cover of National Review  above —

A related figure from Pentagram Design

See, more generally,  Isaac Singer  in this  journal.

Thursday, November 16, 2017

Lost Horizon

Filed under: G-Notes,General,Geometry — m759 @ 11:29 AM

Related material —

The following image in this journal

  .

Tuesday, October 10, 2017

The 35-Year Wait

Filed under: G-Notes,General,Geometry — m759 @ 11:17 AM

From the Web this morning —

A different 35-year wait:

A monograph of August 1976 —

Thirty-five years later, in a post of August 2011, "Coordinated Steps" —

'The Seven Dwarfs and their Diamond Mine

"SEE HEAR READ" — Walt Disney Productions

Some other diamond-mine productions —

 Image -- The cast of 1937's 'King Solomon's Mines' goes back to the future

Monday, January 23, 2017

Believe It or Not

Filed under: General — Tags: — m759 @ 7:11 PM

"It's not a lie if you believe it."

Poster for "Operation Avalanche"

“We keep coming back and coming back
To the real: to the hotel instead of the hymns . . . .

— Wallace Stevens, quoted in posts tagged Portal1937

Thursday, December 8, 2016

Finite Groups and Their Geometric Representations

Filed under: General,Geometry — Tags: — m759 @ 8:06 AM

The title is that of a presentation by Arnold Emch
at the 1928 International Congress of Mathematicians:

See also yesterday's "Emch as a Forerunner of S(5, 8, 24)."

Related material: Diamond Theory in 1937.

Further remarks:  Christmas 2013 and the fact that
759 × 322,560 = the order of the large Mathieu group  M24 .

Friday, September 30, 2016

Desmic Midrash

Filed under: General — Tags: — m759 @ 9:19 AM

The author of the review in the previous post, Dara Horn, supplies
below a midrash on "desmic," a term derived from the Greek desme 
δεσμή , bundle, sheaf, or, in the mathematical sense, pencil —
French faisceau ), which is apparently related to the term desmos , bond 

(The term "desmic," as noted earlier, is relevant to the structure of
Heidegger's Sternwürfel .)

The Horn midrash —

(The "medieval philosopher" here is not the remembered pre-Christian
Ben Sirah (Ecclesiasticus ) but the philosopher being read — Maimonides:  
Guide for the Perplexed , 3:51.)

Here of course "that bond" may be interpreted as corresponding to the
Greek desmos  above, thus also to the desmic  structure of the
stellated octahedron, a sort of three-dimensional Star of David.

See "desmic" in this journal.

Monday, December 21, 2015

ART WARS (continued)

Filed under: General — m759 @ 11:22 AM

Today in History —

"On December 21, 1937, 'Snow White and the Seven Dwarfs'
premiered to a record-breaking audience at the Carthay Circle
Theatre in Los Angeles."

Related material: Today's previous post and the Red Book.

Slouching Towards Christmas (continued)

Filed under: General — m759 @ 11:00 AM

Under the Volcano:
A Bottle, a Door, a Box

Katherine Neville, 'The Magic Circle' excerpt

See also Glory Season (Nov. 12, 2005) and Unique Figure (April 12, 2011).

Update of 11:22 AM —

Today in History —

"On December 21, 1937, 'Snow White and the Seven Dwarfs'
premiered to a record-breaking audience at the Carthay Circle
Theatre in Los Angeles."

Related material:  The Red Book.

Friday, November 27, 2015

Einstein and Geometry

Filed under: General,Geometry — Tags: — m759 @ 2:01 PM

(A Prequel to Dirac and Geometry)

"So Einstein went back to the blackboard.
And on Nov. 25, 1915, he set down
the equation that rules the universe.
As compact and mysterious as a Viking rune,
it describes space-time as a kind of sagging mattress…."

— Dennis Overbye in The New York Times  online,
     November 24, 2015

Some pure  mathematics I prefer to the sagging Viking mattress —

Readings closely related to the above passage —

Thomas Hawkins, "From General Relativity to Group Representations:
the Background to Weyl's Papers of 1925-26
," in Matériaux pour
l'histoire des mathématiques au XXe siècle:
Actes du colloque
à la mémoire de Jean Dieudonné
, Nice, 1996  (Soc. Math.
de France, Paris, 1998), pp. 69-100.

The 19th-century algebraic theory of invariants is discussed
as what Weitzenböck called a guide "through the thicket
of formulas of general relativity."

Wallace Givens, "Tensor Coordinates of Linear Spaces," in
Annals of Mathematics  Second Series, Vol. 38, No. 2, April 1937
pp. 355-385.

Tensors (also used by Einstein in 1915) are related to 
the theory of line complexes in three-dimensional
projective space and to the matrices used by Dirac
in his 1928 work on quantum mechanics.

For those who prefer metaphors to mathematics —

"We acknowledge a theorem's beauty
when we see how the theorem 'fits' in its place,
how it sheds light around itself, like a Lichtung ,
a clearing in the woods." 
— Gian-Carlo Rota, Indiscrete Thoughts ,
Birkhäuser Boston, 1997, page 132

Rota fails to cite the source of his metaphor.
It is Heidegger's 1964 essay, "The End of Philosophy
and the Task of Thinking" —

"The forest clearing [ Lichtung ] is experienced
in contrast to dense forest, called Dickung  
in our older language." 
— Heidegger's Basic Writings 
edited by David Farrell Krell, 
Harper Collins paperback, 1993, page 441

Saturday, September 19, 2015

The Observatory Mystery

Filed under: General — m759 @ 11:00 AM

Part I:        Magic Moonlight

Part II:    To Walk the Night 

Cover from a 1944 edition of
the 1937 novel by William Sloane


Part III:   Sept. 18, 2015, review by Stephen King
                of the works of William Sloane 

Friday, March 27, 2015

The McEvoy Rite

Filed under: General — m759 @ 8:00 PM

Nan Tucker McEvoy, last of founding family
to run Chronicle, dies

By Sam Whiting at SFGate.com, Friday, March 27, 2015 

From the story —

"After graduating from Dominican Convent Upper School 
in San Rafael in 1937, she was discouraged from attending college
by family members who wanted her to be a socialite."

Related material —

A school, a tweet, and a post.

Monday, August 4, 2014

A Wrinkle in Space

Filed under: General,Geometry — Tags: , — m759 @ 10:30 AM

"There is  such a thing as a tesseract." — Madeleine L'Engle

An approach via the Omega Matrix:

http://www.log24.com/log/pix10A/100619-TesseractAnd4x4.gif

See, too, Rosenhain and Göpel as The Shadow Guests .

Tuesday, July 15, 2014

Photo Opportunity

Filed under: General,Geometry — Tags: — m759 @ 2:02 PM

"I need a photo opportunity, I want a shot at redemption.
Don't want to end up a cartoon in a cartoon graveyard."
– Paul Simon

Pinocchio: 'Multiplane Technicolor'

"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color…. The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt  went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space– which he calls the mind or heart of creation– determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less.

 

 

— Wallace Stevens, Harvard College Class of 1901, "The Relations between Poetry and Painting" in The Necessary Angel   (Knopf, 1951)

For background on the planes illustrated above,
see Diamond theory in 1937.

Sunday, January 26, 2014

A Midnight Exorcism

Filed under: General,Geometry — Tags: — m759 @ 12:00 AM

The summoning of the spirit of Bertrand Russell
yesterday by Peter J. Cameron at his weblog
suggests a review of this  weblog’s posts of
Christmas Eve, December 24-25, 2013.

(Recall that Robert D. Carmichael, who, in a book
linked to at midnight last Christmas Eve discusses
some “magic” mathematical structures,
reportedly was trained as a Presbyterian minister.
See also The Presbyterian Exorcist.)

Wednesday, December 25, 2013

Rotating the Facets

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

Previous post

"… her mind rotated the facts…."

Related material— hypercube rotation,* in the context
of rotational symmetries of the Platonic solids:

IMAGE- Count rotational symmetries by rotating facets. Illustrated with 'Plato's Dice.'

"I've heard of affairs that are strictly Platonic"

Song lyric by Leo Robin

* Footnote added on Dec. 26, 2013 —

 See Arnold Emch, "Triple and Multiple Systems, Their Geometric 
 Configurations and Groups
," Trans. Amer. Math. Soc.  31 (1929),
 No. 1, 25–42. 

 On page 42, Emch describes the above method of rotating a
 hypercube's 8 facets (i.e., three-dimensional cubes) to count
 rotational symmetries —

See also Diamond Theory in 1937.

Also on p. 42, Emch mentions work of Carmichael on a
Steiner system with the Mathieu group M11 as automorphism
group, and poses the problem of finding such systems and
groups that are larger. This may have inspired the 1931
discovery by Carmichael of the Steiner system S(5, 8, 24),
which has as automorphisms the Mathieu group M24 .

A Midnight Clear

Filed under: General — Tags: , — m759 @ 12:00 AM

Click image for a meditation.

Saturday, September 21, 2013

Mathematics and Narrative (continued)

Filed under: General,Geometry — Tags: , — m759 @ 1:00 AM

Mathematics:

A review of posts from earlier this month —

Wednesday, September 4, 2013

Moonshine

Filed under: Uncategorized — m759 @ 4:00 PM

Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine."  An example—  the apparent connections
between parts of complex analysis and groups related to the
large Mathieu group M24. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface  (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4-element
subsets of a 16-element 4×4 array.  It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the two-element Galois field.
(See Diamond Theory in 1937.)

Thursday, September 5, 2013

Moonshine II

Filed under: Uncategorized — Tags:  — m759 @ 10:31 AM

(Continued from yesterday)

The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.

The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3-space."  The affine 4-space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."

"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
non-isotropic planes in this affine space over the finite field."

The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)
."

Narrative:

Aooo.

Happy birthday to Stephen King.

Wednesday, September 4, 2013

Moonshine

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

Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine."  An example—  the apparent connections
between parts of complex analysis and groups related to the 
large Mathieu group M24. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface  (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4-element
subsets of a 16-element 4×4 array.  It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the two-element Galois field.
(See Diamond Theory in 1937.) 

A Google search documents the moonshine
relating Rosenhain's and Göpel's 19th-century work
in complex analysis to M24  via the book of Hudson and
the geometry of the 4×4 square.

Tuesday, July 16, 2013

Space Itself

Filed under: General,Geometry — Tags: — m759 @ 10:18 AM

"How do you get young people excited
about space? How do you get them interested
not just in watching movies about space,
or in playing video games set in space
but in space itself?"

Megan Garber in The AtlanticAug. 16, 2012

One approach:

"There is  such a thing as a tesseract" and
Diamond Theory in 1937.

See, too, Baez in this journal.

Friday, January 25, 2013

Where Credit Is Due

Filed under: General — m759 @ 1:00 PM

Harvard's President Faust:

IMAGE- Harvard President Faust at Boston College, a Jesuit institution, on Oct. 10, 2012

Last evening's post Moondance was suggested by a check
in this journal of the date October 10, 2012. That date was
in turn suggested by the date of the above remarks.

Wer immer strebend sich bemüht,
Den können wir erlösen.

Who always striving efforts makes,
For him there is salvation
.

Faust Part 2, Act V, Scene 7: Mountain Gorges.

Saturday, January 5, 2013

Vector Addition in a Finite Field

Filed under: General,Geometry — Tags: , — m759 @ 10:18 AM

The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—

The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4-space over GF(2)—


The same field, again disguised as an affine 4-space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—

The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vector-space structure of the finite
field GF(16).

This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—

(Thanks to June Lester for the 3D (uvw) part of the above figure.)

For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.

For some related narrative, see tesseract  in this journal.

(This post has been added to finitegeometry.org.)

Update of August 9, 2013—

Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.

Update of August 13, 2013—

The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor:  Coxeter’s 1950 hypercube figure from
Self-Dual Configurations and Regular Graphs.”

Monday, August 13, 2012

Raiders of the Lost Tesseract

Filed under: General,Geometry — Tags: — m759 @ 3:33 PM

(An episode of Mathematics and Narrative )

A report on the August 9th opening of Sondheim's Into the Woods

Amy Adams… explained why she decided to take on the role of the Baker’s Wife.

“It’s the ‘Be careful what you wish’ part,” she said. “Since having a child, I’m really aware that we’re all under a social responsibility to understand the consequences of our actions.” —Amanda Gordon at businessweek.com

Related material—

Amy Adams in Sunshine Cleaning  "quickly learns the rules and ropes of her unlikely new market. (For instance, there are products out there specially formulated for cleaning up a 'decomp.')" —David Savage at Cinema Retro

Compare and contrast…

1.  The following item from Walpurgisnacht 2012

IMAGE- Excerpt from 'Unified Approach to Functional Decompositions of Switching Functions,' by Marek A. Perkowski et al., 1995

2.  The six partitions of a tesseract's 16 vertices 
       into four parallel faces in Diamond Theory in 1937

Thursday, October 6, 2011

What’s NeXT?

Filed under: General — m759 @ 1:44 AM

http://www.log24.com/log/pix11C/111006-NeXT-logo.jpg

(Click logo for details.)

NeXT in action:

http://www.log24.com/log/pix11C/111006-NeXT-CERN-500w.jpg

This morning's post Opening Act suggests the following scholium

To Purgatory fire you'll come at last;
And Christ receive your soul.

If ever you gave meat or drink,
Every night and all,
The fire will never make you shrink;
And Christ receive your soul.

See also The Wall Street Journal 's Ice Water in Hell story.

Followup scholium — "Vague but exciting …" —

http://www.log24.com/log/pix11C/111006-NeXT-CERN-Proposal-500w.jpg

Thursday, September 1, 2011

How It Works

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

"Design is how it works." — Steven Jobs (See Symmetry and Design.)

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."
 — "Block Designs," by Andries E. Brouwer

IMAGE- Harvard senior thesis on Mathieu groups, 2010, and supporting material from book 'Design Theory'

The name Carmichael is not to be found in Booher's thesis. In a reference he does  give for the history of S(5,8,24), Carmichael's construction of this design is dated 1937. It should be dated 1931, as the following quotation shows—

From Log24 on Feb. 20, 2010

"The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24."

– R. D. Carmichael, "Tactical Configurations of Rank Two," in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240

Epigraph from Ch. 4 of Design Theory , Vol. I:

"Es is eine alte Geschichte,
 doch bleibt sie immer neu
"
 —Heine (Lyrisches Intermezzo  XXXIX)

See also "Do you like apples?"

Sunday, August 21, 2011

Relativistic Truth

Filed under: General — m759 @ 12:25 PM

Today's online New York Times on the conclusion of the Roman Catholic Church's "World Youth" week—

"At the end of Sunday’s Mass, the pope announced that the next such event would be in Rio de Janeiro in 2013. Until then, he told those at the service, in Portuguese, that they 'will be swimming against the tide in a society with a relativistic culture, which wishes neither to seek nor hold on to the truth.'*"

* Fact check— This agrees with the Vatican Radio version.

Related material: Relativity Blues and Portal to 1937

IMAGE- Hotel Bella Vista as 'Portal del Aguila de Oro'

The "Portal" link above is in honor of the May 2 dies natalis of Salomon Bochner (pdf).
For some background, see yesterday's Castles in the Air and Bochner in this journal.

Saturday, August 20, 2011

Castles in the Air

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

"… the Jews have discovered a way to access a fourth spatial dimension."
— Clifford Pickover, description of his novel Jews in Hyperspace

"If you have built castles in the air, your work need not be lost;
that is where they should be. Now put the foundations under them.”
— Henry David Thoreau

"King Solomon's Mines," 1937

Image -- The cast of 1937's 'King Solomon's Mines' goes back to the future

The image above is an illustration from  "Romancing the Hyperspace," May 4, 2010.

Happy birthday to the late Salomon Bochner.

Sunday, July 10, 2011

Wittgenstein’s Diamond

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

Philosophical Investigations  (1953)

97. Thought is surrounded by a halo.
—Its essence, logic, presents an order,
in fact the a priori order of the world:
that is, the order of possibilities * ,
which must be common to both world and thought.
But this order, it seems, must be
utterly simple . It is prior  to all experience,
must run through all experience;
no empirical cloudiness or uncertainty can be allowed to affect it
——It must rather be of the purest crystal.
But this crystal does not appear as an abstraction;
but as something concrete, indeed, as the most concrete,
as it were the hardest  thing there is
(Tractatus Logico-Philosophicus  No. 5.5563).

— Translation by G.E.M. Anscombe

5.5563

All propositions of our colloquial language
are actually, just as they are, logically completely in order.
That simple thing which we ought to give here is not
a model of the truth but the complete truth itself.

(Our problems are not abstract but perhaps
the most concrete that there are.)

97. Das Denken ist mit einem Nimbus umgeben.
—Sein Wesen, die Logik, stellt eine Ordnung dar,
und zwar die Ordnung a priori der Welt,
d.i. die Ordnung der Möglichkeiten ,
die Welt und Denken gemeinsam sein muß.
Diese Ordnung aber, scheint es, muß
höchst einfach  sein. Sie ist vor  aller Erfahrung;
muß sich durch die ganze Erfahrung hindurchziehen;
ihr selbst darf keine erfahrungsmäßige Trübe oder Unsicherheit anhaften.
——Sie muß vielmehr vom reinsten Kristall sein.
Dieser Kristall aber erscheint nicht als eine Abstraktion;
sondern als etwas Konkretes, ja als das Konkreteste,
gleichsam Härteste . (Log. Phil. Abh.  No. 5.5563.)

See also

Related language in Łukasiewicz (1937)—

http://www.log24.com/log/pix10B/101127-LukasiewiczAdamantine.jpg

* Updates of 9:29 PM ET July 10, 2011—

A  mnemonic  from a course titled "Galois Connections and Modal Logics"—

"Traditionally, there are two modalities, namely, possibility and necessity.
The basic modal operators are usually written box (square) for necessarily
and diamond (diamond) for possibly. Then, for example, diamondP  can be read as
'it is possibly the case that P .'"

See also Intensional Semantics , lecture notes by Kai von Fintel and Irene Heim, MIT, Spring 2007 edition—

"The diamond symbol for possibility is due to C.I. Lewis, first introduced in Lewis & Langford (1932), but he made no use of a symbol for the dual combination ¬¬. The dual symbol was later devised by F.B. Fitch and first appeared in print in 1946 in a paper by his doctoral student Barcan (1946). See footnote 425 of Hughes & Cresswell (1968). Another notation one finds is L for necessity and M for possibility, the latter from the German möglich  ‘possible.’"

Barcan, Ruth C.: 1946. “A Functional Calculus of First Order Based on Strict Implication.” Journal of Symbolic Logic, 11(1): 1–16. URL http://www.jstor.org/pss/2269159.

Hughes, G.E. & Cresswell, M.J.: 1968. An Introduction to Modal Logic. London: Methuen.

Lewis, Clarence Irving & Langford, Cooper Harold: 1932. Symbolic Logic. New York: Century.

Monday, May 2, 2011

Aguila de Oro

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

IMAGE- Hotel Bella Vista as 'Portal del Aguila de Oro'

See also Harvard's Memorial Church in "Ready when you are, C. B."—

IMAGE- Sharon Stone in the Gold Eagle pulpit of Harvard's Memorial Church
HARVARD CRIMSON/ ALEX R. LEVIN

Sharon Stone lectures at
Harvard's Memorial Church

on March 14, 2005…

"Ready when you are, C. B."

Pasaje

Filed under: General — Tags: — m759 @ 8:28 PM

http://www.log24.com/log/pix11A/110502-PostcardsFromCuernavaca-500w.jpg

From Under the Volcano , Chapter II—

Hotel Bella Vista
Gran Baile Noviembre 1938
a Beneficio de la Cruz Roja.
Los Mejores Artistas del radio en accion.
No falte Vd.

From Shining Forth

"What he sees is something real."
— Charles Williams, The Figure of Beatrice

The Vine*

Filed under: General — Tags: , — m759 @ 9:00 AM

See "Nine is a Vine" and "Hereafter" in this journal.

IMAGE- Matt Damon and the perception of doors in 'Hereafter'

As quoted here last October 23

Margaret Atwood on Lewis Hyde's Trickster Makes This World: Mischief, Myth, and Art

"Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159)

What is "the next world"? It might be the Underworld….

The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation  and art  all come from the same ancient root, a word meaning "to join," "to fit," and "to make." (254)  If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist.  Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart.

* April 7, 2005

Thursday, April 21, 2011

Spaghetti Junction

Filed under: General — m759 @ 7:59 PM

Literary remarks for Maundy Thursday—

IMAGE- 'It was a perfectly ordinary night at Christ's high table....'

      — C. P. Snow, foreword to G. H. Hardy's A Mathematician's Apology

Related material—

Emory University press release of January 20th, 2011:

"In 1937, Hans Rademacher found an exact formula for calculating partition values. While the method was a big improvement over Euler's exact formula, it required adding together infinitely many numbers that have infinitely many decimal places. 'These numbers are gruesome,' Ono says….

… The final eureka moment occurred near another Georgia landmark: Spaghetti Junction. Ono and Jan Bruinier were stuck in traffic near the notorious Atlanta interchange. While chatting in the car, they hit upon a way to overcome the infinite complexity of Rademacher's method. They went on to prove a formula that requires only finitely many simple numbers.

'We found a function, that we call P, that is like a magical oracle,' Ono says. 'I can take any number, plug it into P, and instantly calculate the partitions of that number….'"

See also this journal on April 15 and a Google Groups [sage-devel] thread, Ono-Bruinier partition formula. That thread started on April 15 and was last updated this morning.

Friday, March 18, 2011

Defining Configurations*

Filed under: General,Geometry — m759 @ 7:00 PM

The On-Line Encyclopedia of Integer Sequences has an article titled "Number of combinatorial configurations of type (n_3)," by N.J.A. Sloane and D. Glynn.

From that article:

  • DEFINITION: A combinatorial configuration of type (n_3) consists of an (abstract) set of n points together with a set of n triples of points, called lines, such that each point belongs to 3 lines and each line contains 3 points.
  • EXAMPLE: The unique (8_3) configuration consists of the triples 125, 148, 167, 236, 278, 347, 358, 456.

The following corrects the word "unique" in the example.

http://www.log24.com/log/pix11/110320-MoebiusKantorConfig500w.jpg

* This post corrects an earlier post, also numbered 14660 and dated 7 PM March 18, 2011, that was in error.
   The correction was made at about 11:50 AM on March 20, 2011.

_____________________________________________________________

Update of March 21

The problem here is of course with the definition. Sloane and Glynn failed to include in their definition a condition that is common in other definitions of configurations, even abstract or purely "combinatorial" configurations. See, for instance, Configurations of Points and Lines , by Branko Grunbaum (American Mathematical Society, 2009), p. 17—

In the most general sense we shall consider combinatorial (or abstract) configurations; we shall use the term set-configurations as well. In this setting "points" are interpreted as any symbols (usually letters or integers), and "lines" are families of such symbols; "incidence" means that a "point" is an element of a "line". It follows that combinatorial configurations are special kinds of general incidence structures. Occasionally, in order to simplify and clarify the language, for "points" we shall use the term marks, and for "lines" we shall use blocks. The main property of geometric configurations that is preserved in the generalization to set-configurations (and that characterizes such configurations) is that two marks are incident with at most one block, and two blocks with at most one mark.

Whether or not omitting this "at most one" condition from the definition is aesthetically the best choice, it dramatically changes the number  of configurations in the resulting theory, as the above (8_3) examples show.

Update of March 22 (itself updated on March 25)

For further background on configurations, see Dolgachev—

http://www.log24.com/log/pix11/110322-DolgachevIntro.gif

Note that the two examples Dolgachev mentions here, with 16 points and 9 points, are not unrelated to the geometry of 4×4 and 3×3 square arrays. For the Kummer and related 16-point configurations, see section 10.3, "The Three Biplanes of Order 4," in Burkard Polster's A Geometrical Picture Book  (Springer, 1998). See also the 4×4 array described by Gordon Royle in an undated web page and in 1980 by Assmus and Sardi. For the Hesse configuration, see (for instance) the passage from Coxeter quoted in Quaternions in an Affine Galois Plane.

Update of March 27

See the above link to the (16,6) 4×4 array and the (16,6) exercises using this array in R.D. Carmichael's classic Introduction to the Theory of Groups of Finite Order  (1937), pp. 42-43. For a connection of this sort of 4×4 geometry to the geometry of the diamond theorem, read "The 2-subsets of a 6-set are the points of a PG(3,2)" (a note from 1986) in light of R.W.H.T. Hudson's 1905 classic Kummer's Quartic Surface , pages 8-9, 16-17, 44-45, 76-77, 78-79, and 80.

Wednesday, March 9, 2011

Ash Wednesday Surprise

Filed under: General — m759 @ 11:07 AM

IMAGE- Errol Morris- 'The Ashtray'- at The New York Times

 Part 3 of 5  (See also Part 1 and Part 2) begins as follows…

"Incommensurable. It is a strange word. I wondered, why  did Kuhn choose it? What was the attraction? 

Here’s one clue. At the very end of 'The Road Since Structure,' a compendium of essays on Kuhn’s work, there is an interview with three Greek philosophers of science, Aristides Baltas, Kostas Gavroglu and Vassiliki Kindi. Kuhn provides a brief account of the historical origins of his idea. Here is the relevant segment of the interview.

T. KUHN: Look, 'incommensurability' is easy.

V. KINDI: You mean in mathematics?

T. KUHN: …When I was a bright high school mathematician and beginning to learn Calculus, somebody gave me—or maybe I asked for it because I’d heard about it—there was sort of a big two-volume Calculus book by, I can’t remember whom. And then I never really read it. I read the early parts of it. And early on it gives the proof of the irrationality of the square root of 2. And I thought it was beautiful. That was terribly exciting, and I learned what incommensurability was then and there. So, it was all ready for me, I mean, it was a metaphor but it got at nicely what I was after. So, that’s where I got it.

'It was all ready for me.' I thought, 'Wow.' The language was suggestive. I imagined √2 provocatively dressed, its lips rouged. But there was an unexpected surprise. The idea didn’t come from the physical sciences or philosophy or linguistics, but from mathematics ."

A footnote from Morris (no. 29)—

"Those who are familiar with the proof [of irrationality] certainly don’t want me to explain it here; likewise, those who are unfamiliar with it don’t want me to explain it here, either. There are many simple proofs in many histories of mathematics — E.T. Bell, Sir Thomas Heath, Morris Kline, etc., etc. Barry Mazur offers a proof in his book, 'Imagining Numbers (particularly the square root of minus fifteen),' New York, NY: Farrar, Straus and Giroux. 2003, 26ff. And there are two proofs in his essay, 'How Did Theaetetus Prove His Theorem?', available on Mazur’s Harvard Web site."

There may, actually, be a few who do  want the proof. They may consult the sources Morris gives, or the excellent  description by G.H. Hardy in A Mathematician's Apology , or, perhaps best of all for present purposes, the proof as described in a "sort of a big two-volume Calculus book" (perhaps the one Kuhn mentioned)…  See page 6 and page 7 of  Volume One  of Richard Courant's classic Differential and Integral Calculus  (second edition, 1937, reprinted many times through 1970, and again in a Wiley Classics Library Edition in 1988).

Tuesday, February 22, 2011

Gestell

Filed under: General — m759 @ 12:00 PM

From May 11, 2010, an image—

Image-- The 4-dimensional space over the 2-element field

See also the same date in 2005 in light of

Saturday, June 12, 2010

Holy Geometry

Filed under: General,Geometry — m759 @ 10:31 AM

The late mathematician V.I. Arnold was born on this date in 1937.

"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

Light

Image-- AMS site screenshot of V.I. Arnold obituary, June 12, 2010

Darkness

Image-- AMS site screenshot of Martin Gardner tribute, May 25, 2010

Choosing light rather than darkness, we observe Arnold's birthday with a quotation from his 1997 Paris talk 'On Teaching Mathematics.'

"The Jacobi identity (which forces the heights of a triangle to cross at one point) is an experimental fact…."

The "experimental fact" part, perhaps offered with tongue in cheek, is of less interest than the assertion that the Jacobi identity forces the altitude-intersection theorem.

Albert Einstein on that theorem in the "holy geometry book" he read at the age of 12—

"Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which– though by no means evident– could nevertheless be proved with such certainty that any doubt appeared to be out of the question.  This lucidity and certainty made an indescribable impression upon me.”

Arnold's much less  evident assertion about altitudes and the Jacobi identity is discussed in "Arnol'd, Jacobi identity, and orthocenters" (pdf) by Nikolai V. Ivanov.

Ivanov says, without giving a source,  that the altitudes theorem "was known to Euclid." Alexander Bogomolny, on the other hand, says it is "a matter of real wonderment that the fact of the concurrency of altitudes is not mentioned in either Euclid's Elements  or subsequent writings of the Greek scholars. The timing of the first proof is still an open question."

For other remarks on geometry, search this journal for the year of Arnold's birth.

Tuesday, May 11, 2010

Lubtchansky’s Key

Filed under: General — m759 @ 12:00 PM

William Lubtchansky, a cinematographer, was born on October 26, 1937, and died on May 4, 2010.

Yesterday's post included an illustration from this journal on the date of his death.

Here is a Log24 entry from last year on the date of his birth—

Monday, October 26, 2009
The Keys Enigma

Image-- Back Space key from manual typewriter, linking to Babich on Music, Nietzsche, and Heidegger
Image-- Shift Lock key from manual typewriter, linking to Levin's 'The Philosopher's Gaze'

Related material:

Posts of Sept. 21-25

Clicking on the Shift Lock key leads to the following page—

Image-- Page 432 of 'The Philosopher's Gaze'-- Heidegger on Gestell and shining forth

The Philosopher's Gaze,
by David Michael Levin,
University of California Press, 1999

Related images—

Detail from May 4 image:

Image-- The 4-dimensional space over the 2-element field

Holocaust Museum, Washington, DC:

Image-- Holocaust Museum tour group entrance
(http://www.scrapbookpages.com/USHMM/Exterior.html)

See also Lubtchansky's Duelle and
Art Wars for Trotsky's Birthday, 2003.

Monday, May 10, 2010

Requiem for Georgia Brown

Filed under: General — m759 @ 2:45 AM

Image-- Lena Horne in 'Cabin in the Sky'

Paul Robeson in
"King Solomon's Mines," 1937

Image -- The cast of 1937's 'King Solomon's Mines' goes back to the future

The image above is an illustration from
  "Romancing the Hyperspace," May 4, 2010.

This illustration, along with Georgia Brown's
song from "Cabin in the Sky"—
"There's honey in the honeycomb"—
suggests the following picture.

Image-- Tesseract and Hyperspace (the 4-space over GF(2)). Source: Coxeter's 'Twisted Honeycombs'

"What might have been and what has been
   Point to one end, which is always present."
Four Quartets

Tuesday, May 4, 2010

Mathematics and Narrative, continued

Filed under: General,Geometry — Tags: — m759 @ 8:28 PM

Romancing the
Non-Euclidean Hyperspace

Backstory
Mere Geometry, Types of Ambiguity,
Dream Time, and Diamond Theory, 1937

The cast of 1937's 'King Solomon's Mines' goes back to the future

For the 1937 grid, see Diamond Theory, 1937.

The grid is, as Mere Geometry points out, a non-Euclidean hyperspace.

For the diamonds of 2010, see Galois Geometry and Solomon’s Cube.

Monday, May 3, 2010

An Ordinary Evening

Filed under: General,Geometry — Tags: — m759 @ 8:00 PM

“…geometrically organized, with the parts labeled”

— Ursula K. Le Guin on what she calls “the Euclidean utopia

“There is such a thing as a tesseract.”

Madeleine L’Engle

Related material– Diamond Theory, 1937

Monday, December 28, 2009

Brightness at Noon, continued

Filed under: General — m759 @ 12:00 PM

This journal’s Christmas Day entry, Brightness at Noon, was in response to the Orwellian headline “Arthur Koestler, Man of Darkness,” at the top of the online New York Times front page on Christmas morning.

The entry offered, as an example of brightness, some thoughts of Leibniz on his discovery of binary arithmetic.

Related material:

KRAWTCHOUK ENCYCLOPEDIA:
home > welcome > Leibniz

Omnibus ex nihilo ducendis sufficit unum

G W Leibniz

“To make all things from nothing, unity suffices.” So it is written on a medal entitled Imago Creationis and designed by Leibniz to “exhibit to posterity in silver” his discovery of the binary system.

Baron Gottfried Wilhelm von Leibniz (also Leibnitz) 1646-1716. Philosopher and mathematician. Invented calculus independently of Newton. Proposed the metaphysical theory that we live in “the best of all possible worlds.”

He also discovered binary number system and believed in its profound metaphysical significance. He noticed similarity with the ancient Chinese divination system “I Ching.”

We chose him for our patron, for Krawtchuk polynomials can be understood as a sophistication of the simple counting of 0 and 1…

Philip Feinsilver and Jerzy Kocik, 17 July 2001

From Mikhail Krawtchouk: Short Biography

Anyone knowing even a little Soviet history of the thirties can conclude that Krawtchouk could not avoid the Great Terror. During the Orwellian “hours of hatred” in 1937 he was denounced as a “Polish spy,” “bourgeois nationalist,” etc. In 1938, he was arrested and sentenced to 20 years of confinement and 5 years of exile.

Academician Krawtchouk, the author of results which became part of the world’s mathematical knowledge, outstanding lecturer, member of the French, German, and other mathematical societies, died on March 9, 1942, in Kolyma branch of the GULAG (North-Eastern Siberia) more than 6 months short of his 50th birthday.

Incidentally, happy birthday
to John von Neumann.

Sunday, December 20, 2009

The Test

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

Dies Natalis of
Emil Artin

From the September 1953 Bulletin of the American Mathematical Society

Emil Artin, in a review of Éléments de mathématique, by N. Bourbaki, Book II, Algebra, Chaps. I-VII–

"We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt he must always fail. Mathematics is logical to be sure; each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that its perception should be instantaneous. We all have experienced on some rare occasions the feeling of elation in realizing that we have enabled our listeners to see at a moment's glance the whole architecture and all its ramifications. How can this be achieved? Clinging stubbornly to the logical sequence inhibits the visualization of the whole, and yet this logical structure must predominate or chaos would result."

Art Versus Chaos

http://www.log24.com/log/pix09A/091220-ForakisHypercube.jpg
From an exhibit,
"Reimagining Space
"

The above tesseract (4-D hypercube)
sculpted in 1967 by Peter Forakis
provides an example of what Artin
called "the visualization of the whole."

For related mathematical details see
Diamond Theory in 1937.

"'The test?' I faltered, staring at the thing.
'Yes, to determine whether you can live
in the fourth dimension or only die in it.'"
Fritz Leiber, 1959

See also the Log24 entry for
Nov. 26,  2009, the date that
Forakis died.

"There is such a thing
as a tesseract."
Madeleine L'Engle, 1962

Wednesday, June 17, 2009

Wednesday June 17, 2009

Filed under: General — Tags: — m759 @ 4:30 AM

Back to the Real

Colum McCann on yesterday’s history:

“Fiction gives us access to a very real history.”

The Associated Press thought for today:

“Journalism allows its readers to witness history; fiction gives its readers an opportunity to live it.”

— John Hersey, American author (born on this date in 1914, died 1993).

From John Hersey’s The Child Buyer (1960):

“I was wondering about that this morning… About forgetting. I’ve always had an idea that each memory was a kind of picture, an insubstantial picture. I’ve thought of it as suddenly coming into your mind when you need it, something you’ve seen, something you’ve heard, then it may stay awhile, or else it flies out, then maybe it comes back another time…. If all the pictures went out, if I forgot everything, where would they go? Just out into the air? Into the sky? Back home around my bed, where my dreams stay?”

“We keep coming back and coming back
To the real: to the hotel instead of the hymns….”

— Wallace Stevens

Hotel Bella Vista, Cuernavaca, Morelos, Mexico

Postcard from eBay
From Under the Volcano, by Malcolm Lowry, 1947, Chapter I: 

Faustus is gone: regard his hellish fall —
Shaken, M. Laruelle replaced the book on the table… he reached to the floor for a folded sheet of paper that had fluttered out of it. He picked the paper up between two fingers and unfolded it, turning it over. Hotel Bella Vista, he read. There were really two sheets of uncommonly thin hotel notepaper….

I sit now in a little room off the bar at four-thirty in the morning drinking ochas and then mescal and writing this on some Bella Vista notepaper I filched the other night…. But this is worst of all, to feel your soul dying. I wonder if it is because to-night my soul has really died that I feel at the moment something like peace. Or is it because right through hell there is a path, as Blake well knew, and though I may not take it, sometimes lately in dreams I have been able to see it? …And this is how I sometimes think of myself, as a great explorer who has discovered some extraordinary land from which he can never return to give his knowledge to the world: but the name of this land is hell. It is not Mexico of course but in the heart.

Saturday, December 13, 2008

Saturday December 13, 2008

Filed under: General — m759 @ 1:06 PM

The Shining
of Dec. 13

continued from
Dec. 13, 2003

“There is a place for a hint
somewhere of a big agent
to complete the picture.”

Notes for an unfinished novel,
The Last Tycoon,
by F. Scott Fitzgerald

Internet Movie Database
Filmography:William Grady

The Good Earth (1937)
casting: Chinese extras
(uncredited)

A Place for a Hint:

http://www.log24.com/log/pix08A/081213-Tea2.jpg(From the book Tangram)

See also
yesterday’s entries
as well as…

Serpent’s Eyes Shine,
Alice’s Tea Party,
Janet’s Tea Party,
Hollywood Memory,
and
Hope of Heaven.

“… it’s going to be
accomplished in steps,
this establishment of
the Talented
in the scheme of things.”

Anne McCaffrey

Monday, December 1, 2008

Monday December 1, 2008

Filed under: General — m759 @ 12:00 PM
Pictures at
an Exhibition

Day Without Art:

Day Without Art logo: X'd-out frame

and therefore…

Art:

Art logo: frame not X'd out

From Braque’s birthday, 2006:

“The senses deform, the mind forms. Work to perfect the mind. There is no certitude but in what the mind conceives.”

— Georges Braque,
   Reflections on Painting, 1917

Those who wish to follow Braque’s advice may try the following exercise from a book first published in 1937:

Carmichael on groups, exercise, p. 440

Hint: See the following
construction of a tesseract:

Point, line, square, cube, tesseract
From a page by Bryan Clair

For a different view
of the square and cube
see yesterday’s entry
Abstraction and Faith.

Saturday, April 19, 2008

Saturday April 19, 2008

Filed under: General,Geometry — m759 @ 5:01 AM
A Midrash for Benedict

On April 16, the Pope’s birthday, the evening lottery number in Pennsylvania was 441. The Log24 entries of April 17 and April 18 supplied commentaries based on 441’s incarnation as a page number in an edition of Heidegger’s writings.  Here is a related commentary on a different incarnation of 441.  (For a context that includes both today’s commentary and those of April 17 and 18, see Gian-Carlo Rota– a Heidegger scholar as well as a mathematician– on mathematical Lichtung.)

From R. D. Carmichael, Introduction to the Theory of Groups of Finite Order (Boston, Ginn and Co., 1937)– an exercise from the final page, 441, of the final chapter, “Tactical Configurations”–

“23. Let G be a multiply transitive group of degree n whose degree of transitivity is k; and let G have the property that a set S of m elements exists in G such that when k of the elements S are changed by a permutation of G into k of these elements, then all these m elements are permuted among themselves; moreover, let G have the property P, namely, that the identity is the only element in G which leaves fixed the nm elements not in S.  Then show that G permutes the m elements S into

n(n -1) … (nk + 1)
____________________

m(m – 1) … (mk + 1)

sets of m elements each, these sets forming a configuration having the property that any (whatever) set of k elements appears in one and just one of these sets of m elements each. Discuss necessary conditions on m, n, k in order that the foregoing conditions may be realized. Exhibit groups illustrating the theorem.”

This exercise concerns an important mathematical structure said to have been discovered independently by the American Carmichael and by the German Ernst Witt.

For some perhaps more comprehensible material from the preceding page in Carmichael– 440– see Diamond Theory in 1937.

Tuesday, March 4, 2008

Tuesday March 4, 2008

Filed under: General,Geometry — m759 @ 1:00 PM
… And for a
    Swiftly Tilting
       Shadowed Planet …

Wm. F. Buckley as Archimedes, moving the world with a giant pen as lever. The pen's point is applied to southern South America.
John Trever, Albuquerque Journal, 2/29/08

The pen's point:

Log24, Dec. 11, 2006

SINGER, ISAAC:
"Are Children the
Ultimate Literary Critics?"
— Top of the News 29
(Nov. 1972): 32-36.

"Sets forth his own aims in writing for children and laments 'slice of life' and chaos in children's literature. Maintains that children like good plots, logic, and clarity, and that they have a concern for 'so-called eternal questions.'"

An Annotated Listing
of Criticism
by Linnea Hendrickson

"She returned the smile, then looked across the room to her youngest brother, Charles Wallace, and to their father, who were deep in concentration, bent over the model they were building of a tesseract: the square squared, and squared again: a construction of the dimension of time."

A Swiftly Tilting Planet,
by Madeleine L'Engle

Cover of 'A Swiftly Tilting Planet' and picture of tesseract

For "the dimension of time,"
see A Fold in Time,
Time Fold, and
Diamond Theory in 1937
 
A Swiftly Tilting Planet  is a fantasy for children set partly in Vespugia, a fictional country bordered by Chile and Argentina.

Tuesday, February 26, 2008

Tuesday February 26, 2008

Filed under: General,Geometry — Tags: — m759 @ 7:00 PM
The Just Word

The title of the previous entry, "Where Entertainment is God," comes (via Log24, Nov. 26, 2004) from Frank Rich.

The previous entry dealt, in part, with a dead Jesuit whose obituary appears in today's Los Angeles Times.  The online obituaries page places the Jesuit, without a photo, beneath a picture of a dead sitcom writer and to the left of a picture of a dead guru.

From the obituary proper:

Walter J. Burghardt, alleged preacher of 'the just word'

The obituary does not say
exactly what "the just word" is.
 

"Walter John Burghardt was born July 10, 1914, in New York, the son of immigrants from what is now Poland. He entered a Jesuit seminary in Poughkeepsie, N.Y., at 16, and in 1937 received a master's degree from Woodstock College in Maryland. He was ordained in 1941." He died, by the way, on Saturday, Feb. 16, 2008.

The reference to Woodstock College brings to mind a fellow Jesuit, Joseph T. Clark, who wrote a book on logic published by that college.

From a review of the book:

"In order to show that Aristotelian logicians were at least vaguely aware of a kind of analogy or possible isomorphism between logical relations and mathematical relations, Father Clark seizes at one place (p. 8) upon the fact that Aristotle uses the word, 'figure' (schema), in describing the syllogism and concludes from this that 'it is obvious that the schema of the syllogism is to serve the logician precisely as the figure serves the geometer.' On the face of it, this strikes one as a bit far fetched…."

Henry Veatch in Speculum, Vol. 29, No. 2, Part 1 (Apr., 1954), pp. 266-268 (review of Conventional Logic and Modern Logic: A Prelude to Transition (1952), by Joseph T. Clark, Society of Jesus)
 

Perhaps the just word is,
as above, "schema."

Related material:

The Geometry of Logic

Tuesday, November 6, 2007

Tuesday November 6, 2007

Filed under: General — m759 @ 12:25 PM
The Third Person

Of Modern Art


The New York Times
November 6, 2007

More on the Career of
the Genius Who Boldly
Compared Himself to God

By MICHIKO KAKUTANI

“Picasso… once said…

‘… No wonder his [Picasso’s] style is so ambiguous. It’s like God’s. God is really only another artist. He invented the giraffe, the elephant and the cat. He has no real style. He just keeps on trying other things. The same with this sculptor….’

The comparison to God, like the use of the third person, was deliberate, of course.”

Of Modern Poetry

The poem of the mind
    in the act of finding
What will suffice ….
                            … It has
To construct a new stage.
    It has to be on that stage,
And, like an insatiable actor,
    slowly and
With meditation, speak words
    that in the ear,
In the delicatest ear
    of the mind, repeat,
Exactly, that which it
    wants to hear, at the sound
Of which, an invisible
    audience listens,
Not to the play, but to
    itself, expressed
In an emotion as of
    two people, as of two
Emotions becoming one.
   The actor is
A metaphysician in the dark….

— Wallace Stevens in
    Parts of a World, 1942


Of Modern Metaphysics

“For every work [or act] of creation is threefold, an earthly trinity to match the heavenly.

First, [not in time, but merely in order of enumeration] there is the Creative Idea, passionless, timeless, beholding the whole work complete at once, the end in the beginning: and this is the image of the Father.

Second, there is the Creative Energy [or Activity] begotten of that idea, working in time from the beginning to the end, with sweat and passion, being incarnate in the bonds of matter: and this is the image of the Word.

Third, there is the Creative Power, the meaning of the work and its response in the lively soul: and this is the image of the indwelling Spirit.

And these three are one, each equally in itself the whole work, whereof none can exist without other: and this is the image of the Trinity.”

— Concluding speech of St. Michael the Archangel in a 1937 play, “The Zeal of Thy House,” by Dorothy Sayers, as quoted in her 1941 book The Mind of the Maker. That entire book was, she wrote, an expansion of St. Michael’s speech.

Related material:

Thursday, July 26, 2007

Thursday July 26, 2007

Filed under: General — m759 @ 4:00 AM
The Varieties
of Religious Experience

PA Lottery July 25, 2007: Mid-day 057, Evening 225

In memory of
author George Tabori
(see previous entry):

57:

“The author takes the place of the omniscient narrator. He heightens the tension by using striking dialogue. To decrease the tension he uses some light forms of comedy, like the commands for the Dobermans of the little boy: ‘Ketchup’ for retreating, ‘Pickles’ for attacking, and ‘Mustard’ for killing.”

Menno Mertens  
on Ira Levin’s
The Boys from Brazil
225:
 
George Tabori
Log24 on 2/25, 2007:

“I caught the sudden look
of some dead master….”

— T. S. Eliot, Four Quartets

Monday, December 11, 2006

Monday December 11, 2006

Filed under: General,Geometry — m759 @ 7:20 AM
Geometry and Death

J. G. Ballard on “the architecture of death“:

“… a huge system of German fortifications that included the Siegfried line, submarine pens and huge flak towers that threatened the surrounding land like lines of Teutonic knights. Almost all had survived the war and seemed to be waiting for the next one, left behind by a race of warrior scientists obsessed with geometry and death.”

The Guardian, March 20, 2006

Edward Hirsch on Lorca:

“For him, writing is a struggle both with geometry and death.”

— “The Duende,” American Poetry Review, July/August 1999

“Rosenblum writes with
absolute intellectual honesty,
and the effect is sheer liberation….
The disposition of the material is
a model of logic and clarity.”

Harper’s Magazine review
quoted on back cover of
Cubism and Twentieth-Century Art,
by Robert Rosenblum
(Abrams paperback, 2001)

SINGER, ISAAC:
“Are Children the Ultimate Literary Critics?”
 — Top of the News 29 (Nov. 1972): 32-36.
“Sets forth his own aims in writing for children
 and laments ‘slice of life’ and chaos in
children’s literature. Maintains that children
like good plots, logic, and clarity,
and that they have a concern for
‘so-called eternal questions.'”

An Annotated Listing of Criticism
by Linnea Hendrickson

“She returned the smile, then looked
across the room to her youngest brother,
Charles Wallace, and to their father,
who were deep in concentration, bent
over the model they were building
of a tesseract: the square squared,
and squared again: a construction
of the dimension of time.”

A Swiftly Tilting Planet,
by Madeleine L’Engle

The image “http://www.log24.com/log/pix06B/061211-Swiftly2.gif” cannot be displayed, because it contains errors.

For “the dimension of time,”
see A Fold in Time,
Time Fold, and
Diamond Theory in 1937

A Swiftly Tilting Planet is a fantasy for children set partly in Vespugia, a fictional country bordered by Chile and Argentina.

For a more adult audience —

In memory of General Augusto Pinochet, who died yesterday in Santiago, Chile, a quotation from Federico Garcia Lorca‘s lecture on “the Duende” (Buenos Aires, Argentina, 1933):

“… Philip of Austria… longing to discover the Muse and the Angel in theology, found himself imprisoned by the Duende of cold ardors in that masterwork of the Escorial, where geometry abuts with a dream and the Duende wears the mask of the Muse for the eternal chastisement of the great king.”


Perhaps. Or perhaps Philip, “the lonely
hermit of the Escorial,” is less lonely now.

Monday, November 13, 2006

Monday November 13, 2006

Filed under: General — m759 @ 8:23 PM
Cognitive Blend:

Casino Royale
and
Time in the Rock

PA lottery Nov. 13, 2006: Mid-day 726, Evening 329
 
In today’s cognitive blend,
the role of Casino Royale
is played by the
Pennsylvania Lottery,
which points to 7/26,
Venus at St. Anne’s
(title of the closing chapter
of That Hideous Strength).

The role of
Time in the Rock
is played by a
Log24 entry of 3/29,
Diamond Theory in 1937.

There is such a thing
as a tesseract.

Tuesday, November 7, 2006

Tuesday November 7, 2006

Filed under: General — m759 @ 9:00 PM
Mate.
 
“What is called ‘losing’ in chess
may constitute winning
in another game.”
 
— Ludwig Wittgenstein,
Remarks on the
Foundations of Mathematics,
rev. ed., MIT Press, 1978–
Appendix III, paragraph 8,
said to have been written
on September 23, 1937
 
PA lottery Nov. 7, 2006: Mid-day 023, Evening 666

For clues to interpreting
today’s Keystone State
mid-day lottery number,
023, see
The Prime Cut Gospel.

For clues to interpreting
today’s Keystone State
evening lottery number,
666, see
the “Apocalypse Now”
quotations on
All Saints’ Day, 2006.

Tuesday, October 31, 2006

Tuesday October 31, 2006

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 PM
To Announce a Faith

From 7/07, an art review from The New York Times:

Endgame Art?
It's Borrow, Sample and Multiply
in an Exhibition at Bard College

"The show has an endgame, end-time mood….

I would call all these strategies fear of form…. the dismissal of originality is perhaps the oldest ploy in the postmodern playbook. To call yourself an artist at all is by definition to announce a faith, however unacknowledged, in some form of originality, first for yourself, second, perhaps, for the rest of us.

Fear of form above all means fear of compression– of an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible."

— Roberta Smith

It is doubtful that Smith
 would consider the
following "found" art an
example of originality.

It nevertheless does
"announce a faith."


The image “http://www.log24.com/log/pix06A/061031-PAlottery2.jpg” cannot be displayed, because it contains errors.


"First for yourself"

Today's mid-day
Pennsylvania number:
707

See Log24 on 7/07
and the above review.
 

"Second, perhaps,
for the rest of us"

Today's evening
Pennsylvania number:
384

This number is an
example of what the
reviewer calls "compression"–

"an artistic focus that condenses
 experiences, ideas and feelings
into something
whole, committed
 and visually comprehensible."

"Experiences"

See (for instance)

Joan Didion's writings
(1160 pages, 2.35 pounds)
on "the shifting phantasmagoria
which is our actual experience."

"Ideas"

See Plato.

"Feelings"

See A Wrinkle in Time.

"Whole"

The automorphisms
of the tesseract
form a group
of order 384.

"Committed"

See the discussions of
groups of degree 16 in
R. D. Carmichael's classic
Introduction to the Theory
of Groups of Finite Order
.

"Visually comprehensible"

See "Diamond Theory in 1937,"
an excerpt from which
is shown below.

The image “http://www.log24.com/theory/images/Carmichael440abbrev.gif” cannot be displayed, because it contains errors.

The "faith" announced by
the above lottery numbers
on All Hallows' Eve is
perhaps that of the artist
Madeleine L'Engle:

"There is such a thing
as a tesseract.
"

Saturday, May 13, 2006

Saturday May 13, 2006

Filed under: General — m759 @ 4:00 PM

ART WARS continued…

A Fold in Time

From May 13, Braque’s birthday, 2003:


Braque


Above: Braque and tesseract

“The senses deform, the mind forms.  Work to perfect the mind.  There is no certitude but in what the mind conceives.”

— Georges Braque, Reflections on Painting, 1917

Those who wish to follow Braque’s advice may try the following exercise from a book first published in 1937:

The image “http://www.log24.com/theory/images/Carmichael440ex.gif” cannot be displayed, because it contains errors.

Hint: See the above picture of
Braque and the construction of
a tesseract.

Related material:

Storyline and Time Fold
(both of Oct. 10, 2003),
and the following–

“Time, for L’Engle, is accordion-pleated. She elaborated, ‘When you bring a sheet off the line, you can’t handle it until it’s folded, and in a sense, I think, the universe can’t exist until it’s folded– or it’s a story without a book.'”

Cynthia Zarin on Madeleine L’Engle,
“The Storyteller,” in The New Yorker,
issue dated April 12, 2004

Friday, May 12, 2006

Friday May 12, 2006

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

"Does the word 'tesseract'
mean anything to you?"
— Robert A. Heinlein in
The Number of the Beast
(1980)

My reply–

Part I:

The image “http://www.log24.com/log/pix06A/WrinkleInTime1A.jpg” cannot be displayed, because it contains errors.

A Wrinkle in Time, by
Madeleine L'Engle
(first published in 1962)

Part II:

Diamond Theory in 1937
and
Geometry of the 4×4 Square

Part III:

Catholic Schools Sermon

Conclusion:
 

"Wells and trees were dedicated to saints.  But the offerings at many wells and trees were to something other than the saint; had it not been so they would not have been, as we find they often were, forbidden.  Within this double and intertwined life existed those other capacities, of which we know more now, but of which we still know little– clairvoyance, clairaudience, foresight, telepathy."

— Charles Williams, Witchcraft, Faber and Faber, London, 1941

Related material:

A New Yorker profile of Madeleine L'Engle from April 2004, which I found tonight online for the first time.  For a related reflection on truth, stories, and values, see Saint's Day.  For a wider context, see the Log24 entries of February 1-15, 2003 and February 1-15, 2006.
 

Friday, November 25, 2005

Friday November 25, 2005

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

Holy Geometry

What was “the holy geometry book” (“das heilige Geometrie-Büchlein,” p. 10 in the Schilpp book below) that so impressed the young Albert Einstein?

“At the age of 12 I experienced a second wonder of a totally different nature: in a little book dealing with Euclidian plane geometry, which came into my hands at the beginning of a schoolyear.  Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which– though by no means evident– could nevertheless be proved with such certainty that any doubt appeared to be out of the question.  This lucidity and certainty made an indescribable impression upon me.”

(“Im Alter von 12 Jahren erlebte ich ein zweites Wunder ganz verschiedener Art: An einem Büchlein über Euklidische Geometrie der Ebene, das ich am Anfang eines Schuljahres in die Hand bekam.  Da waren Aussagen wie z.B. das Sich-Schneiden der drei Höhen eines Dreieckes in einem Punkt, die– obwohl an sich keineswegs evident– doch mit solcher Sicherheit bewiesen werden konnten, dass ein Zweifel ausgeschlossen zu sein schien.  Diese Klarheit und Sicherheit machte einen unbeschreiblichen Eindruck auf mich.”)

— Albert Einstein, Autobiographical Notes, pages 8 and 9 in Albert Einstein: Philosopher-Scientist, ed. by Paul A. Schilpp

From a website by Hans-Josef Küpper:

“Today it cannot be said with certainty which book is Einstein’s ‘holy geometry book.’  There are three different titles that come into question:

Theodor Spieker, 1890
Lehrbuch der ebenen Geometrie. Mit Übungsaufgaben für höhere Lehranstalten.

Heinrich Borchert Lübsen, 1870
Ausführliches Lehrbuch der ebenen und sphärischen Trigonometrie. Zum Selbstunterricht. Mit Rücksicht auf die Zwecke des praktischen Lebens.

Adolf Sickenberger, 1888
Leitfaden der elementaren Mathematik.

Young Albert Einstein owned all of these three books. The book by T. Spieker was given to him by Max Talmud (later: Talmey), a Jewish medic. The book by H. B. Lübsen was from the library of his uncle Jakob Einstein and the one of A. Sickenberger was from his parents.”

Küpper does not state clearly his source for the geometry-book information.

According to Banesh Hoffman and Helen Dukas in Albert Einstein, Creator and Rebel, the holy geometry book was Lehrbuch der Geometrie zum Gebrauch an höheren Lehranstalten, by Eduard Heis (Catholic astronomer and textbook writer) and Thomas Joseph Eschweiler.

An argument for Sickenberger from The Young Einstein: The Advent of Relativity (pdf), by Lewis Pyenson, published by Adam Hilger Ltd., 1985:

   Throughout Einstein’s five and a half years at the Luitpold Gymnasium, he was taught mathematics from one or another edition of the separately published parts of Sickenberger’s Textbook of Elementary Mathematics.  When it first appeared in 1888 the book constituted a major contribution to reform pedagogy.  Sickenberger based his book on twenty years of experience that in his view necessarily took precedence over ‘theoretical doubts and systematic scruples.’  At the same time Sickenberger made much use of the recent pedagogical literature, especially that published in the pages of Immanuel Carl Volkmar Hoffmann’s Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht, the leading pedagogical mathematics journal of the day.  Following in the tradition of the reform movement, he sought to present everything in the simplest, most intuitive way possible.  He opposed introducing scientific rigour and higher approaches in an elementary text.  He emphasised that he would follow neither the synthesis of Euclidean geometry nor the so-called analytical-genetic approach.  He opted for a great deal of freedom in the form of presentation because he believed that a textbook was no more than a crutch for oral instruction.  The spoken word, in Sickenberger’s view, could infuse life into the dead forms of the printed text.  Too often, he insisted in the preface to his text, mathematics was seen and valued ‘as the pure science of reason.’  In reality, he continued, mathematics was also ‘an essential tool for daily work.’  In view of the practical dimension of mathematics Sickenberger sought most of all to present basic propositions clearly rather than to arrive at formal conciseness.   Numerous examples took the place of long, complicated, and boring generalities.  In addition to the usual rules of arithmetic Sickenberger introduced diophantine equations.  To solve three linear, homogeneous, first-order equations with three unknowns he specified determinants and determinant algebra.  Then he went on to quadratic equations and logarithms.  In the second part of his book, Sickenberger treated plane geometry.
     According to a biography of Einstein written by his step-son-in-law, Rudolf Kayser– one that the theoretical physicist described as ‘duly accurate’– when he was twelve years old Einstein fell into possession of the ‘small geometry book’ used in the Luitpold Gymnasium before this subject was formally presented to him.  Einstein corroborated Kayser’s passage in autobiographical notes of 1949, when he described how at the age of twelve ‘a little book dealing with Euclidean plane geometry’ came into his hands ‘at the beginning of a school year.’  The ‘lucidity and certainty’ of plane geometry according to this ‘holy geometry booklet’ made, Einstein wrote, ‘an indescribable impression on me.’  Einstein saw here what he found in other texts that he enjoyed: it was ‘not too particular’ in logical rigour but ‘made up for this by permitting the main thoughts to stand out clearly and synoptically.’  Upon working his way through this text, Einstein was then presented with one of the many editions of Theodor Spieker’s geometry by Max Talmey, a medical student at the University of Munich who dined with the Einsteins and who was young Einstein’s friend when Einstein was between the ages of ten and fifteen.  We can only infer from Einstein’s retrospective judgment that the first geometry book exerted an impact greater than that produced by Spieker’s treatment, by the popular science expositions of Aaron Bernstein and Ludwig Büchner also given to him by Talmey, or by the texts of Heinrich Borchert Lübsen from which Einstein had by the age of fourteen taught himself differential and integral calculus.
     Which text constituted the ‘holy geometry booklet’?  In his will Einstein gave ‘all his books’ to his long-time secretary Helen Dukas.  Present in this collection are three bearing the signature ‘J Einstein’: a logarithmic and trigonometric handbook, a textbook on analysis, and an introduction to infinitesimal calculus.  The signature is that of Einstein’s father’s brother Jakob, a business partner and member of Einstein’s household in Ulm and Munich.  He presented the books to his nephew Albert.  A fourth book in Miss Dukas’s collection, which does not bear Jakob Einstein’s name, is the second part of a textbook on geometry, a work of astronomer Eduard Heis’s which was rewritten after his death by the Cologne schoolteacher Thomas Joseph Eschweiler.  Without offering reasons for his choice Banesh Hoffmann has recently identified Heis and Eschweiler’s text as the geometry book that made such an impression on Einstein.  Yet, assuming that Kayser’s unambiguous reporting is correct, it is far more likely that the geometrical part of Sickenberger’s text was what Einstein referred to in his autobiographical notes.  Sickenberger’s exposition was published seven years after that of Heis and Eschweiler, and unlike the latter it appeared with a Munich press.  Because it was used in the Luitpold Gymnasium, copies would have been readily available to Uncle Jakob or to whoever first acquainted Einstein with Euclidean geometry.”

What might be the modern version of a “holy geometry book”?

I suggest the following,
first published in 1940:

The image “http://www.log24.com/log/pix05B/BasicGeometry.gif” cannot be displayed, because it contains errors.

Click on picture for details.

Saturday, June 12, 2004

Saturday June 12, 2004

Filed under: General — m759 @ 3:31 PM

Ray Charles with Nancy and Ronald Reagan

“Who’s got the last laugh now?”

— “They All Laughed,”
words by Ira Gershwin,
music by George Gershwin,
from the 1937 film Shall We Dance
(sung by Ginger Rogers,
then danced by
Fred Astaire and Ginger Rogers)

Ray Charles with Nancy and Ronald Reagan

See also
the entry of June 4 last year,
The Four Last Things.

Sunday, March 14, 2004

Sunday March 14, 2004

Filed under: General,Geometry — m759 @ 3:28 PM

Clarity and Certainty

“At the age of 12 I experienced a second wonder of a totally different nature: in a little book* dealing with Euclidean plane geometry, which came into my hands at the beginning of a schoolyear. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which — though by no means evident — could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty [Klarheit und Sicherheit] made an indescribable impression upon me….  For example I remember that an uncle told me the Pythagorean theorem before the holy geometry booklet* had come into my hands. After much effort I succeeded in ‘proving’ this theorem on the basis of the similarity of triangles … for anyone who experiences [these feelings] for the first time, it is marvellous enough that man is capable at all to reach such a degree of certainty and purity [Sicherheit und Reinheit] in pure thinking as the Greeks showed us for the first time to be possible in geometry.”

— from “Autobiographical Notes” in Albert Einstein: Philosopher-Scientist, edited by Paul Arthur Schilpp

“Although our intellect always longs for clarity and certainty, our nature often finds uncertainty fascinating.”

— Carl von Clausewitz at Quotes by Clausewitz

For clarity and certainty, consult All About Altitudes (and be sure to click the “pop it up” button).

For murkiness and uncertainty, consult The Fog of War.

Happy birthday, Albert.

* Einstein’s “holy geometry booklet” was, according to Banesh Hoffman, Lehrbuch der Geometrie zum Gebrauch an höheren Lehranstalten, by Eduard Heis (Catholic astronomer and textbook writer) and Thomas Joseph Eschweiler.

Sunday, May 25, 2003

Sunday May 25, 2003

Filed under: General,Geometry — Tags: — m759 @ 7:11 PM

ART WARS

Mental Health Month, Day 25:

Matrix of the Death God

Having dealt yesterday with the Death Goddess Sarah, we turn today to the Death God Abraham.  (See Jacques Derrida, The Gift of Death, University of Chicago Press, 1996.)  For a lengthy list of pictures of this damned homicidal lunatic about to murder his son, see The Text This Week.

 

See, too, The Matrix of Abraham, illustrated below.  This is taken from a book by R. M. Abraham, Diversions and Pastimes, published by Constable and Company, London, in 1933.

The Matrix of Abraham

A summary of the religious import of the above from Princeton University Press:

“Moslems of the Middle Ages were fascinated by pandiagonal squares with 1 in the center…. The Moslems thought of the central 1 as being symbolic of the unity of Allah.  Indeed, they were so awed by that symbol that they often left blank the central cell on which the 1 should be positioned.”

— Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton U. Press, 2002, pp. 71-72

Other appearances of this religious icon on the Web:

On Linguistic Creation

Picasso’s Birthday

A less religious approach to the icon may be found on page 393 of R. D. Carmichael’s Introduction to the Theory of Groups of Finite Order (Ginn, Boston, 1937, reprinted by Dover, 1956).

This matrix did not originate with Abraham but, unlike Neo, I have not yet found its Architect.

Friday, May 23, 2003

Friday May 23, 2003

Filed under: General — m759 @ 7:23 PM

Mental Health Month, Day 23:

The Prime Cut Gospel

On Christmas Day, 1949,
Mary Elizabeth Spacek was born in Texas.

Lee Marvin, Sissy Spacek in “Prime Cut”

Exercises for Mental Health Month:

Read this discussion of the phrase, suggested by Spacek’s date of birth, “God’s gift to men.”

Read this discussion of the phrase “the same yesterday, today, and forever,” suggested by the previous reading.

Read the more interesting of these discussions of the phrase “the eternal in the temporal.”

Read this discussion of eternal, or “necessary,” truths versus other sorts of alleged “truths.”

Read this discussion of unimportant mathematical properties of the prime number 23.

Read these discussions of important properties of 23:

  • R. D. Carmichael’s 1937 discussion of the linear fractional group modulo 23 in 

Introduction to the Theory of Groups of Finite Order, Ginn, Boston, 1937 (reprinted by Dover in 1956), final chapter, “Tactical Configurations,” and

  • Conway’s 1969 discussion of the same group in    

J. H. Conway, “Three Lectures on Exceptional Groups,” pp. 215-247 in Finite Simple Groups (Oxford, 1969), edited by M. B. Powell and G. Higman, Academic Press, London, 1971….. Reprinted as Ch. 10 in Sphere Packings, Lattices, and Groups 

Read this discussion of what might be called “contingent,” or “literary,” properties of the number 23. 

Read also the more interesting of  these discussions of the phrase “the 23 enigma.”

Having thus acquired some familiarity with both contingent and necessary properties of 23…

Read this discussion of Aquinas’s third proof of the existence of God.

Note that the classic Spacek film “Prime Cut” was released in 1972, the year that Spacek turned 23:

1949
+ 23


1972
 
Essay question:  
 
If Jesus was God’s gift to man, and (as many men would agree) so was the young Sissy Spacek (also born on Christmas Day), was young Sissy’s existence in her 23rd year contingent or necessary?  If the latter, should she be recognized as a Person of the Trinity? Quaternity? N-ity?
 
Talk amongst yourselves.

Saturday, April 12, 2003

Saturday April 12, 2003

Filed under: General — m759 @ 2:23 PM

2:23 PM
Sequel
to the previous two entries

“This world is not conclusion;
A sequel stands beyond….”
— Emily Dickinson

Today’s birthday: dancer/actress Ann Miller.

“In 1937, she was discovered by Lucille Ball….”

Lucille Ball, Desi Arnaz,
and Ann Miller, cast photo
from Too Many Girls (1940)

“Just goes to show star quality shines through….”
— Website on Too Many Girls 

“It’ll shine when it shines.”
— Folk saying, epigraph to The Shining

“Shine on, you crazy diamond.”
Pink Floyd

“Well we all shine on…”
— John Lennon, “Instant Karma

Saturday, December 28, 2002

Saturday December 28, 2002

Filed under: General — m759 @ 12:00 AM

On This Date


    Kylie

In 1937, composer
Maurice Ravel died.

Our site music for today
is Ravel’s classic, “
Bolero.”

For “Bolero” purposes, some may prefer Kylie Minogue’s rendition of “Locomotion.”

Zen meditation: “Kylie Eleison!”

(For evidence that this is a valid Japanese religious exclamation, click here.)

Saturday, December 21, 2002

Saturday December 21, 2002

Filed under: General — m759 @ 12:00 AM

Nightmare Alley

Tonight’s site music in the garden of good and evil is “Hooray for Hollywood,” with lyrics by Johnny Mercer:

Hooray for Hollywood.
You may be homely in your neighborhood,
But if you think you can be an actor,
see Mr. Factor,
he’d make a monkey look good.
Within a half an hour,
you look like Tyrone Power!
Hooray for Hollywood!

 

From Pif magazine:

Nightmare Alley (1947)
Directed by Edmund Goulding
Reviewed by Nick Burton

“Edmund Goulding’s film of William Lindsay Gresham’s 1946 novel Nightmare Alley may just be the great forgotten American film; it is certainly the darkest film that came from the Hollywood studio system in the ’40s….

A never better Tyrone Power stars as Stan Carlisle, a small-time carny shill….  Stan shills for mind reader Zeena…. The… pretty ‘electric girl’…   tells Stan that Zeena… had a ‘code’ for the mind-reading act… Stan… decides to seduce… Zeena in hopes of luring the code from her.”

The rest of this review is well worth reading, though less relevant to my present theme — that of my 

Sermon for St. Patrick’s Day,

which points out that the article on “nothing” is on page 265 of The Oxford Dictionary of Philosophy. (This is also the theme of yesterday’s journal entry “Last-Minute Shopping.”) Here is another work that prominently features “nothing” on page 265… As it happens, this is a web page describing a mind-reading act, titled simply

Page 265

“Imagine this: A spectator is invited to take a readable and 100% examinable, 400 page, 160,000 word novel, open it to any page and think of any word on that page. Without touching the book or approaching the spectator, you reveal the word in the simplest, most startlingly direct manner ever! It truly must be seen to be believed.

The ultimate any-word-on-any-page method that makes all other book tests obsolete….

All pages are different.

Nothing is written down.

There are no stooges of any kind. Everything may be examined….

 ‘Throw away your Key. This is direct mindreading at its best.'”

From Finnegans Wake, page 265:

“…the winnerful wonnerful wanders off, with hedges of ivy and

hollywood
 
and bower of mistletoe….”

Hooray.

Mercer’s lyrics are from the 1937 film Hollywood Hotel.”  For a somewhat more in-depth look at Hollywood, hotels of this period, and mind-reading, see

Shining Forth.

Friday, November 29, 2002

Friday November 29, 2002

Filed under: General,Geometry — Tags: — m759 @ 1:06 PM

A Logocentric Archetype

Today we examine the relativist, nominalist, leftist, nihilist, despairing, depressing, absurd, and abominable work of Samuel Beckett, darling of the postmodernists.

One lens through which to view Beckett is an essay by Jennifer Martin, "Beckettian Drama as Protest: A Postmodern Examination of the 'Delogocentering' of Language." Martin begins her essay with two quotations: one from the contemptible French twerp Jacques Derrida, and one from Beckett's masterpiece of stupidity, Molloy. For a logocentric deconstruction of Derrida, see my note, "The Shining of May 29," which demonstrates how Derrida attempts to convert a rather important mathematical result to his brand of nauseating and pretentious nonsense, and of course gets it wrong. For a logocentric deconstruction of Molloy, consider the following passage:

"I took advantage of being at the seaside to lay in a store of sucking-stones. They were pebbles but I call them stones…. I distributed them equally among my four pockets, and sucked them turn and turn about. This raised a problem which I first solved in the following way. I had say sixteen stones, four in each of my four pockets these being the two pockets of my trousers and the two pockets of my greatcoat. Taking a stone from the right pocket of my greatcoat, and putting it in my mouth, I replaced it in the right pocket of my greatcoat by a stone from the right pocket of my trousers, which I replaced by a stone from the left pocket of my trousers, which I replaced by a stone from the left pocket of my greatcoat, which I replaced by the stone which was in my mouth, as soon as I had finished sucking it. Thus there were still four stones in each of my four pockets, but not quite the same stones….But this solution did not satisfy me fully. For it did not escape me that, by an extraordinary hazard, the four stones circulating thus might always be the same four."

Beckett is describing, in great detail, how a damned moron might approach the extraordinarily beautiful mathematical discipline known as group theory, founded by the French anticleric and leftist Evariste Galois. Disciples of Derrida may play at mimicking the politics of Galois, but will never come close to imitating his genius. For a worthwhile discussion of permutation groups acting on a set of 16 elements, see R. D. Carmichael's masterly work, Introduction to the Theory of Groups of Finite Order, Ginn, Boston, 1937, reprinted by Dover, New York, 1956.

There are at least two ways of approaching permutations on 16 elements in what Pascal calls "l'esprit géométrique." My website Diamond Theory discusses the action of the affine group in a four-dimensional finite geometry of 16 points. For a four-dimensional euclidean hypercube, or tesseract, with 16 vertices, see the highly logocentric movable illustration by Harry J. Smith. The concept of a tesseract was made famous, though seen through a glass darkly, by the Christian writer Madeleine L'Engle in her novel for children and young adults, A Wrinkle in Tme.

This tesseract may serve as an archetype for what Pascal, Simone Weil (see my earlier notes), Harry J. Smith, and Madeleine L'Engle might, borrowing their enemies' language, call their "logocentric" philosophy.

For a more literary antidote to postmodernist nihilism, see Archetypal Theory and Criticism, by Glen R. Gill.

For a discussion of the full range of meaning of the word "logos," which has rational as well as religious connotations, click here.

Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: , — m759 @ 10:13 PM
 

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.
We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

Example:


For an animated version, click here.

Remarks:

Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).


The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry

Walsh Functions

Inscapes

The Diamond Theory of Truth

Geometry of the I Ching

Solomon's Cube and The Eightfold Way

Crystal and Dragon in Diamond Theory

The Form, the Pattern

The Grid of Time

Block Designs

Finite Relativity

Theme and Variations

Models of Finite Geometries

Quilt Geometry

Pattern Groups

The Fano Plane Revisualized,
or the Eightfold Cube

The Miracle Octad Generator

Kaleidoscope

Visualizing GL(2,p)

Jung's Imago

Author's home page

AMS Mathematics Subject Classification:

20B25 (Group theory and generalizations :: Permutation groups :: Finite automorphism groups of algebraic, geometric, or combinatorial structures)

05B25 (Combinatorics :: Designs and configurations :: Finite geometries)

51E20 (Geometry :: Finite geometry and special incidence structures :: Combinatorial structures in finite projective spaces)




Creative Commons License
This work is licensed under a
Creative Commons Attribution-NonCommercial-NoDerivs 2.5 License
.




Page created Jan. 6, 2006, by Steven H. Cullinane      diamondtheorem.com

 

Initial Xanga entry.  Updated Nov. 18, 2006.

Friday, July 19, 2002

Old Xanga post numbers

Filed under: — m759 @ 9:22 PM

This WordPress page from 9:22 PM ET on Aug. 17, 2016,
gives id numbers of old Xanga posts for Log24 and user m759.

It is backdated to July 19, 2002, the day before the first post in
this WordPress weblog, so it will not appear before other posts
in searches of the weblog.

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Monday, May 19, 2003

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Sunday, May 18, 2003

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Friday, May 16, 2003

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Thursday, May 15, 2003

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Thursday, May 15, 2003

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Wednesday, May 14, 2003

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Tuesday, May 13, 2003

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Monday, May 12, 2003

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Sunday, May 11, 2003

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Friday, May 09, 2003

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Friday, May 09, 2003

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Friday, May 02, 2003

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Thursday, May 01, 2003

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Sunday, June 29, 2003

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Friday, June 27, 2003

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Thursday, June 26, 2003

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Monday, June 23, 2003

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Sunday, June 22, 2003

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Friday, June 20, 2003

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Wednesday, June 18, 2003

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Tuesday, June 17, 2003

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Monday, June 16, 2003

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Sunday, June 15, 2003

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Saturday, June 14, 2003

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Wednesday, June 11, 2003

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Tuesday, June 10, 2003

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Wednesday, June 04, 2003

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Tuesday, June 03, 2003

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Monday, June 02, 2003

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Sunday, June 01, 2003

<strong><font size="3">Thursday, May 29,
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Thursday, July 31, 2003

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Wednesday, July 30, 2003

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Monday, July 28, 2003

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Monday, July 28, 2003

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Sunday, July 27, 2003

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Saturday, July 26, 2003

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Friday, July 25, 2003

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Friday, July 18, 2003

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Sunday, July 13, 2003

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Saturday, July 12, 2003

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Friday, July 11, 2003

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Saturday, August 30, 2003

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Wednesday, August 27, 2003

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Monday, August 25, 2003

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Sunday, August 24, 2003

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Saturday, August 23, 2003

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Tuesday, August 19, 2003

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Monday, August 18, 2003

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Sunday, August 17, 2003

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Tuesday, August 12, 2003

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Saturday, August 09, 2003

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Friday, August 08, 2003

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Tuesday, August 05, 2003

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Monday, August 04, 2003

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Sunday, August 03, 2003

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Friday, August 01, 2003

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Friday, August 01, 2003

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Tuesday, September 30, 2003

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Monday, September 29, 2003

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Sunday, September 28, 2003

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Friday, September 26, 2003

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Thursday, September 25, 2003

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Tuesday, September 23, 2003

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Friday, September 19, 2003

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Monday, September 15, 2003

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Sunday, September 14, 2003

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Saturday, September 13, 2003

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Tuesday, September 09, 2003

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Monday, September 08, 2003

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Monday, September 08, 2003

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Sunday, September 07, 2003

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Saturday, September 06, 2003

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Friday, September 05, 2003

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Thursday, September 04, 2003

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Friday, October 31, 2003

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Wednesday, October 29, 2003

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Tuesday, October 28, 2003

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Sunday, October 26, 2003

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Saturday, October 25, 2003

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Friday, October 24, 2003

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Thursday, October 16, 2003

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Wednesday, October 15, 2003

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Tuesday, October 14, 2003

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Sunday, October 12, 2003

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Saturday, October 11, 2003

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Friday, October 10, 2003

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Wednesday, October 08, 2003

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Tuesday, October 07, 2003

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Sunday, October 05, 2003

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Saturday, October 04, 2003

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Friday, October 03, 2003

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Thursday, October 02, 2003

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Sunday, November 30, 2003

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Saturday, November 29, 2003

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Friday, November 28, 2003

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Tuesday, November 25, 2003

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Monday, November 24, 2003

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Sunday, November 23, 2003

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Saturday, November 22, 2003

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Thursday, November 20, 2003

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Monday, November 17, 2003

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Sunday, November 16, 2003

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Saturday, November 15, 2003

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Friday, November 14, 2003

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Thursday, November 13, 2003

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Wednesday, November 12, 2003

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Tuesday, November 11, 2003

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Sunday, November 09, 2003

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Friday, November 07, 2003

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Thursday, November 06, 2003

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Wednesday, November 05, 2003

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Tuesday, November 04, 2003

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Monday, November 03, 2003

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Saturday, November 01, 2003

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Wednesday, December 31, 2003

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Saturday, December 27, 2003

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Friday, December 26, 2003

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Monday, December 22, 2003

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Monday, December 22, 2003

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Sunday, December 21, 2003

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Saturday, December 20, 2003

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Saturday, December 20, 2003

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Friday, December 19, 2003

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Wednesday, December 17, 2003

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Tuesday, December 16, 2003

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Sunday, December 14, 2003

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Saturday, December 13, 2003

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Friday, December 12, 2003

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Thursday, December 11, 2003

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Wednesday, December 10, 2003

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Tuesday, December 09, 2003

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Monday, December 08, 2003

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Sunday, December 07, 2003

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Sunday, December 07, 2003

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Friday, December 05, 2003

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Saturday, January 31, 2004

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Thursday, January 29, 2004

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Wednesday, January 28, 2004

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Monday, January 26, 2004

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Monday, January 26, 2004

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Saturday, January 24, 2004

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Thursday, January 22, 2004

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Wednesday, January 21, 2004

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Tuesday, January 20, 2004

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Sunday, January 18, 2004

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Saturday, January 17, 2004

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Friday, January 16, 2004

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Wednesday, January 14, 2004

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Tuesday, January 13, 2004

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Sunday, January 11, 2004

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Friday, January 09, 2004

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Thursday, January 08, 2004

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Tuesday, January 06, 2004

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Monday, January 05, 2004

Sunday, January 04, 2004

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Sunday, January 04, 2004

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Friday, January 02, 2004

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Thursday, January 01, 2004

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Sunday, February 29, 2004

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Saturday, February 28, 2004

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Thursday, February 26, 2004

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Wednesday, February 25, 2004

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Tuesday, February 24, 2004

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Monday, February 23, 2004

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Sunday, February 22, 2004

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Friday, February 20, 2004

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Thursday, February 19, 2004

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Wednesday, February 18, 2004

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Wednesday, February 18, 2004

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Tuesday, February 17, 2004

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Monday, February 16, 2004

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Saturday, February 14, 2004

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Thursday, February 12, 2004

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Wednesday, February 11, 2004

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Tuesday, February 10, 2004

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Monday, February 09, 2004

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Sunday, February 08, 2004

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Saturday, February 07, 2004

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Friday, February 06, 2004

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Thursday, February 05, 2004

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Tuesday, February 03, 2004

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Tuesday, February 03, 2004

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Monday, February 02, 2004

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Sunday, February 01, 2004

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Wednesday, March 31, 2004

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Tuesday, March 30, 2004

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Tuesday, March 30, 2004

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Sunday, March 28, 2004

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Saturday, March 27, 2004

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Tuesday, March 23, 2004

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Monday, March 22, 2004

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Sunday, March 21, 2004

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Saturday, March 20, 2004

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Friday, March 19, 2004

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Thursday, March 18, 2004

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Wednesday, March 17, 2004

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Tuesday, March 16, 2004

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Monday, March 15, 2004

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Monday, March 15, 2004

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Sunday, March 14, 2004

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Saturday, March 13, 2004

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Saturday, March 13, 2004

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Friday, March 12, 2004

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Thursday, March 11, 2004

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Thursday, March 11, 2004

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Wednesday, March 10, 2004

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Sunday, March 07, 2004

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Saturday, March 06, 2004

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Friday, March 05, 2004

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Thursday, March 04, 2004

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Wednesday, March 03, 2004

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Wednesday, March 03, 2004

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Tuesday, March 02, 2004

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Friday, April 30, 2004

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Thursday, April 29, 2004

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Wednesday, April 28, 2004

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Tuesday, April 27, 2004

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Tuesday, April 27, 2004

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Monday, April 26, 2004

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Sunday, April 25, 2004

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Friday, April 23, 2004

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Thursday, April 22, 2004

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Wednesday, April 21, 2004

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Tuesday, April 20, 2004

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Monday, April 19, 2004

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Sunday, April 18, 2004

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Friday, April 16, 2004

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Thursday, April 15, 2004

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Wednesday, April 14, 2004

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Tuesday, April 13, 2004

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Sunday, April 11, 2004

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Saturday, April 10, 2004

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Friday, April 09, 2004

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Friday, April 09, 2004

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Thursday, April 08, 2004

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Wednesday, April 07, 2004

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Tuesday, April 06, 2004

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Monday, April 05, 2004

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Sunday, April 04, 2004

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Friday, April 02, 2004

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Thursday, April 01, 2004

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Monday, May 31, 2004

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Thursday, May 27, 2004

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Saturday, May 22, 2004

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Friday, May 21, 2004

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Friday, May 21, 2004

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Thursday, May 20, 2004

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Wednesday, May 19, 2004

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Saturday, May 15, 2004

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Saturday, May 15, 2004

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Friday, May 14, 2004

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Thursday, May 13, 2004

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Wednesday, May 12, 2004

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Monday, May 10, 2004

Sunday, May 09, 2004

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Saturday, May 08, 2004

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Saturday, May 08, 2004

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Friday, May 07, 2004

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Wednesday, May 05, 2004

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Tuesday, May 04, 2004

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Monday, May 03, 2004

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Sunday, May 02, 2004

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Saturday, May 01, 2004

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Saturday, May 01, 2004

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Tuesday, June 29, 2004

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Sunday, June 27, 2004

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Saturday, June 26, 2004

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Friday, June 25, 2004

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Tuesday, June 22, 2004

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Thursday, June 17, 2004

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Wednesday, June 16, 2004

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Tuesday, June 15, 2004

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Sunday, June 13, 2004

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Saturday, June 12, 2004

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Friday, June 11, 2004

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Tuesday, June 08, 2004

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Sunday, June 06, 2004

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Sunday, June 06, 2004

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Saturday, June 05, 2004

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Friday, June 04, 2004

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Thursday, June 03, 2004

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Tuesday, June 01, 2004

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Tuesday, June 01, 2004

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Sunday, July 11, 2004

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<a name="3" target="_new"></a><big><b><font
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Tuesday, August 09, 2005

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Friday, September 30, 2005

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Thursday, September 29, 2005

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Tuesday, September 27, 2005

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Sunday, September 25, 2005

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Thursday, September 22, 2005

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Wednesday, September 21, 2005

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Monday, September 19, 2005

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Tuesday, September 13, 2005

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Saturday, September 10, 2005

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Saturday, October 22, 2005

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Thursday, October 20, 2005

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Thursday, October 13, 2005

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Monday, October 10, 2005

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Thursday, October 06, 2005

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Wednesday, November 30, 2005

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Tuesday, November 29, 2005

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Friday, November 25, 2005

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Friday, November 18, 2005

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Wednesday, November 09, 2005

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Tuesday, November 08, 2005

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Sunday, November 06, 2005

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Thursday, November 03, 2005

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Wednesday, November 02, 2005

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Tuesday, November 01, 2005

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Saturday, December 31, 2005

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Sunday, December 25, 2005

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Wednesday, December 21, 2005

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Tuesday, February 28, 2006

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Monday, February 27, 2006

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Thursday, February 02, 2006

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Wednesday, February 01, 2006

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Friday, March 31, 2006

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Saturday, March 18, 2006

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Tuesday, March 14, 2006

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Friday, March 10, 2006

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Saturday, March 04, 2006

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Wednesday, March 01, 2006

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Sunday, April 30, 2006

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Friday, April 28, 2006

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Wednesday, April 26, 2006

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Tuesday, April 18, 2006

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Monday, April 17, 2006

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Wednesday, April 12, 2006

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Saturday, April 01, 2006

Wednesday, May 31, 2006

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Monday, May 29, 2006

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Monday, May 29, 2006

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Sunday, May 28, 2006

<a
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March 22, 2006</a><br>
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 490235342 4:29 PM

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Friday, May 26, 2006

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Wednesday, May 24, 2006

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Tuesday, May 23, 2006

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Monday, May 22, 2006

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Sunday, May 21, 2006

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Friday, May 19, 2006

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Thursday, May 18, 2006

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Friday, May 12, 2006

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Wednesday, May 10, 2006

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Monday, May 08, 2006

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Sunday, May 07, 2006

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Saturday, May 06, 2006

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Friday, May 05, 2006

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Thursday, May 04, 2006

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Wednesday, May 03, 2006

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Friday, June 30, 2006

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Thursday, June 29, 2006

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Tuesday, June 27, 2006

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Monday, June 26, 2006

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Thursday, June 15, 2006

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Wednesday, June 14, 2006

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Friday, June 02, 2006

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Thursday, June 01, 2006

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Monday, July 31, 2006

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Saturday, August 19, 2006

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Thursday, August 17, 2006

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Monday, August 14, 2006

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Monday, August 14, 2006

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Sunday, August 13, 2006

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Friday, August 11, 2006

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Friday, August 11, 2006

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Wednesday, August 09, 2006

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Sunday, August 06, 2006

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Saturday, August 05, 2006

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Wednesday, August 02, 2006

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Sunday, October 01, 2006

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Saturday, September 30, 2006

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Friday, September 29, 2006

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Thursday, September 28, 2006

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Tuesday, September 26, 2006

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Saturday, September 23, 2006

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Friday, September 15, 2006

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Thursday, September 14, 2006

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Wednesday, September 13, 2006

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Monday, September 11, 2006

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Thursday, September 07, 2006

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Wednesday, September 06, 2006

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Sunday, September 03, 2006

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Saturday, September 02, 2006

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Tuesday, October 31, 2006

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Monday, October 30, 2006

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Sunday, October 29, 2006

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Saturday, October 28, 2006

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Thursday, October 26, 2006

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Wednesday, October 25, 2006

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Tuesday, October 24, 2006

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Thursday, October 19, 2006

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Wednesday, October 18, 2006

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Tuesday, October 17, 2006

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Monday, October 16, 2006

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Sunday, October 15, 2006

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Saturday, October 14, 2006

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Saturday, October 14, 2006

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Friday, October 13, 2006

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Thursday, October 12, 2006

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Wednesday, October 11, 2006

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Tuesday, October 10, 2006

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Tuesday, October 10, 2006

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Monday, October 09, 2006

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Sunday, October 08, 2006

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Saturday, October 07, 2006

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Friday, October 06, 2006

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Friday, October 06, 2006

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Thursday, October 05, 2006

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Wednesday, October 04, 2006

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Tuesday, October 03, 2006

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Monday, October 02, 2006

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Sunday, October 01, 2006

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Tuesday, November 28, 2006

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Monday, November 27, 2006

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Sunday, November 26, 2006

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Saturday, November 25, 2006

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Friday, November 24, 2006

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Wednesday, November 22, 2006

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Tuesday, November 21, 2006

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Monday, November 20, 2006

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Sunday, November 19, 2006

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Saturday, November 18, 2006

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Thursday, November 16, 2006

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Thursday, November 16, 2006

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Wednesday, November 15, 2006

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Monday, November 13, 2006

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Sunday, November 12, 2006

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Sunday, November 12, 2006

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Friday, November 10, 2006

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Friday, November 10, 2006

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Thursday, November 09, 2006

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Tuesday, November 07, 2006

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Tuesday, November 07, 2006

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Monday, November 06, 2006

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Sunday, November 05, 2006

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Saturday, November 04, 2006

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Friday, November 03, 2006

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Thursday, November 02, 2006

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Wednesday, November 01, 2006

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Sunday, December 31, 2006

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Friday, December 29, 2006

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Wednesday, December 27, 2006

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Tuesday, December 26, 2006

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Monday, December 25, 2006

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Sunday, December 24, 2006

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Saturday, December 23, 2006

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Thursday, December 21, 2006

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Wednesday, December 20, 2006

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Tuesday, December 19, 2006

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Monday, December 18, 2006

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Saturday, December 16, 2006

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Friday, December 15, 2006

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Thursday, December 14, 2006

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Wednesday, December 13, 2006

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Tuesday, December 12, 2006

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Monday, December 11, 2006

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Sunday, December 10, 2006

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Sunday, December 10, 2006

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Saturday, December 09, 2006

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Friday, December 08, 2006

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Thursday, December 07, 2006

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Wednesday, December 06, 2006

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Tuesday, December 05, 2006

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Monday, December 04, 2006

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Sunday, December 03, 2006

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Saturday, December 02, 2006

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Saturday, December 02, 2006

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Friday, December 01, 2006

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Wednesday, January 31, 2007

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Monday, January 29, 2007

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Saturday, January 27, 2007

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Friday, January 26, 2007

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Wednesday, January 24, 2007

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Tuesday, January 23, 2007

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Monday, January 22, 2007

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Sunday, January 21, 2007

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Saturday, January 20, 2007

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Friday, January 19, 2007

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Tuesday, January 09, 2007

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Sunday, January 07, 2007

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Saturday, January 06, 2007

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Friday, January 05, 2007

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Thursday, January 04, 2007

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Wednesday, January 03, 2007

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Tuesday, January 02, 2007

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Monday, January 01, 2007

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Wednesday, February 28, 2007

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Tuesday, February 27, 2007

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Monday, February 26, 2007

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Sunday, February 25, 2007

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Sunday, February 25, 2007

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Tuesday, February 20, 2007

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Sunday, February 18, 2007

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Saturday, February 17, 2007

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Friday, February 16, 2007

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Thursday, February 15, 2007

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Tuesday, February 13, 2007

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Monday, February 12, 2007

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Sunday, February 11, 2007

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Saturday, February 10, 2007

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Friday, February 09, 2007

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Wednesday, February 07, 2007

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Tuesday, February 06, 2007

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Sunday, February 04, 2007

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Saturday, February 03, 2007

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Saturday, February 03, 2007

Friday, February 02, 2007

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Thursday, February 01, 2007

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Saturday, March 31, 2007

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Friday, March 30, 2007

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Wednesday, March 28, 2007

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Saturday, March 24, 2007

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Thursday, March 22, 2007

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Wednesday, March 21, 2007

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Wednesday, March 21, 2007

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Monday, March 19, 2007

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Sunday, March 18, 2007

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Saturday, March 17, 2007

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Friday, March 16, 2007

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Thursday, March 15, 2007

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Wednesday, March 14, 2007

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Saturday, March 10, 2007

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Friday, March 09, 2007

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Thursday, March 08, 2007

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Wednesday, March 07, 2007

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Sunday, March 04, 2007

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Saturday, March 03, 2007

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Friday, March 02, 2007

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Thursday, March 01, 2007

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Monday, April 30, 2007

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Sunday, April 29, 2007

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Saturday, April 28, 2007

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Friday, April 27, 2007

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Wednesday, April 25, 2007

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Monday, April 23, 2007

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Monday, April 23, 2007

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Sunday, April 22, 2007

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Friday, April 20, 2007

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Friday, April 20, 2007

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Thursday, April 19, 2007

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Wednesday, April 18, 2007

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Monday, April 16, 2007

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Sunday, April 15, 2007

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Sunday, April 15, 2007

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Saturday, April 14, 2007

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Saturday, April 14, 2007

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Friday, April 13, 2007

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Thursday, April 12, 2007

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Wednesday, April 11, 2007

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Monday, April 09, 2007

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Monday, April 09, 2007

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Sunday, April 08, 2007

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Saturday, April 07, 2007

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Friday, April 06, 2007

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Thursday, April 05, 2007

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Wednesday, April 04, 2007

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Tuesday, April 03, 2007

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Monday, April 02, 2007

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Sunday, April 01, 2007

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Thursday, May 31, 2007

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Wednesday, May 30, 2007

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Monday, May 28, 2007

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Sunday, May 27, 2007

Saturday, May 26, 2007

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Friday, May 25, 2007

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Wednesday, May 23, 2007

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Tuesday, May 22, 2007

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Monday, May 21, 2007

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Monday, May 21, 2007

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Sunday, May 20, 2007

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Saturday, May 19, 2007

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Friday, May 18, 2007

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Thursday, May 17, 2007

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Wednesday, May 16, 2007

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Tuesday, May 15, 2007

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Monday, May 14, 2007

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Sunday, May 13, 2007

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Sunday, May 13, 2007

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Saturday, May 12, 2007

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Friday, May 11, 2007

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Friday, May 11, 2007

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Thursday, May 10, 2007

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Wednesday, May 09, 2007

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Tuesday, May 08, 2007

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Monday, May 07, 2007

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Sunday, May 06, 2007

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Sunday, May 06, 2007

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Friday, May 04, 2007

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Thursday, May 03, 2007

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Tuesday, May 01, 2007

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Saturday, June 30, 2007

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Thursday, June 28, 2007

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Wednesday, June 27, 2007

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Monday, June 25, 2007

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Sunday, June 24, 2007

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Saturday, June 23, 2007

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Friday, June 22, 2007

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Thursday, June 21, 2007

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Wednesday, June 20, 2007

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Tuesday, June 19, 2007

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Monday, June 18, 2007

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Monday, June 18, 2007

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Sunday, June 17, 2007

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Saturday, June 16, 2007

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Friday, June 15, 2007

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Thursday, June 14, 2007

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Wednesday, June 13, 2007

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Tuesday, June 12, 2007

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Monday, June 11, 2007

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Sunday, June 10, 2007

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Saturday, June 09, 2007

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Friday, June 08, 2007

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Thursday, June 07, 2007

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Thursday, June 07, 2007

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Wednesday, June 06, 2007

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Tuesday, June 05, 2007

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Sunday, June 03, 2007

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Saturday, June 02, 2007

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Tuesday, July 31, 2007

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Monday, July 30, 2007

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Sunday, July 29, 2007

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Saturday, July 28, 2007

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Thursday, July 26, 2007

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Monday, July 23, 2007

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Saturday, July 21, 2007

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Thursday, July 19, 2007

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Wednesday, July 18, 2007

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Saturday, July 14, 2007

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Tuesday, July 10, 2007

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Monday, July 09, 2007

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Saturday, July 07, 2007

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Tuesday, July 03, 2007

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Monday, July 02, 2007

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Monday, July 02, 2007

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Sunday, July 01, 2007

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<span>
<div class="blogheader">Friday, August 31, 2007

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Wednesday, August 22, 2007

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Tuesday, August 21, 2007

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Monday, August 20, 2007

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Sunday, August 19, 2007

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Saturday, August 18, 2007

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Tuesday, August 14, 2007

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Monday, August 13, 2007

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Sunday, August 12, 2007

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