(Continued from Part I and Part II.)
The paper excerpted below supplies some badly needed technical
background for the Wikipedia article on functional decomposition.
The preprint above gives the precise definitions and technical references
that are completely absent from Wikipedia's Functional decomposition.
For some related material on 4×4 arrays like those in the above figure
see Decomposition Part I and Geometry of the 4×4 Square.
Compare and contrast
1. The following excerpt from Wikipedia—
2. A webpage subtitled "Function Decomposition Over a Finite Field."
Related material—
“Birthday, death-day — what day is not both?”
— John Updike, The New Yorker dated August 5, 2002, page 63
Today's date— Poincaré's birth, Wittgenstein's death.
A Saint for Clark University—
Today is also the birth date of William Edward Story,
a mathematician who taught at Clark University
in Worcester, Mass.
Story's date of death was April 10, 1930.
See the Log24 posts for that date in 2012.
“Oh, Sara!” she whispered joyfully. “It is like a story!”
“It is a story,” said Sara. “Everything's a story.
You are a story— I am a story.”
— Frances Hodgson Burnett, A Little Princess
The last paragraph of the previous post
(as updated at about 7:20 PM today)
suggests a search for the phrase
"play and interplay" that yields…
"He had accepted the world as the world,
but now he was comprehending the
organization of it, the play and interplay
of force and matter."
— Martin Eden by Jack London
This in turn suggests a review of the film "Queen to Play" —
(Background: Nabokov + Patterns.)
The review announces showings of the film at Clark University
in Worcester, Mass., on Sunday, October 30, 2011.
See also this journal on that date— "The Idea Idea"— and
references to a knight figure from today's date in 1985.
A Log24 post, "Bridal Birthday," one year ago today linked to
"The Discrete and the Continuous," a brief essay by David Deutsch.
From that essay—
"The idea of quantization—
the discreteness of physical quantities—
turned out to be immensely fruitful."
Deutsch's "idea of quantization" also appears in
the April 12 Log24 post Mythopoetic—
|
"Is Space Digital?" — Cover story, Scientific American "The idea that space may be digital — Physicist Sabine Hossenfelder "A quantization of space/time — Peter Woit in a comment |
It seems some clarification is in order.
Hossenfelder's "The idea that space may be digital"
and Woit's "a quantization of space/time" may not
refer to the same thing.
Scientific American on the concept of digital space—
"Space may not be smooth and continuous.
Instead it may be digital, composed of tiny bits."
Wikipedia on the concept of quantization—
Causal sets, loop quantum gravity, string theory,
and black hole thermodynamics all predict
a quantized spacetime….
For a purely mathematical approach to the
continuous-vs.-discrete issue, see
Finite Geometry and Physical Space.
The physics there is somewhat tongue-in-cheek,
but the geometry is serious.The issue there is not
continuous-vs.-discrete physics , but rather
Euclidean-vs.-Galois geometry .
Both sorts of geometry are of course valid.
Euclidean geometry has long been applied to
physics; Galois geometry has not. The cited
webpage describes the interplay of both sorts
of geometry— Euclidean and Galois, continuous
and discrete— within physical space— if not
within the space of physics.
This post was suggested by Paradigms Lost
(a post cited here a year ago today),
by David Weinberger's recent essay "Shift Happens,"
and by today's opening of "The Raven."
David Weinberger in The Chronicle of Higher Education , April 22—
"… Kuhn was trying to understand how Aristotle could be such a brilliant natural scientist except when it came to understanding motion. Aristotle's idea that stones fall and fire rises because they're trying to get to their natural places seems like a simpleton's animism.
Then it became clear to Kuhn all at once. Ever since Newton, we in the West have thought movement changes an object's position in neutral space but does not change the object itself. For Aristotle, a change in position was a change in a quality of the object, and qualitative change tended toward an asymmetric actualization of potential: an acorn becomes an oak, but an oak never becomes an acorn. Motion likewise expressed a tendency for things to actualize their essence by moving to their proper place. With that, 'another initially strange part of Aristotelian doctrine begins to fall into place,' Kuhn wrote in The Road Since Structure ."
Dr. John Raven (of Raven's Progressive Matrices)—
"… these tools cannot be immediately applied within our current workplaces, educational systems, and public management systems because the operation of these systems is determined, not by personal developmental or societal needs, but by a range of latent, rarely discussed, and hard to influence sociological forces.
But this is not a cry of despair: It points to another topic which has been widely neglected by psychologists: It tells us that human behaviour is not mainly determined by internal properties— such as talents, attitudes, and values— but by external social forces. Such a transformation in psychological thinking and theorising is as great as the transformation Newton introduced into physics by noting that the movement of inanimate objects is not determined by internal, 'animistic,' properties of the objects but by invisible external forces which act upon them— invisible forces that can nevertheless be mapped, measured, and harnessed to do useful work for humankind.
So this brings us to our fourth conceptualisation and measurement topic: How are these social forces to be conceptualised, mapped, measured, and harnessed in a manner analogous to the way in which Newton made it possible to harness the destructive forces of the wind and the waves to enable sailing boats to get to their destinations?"
Before Newton, boats never arrived?
"These lowbrow, popular cultural references bring the possibilities
of interpretation down to an everyday level, forcing us to acknowledge
that not every painting that looks like a splatter is necessarily a
homage/anti-homage to Jackson Pollock."
— Stina Högqvist, review of the art of Josefine Lyche
The 3×3×3 Galois Cube
Backstory— The Talented, from April 26 last year,
and Atlas Shrugged, from April 27 last year.
(Mythopoetic continued)
Voice of America today—
Thousands of Norwegians Defy Confessed Killer Breivik in Song
"The demonstrators waved roses and flags
Thursday as they and Norwegian folk singer
Lillebjoern Nilsen sang an adaptation
of the children's song, 'My Rainbow Race,'
which Breivik in court last week called
an example of Marxist brainwashing."
[See also PETE SEEGER AND LILLEBJØRN NILSEN.
Click on the image below for Seeger's original version.]
Liberia Reacts to Taylor Conviction With Mixed Emotions
"As the verdict was read out, a rainbow was seen
in the sky, encircling the sun. For many Liberians,
superstition is a part of life. The rainbow heralded
a new era, they said, beginning with the verdict of Taylor."
["You're not the only one… with mixed emotions."]
Two narratives in memory of my seventh-grade
history teacher, who died on January 27, 2012—
1. On the history of Liberia
(subject of a paper I wrote in seventh grade), and
2. Comic-book history (from the above date)
A meditation suggested by the April 20 post Complex Reflection
and by the life and April 20 death of a scientist who worked
at Los Alamos (home of the Monte Carlo method) and at
the Santa Fe Institute (home of complexity theory).
A search for 286 in this journal yields "Yet Another Cartoon Graveyard."
That June 1, 2008, post linked to poem 286 in a 1919 anthology.
Here is that poem, together with poem 823.
Together, these poems may be regarded as a meditation on
Simone Weil and her brother André Weil or,
more abstractly, on Love and Death.
Happy birthday to Al Pacino.
(Not Olympus )
(Continued from Mythopoetic, a post of April 12)
This post was suggested by a 2010 film about fictional Olympians,
by today's New York Times obituaries, and by a bar brawl at the Olympian
New York Athletic Club in "the wee hours of April 13."
Rick Riordan in the image below advertises another large
"Demigod Gathering" on October 12, 2010.
The Riordan image is from a post at ComposersCave.com
made on October 3, 2010.
Applying Jung's principle of synchronicity to this demigod material,
we find the October 3, 2010, Log24 post Search for the Basic Picture
and the October 12, 2010, Log24 post King Solomon's Mind.
Note that the latter October date is that of the traditional
Columbus Day, and that the 2010 film of The Lightning Thief
was directed by Chris Columbus of 1492 Pictures.
The film's release was earlier in 2010, on February 12.
See also —
"Pardon me. J'adoube."
— The Consul as he fastens his fly in Under the Volcano ,
the Garden of Eden scene
Suggested by a piece in the
Melbourne Herald Sun dated August 1, 2007—
A link from Log24 on that date—
Show Business
according to Fritz Leiber:
“Sid thinks you’re ready for
some of the smaller parts.”
In memory of Mike Wallace—
See also Knight Moves.
"… this notion of ‘depth’ is an elusive one
even for a mathematician who can recognize it…."
— G. H. Hardy, A Mathematician's Apology
In Geometry and the Imagination , Hilbert and Cohn-Vossen
describe the Brianchon-Pascal configuration of 9 points
and 9 lines, with 3 points on each line and 3 lines through
each point, as being "the most important configuration of all geometry."
The Brianchon-Pascal configuration is also known as the Pappus configuration—
Some background …
"The Theorem of Pappus: A Bridge Between Algebra and Geometry"
Elena Anne Marchisotto
The American Mathematical Monthly
Vol. 109, No. 6 (Jun. – Jul., 2002), pp. 497-516
Click image for some context from MIT.
Background from The New York Times— "Can You Make Yourself Smarter?"
See also "Plan 9" in this journal.
"Every multiplayer game, both free-for-all and team based,
will begin with a Mexican standoff." —Wikipedia
"The Cardinal seemed a little preoccupied today."
In "Contact," Dr. Arroway is shown the key to the Primer—
In this journal, fictional symbologist Robert Langdon is shown a cube—
"Confusion is nothing new." — Song lyric
"So the sundering we sense between nature and culture
lies not like a canyon outside us, but splits our being
at its most intimate depths the way mind breaks off from body.
It is still another version of that bitter bifurcation
long ago decreed— our expulsion from Eden…."
— William H. Gass in Finding a Form ,
Cornell U. Press paperback, 1997, page 138
See also…
For another bitter bifurcation, see La Despedida .
— Title of a 1975 film. The late Martin Poll (previous post)
was the executive producer.
See also "A Corpse Will Be Transported by Express."
Related material—
From Deadline Hollywood—
A film producer's death "between Friday night and early Saturday morning,"
April 13-14, 2012—
| R.I.P. Martin Poll
By THE DEADLINE TEAM | Sunday April 15, 2012 @ 7:36 pm PDT Veteran movie and TV producer Martin Poll died between Friday night and early Saturday morning of natural causes at a care facility on the Upper Westside in New York City. He was 89. |
See also the post linked to on the afternoon of Friday the 13th of April—
"All the saints have powers." — Cardinal Marchisano.
Happy birthday, James McAvoy (at left below in X-Men: First Class ).
(Continued from October First, 2010)
Today's Birthdays:
Britain's Queen Elizabeth II is 86.
Actress-comedian-writer Elaine May is 80.
Actor Charles Grodin is 77.
— AP, Today in History, April 21, 2012
Found in a Google search today—
(Click image for videos.)
I hadn't known until today that youtube.com/blogs/m759 exists.
The Arduino link under "More Channels" suggests a site
that may interest some future Arroway or Hadden.
Yesterday's post in memory of Octavio Paz—
|
… the free-standing, two-sided “Life-Death Figure,” |
An earlier post yesterday, Fashion Notes, linked to a Sting video—
|
From "Loo Ree," by Zenna Henderson "It's so hard to explain–" "Oh, foof!" I cried defiantly, taking off my glasses and, smearing the tears across both lenses with a tattered Kleenex. "So I'm a dope, a moron! If I can explain protective coloration to my six-year-olds and the interdependence of man and animals, you can tell me something of what the score is!" I scrubbed the back of my hand across my blurry eyes. "If you have to, start out 'Once upon a time."' I sat down– hard. Loo Ree smiled and sat down, too. "Don't cry, teacher. Teachers aren't supposed to have tears." "I know it," I sniffed. "A little less than human-that's us." "A little more than human, sometimes." Loo Ree corrected gently. "Well then, you must understand that I'll have to simplify. You will have to dress the bare bones of the explanation according to your capabilities. "Once upon a time there was a classroom. Oh, cosmic in size, but so like yours that you would smile in recognition if you could see it all. And somewhere in the classroom something was wrong. Not the whispering and murmuring– that's usual. Not the pinching and poking and tattling that goes on until you get so you don't even hear it." I nodded. How well I knew. "It wasn't even the sudden blow across the aisle or the unexpected wrestling match in the back of the room. That happens often, too. But something else was wrong. It was an undercurrent, a stealthy, sly sort of thing that has to be caught early or it disrupts the whole classroom and tarnishes the children with a darkness that will never quite rub off. "The teacher could feel it –as all good teachers can– and she spoke to the principal. He, being a good principal, immediately saw the urgency of the matter and also saw that it was beyond him, so he called in an Expert." "You?" I asked, feeling quite bright because I had followed the analogy so far. Loo Ree smiled. "Well, I'm part of the Expert." |
"If you have to, start out 'Once upon a time.'"
Yesterday's Paz post was at 6:48 PM EDT.
For the autistic, here is some related mathematics.
Yesterday's Fashion Notes post was at 1:06 PM EDT.
A related chronological note from Rolling Stone yesterday—
"Levon Helm, singer and drummer for the Band,
died on April 19th in New York of throat cancer.
He was 71.
"He passed away peacefully at 1:30 this afternoon…."
Helm and The Band performing "The Weight"—
"I pulled into Nazareth, I was a-feelin' 'bout half past dead…"
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