Extract from
https://rjlipton.wordpress.com/2019/06/29/a-prime-breakthrough/
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What GORZ prove is this:
Theorem 1 For each {d} and almost all {n}, the polynomial J(n,
d)(x) is hyperbolic.
Of course, “almost all” means that there at most a finite number of
polynomials that are not hyperbolic.
>>
The quotation raises a number of questions, such as
What inspired its author to decorate an initial straightforward,
unambiguous quotation with the subsequent elementary howler in the
first place?
And how did this piffle manage to earn a link ("Gödel’s Lost Letter
and P=NP" in the left-hand column) from
https://terrytao.wordpress.com/2019/06/29/symmetric-functions-in-a-fraction…
in the second place?
In contrast TT's own contribution is informative and provocative,
even for a reader (like myself) who manages to grasp approximately 10%
of its contents. I'm a sucker for investigations that manage to
extend what are apparently inherently functions of natural numbers to
the complex domain!
Fred Lunnon
ITEM 4269604 (History of Science). Circa 1820, Sengai Gibon drew
a famous calligraphy often referred to as "Circle, Triangle, Square":
https://terebess.hu/zen/sengai.html
Compare with the following algebraic plane curves:
1 = X^2 + Y^2
1 = X^2 + Y^2 - sqrt(4/27)*X^3
1 = X^2 + Y^2 - (1/4)*X^4
1 = X^2 + Y^2 - sqrt(4/27)*(3*Y^2*X - X^3)
1 = X^2 + Y^2 - X^2*Y^2
Did Sengai forget to draw a monogon and a digon?
ITEM 4269605 (a minor gripe). The title "Circle, Triangle, Square"
is not canonical. The same drawing is sometimes referred to as
"The Universe". But how could one conceive of a universe without
"Hotaru"? Did poor Sengai forget the Kanji for "Hotaru" as well?
Or should we interpret Sengai's minimalism as a prognostication
of future bio-devastation ( i.e. of "the impermanence of all
conditioned phenomena" ) ?
ITEM 4269606 ( On Impatience ). First, learn how to expand
the following nomes:
https://oeis.org/draft/A308835https://oeis.org/draft/A308836https://oeis.org/draft/A308837
After: https://oeis.org/A005797
The same technique applies to either of the following differential
equations:
0 = (x-1)*(x-2)*T + d/dx( (x-3)*(x-4)*(x-5)*(x-6)*dT/dx)
0 = 15*(84 - 197*x + 78*x^2)*T
+ d/dx(4*(x-1)*(2*x-1)*(9*x-29) (9*x-2)*dT/dx)
One is poorly-motivated, might as well be random, thus
analysis of the nomes at any of the four singular points does
not lead to much enlightenment. In particular, it does not even
seem possible to integerize the expansion coefficients.
As opposed, the second differential equation comes from a
geometric precursor. It appears that the nome expansions
around each of four singular points are integral after a scale
transformation of the x variable:
2/9: 0, 1, 11742, 156203784, 2284549324448 ...
1/2: 0, 1, 304, 618436, 584357184 ...
1: 0, 1, 2152, 6629044, 21558540128 ...
29/9: 0, 1, 622056, 365686615476, 208699680038836576 ...
The first one hundred terms have been quickly checked.
PROBLEM: How to prove integrality? (not obvious to me).
PROBLEM: WTF (What's The Function)? (Hint: Try searching
through x-variate sphere curve geometries with local dihedral
symmetry)
PROBLEM: Once you've succeeded in finding something: Is it
possible to use the nome variables to define pseudo-elliptic
time parameterizations of the disjoint submanifolds?
ITEM 4269607 ( On patience ). After many seasons of study
in very dim light, a hotaru lantern takes definite form. Even
more work remains to be done.
PROBLEM: How do we construct useful objects in the material
world? If we need to make a new lantern, what type of paper
should we use?
Cheers,
Brad
