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On the Shapes of Surfaces and the Solutions to DEs

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Expand view Topic review: On the Shapes of Surfaces and the Solutions to DEs

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Thu Mar 16, 2023 6:57 am

FYI: Prof. Mircea Mustaţă

Go Blue! :D

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Wed Mar 15, 2023 10:21 am

FYI: RATIONAL LINEAR SUBSPACES OF HYPERSURFACES OVER FINITE FIELDS.

Enjoy! :)

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Sat Jan 28, 2023 7:33 pm

FYI: Prof. Bhargav Bhatt;

Some Published Papers by Prof. Bhargav Bhatt.

Enjoy! :)

Go Blue!

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Tue Nov 29, 2022 4:53 pm

FYI: The Number of Curves of Genus Two with Elliptic Differentials;

Helpful Article: Mathematical theorem used to crack US government encryption algorithm.

Enjoy! :)

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Mon Nov 21, 2022 6:59 pm

FYI: He, June Huh, Dropped Out to Become a Poet. Now He’s Won a Fields Medal. Go Blue! :D

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Mon Nov 21, 2022 6:33 pm

FYI: The Work of Prof. June Huh Go Blue! :D

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Thu Sep 08, 2022 5:28 am

Guest wrote:Final Remark: We have decided to write and publish a manuscript on our solution to David Hilbert's Tenth Problem. We hope to complete our work by January 6, 2023.

Manuscript Reference Link: https://theory-of-energy.org/2021/11/22/david-hilberts-tenth-problem-and-the-art-and-science-of-problem-solving/.

Dave.

Go Blue! :D

P.S. The initial work will be made public...


Oops! I cannot complete the difficult work, but I expect the definitive work will be completed or has been completed by the top experts in algebraic geometry/number theory, etc. Good luck! :)

Dave.

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Mon Nov 22, 2021 4:52 am

Final Remark: We have decided to write and publish a manuscript on our solution to David Hilbert's Tenth Problem. We hope to complete our work by January 6, 2023.

Manuscript Reference Link: https://theory-of-energy.org/2021/11/22/david-hilberts-tenth-problem-and-the-art-and-science-of-problem-solving/.

Dave.

Go Blue! :D

P.S. The initial work will be made public...
Attachments
hilbert's tenth problem.jpg
Hilbert's Tenth Problem is a very beautiful and important problem! Amen! :-)
hilbert's tenth problem.jpg (138.44 KiB) Viewed 395 times

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Sun Nov 21, 2021 4:49 pm

Remark: We believe we have sufficient math resources (excellent published papers on DEs or related topics, ideas, methods, etc.) to solve Hilbert's Tenth Problem either positively or negatively soon (a year or less). And we hope the problem has a positive answer (a DE algorithm).

Good Luck! :)

Go Blue! :D

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Sat Nov 06, 2021 3:28 pm

FYI: 'DIOPHANTINE PROBLEMS IN MANY VARIABLES: THE ROLE OF ADDITIVE NUMBER THEORY' by Prof. T. D. Wooley,

https://www.math.purdue.edu/~twooley/publ/1999%20dpv.pdf.

Enjoy! :)

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Wed Nov 03, 2021 9:26 pm

Question: Does this stuff (previous posts) makes sense?

We hope so! Please share your comments, corrections, solutions, etc.

Thank you!

Go Blue! :)
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making sense....jpg
making sense....jpg (15.45 KiB) Viewed 458 times

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Thu Oct 28, 2021 7:32 pm

Guest wrote:Remark: On the shape of space as defined by T, we may want to consider the 'generic' hypercube initially since we must establish limits or bounds for all possible solutions of T on a multidimensional surface...

https://en.m.wikipedia.org/wiki/Hypercube


FYI: 'Counting Rational Points on Algebraic Varieties' by Prof. D.R. Heath-Brown,

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.155.5182&rep=rep1&type=pdf.

Enjoy! :)

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Wed Oct 27, 2021 10:25 pm

Remark: On the shape of space as defined by T, we may want to consider the 'generic' hypercube initially since we must establish limits or bounds for all possible solutions of T on a multidimensional surface...

https://en.m.wikipedia.org/wiki/Hypercube

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Wed Oct 27, 2021 6:07 pm

We let [tex]\Omega[/tex] represent the set of all integral solutions for our Diophantine equation, and we assume card([tex]\Omega[/tex]) [tex]\ge 1[/tex].

We know X is an integral solution, but we want to find more solutions. How do we proceed?

Solve T([tex]\varphi_{1}x_{1 }[/tex], [tex]\varphi_{2}x_{2 }[/tex], [tex]\varphi_{3}x_{3 }[/tex], ..., [tex]\varphi_{n}x_{n }[/tex]) = k

for some [tex]\varphi_{i} \in \mathbb{Q}[/tex]...

Remark: This is a difficult problem. And we are clueless!

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Wed Oct 27, 2021 5:47 pm

[tex]\varphi_{i} \in \mathbb{Q}[/tex]

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Wed Oct 27, 2021 5:43 pm

Remark: Oops! Zero is a rational number.

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Wed Oct 27, 2021 5:27 pm

Minor Update: [tex]\varphi_{i} \in \mathbb{Q} \cup[/tex] {0}

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Wed Oct 27, 2021 5:19 pm

Guest wrote:Question: Can geometry and algebra help us find other solutions (if they exist) to the Diophantine equation, T(X) = k?

Remark: We assume T(X) = k has at least one integral solution.


A Tentative Claim :idea: :

If there is more than one distinct integral solution, X, for our Diophantine equation, then we claim [tex]\varphi \bigotimes X \in \mathbb{Z^{n}}[/tex] and

T([tex]\varphi \bigotimes X[/tex]) = T([tex]\varphi_{1}x_{1 }[/tex], [tex]\varphi_{2}x_{2 }[/tex], [tex]\varphi_{3}x_{3 }[/tex], ..., [tex]\varphi_{n}x_{n }[/tex]) = k

for some [tex]\varphi_{i} \in \mathbb{Q} \cup {0}[/tex] such that [tex]\varphi_{i} \ne 1[/tex] for [tex]1 \le i \le n[/tex].

It's worth a try. Right? :|

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Wed Oct 27, 2021 12:54 am

Question: Can geometry and algebra help us find other solutions (if they exist) to the Diophantine equation, T(X) = k?

Remark: We assume T(X) = k has at least one integral solution.

Re: On the Shapes of Surfaces and the Solutions to DEs

Post by Guest » Sat Oct 23, 2021 2:45 pm

FYI: 'Hilbert metric',

https://en.m.wikipedia.org/wiki/Hilbert_metric;

'Birkhoff's version of Hilbert's metric and its applications in analysis',

https://www.researchgate.net/publication/236578464_Birkhoff's_version_of_Hilbert's_metric_and_its_applications_in_analysis.
Attachments
an ellipsoid with the Hilbert metric.png
an ellipsoid with the Hilbert metric.png (12.61 KiB) Viewed 345 times

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