On the Shapes of Surfaces and the Solutions to DEs

[Guest]

Remark: Of course the function T maps or projects the multidimensional vector X (point X) to [tex]k \in \mathbb{Z}[/tex] , a flat surface.

[Guest]

Remark: The Hilbert metric may help us to determine the solvability of T(X) = k (a general Diophantine equation).

[Guest]

There may be solutions galore, [tex]X \in \mathbb{R^{n}}[/tex], for the Diophantine equation, T(X) = k. But there could also be no solutions, [tex]X \in \mathbb{Z^{n}}[/tex], such that T(X) = k for n > 3.

Go figure!

Exhaustive Search:

We assume T(X) = k and [tex]X \in \mathbb{R^{n}}[/tex] such [tex]X \notin \mathbb{Z^{n}}[/tex] for n > 3.

Let [tex]X + \triangle X \ne X[/tex] such that T([tex]X + \triangle X[/tex]) = k where [tex]\triangle X \in \mathbb{R^{n}}[/tex].

Does the sum, [tex]X + \triangle X \in \mathbb{Z^{n}}[/tex]?

It's time to do some computing. Good Luck!

While there are some or all [tex]x_{i } \in X[/tex] that are not integers, we try to adjust [tex]\triangle x_{i } \in \triangle X \in \mathbb{R}[/tex] so that [tex]x_{i } + \triangle x_{i } \in \mathbb{Z}[/tex].

And me must solve T([tex]X + \triangle X[/tex]) = k.

We repeat the process until [tex]X + \triangle X \in \mathbb{Z^{n}}[/tex].

Does the process determines a solution or no solution?

[Guest]

Remark: ... While there are some or all [tex]x_{i } \in X[/tex] that are not integers, we try to adjust [tex]\triangle x_{i } \in \triangle X \in \mathbb{R^{n}}[/tex] so that [tex]x_{i } + \triangle x_{i } \in \mathbb{Z}[/tex].

[Guest]

Remark: Of course the function T maps or projects the multidimensional vector X (point X) to [tex]k \in \mathbb{Z}[/tex] , a flat surface.

Oops! Flat surface, k, is wrong!


Consider the equation, [tex]x^{3 } + y^{3 } + z^{3 } = k[/tex].

It's graph is a spherical surface of radius, k.

[Guest]

...Radius [tex]k^{ \frac{1}{3} }[/tex] ...

[Guest]

Remark: Of course the function T maps or projects the multidimensional vector X (point X) to [tex]k \in \mathbb{Z}[/tex] , a flat surface.

Oops! Flat surface, k, is generally wrong!


Consider the equation, [tex]x^{2} + y^{2 } + z^{2 } = k[/tex].

It's graph is a spherical surface with radius, [tex]k^{ \frac{1}{3} }[/tex].

https://www.wolframalpha.com/input/?i=plot+x%5E2+%2B++y%5E2%2B++z%5E2+%3D+7%5E2

[Guest]

Oops! Radius is [tex]\sqrt{k}[/tex].

[Guest]

Remark: We apologize for our mistakes...

In addition to our possible use of the Hilbert metric, we must discover how the integral constant, k, determines the minimum and maximum integral values for all variables in T(X) = k.

We are hopeful that Hilbert's tenth problem has a positive solution, but is difficult work.

[Guest]

Remark: We believe the solution (positive or negative) to Hilbert's tenth problem requires some knowledge of number theory, algebra, geometry, analysis, decision or complexity theory, and algorithms.

It is difficult work. But we can solve the problem. And we will solve the problem!

Good luck!

[Guest]

Use Randomness:

Suppose we want to solve a very complex/difficult, DE,
T(X) = k.

Initially we select a random integral vector, [tex]X_{0}[/tex], and we have T([tex]X_{0 }[/tex]) = [tex]k_{0 }[/tex] with the mostly case, [tex]k_{0 } \ne k[/tex].

We become more selective so that the our latest guess, ([tex]X_{m}[/tex], [tex]k_{m }[/tex]) is less random and more close to the unknown solution, X.

We foresee two possible (good) cases before we begin our final exhaustive search to solve T(X)= k:

1. [tex]X_{below}[/tex] with [tex].9k < k_{below } < .99k[/tex] ;

2. [tex]X_{above}[/tex], with [tex]1.01k < k_{above } < 1.1k[/tex].

Remark: If we can achieve those outcomes, 1 and 2, then we should be very confident we can solve our problem, positively (integral solution, X) or negatively (no integral solution, X).

[Guest]

... Initially we select a random integral vector, [tex]X_{0}[/tex], and we have T([tex]X_{0 }[/tex]) = [tex]k_{0 }[/tex] with the most likely case, [tex]k_{0 } \ne k[/tex]...

[Guest]

Remark: Hmm. Theory like technology is great when it works in practice (application).

Those two good/very optimistic outcomes, 1 and 2, in a previous post would go a long way to establish the very important halting criterion for our solution search.

[Guest]

Important Remark: The quasi randomness method/idea is not foolproof! Why?

We must also use other ideas/methods in conjunction with it.

[Guest]

Remark: Our main concern is to solve a special case of T(X) = k, Hilbert's tenth problem.

[Guest]

Remark: Hmm. Theory like technology is great when it works in practice (application).

Those two good/very optimistic outcomes, 1 and 2, in a previous post would go a long way to establish the very important halting criterion for our solution search.

We may expect a convergence to an integral solution, but we will accept a non convergent oscillating result (no integral solution). And we will not overlook the crucial and generally abundant non pure integral solutions. They are quite helpful in our computations and reasoning...

Hmm. There could be a deep and undiscovered theorem on the solvability of general DEs. Who knows?

Good luck!

[Guest]

A Personal Remark: We, humans, live in an increasingly political/competitive/dangerous world with too much misinformation/disinformation/secrets/censorship/DOS...

But the open search for truth/understanding must remain a vital priority for all.

Yes, I am a flawed human, and I am my harshest critic. And I strive to be the first to admit my mistakes/ignorance and to correct them once they are detected. And I am also critical of others (their work) too.

I want to be fair and honest. And I generally have good intentions with a strong faith in Lord GOD. Amen!

Dave.

[Guest]

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.

[Guest]

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.

[Guest]

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 :

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?