Linked Questions

460
votes
2answers
40k views

Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?

Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}$ such that $f\colon\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?
60
votes
3answers
4k views

Are there infinitely many integer-valued polynomials dominated by $1.9^n$ on all of $\mathbb{N}$?

The original post is below. Question 1 was solved in the negative by David Speyer, and the title has now been changed to reflect Question 2, which turned out to be the more difficult one. A bounty of ...
64
votes
2answers
5k views

Barrelled, bornological, ultrabornological, semi-reflexive, … how are these used?

I'm not a functional analyst (though I like to pretend that I am from time to time) but I use it and I think it's a great subject. But whenever I read about locally convex topological vector spaces, ...
19
votes
6answers
2k views

Consistency strength needed for applied mathematics

Given that we can never proof the consistency of a theory for the foundations of mathematics in a weaker system, one could seriously doubt whether any of the commonly used foundational frameworks (ZFC ...
31
votes
2answers
7k views

The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis? Please see this related post and also the following post.. For Mathematical ...
10
votes
5answers
3k views

Applications of functional analysis beyond analysis(towards algebra, geometry, number theory…) [closed]

So far, We have seen the applications of functional analysis in PDE, probability and many areas in applied mathematics. On the other hand, methods of algebraic topology are introduced to functional ...
12
votes
2answers
1k views

Questions about categorification (& combinatorial simplification of the Russian approach to Lusztig's conjectures, in zero & positive characteristic) [closed]

This is a question about the proofs of Kazhdan-Lusztig's conjectures for category $\mathcal{O}$ using higher representation theory (avoiding Beilinson-Bernstein's geometric localization theory). ...
18
votes
3answers
4k views

Number of unique determinants for an NxN (0,1)-matrix

I'm interested in bounds for the number of unique determinants of NxN (0,1)-matrices. Obviously some of these matrices will be singular and therefore will trivially have zero determinant. While it ...
15
votes
1answer
2k views

Values of zeta at odd positive integers and Borel's computations

Someone recently quoted to me this recent article that claims to prove that $\zeta(2n+1) \notin (2\pi )^{2n+1} \mathbb{Q}$. I always assumed this was well known. More precisely I thought this result ...
10
votes
2answers
972 views

Elliptic operators corresponds to non vanishing vector fields

Added, June 19, 2019: The main motivation of this post is to associate an index to differential operator associated to a dynamical system such that the index has an interesting ...
24
votes
2answers
1k views

Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map $D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the differential operator ...
10
votes
3answers
2k views

Limit cycles as closed geodesics(in negatively or positively curved space)

EDIT: Here is a related post which concern quadratic vector fields rather than Van der pol equation. In this linked post we see that the convexity of limit cycle play a crucial role. On ...
7
votes
0answers
3k views

Is the conjecture A+B=C following correct?

Is the conjecture on A+B=C following correct ? Conjecture: Let $A, B, C$ be three positive integer numbers such that $A+B=C$ with $\gcd(A, B, C) = 1$. By Fundamental theorem of arithmetic we write: ...
6
votes
1answer
547 views

Hilbert 16th problem via hyperbolic geometry

More than 16 years ago, I heard from someone that he thinks that there is a possible relation between Hilbert's 16th problem(for $n=2$) and Hyperbolic geometry. He says that a ...
6
votes
2answers
1k views

The adjoint operators as elliptic operators

Edit: It seems that the link "https://cms.math.ca/Events/Toulouse2004/abs/ss7.html#lt" which contains a talk by Loic Teyssier about homological equations and vanishing cycles is temporally ...

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