相关论文: Some Questions around The Hilbert 16th Problem

Thanks to the interest of many people, a mistake has been found in our way of counting limit cycles. We are working on a new version.

Starting from a Pfaffian equation in dimension $N$ and focusing on compact solutions for it, we place in perspective the variational method used in [29] to solve Hilbert's 16th problem. In addition to exploring how this viewpoint can help…

The main purpose of the present article is to give some new Hilbert's sum type inequalities, which in special cases yield the classical Hilbert's inequalities. Our results provide some new estimates to these types of inequalities.

In this paper, we will continue the investigation of Waring's problem, and give further improvements.

In this note we briefly survey and propose some open problems related to isoparametric theory.

Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into either a problem involving a set of infinitely coupled differential equations or a problem involving a Shr\"odinger propagator…

Considering some parameters and by means of an inequality of Hadamard, we derive general half-discrete Hilbert-type inequalities. Then we highlight some special cases.

In this paper, we will solve the Reifenberg Plateau Problem in Hilbert space.

Fractionally-quadratic transformations which reduce any two-dimensional quadratic system to the special Lienard equation are introduced. Existence criteria of cycles are obtained.

Some symmetry problems are formulated and solved. New simple proofs are given for the earlier studied symmetry problems.

A complete characterization of near subnormality for bilateral weighted shifts is obtained. As an application of the main results, many new answers to the Hilbert space problem 160 are presented at the end of the paper.

I discuss some recent work linking certain aspects of the second part of Hilbert's 16th problem to the theory of \hbox{o-minimality}. These notes are adapted from a lecture I gave in the Jour fixe seminar series at the Zukunfts\-kolleg of…

We proposed an algorithm that covers some cases of Hamilton Circuit Problem.

Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into problems involving either a set of infinitely-coupled non-linear differential equations or a class of linear Schr\"odinger…

In this survey article we revisit Hilbert's $19^{\text{th}}$ problem concerning the regularity of minimizers of variational integrals. We first discuss the classical theory (that is, the statement and resolution of Hilbert's problem in all…

Some Open Problems Concerning Orthogonal Polynomials.

We give new examples of weak Hilbert spaces.

We study a family of Riemannian problems on the Heisenberg group that tends to the sub-Riemannian problem on this group.

In this paper we propose a conjecture concerning partial sums of an arbitrary finite subset of an abelian group, that naturally arises investigating simple Heffter systems. Then, we show its connection with related open problems and we…

The thesis concerns Hilbert schemes of points and apart from mathematical results, contains small open problems and history sections, see the introduction for more details. The thesis has not been edited since 2017, see first page for more…