Related papers: On Erd\"os--Szekeres problem and related problems

We explore a question related to the celebrated Erd\H{o}s-Szekeres Theorem and develop a geometric approach to answer it. Our main object of study is the Erd\H{o}s-Szekeres tableau, or EST, of a number sequence. An EST is the sequence of…

We survey classical and recent results on exponents of Diophantine approximation. We give only a few proofs and highlight several open problems.

The results of the Sudbury experiments have seemingly solved the solar neutrino problem. However, we must remain open to the possibility that these results may be subjected to new theoretical interpretations in the future. Should such a…

Recently we have obtained two simple proofs of Sharkovsky's theorem, one with directed graphs [7] and the other without [8]. In this note, we present yet more simple proofs of Sharkovsky's theorem.

This is an appendix to our paper "An update of the Hirsch Conjecture" (arXiv:0907.1186), containing proofs of some of the results and comments that were omitted in it.

This paper investigates several classical and novel variations of the Erd\H{o}s--Szekeres problem, including multicolored point sets, convex hexagons with a given number of interior points, and polygons with constraints on edge colors. We…

We give a survey on recent developments on nonlinear Schr\"odinger equations with dissipative structure based on the authors' recent works.

The Simes inequality has received considerable attention recently because of its close connection to some important multiple hypothesis testing procedures. We revisit in this article an old result on this inequality to clarify and…

In the paper based on the question of Zhang and L\"{u}[15], we present one theorem which will improve and extend the results of Banerjee-Majumder [2] and a recent result of Li-Huang [9].

We prove existence and uniqueness of solutions for an entropic version of the semi-geostrophic equations. We also establish convergence to a weak solution of the semi-geostrophic equations as the entropic parameter vanishes. Convergence is…

This is a very brief report on recent developments on the Dirichlet problem for the minimal surface system and minimal cones in Euclidean spaces. We shall mainly focus on two directions: (1) Further systematic developments after…

We give a new proof of Tietze Theorem on the convergence of infinite semi-regular continued fractions.

In this note we give a detailed proof of certain results on geometry of numbers in the $S$-adic case. These results are well-known to experts, so the aim here is to provide a convenient reference for the people who need to use them.

We give a short proof that essentially all questions concerning singularities of Richardson varieties reduce to corresponding questions about Schubert varieties. Consequently, we quickly deduce some new and previously known results.

The aim of this short note is to give counterexamples to two results by D. Y. Gao [5, Th. 16], [4, Th. 2] and to improve a related result by S.-C. Fang, D. Y. Gao, R.-L. Sheu and S.-Y. Wu [1, Th. 3].

In this paper we give a complete proof of the Brumer-Stark conjecture over $\mathbf{Z}$.

We prove the existence of smooth solutions to the Gross-Pitaevskii equation on $\mathbf{R}^3$ that feature arbitrarily complex quantum vortex reconnections. We can track the evolution of the vortices during the whole process. This permits…

Recently we have reanalyzed the consistency of the solutions of the space fractional Schr\"odinger equation found in a piecewise manner, and showed that an exact and a proper treatment of the relevant integrals prove that they are…

By some new recursive algorithms, in this paper, we will give some improvements on Waring's problem.

We construct explicit equations of Cartwright-Steger and related surfaces.