Related papers: Problems concerning Diophantine exponents of latti…
The goal of this survey is to discuss the Quantitative non-Divergence estimate on the space of lattices and present a selection of its applications. The topics covered include extremal manifolds, Khintchine-Groshev type theorems, rational…
Some topics from recent progresses in lattice QCD are reviewed.
We consider some lattices and look at discrete Laplacians on these lattices. In particular we look at solutions of the equation $\triangle(1)\phi = \triangle(2)Z$ where $\triangle(1)$ and $\triangle(2)$ are two such laplacians on the same…
We investigate matrix problems arising in the study of Majorana Fermions on a lattice.
This paper is a sequel to our previous paper arXiv:1105.1554, where we defined two types of intermediate Diophantine exponents, connected them to Schmidt exponents and split Dyson's transference inequality into a chain of inequalities for…
This paper investigates the exponential Diophantine equation of the form $a^x+b=c^y$, where $a, b, c$ are given positive integers with $a,c \ge 2$, and $x,y$ are positive integer unknowns. We define this form as a "Type-I transcendental…
In this paper, we study some properties of Whitney numbers of Dowling lattices and related polynomials. We answer the following question: there is relation between Stirling and Eulerian polynomials. Can we find a new relation between…
I review the current status of lattice QCD results. I concentrate on new analytical developments and on numerical results relevant to phenomenology.
In this article we give a survey on open problems and conjectures concerning L^2-invariants. We cover the whole portfolio and not only certain aspects as they are considered in the previous more specialized (and within their scope more…
In this paper we consider a multiparametric version of Wolfgang Schmidt and Leonard Summerer's parametric geometry of numbers. We apply this approach in two settings: the first one concerns weighted Diophantine approximation, the second one…
This is a survey article describing some recent results at the interface of homogeneous dynamics and Diophantine approximation.
This paper is a survey of recent results on comparison of various fractional Laplacians prepared for Proceedings of ICM2022.
This paper investigates a generalized version of Diophantine tuples in finite fields. Applying Shparlinski's method, we obtain power-saving results on the number of such tuples.
We consider the number of visible lattice points under the assumption of the Extended Lindel\"{o}f Hypothesis. We get a relation between visible lattice points and the Extended Lindel\"{o}f Hypothesis. And we also get a relation between…
This note is designed to show some classes of differential-difference equations admitting Lax representation which generalize evolutionary equations known in the literature.
Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is…
Consider the equation $q_1\alpha^{x_1}+\dots+q_k\alpha^{x_k} = q$, with constants $\alpha \in \overline{\mathbb{Q}} \setminus \{0,1\}$, $q_1,\ldots,q_k,q\in\overline{\mathbb{Q}}$ and unknowns $x_1,\ldots,x_k$, referred to in this paper as…
Two-, three- and four-dimensional representations of Penrose tilings of the plane are described. The vertices that occur in these representations lie on lattices. Symmetries and methods of visualizing these representations are discussed.…
We study some Diophantine problems related to triangles with two given integral sides. We solve two problems posed by Zolt\'an Bertalan and we also provide some generalization.
We introduce a concept of multiplicity lattices of 2-multiarrangements, determine the combinatorics and geometry of that lattice, and give a criterion and method to construct a basis for derivation modules effectively.