Related papers: Solutions to some problems on unique representatio…
I discuss computational challenges in the relativistic few-nucleon problem and the resolution of some of these challenges. I also discuss the outlook for the future.
In this article, we represent an even Gaussian integer with sufficiently large norm as a sum of a Gaussian prime and a Gaussian integer with at most two Gaussian prime factors akin to Chen in the rational case.
We present some variations on some of the main open problems on character degrees. We collect some of the methods that have proven to be very useful to work on these problems. These methods are also useful to solve certain problems on zeros…
Representation theory is shown to be incomplete in terms of enumerating all integrable limits of quantum systems. As a consequence, one can find exactly solvable Hamiltonians which have apparently strongly broken symmetry. The number of…
We prove uniform bounds for the Petersson norm of the cuspidal part of the theta series. This gives an improved asymptotic formula for the number of representations by a quadratic form. As an application, we show that every integer $n \neq…
We consider the question whether one can construct an embedded resolution of singularities of a singular variety $X\subset \textbf{A}^n$ from the data of the irreducible components of the spaces of jets (of $X$) centered at the singular…
There exists "a square problem": in a unit square is there a point with four rational distances to the vertices? This problem is still regarded as unproved. Yang showed proofs for several special cases of the square problem. By the…
The problem of construction of irreducible representations of quantum $A^q_n$ algebras is solved at the level of explicit integration of the linear (inhomogeneous) system in finite differences in the n-dimensional space. The general…
A new computational idea for continuum quantum field theories is outlined. This approach is based on the lattice source Galerkin methods developed by Garcia, Guralnik and Lawson. The method has many promising features including treating…
Chen and Chv\'atal introduced the notion of lines in hypergraphs; they proved that every 3-uniform hypergraph with $n$ vertices either has a line that consists of all $n$ vertices or else has at least $\log_2 n$ distinct lines. We improve…
A new approach to the theory of polynomial solutions of q - difference equations is proposed. The approach is based on the representation theory of simple Lie algebras and their q - deformations and is presented here for U_q(sl(n)). First a…
The polynomial Pell equation is \[P^2 - D Q^2 = 1\] where $D$ is a given integer polynomial and the solutions $P, Q$ must be integer polynomials. A classical paper of Nathanson \cite{Nat} solved it when $D(x) = x^2 + d$. We show that the…
We derive several explicit formulae for finding infinitely many solutions of the equation $AXA=XAX$, when $A$ is singular. We start by splitting the equation into a couple of linear matrix equations and then show how the projectors…
A set $A$ of nonnegative integers is an asymptotic basis of order $h$ if every sufficiently large integer can be represented as the sum of $h$ integers (not necessarily distinct) of $A$. An asymptotic basis $A$ of order $h$ is minimal if no…
We study several variations of line segment covering problem with axis-parallel unit squares in $I\!\!R^2$. A set $S$ of $n$ line segments is given. The objective is to find the minimum number of axis-parallel unit squares which cover at…
This paper presents two approaches to reducing problems on $2$-cycles on a smooth cubic hypersurface $X$ over an algebraically closed field of characteristic $\neq 2$, to problems on $1$-cycles on its variety of lines $F(X)$. The first one…
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…
Many real life problems can be reduced to the solution of a complex exponentials approximation problem which is usually ill posed. Recently a new transform for solving this problem, formulated as a specific moments problem in the plane, has…
A Grobner basis-based algorithm for solving the Frobenius Instance Problem is presented, and this leads to an algorithm for solving the Frobenius Problem that can handle numbers with thousands of digits. Connections to irreducible…
The embedded Nash problem for a hypersurface in a smooth algebraic variety, is to characterize geometrically the maximal irreducible families of arcs with fixed order of contact along the hypersurface. We show that divisors on minimal…