Related papers: Arithmetic properties of Cantor sets involving non…

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…

We prove the following statement. Let $f\in\mathbb{R}[x_1,\ldots,x_d]$, for some $d\ge 3$, and assume that $f$ depends non-trivially in each of $x_1,\ldots,x_d$. Then one of the following holds. (i) For every finite sets…

We study experimentally systems of orthogonal polynomials with respect to self-similar measures. When the support of the measure is a Cantor set, we observe some interesting properties of the polynomials, both on the Cantor set and in the…

A polynomial representation of a convex d-polytope P is a finite set \{p_1(x),...,p_n(x)\} of polynomials over E^d such that P=\setcond{x \in \E^d}{p_1(x) \ge 0 {for every} 1 \le i \le n}. By s(d,P) we denote the least possible number of…

We show that every (possibly unbounded) convex polygon $P$ in $R^2$ with $m$ edges can be represented by inequalities $p_1 \ge 0,...,p_n \ge 0,$ where the $p_i$'s are products of at most $k$ affine functions each vanishing on an edge of $P$…

We consider the problem of classifying all positive-definite integer-valued quadratic forms that represent all positive odd integers. Kaplansky considered this problem for ternary forms, giving a list of 23 candidates, and proving that 19…

Integrals involving derivatives of Legendre polynomials frequently arise in applications ranging from multipole expansions for processes involving electromagnetic probes to spectral methods in numerical physics. Despite their practical…

In math/0702157, arXiv:0712.4185, we investigated monic multivariate non-commutative orthogonal polynomials, their recursions, states of orthogonality, and corresponding continued fraction expansions. In this note, we collect a number of…

We lift the constraint of a diagonal representation of the Hamiltonian by searching for square integrable bases that support an infinite tridiagonal matrix representation of the wave operator. The class of solutions obtained as such…

The Lax pair representation in Fourier space is used to obtain a list of integrable scalar evolutionary equations with quadratic nonlinearity. The famous systems of this type such as KdV, intermediate long-wave equation (ILW), Camassa-Holm…

We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers $x \in (0,1]$ with the following property is comeager: for all integers $b\ge 2$ and $k\ge 1$, the sequence of…

A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form $x$, $x+d$, $x+d^2$. We obtain a multidimensional version of this result, which can be regarded as a first step towards…

We consider finite dimensional representations of the dihedral group $D_{2p}$ over an algebraically closed field of characteristic two where $p$ is an odd integer and study the degrees of generating and separating polynomials in the…

Zeckendorf's Theorem says that for all $k \geq 3$, every nonnegative integer has a unique $k$-Zeckendorf representation as a sum of distinct $k$-bonacci numbers, where no $k$ consecutive $k$-bonacci numbers are present in the…

It is shown that rational extensions of the isotropic Dunkl oscillator in the plane can be obtained by adding some terms either to the radial equation or to the angular one obtained in the polar coordinates approach. In the former case, the…

We investigate the average distribution of primes represented by positive definite integral binary quadratic forms, the average being taken over negative fundamental discriminants in long ranges. In particular, we prove corresponding…

For the family $P:=x^n+a_1x^{n-1}+\cdots +a_n$ of complex polynomials in the variable $x$ we study its {\em discriminant} $R:=$Res$(P,P',x)$, $R\in \mathbb{C}[a]$, $a=(a_1,\ldots ,a_n)$. When $R$ is regarded as a polynomial in $a_k$, one…

We study the ring generated over a field of characteristic 0 by noncommuting indeterminates {x_1,x_2,...,x_n} subject only to the relations x_i\sigma_k=\sigma_k x_i, for i,k=1,2,...,n, and their consequences, where \sigma_k…

Let k>1 be an integer and let p be a prime. We show that if $p^a\le k<2p^a$ or $k=p^aq+1$ (with 2q<p) for some a=1,2,..., then the set {\binom{n}{k}: n=0,1,2,...} is dense in the ring Z_p of p-adic integers, i.e., it contains a complete…

A number of the form $x(x+1)/2$ where $x$ is an integer is called a triangular number. Suppose, $N(a_1,\cdots,a_k;n)$ and $T(a_1,\cdots,a_k;n)$ denote the number of ways $n$ can be expressed as $\sum_{i=1}^k a_ix_i^2$ and $\sum_{i=1}^k…