Related papers: Pendulum: separatrix splitting

A form in a polynomial ring over a field is said to be homaloidal if its polar map is a Cremona map, i.e., if the rational map defined by the partial derivatives of the form has an inverse rational map. The object of this work is the search…

We prove a sharp upper bound on the number of distinct columns of a totally unimodular matrix with column sums $1$ improving upon Heller's classical bound. The proof uses Seymour's decomposition theorem. Such matrices are closely related to…

We obtain an asymptotic formula for the number of integer $2\times 2$ matrices that have determinant $\Delta$ and whose absolute values of the entries are at most $H$. The result holds uniformly for a large range of $\Delta$ with respect to…

There is a description of the torsion product of two modules in terms of generators and relations given by Eilenberg and Mac Lane. With some additional data on the chain complexes there is a splitting of the map in the Kunneth formula in…

Given a sequence of real rooted polynomials $\{p_n\}_{n\geq 1}$ with a fixed asymptotic root distribution, we study the asymptotic root distribution of the repeated polar derivatives of this sequence. This limiting distribution can be seen…

We give a formula for the determinant of an $n\times n$ matrix with entries from a commutative ring with unit. The formula can be evaluated by a "straight-line program" performing only additions, subtractions and multiplications of ring…

It is shown that a trivial version of polarization is sufficient to produce separating systems of polynomial invariants: if two points in the direct sum of the $G$--modules $W$ and $m$ copies of $V$ can be separated by polynomial…

We give a new proof for the well-known Blaschke--Petkantschin formula which is based on the polar decomposition of rectangular matrices and may be of interest in random matrix theory.

Generalizations of oscillator and Coulomb models are discussed via introduction of holomorphic coordinates. Complex Euclidean analogue of the Smorodinsky-Winternitz system is introduced and studied. Complex projective analogue of…

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do…

We introduce and extend the outer product and contractive product of tensors and matrices, and present some identities in terms of these products. We offer tensor expressions of derivatives of tensors, focus on the tensor forms of…

We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of…

In the first five sections, we deal with the class of probability measures with asymptotically periodic Verblunsky coefficients of p-type bounded variation. The goal is to investigate the perturbation of the Verblunsky coefficients when we…

This paper presents the first in a series of results that allow us to develop a theory providing finer control over the complexity of normalisation, and in particular of cut elimination. By considering atoms as self-dual non-commutative…

Let $\{a_k\}$ be a sequence of real numbers defined by an $m$th order linear homogenous recurrence relation. In this paper we obtain a determinant formula for the circulant matrix $A=circ(a_1, a_2, \cdots, a_n)$, providing a generalization…

The `classical' model for a massive spinning particle, which was recently proposed, is derived from the isotropic rotator model. Through this derivation, we note that the spin can be understood as the relativistic extension of the isotropic…

The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such…

In this paper we generalize and improve a method for calculating the period of a classical oscillator and other integrals of physical interest, which was recently developed by some of the authors. We derive analytical expressions that prove…

We show that the Politzer theorem on the equations of motion implies approximate constraints on the quark correlator, restricting considerably, for sufficiently large Q^2, the number of independent distribution functions that characterize…

Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $n\le N$ and integer parts of polynomials evaluated at $n$. The error…