Related papers: On the Jones Polynomial
To the best of our knowledge this paper is the first attempt to introduce and study polynomial interpolation of the polynomial data given on arbitrary varieties. In the first part of the paper we present results on the solvability of such…
The aim of this paper is to represent any polynomial in terms of the degenerate Frobenius-Euler polynomials and more generally of the higher-order degenerate Frobenius-Euler polynomials. We derive explicit formulas with the help of umbral…
We formulate several polynomial identities. One side of these identities has a nice simple form. Whereas the other has a form of a polynomial whose coefficients contain binomial coefficients double factorials or (and) rising factorials. The…
In this paper we consider the notions of binomial thinning, binomial mixing, their generalizations, certain interplay between them, associated limit theorems and provide various examples.
Brief Description: The book provides a unique highly self-contained text introducing the reader to the classical and modern theory of polyanalytic functions and their generalizations. This is a subbranch of complex analysis of several…
In a series of papers published in this Journal (J. Math. Phys.), a discussion was started on the significance of a new definition of projective representations in quaternionic Hilbert spaces. The present paper gives what we believe is a…
It is a well known result from Thistlethwaite that the Jones polynomial of a non-split alternating link is alternating. We find the right generalization of this result to the case of non-split alternating tangles. More specifically: the…
In this paper we revisit the work of E.T. Bell concerning partition polynomials in order to introduce the reciprocal partition polynomials. We give their explicit formulas and apply the result to compute closed formulae for some well-known…
We establish a characterization of adequate knots in terms of the degree of their colored Jones polynomial. We show that, assuming the Strong Slope conjecture, our characterization can be reformulated in terms of "Jones slopes" of knots and…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
The work in this paper is to initiate a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by certain economically compact structure has a multilinear monomial in its…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
A polynomial triangle is an array whose inputs are the coefficients in integral powers of a polynomial. Although polynomial coefficients have appeared in several works, there is no systematic treatise on this topic. In this paper we plan to…
This monograph is a study of the category of polynomial endofunctors on the category of sets and its applications to modeling interaction protocols and dynamical systems. We assume basic categorical background and build the categorical…
In the preprint we present an outline of the one dimensional version of topological Galois theory. The theory studies topological obstruction to solvability of equations "in finite terms" (i.e. to their solvabilty by radicals, by elementary…
In this paper we introduce and discuss some classes of orthogonal polynomials in several non-commuting variables. The emphasis is on a non-commutative version of the orthogonal polynomials on the real line. We introduce recurrence equations…
This paper gives a generalization of the AJL algorithm and unitary braid group representation for quantum computation of the Jones polynomial to continuous ranges of values on the unit circle of the Jones parameter. We show that our…
This is an expository survey of the Jacobian problem for the class of Pluriharmonic functions.
Techniques for the evaluation of complex polynomials with one and two variables are introduced. Polynomials arise in may areas such as control systems, image and signal processing, coding theory, electrical networks, etc., and their…
Comment on ``Gibbs Sampling, Exponential Families, and Orthogonal Polynomials'' [arXiv:0808.3852]