Related papers: String-Orthogonal Polynomials, String equations, a…
A class of two-dimensional field theories with exponential interactions is introduced. The interaction depends on two ``coupling'' matrices and is sufficiently general to include all Toda field theories existing in the literature. Lie point…
Closely following recent ideas of J. Borcea, we discuss various modifications and relaxations of Sendov's conjecture about the location of critical points of a polynomial with complex coefficients. The resulting open problems are formulated…
In a previous paper we have introduced the ortho-homological triangles, which are triangles that are orthological and homological simultaneously. In this article we call attention to two remarkable ortho-homological triangles (the given…
We begin a search for nonsupersymmetric/supersymmetric dual string pairs by constructing candidate critical nonsupersymmetric strings as solitons in supersymmetric string theories. Using orbifold techniques, one can construct cosmic string…
We consider thorny spheres, that is 2-dimensional compact surfaces which are everywhere locally isometric to a round sphere $S^2$ except for a finite number of isolated points where they have conical singularities. We use thorny spheres to…
A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…
We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an…
We show how one can get solitonic strings in a six-dimensional (2,0) supersymmetric theory by incorporating a nonlinear interaction term. We derive a zero force condition between parallel strings, and compute a metric on a moduli space…
I discuss several aspects of strings as unified theories. After recalling the difficulties of the simplest supersymmetric grand unification schemes I emphasize the distinct features of string unification. An important role in constraining…
In this paper, we extend our investigation into semiclassical multiple discrete orthogonal polynomials by considering an arbitrary number of weights. We derive multiple versions of the Toda equations and the Laguerre-Freud equations for the…
Classically, a single weight on an interval of the real line leads to moments, orthogonal polynomials and tridiagonal matrices. Appropriately deforming this weight with times t=(t_1,t_2,...), leads to the standard Toda lattice and…
Chapters : 1. Introduction to electric-magnetic duality 2. Classical duality in bosonic brane electrodynamics 3. Massless spin two gauge theory 4. Duality-symmetric actions and chiral forms 5. BRST quantization of duality-symmetric…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss--Borel factorization of two, left and a right, Cantero-Morales-Velazquez block moment matrices, which are…
A class of cross-shaped difference operators on a two dimensional lattice is introduced. The main feature of the operators in this class is that their formal eigenvectors consist of multiple orthogonal polynomials. In other words, this…
The noncommutative analogues of the nonisospectral Toda and Lotka-Volterra lattices are proposed and studied by performing nonisopectral deformations on the matrix orthogonal polynomials and matrix symmetric orthogonal polynomials without…
This is an expository article which describes one approach to the construction and classification of harmonic tori "of finite type", namely, via their ring of polynomial Killing fields. To keep the discussion focussed, the first section is…
By using a generalization of Sturm-Liouville problems in $q$-difference spaces, a class of symmetric $q$-orthogonal polynomials with four free parameters is introduced. The standard properties of these polynomials, such as a second order…
We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice $\mathbb{Z}^2$. We first prove a formula on the rotation number of a…
We address the novel structures arising in quantum and string integrable theories, as well as construct methods to obtain them and provide further analysis. Specifically, we implement the automorphic symmetries on periodic lattice systems…