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We solve direct and inverse problems for two-dimensional (quasi) canonical systems related to exponential polynomials of a specific but sufficiently general type. The approach to the inverse problem in this paper provides an interpretation…
In this contribution, discrete semiclassical orthogonal polynomials of class $s\leq2$ are studied. By considering all possible solutions of the Pearson equation, we obtain the canonical families in each class. We also consider limit…
We present a method for nonlinear parametric optimization based on algebraic geometry. The problem to be studied, which arises in optimal control, is to minimize a polynomial function with parameters subject to semialgebraic constraints.…
In this paper we considerably extend the class of known $\alpha$-minimizing hypercones using sub-calibration methods. Indeed, the improvement of previous results follows from a careful analysis of special cubic and quartic polynomials.
We present integral representations of solutions to division problems involving matrices of polynomials in several complex variables. We also find estimates of the polynomial degree of the solutions by means of careful degree estimates of…
The main objective of this paper is to introduce an algorithm for solving fractional and classical differential equations based on a new generalized fractional power series. The algorithm relies on expanding the solution of an FDE or an ODE…
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…
Let $f, f_1, \ldots, f_\nV$ be polynomials with rational coefficients in the indeterminates $\bfX=X_1, \ldots, X_n$ of maximum degree $D$ and $V$ be the set of common complex solutions of $\F=(f_1,\ldots, f_\nV)$. We give an algorithm…
There exist several methods for computing exact solutions of algebraic differential equations. Most of the methods, however, do not ensure existence and uniqueness of the solutions and might fail after several steps, or are restricted to…
A class of bivariate infinite series solutions of the elliptic and hyperbolic Kepler equations is described, adding to the handful of 1-D series that have been found throughout the centuries. This result is based on an iterative procedure…
Quantum canonical transformations are used to derive the integral representations and Kummer solutions of the confluent hypergeometric and hypergeometric equations. Integral representations of the solutions of the non-periodic three body…
Gradient-based (a.k.a. `first order') optimization algorithms are routinely used to solve large scale non-convex problems. Yet, it is generally hard to predict their effectiveness. In order to gain insight into this question, we revisit the…
This paper presents a canonical dual method for solving a quadratic discrete value selection problem subjected to inequality constraints. The problem is first transformed into a problem with quadratic objective and 0-1 integer variables.…
This paper mainly addresses the extrema of a nonconvex functional with double-well potential in higher dimensions through the approach of nonlinear partial differential equations. Based on the canonical duality method, the corresponding…
This paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of Moment-SOS relaxations to solve it globally. Under some assumptions, we prove…
In this paper, we give an algorithm for finding general rational solutions of a given first-order ODE with parametric coefficients that occur rationally. We present an analysis, complete modulo Hilbert's irreducibility problem, of the…
We consider the problem of efficiently solving a system of $n$ non-linear equations in ${\mathbb R}^d$. Addressing Smale's 17th problem stated in 1998, we consider a setting whereby the $n$ equations are random homogeneous polynomials of…
Nonlinear second-order ordinary differential equations are common in various fields of science, such as physics, mechanics and biology. Here we provide a new family of integrable second-order ordinary differential equations by considering…
We generalize the differential dimension polynomial from prime differential ideals to characterizable differential ideals. Its computation is algorithmic, its degree and leading coefficient remain differential birational invariants, and it…
Multivariate polynomial optimization is a prevalent model for a number of engineering problems. From a mathematical viewpoint, polynomial optimization is challenging because it is non-convex. The Lasserre's theory, based on semidefinite…