Related papers: Multidimensional Realization Theory and Polynomial…
Using the notions and tools from realization in the sense of systems theory, we establish an explicit and new realization formula for families of infinite products of rational matrix-functions of a single complex variable. Our realizations…
We consider numerical certification of approximate solutions to a system of polynomial equations with more equations than unknowns by first certifying solutions to a square subsystem. We give several approaches that certifiably select which…
We consider the additive superimposition of an extensive number of independent Euclidean Random Matrices in the high-density regime. The resolvent is computed with techniques from free probability theory, as well as with the replica method…
We focus on two central themes in this dissertation. The first one is on decomposing polytopes and polynomials in ways that allow us to perform nonlinear optimization. We start off by explaining important results on decomposing a polytope…
Motivated by the theory of self-duality which provides a variational formulation and resolution for non self-adjoint partial differential equations \cite{G1, G2}, we propose new templates for solving large non-symmetric linear systems. The…
It is well known that zeros and poles of a single-input, single-output system in the transfer function form are the roots of the transfer function's numerator and the denominator polynomial, respectively. However, in the state-space form,…
Recently the authors presented a matrix representation approach to real Appell polynomials essentially determined by a nilpotent matrix with natural number entries. It allows to consider a set of real Appell polynomials as solution of a…
The monodromy group is an invariant for parameterized systems of polynomial equations that encodes structure of the solutions over the parameter space. Since the structure of real solutions over real parameter spaces are of interest in many…
A method of solving Maxwell equations in a vicinity of a multipole particle (moving along an arbitrary trajectory) is proposed. The method is based on a geometric construction of a trajectory-adapted coordinate system, which simplifies…
In this paper, we present a new algorithm for computing the linear recurrence relations of multi-dimensional sequences. Existing algorithms for computing these relations arise in computational algebra and include constructing structured…
By using the squared slack variables technique, we demonstrate that the solution set of a general polynomial complementarity problem is the image, under a specific projection, of the set of real zeroes of a system of polynomials. This paper…
Based on the matrix expression of general nonlinear numerical analogues presented by the present author, this paper proposes a novel philosophy of nonlinear computation and analysis. The nonlinear problems are considered an ill-posed linear…
This paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This construction uses a single linear differential form defined from the…
This article conducts a large dimensional study of a simple yet quite versatile classification model, encompassing at once multi-task and semi-supervised learning, and taking into account uncertain labeling. Using tools from random matrix…
In the context of data-driven control of nonlinear systems, many approaches lack of rigorous guarantees, call for nonconvex optimization, or require knowledge of a function basis containing the system dynamics. To tackle these drawbacks, we…
We investigate the conditions under which systems of two differential eigenvalue equations are quasi exactly solvable. These systems reveal a rich set of algebraic structures. Some of them are explicitely described. An exemple of quasi…
The graphical realization of a given degree sequence and given partition adjacency matrix simultaneously is a relevant problem in data driven modeling of networks. Here we formulate common generalizations of this problem and the Exact…
This paper studies a class of complex-valued linear systems whose state evolution dependents on both the state vector and its conjugate. The complex-valued linear system comes from linear dynamical quantum control theory and is also…
The quest for simplification in physics drives the exploration of concise mathematical representations for complex systems. This Dissertation focuses on the concept of dimensionality reduction as a means to obtain low-dimensional…
Let $f$ be a polynomial system consisting of $n$ polynomials $f_1,\cdots, f_n$ in $n$ variables $x_1,\cdots, x_n$, with coefficients in $\mathbb{Q}$ and let $\langle f\rangle$ be the ideal generated by $f$. Such a polynomial system, which…