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The purpose of this study is twofold. First, we revisit a shape optimization reformulation of a prototypical shape inverse problem and briefly propose a simple yet efficient numerical approach for solving the corresponding minimization…
This paper addresses problems on the structural design of control systems taking explicitly into consideration the possible application to large-scale systems. We provide an efficient and unified framework to solve the following major…
We study the closure of the projection of the (nonconvex) cone of rank restricted positive semidefinite matrices onto subsets of the matrix entries. This defines the feasible sets for semidefinite completion problems with restrictions on…
In this paper we prove a strong version of the Hilbert Nullstellensatz in the ring $\mathbb H[q_1,\ldots,q_n]$ of slice regular polynomials in several quaternionic variables. Our proof deeply depends on a detailed analysis of the common…
This paper concerns the \textbf{abstract geometry of numbers}: namely the pursuit of certain aspects of geometry of numbers over a suitable class of normed domains. (The standard geometry of numbers is then viewed as geometry of numbers…
We deal with equations over free semilattice of infinite rank and prove that any infinite consistent system of equations is equivalent to its finite subsystem. Moreover, we describe irreducible algebraic sets and solve some algorithmic…
In this paper we study the existence and regularity of stable manifolds associated to fixed points of parabolic type in the differentiable and analytic cases, using the parametrization method. The parametrization method relies on a suitable…
We study point-line configurations through the lens of projective geometry and matroid theory. Our focus is on their realisation spaces, where we introduce the concepts of liftable and quasi-liftable configurations, exploring cases in which…
The $P_1$--nonconforming quadrilateral finite element space with periodic boundary condition is investigated. The dimension and basis for the space are characterized with the concept of minimally essential discrete boundary conditions. We…
Rosenbrock's theorem on polynomial system matrices is a classical result in linear systems theory that relates the Smith-McMillan form of a rational matrix $G$ with the Smith forms of an irreducible polynomial system matrix $P$ giving rise…
This paper is devoted to the construction of polynomial 2-surfaces which possess a polynomial area element. In particular we study these surfaces in the Euclidean space $\mathbb R^3$ (where they are equivalent to the PN surfaces) and in the…
This paper investigates the absolute values on $\mathbb{Z}$ valued in the upper reals (i.e. reals for which only a right Dedekind section is given). These necessarily include multiplicative seminorms corresponding to the finite prime fields…
Let $K$ be a global field and let $Z$ be a geometrically irreducible algebraic variety defined over $K$. We show that if a big set $S\subseteq Z$ of rational points of bounded height occupies few residue classes modulo $\mathfrak{p}$ for…
We study several variations of line segment covering problem with axis-parallel unit squares in $I\!\!R^2$. A set $S$ of $n$ line segments is given. The objective is to find the minimum number of axis-parallel unit squares which cover at…
Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in C^N (N greater than or equal to 2), of codimension one or more in C^N, and endowed with the induced CR structure. Assuming that the tangential…
In this paper we prove a new zero-free region for the partition function of the hard-core model, that is, the independence polynomials of graphs with largest degree $\Delta$. This new domain contains the half disk $$D=\left\{ \lambda \in…
Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically,…
We are concerned with the dependence of the lowest positive eigenvalue of the Dirac operator on the geometry of rectangles, subject to infinite-mass boundary conditions. We conjecture that the square is a global minimiser both under the…
We study the convex relaxation of a polynomial optimization problem, maximizing a product of linear forms over the complex sphere. We show that this convex program is also a relaxation of the permanent of Hermitian positive semidefinite…
We expand the class of linear symmetric equations for which large sets with no non-trivial solutions are known. Our idea is based on first finding a small set with no solutions and then enlarging it to arbitrary size using a…