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Geometric modeling of multivariate reliability polynomials is based on algebraic hypersurfaces, constant level sets, rulings etc. The solved basic problems are: (i) find the reliability polynomial using the Maple and Matlab software…
Certain problems in quadratic minimization can be reduced to finding the point $x$ of a polyhedron ${ P}$ that minimizes the distance $\|x-p\|$ for some $p\notin { P}$. This amounts to a search for the appropriate face $F$ of ${ P}$ for…
Based on the partition of parameter space, two algorithms for computing the rational univariate representation of zero-dimensional ideals with parameters are presented in the paper. Unlike the rational univariate representation of…
One dimensional metrical geometry may be developed in either an affine or projective setting over a general field using only algebraic ideas and quadratic forms. Some basic results of universal geometry are already present in this…
In this paper, we present a new approach to linearizing zero-one quadratic minimization problem which has many applications in computer science and communications. Our algorithm is based on the observation that the quadratic term of…
We consider a class of optimization problems that involve determining the maximum value that a function in a particular class can attain subject to a collection of difference constraints. We show that a particular linear programming…
The problem of finding the distance from a given $n \times n$ matrix polynomial of degree $k$ to the set of matrix polynomials having the elementary divisor $(\lambda-\lambda_0)^j, \, j \geqslant r,$ for a fixed scalar $\lambda_0$ and $2…
We study the rational approximation properties of special manifolds defined by a set of polynomials with rational coefficients. Mostly we will assume the case of all polynomials to depend on only one variable. In this case the manifold can…
This work concerns the distance in 2-norm from a matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Perturbations are allowed only on the constant coefficient…
We consider the number of configurations of a surface in two dimensions that has a prescribed length and encloses a prescribed perimeter with respect to a baseline. An approximate analytical treatment in a semi--continuum compares…
We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes,…
The paper is focused on the four-dimensional visualization of hypersurfaces represented by implicit equations without their parametrization. We describe a general method to find shadow boundaries in an arbitrary dimension and apply it in a…
Fix two integers $1\leq d<e$. We study the birational geometry of a parameter space for pairs of homogeneous polynomials of degrees $d$ and $e$ in two variables (in which the higher degree polynomial is well defined only up to a multiple of…
We propose an algorithm based on Newton's method and subdivision for finding all zeros of a polynomial system in a bounded region of the plane. This algorithm can be used to find the intersections between a line and a surface, which has…
The main goal of the paper is the discussion of a deeper interaction between matrix theory over polynomial rings over a field and typical methods of commutative algebra and related algebraic geometry. This is intended in the sense of…
We investigate the manifold $\cal{M}$ of (real) quadratic forms in n > 1 variables having a multiple eigenvalue. In addition to known facts, we prove that 1) $\cal{M}$ is irreducible, 2) in the case of n = 3, scalar matrices and only them…
We introduce a new technique for solving uni-parametric versions of linear programs, convex quadratic programs, and linear complementarity problems in which a single parameter is permitted to be present in any of the input data. We…
A comprehensive analysis of the morphology of the solution space for a special type of quadratic quaternion equation is presented. This equation, which arises in a surface construction problem, incorporates linear terms in a quaternion…
Consider the problem of minimizing a quadratic objective subject to quadratic equations. We study the semialgebraic region of objective functions for which this problem is solved by its semidefinite relaxation. For the Euclidean distance…
Choose a polynomial in three variables with not more than three or four monomials of moderate degree. Take simple coefficients as 1 and -1. Then draw a picture of the solution variety in real three space using a ray-tracing program like…