A polytope is the bounded intersection of a nite set of halfspaces of R d. Every polytope can also be represented as the convex hull conv V of its vertices (or extreme points) V. The convex hull problem is to convert from the vertex representation to the halfspace representation or (equivalently by geometric duality) vice-versa. Given an ordering v 1 : : : v n of the input vertices, after some ...

Let C1, ..., Cd+1 be d+ 1 point sets in R, each containing the origin in its convex hull. A subset C of ⋃d+1i=1 Ciis called a colorful choice (or rainbow) for C1, . . . ,Cd+1, if it contains exactly one point from each set Ci. Thecolorful Carathéodory theorem states that there always exists a colorful choice for C1, . . . , Cd+1 that has theorigin in its convex hull. Thi...

Using a nonlinear integral characterization of orthogonal polynomials in the complex plane, we provide a simple method for deducing a weak form of relative asymptotics exterior to the convex hull of the common support of the generating measures. The simplicity of the approach makes it a natural precursor for the presentation of Szegő theory. Dedicated to Guillermo López Lagomasino on the occasi...

The Choquet integral w.r.t. a capacity is versatile tool commonly used in decision making. Its practical identification requires, however, to solve an optimization problem with exponentially many variables and constraints. introduction of k-additive capacities, through the use Möbius transform, permits reduce number polynomial size, but leaves constraints exponential. When k=2, vertices set 2-a...

A Class of new methods based on a septic non-polynomial spline function for the numerical solution one-dimensional Bratu's problem are presented. The local truncation errors and the methods of order 2th, 4th, 6th, 8th, 10th, and 12th, are obtained. The inverse of some band matrixes are obtained which are required in proving the convergence analysis of the presented method. Associated boundary f...

The main purpose of this paper is to find fuzzy root of fuzzy polynomials (if exists) by using Newton-Raphson method. The proposed numerical method has capability to solve fuzzy polynomials as well as algebric ones. For this purpose, by using parametric form of fuzzy coefficients of fuzzy polynomial and Newton-Rphson method we can find its fuzzy roots. Finally, we illustrate our approach by nu...

A polynomial ideal I ⊆ R[x] is THk-exact if every linear polynomial that is non-negative over VR(I), the real variety of I , is a sum of squares of polynomials of degree at most k modulo I . Lovász recognized that a graph is perfect if and only if the vanishing ideal of the characteristic vectors of its stable sets is TH1-exact, and asked for a characterization of ideals which are TH1-exact. We...

This paper is concerned with the generation of tight equivalent representations for mixedinteger zero-one programming problems. For the linear case, we propose a technique which first converts the problem into a nonlinear, polynomial mixed-integer zero-one problem by multiplying the constraints with some suitable d-degree polynomial factors involving the n binary variables, for any given d E (0...