Abstract. We consider the class of semi-stable solutions to semilinear equations −∆u = f(u) in a bounded smooth domain Ω of R (with Ω convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions n ≤ 4, we establish an priori L∞ bound which holds for every semi-stable solution and every nonlinearity f . This estimate leads to the boundedness o...

A real polynomial system with support W ⊂ Zn is called extremal if all its complex solutions are positive solutions. A support W having n + 2 elements is called a circuit. We previously showed that the number of nondegenerate positive solutions of a system supported on a circuit W ⊂ Zn is at most m(W)+1, where m(W) ≤ n is the degeneracy index of W. We prove that if a circuit W ⊂ Zn supports an ...

 In this paper, some basic results concerning strict, nonstrict inequalities, local existence theorem and differential inequalities  have been proved for an IVP of first order hybrid  random differential equations with the linear perturbation of second type. A comparison theorem is proved and  applied to prove the uniqueness of random solution for the considered perturbed random differential eq...

We show that a class of extremal four-dimensional supersymmetric " highbranes " , i.e. string and domain wall solutions, can be interpreted as intersections of four ten-dimensional Dirichlet branes. These d = 4 solutions are related, via T-duality in ten dimensions, to the four-dimensional extremal Maxwell/scalar black holes that are characterized by a scalar coupling parameter a with a = 0, 1/...

We show that a class of extremal four-dimensional supersymmetric " highbranes " , i.e. string and domain wall solutions, can be interpreted as intersections of four ten-dimensional Dirichlet branes. These d = 4 solutions are related, via T-duality in ten dimensions, to the four-dimensional extremal Maxwell/scalar black holes that are characterized by a scalar coupling parameter a with a = 0, 1/...

We study the existence and computation of extremal solutions of a system of inequations defined over lattices. Using the Knaster-Tarski fixed point theorem, we obtain sufficient conditions for the existence of supremal as well as infimal solution of a given system of inequations. Iterative techniques are presented for the computation of the extremal solutions whenever they exist, and conditions...

Extremal Combinatorics is an area in Discrete Mathematics that has developed spectacularly during the last decades. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers in Extremal Graph Theory, Extremal Finite Set Theory and Combinatorial Geometry. This is not meant to be a comprehensive...