The Sylvester-Gallai theorem asserts that every finite set S of points in two-dimensional Euclidean space includes two points, a and b, such that either there is no other point in S is on the line ab, or the line ab contains all the points in S. V. Chvátal extended the notion of lines to arbitrary metric spaces and made a conjecture that generalizes the Sylvester-Gallai theorem. In the present ...

We bring together several new results related to the classical Sylvester-Gallai Theorem and its recently noted sharp dual. In 1951 Dirac and Motzkin conjectured that a configuration of n not all collinear points must admit at least n/2 ordinary connecting lines. There are two known counterexamples, when n = 7 and n = 13. We provide a construction that yields both counterexamples and show that t...

Exponential time integrators are well-established discretization methods for semilinear systems of ordinary differential equations. These use ? ? functions, which matrix functions related to the exponential. This work introduces an algorithm speed up computation function action over vectors two-dimensional (2D) matrices expressed as a Kronecker sum. For that, we present auxiliary exponential-re...

An interpolation approach is pursued to solve a bilateral matrix polynomial equation frequently arising in control and signal processing. It results in eecient and numerically reliable resolution methods. Abstract New numerical procedures are proposed to solve the symmetric matrix polynomial equation A T (?s)X(s) + X T (?s)A(s) = 2B(s) that is frequently encountered in control and signal proces...

We give a new structure theorem for subresultants precising their gap structure and derive from it a new algorithm for computing them. If d is a bound on the degrees and τ a bound on the bitsize of the minors extracted from Sylvester matrix, our algorithm has O(d2) arithmetic operations and size of intermediate computations 2τ . The key idea is to precise the relations between the successive Sy...

Sylvester equation is often applied to various fields, such as mathematics and control systems due its importance. Zeroing neural network (ZNN), a systematic design method for time-variant problems, has been proved be effective on solving in the ideal conditions. In this paper, order realize predefined-time convergence of ZNN model modify robustness, two new noise-tolerant ZNNs (NNTZNNs) are es...

We will consider the inverse conductivity problem or Calderón problem [11] in three dimensions (and higher). The fundamental paper is the work of Sylvester and Uhlmann [23] which gives global uniqueness in the inverse conductivity problem in three dimensions. In this problem we want to find a coefficient in an elliptic differential operator The first step will be to recover the coefficient at t...

Reverse analyses of three proofs of the Sylvester-Gallai theorem lead to three different and incompatible axiom systems. In particular, we show that proofs respecting the purity of the method, using only notions considered to be part of the statement of the theorem to be proved, are not always the simplest, as they may require axioms which proofs using extraneous predicates do not rely upon.

The global FOM and GMRES algorithms are among the effective methods to solve Sylvester matrix equations. In this paper, we study these algorithms in the case that the coefficient matrices are real symmetric (real symmetric positive definite) and extract two CG-type algorithms for solving generalized Sylvester matrix equations. The proposed methods are iterative projection methods onto matrix Kr...