Solution Of Mx Research Articles

STP method for solving the minimal norm Toeplitz solutions of MX-~XN=GY+R

STP method for solving the minimal norm Toeplitz solutions of MX-~XN=GY+R

A proof of Green's conjecture regarding the removal properties of sets of linear equations

A system of ℓ linear equations in p unknowns Mx = b is said to have the removal property if every set S ⊆ {1, …, n} that contains o(np−ℓ) solutions of Mx = b can be turned into a set S′ containing no solution of Mx = b by the removal of o(n) elements. Green (Geom. Funct. Anal. 15 (2005) 340–376) proved that a single homogenous linear equation always has the removal property, and conjectured that every set of homogenous linear equations has the removal property. In this paper we confirm Green's conjecture by showing that every set of linear equations (even non-homogenous) has the removal property. We also discuss some applications of our result in theoretical computer science and, in particular, use it to resolve a conjecture of Bhattacharyya, Chen, Sudan and Xie related to algorithms for testing properties of boolean functions.

Thermodynamics of 1:1 electrolyte mixtures in aquo-1,4-dioxane solvent system: molar excess enthalpy

Molar excess enthalpy data at 308.15 K for 1 M binary solutions of MX + NY (or NX), MY + NX (or NY), M(or N)X + M(or N)Y (where M is Na +, N is K +, X is Cl −, Y is Br −) electrolytes in 1,4-dioxane (D) + water ( x D= 0.0142, dielectric constant 70 at 308.15 K) solvent (S) system have been determined using an LKB flow microcalorimeter. The data, which differ appreciably from the corresponding data in water and which depend, unlike those in water, on the nature of the common cation and common anion, do not support Young's cross square rule; this failure has been attributed to the dependence of the hydration of the ions of the MX + NX or MX + NY, etc., 1:1 electrolyte mixtures in S on the nature of (i) the common cation and common anion and (ii) the mixture of anions and of cations. The H E data of the present systems have been attributed to both the coulombic interactions and the asymmetric effects, and this has been substantiated by an analysis of the data in terms of Friedman's cluster theory. The importance of the virial theory of Pitzer in the determination of the relative magnitudes of the coulombic and asymmetric interactions on H E data, is also highlighted.