The algebraic operations of addition, intersection, elimination, and quotient are fundamental to computational algebraic geometry. This article describes how to perform these operations on homogeneous ideals using Macaulay dual spaces. If F is a polynomial system with finitely many solutions, these operations are used to compute the homogenization of the ideal generated by F which, in particula...

We present in this work a generalization of the solution of Gorenstein and Yang for a consistent thermodynamics for systems with a temperature dependent Hamiltonian. We show that there is a large class of solutions, work out three particular ones, and discuss their physical relevance. We apply the particular solutions for an ideal gas of quasi-gluons, and compare the calculation to lattice and ...

This paper deals with problems of impulse control which allow control inputs consisting not only of delta functions but also of their higher derivatives (impulses of higher order). The controls are sought for in the form of feedback strategies which leads to the application of respective generalized dynamic programming techniques, where the role of traditional Hamilton–Jacobi–Bellman equations ...

Inspired by a question of Lovász, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lovász’s theta body of the graph. We prove that theta bodies are, up to closure, a version of Lasserre’s rela...

We obtain an explicit representation for quasiradial γ-harmonic functions, which shows that these functions have essentially algebraic nature. In particular, we give a complete description of all γ which admit algebraic quasiradial solutions. Unlike the cases γ = ∞ and γ = 1, only finitely many algebraic solutions is shown to exist for any fixed |γ| > 1. Moreover, there is a special extremal se...

Hydration numbers of calcium ions are determined from extensive measurements of colligative properties of water solutions of calcium salts. The hydration numbers reported refer to the average number of water molecules that are bound sufficiently strongly to calcium ions so as to be removed from the solvent and become part of the solute. Contrary to common descriptions of deviations from ideal b...

Inspired by a question of Lovász, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lovász’s theta body of the graph. We prove that theta bodies are, up to closure, a version of Lasserre’s rela...

The purpose of this note is to record Hensel’s Lemma and a few of its immediate applications for my Math 831 class. At the end of the 19th century Hensel introduced the notion of completion into a number theoretic context in order to bring to bear techniques of analysis on some purely algebraic problems in number theory. One of his most crucial discoveries (in modern terms) was that when a ring...

let $s$ be an ordered semigroup. a fuzzy subset of $s$ is anarbitrary mapping   from $s$ into $[0,1]$, where $[0,1]$ is theusual interval of real numbers. in this paper,  the concept of fuzzygeneralized bi-ideals of an ordered semigroup $s$ is introduced.regular ordered semigroups are characterized by means of fuzzy leftideals, fuzzy right ideals and fuzzy (generalized) bi-ideals.finally, two m...