Tropical Limit in Statistical Physics

Tropical limit for macroscopic systems in equilibrium defined as the formal limit of Boltzmann constant k going to 0 is discussed. It is shown that such tropical limit is well-adapted to analyse properties of systems with highly degenerated energy levels, particularly of frustrated systems like spin ice and spin glasses. Tropical free energy is a piecewise linear function of temperature, tropical entropy is a piecewise constant function and the system has energy for which tropical Gibbs' probability has maximum. Properties of systems in the points of jump of entropy are studied. Systems with finite and infinitely many energy levels and phenomena of limiting temperatures are discussed.Comment: 16 pages, 6 figure

Minor Invertible Products Assignment and sparse hyperdeterminants

We consider an extension of Minor Assignment Problems derived from the determinantal expansion of matrix products, under the condition that the terms of the expansion are units of C(t). This restriction places constraints on the sparsity and the factorisation properties of a family of hyperdeterminants derived from Grassmann-Plucker relations. We find minimal conditions guaranteeing that allowed assignments returning a determinantal expansion are the trivial ones, i.e. those induced by the action of a diagonal matrix of monomials on a pair of constant matrices. Counterexamples are provided when such conditions do not hold.Comment: 15 pages, improved exposition, minor typos corrected. Comments are welcom

Bethe Ansatz techniques for quasilocal charges in quench problems of integrable models

Diamo un' introduzione sui due metodi fondamentali per risolvere sistemi integrabili in una dimensione: il Coordinate Bethe Ansatz e l'Algebraic Bethe Ansatz. Concentriamo particolarmente la nostra attenzione su due catene di Heisenberg ampiamente studiate: il modello XXX e il modello XXZ. Successivamente presentiamo una breve panoramica nell'ambito del quench (in inglese il verbo to quench significa spegnere), che ha portato negli ultimi dieci anni a grandi progressi nel campo della fisica del non-equilibrio. Le tecniche fondamentali usate per implementare l'idea del quench sono l' Ensemble di Gibbs Generalizzato e l'Azione Quench. Affrontiamo con più dettagli la tecnica dell' Ensemble di Gibbs Generalizzato, costruito attraverso l' insieme infinito di cariche conservate locali di un sistema integrabile. Recentemente la tecnica dell' Ensemble di Gibbs Generalizzato è stata migliorata, aggiungendo all'insieme di cariche conservate locali, le cosiddette cariche conservate quasilocali. Proponiamo infine un procedimento per costruire un insieme di cariche quasilocali per il modello XXZ nel regime critico, detto anche senza gap (salto), attraverso il cosiddetto Y-system, ovvero un sistema gerarchico di equazioni, che appare spesso nel contesto delle teorie integrabili

A reclaimer scheduling problem arising in coal stockyard management

We study a number of variants of an abstract scheduling problem inspired by the scheduling of reclaimers in the stockyard of a coal export terminal. We analyze the complexity of each of the variants, providing complexity proofs for some and polynomial algorithms for others. For one, especially interesting variant, we also develop a constant factor approximation algorithm.Comment: 26 page

Nursing Home Quality as a Public Good

There has been much debate among economists about whether nursing home quality is a public good across Medicaid and private-pay patients within a common facility. However, there has been only limited empirical work addressing this issue. Using a unique individual level panel of residents of nursing homes from seven states, we exploit both within-facility and within-patient variation in payer source and quality to examine this issue. We also test the robustness of these results across states with different Medicaid and private-pay rate differentials. Across our various identification strategies, the results generally support the idea that quality is a public good within nursing homes. That is, within a common nursing home, there is very little evidence to suggest that Medicaid-funded residents receive consistently lower quality care relative to their private-paying counterparts.

Sparse deformations of determinant expansions and extensionality of set functions

We introduce an algebraic criterion for the reduction of Z^d-valued maps of subsets to corresponding Z^d-valued maps of elements of a given set. The algebraic structure is given by the expansion of the determinant of two matrices, one of which is generic, where each term of the expansion is deformed through a monomial factor. It is proved that, in broad generality, the deformation is integrable, namely, it is induced by the action of a diagonal matrix on the rows of the generic matrix. This framework allows to extend previous results in the study of signed exponential sums and their applications in statistical physics (statistical amoebas of partition functions), tropical geometry, and integrable systems (soliton solutions of the Kadomtsev-Petviashvili II equation). Furthermore, the hypotheses entailing the integrability of these deformations provide new connections between sparse polynomials, element-set relations, and hyperdeterminants and their factorisation over the ring of Laurent polynomials.Comment: 40 page