Integral closure of rings of integer-valued polynomials on algebras

Let $D$ be an integrally closed domain with quotient field $K$. Let $A$ be a torsion-free $D$-algebra that is finitely generated as a $D$-module. For every $a$ in $A$ we consider its minimal polynomial $\mu_a(X)\in D[X]$, i.e. the monic polynomial of least degree such that $\mu_a(a)=0$. The ring ${\rm Int}_K(A)$ consists of polynomials in $K[X]$ that send elements of $A$ back to $A$ under evaluation. If $D$ has finite residue rings, we show that the integral closure of ${\rm Int}_K(A)$ is the ring of polynomials in $K[X]$ which map the roots in an algebraic closure of $K$ of all the $\mu_a(X)$, $a\in A$, into elements that are integral over $D$. The result is obtained by identifying $A$ with a $D$-subalgebra of the matrix algebra $M_n(K)$ for some $n$ and then considering polynomials which map a matrix to a matrix integral over $D$. We also obtain information about polynomially dense subsets of these rings of polynomials.Comment: Keywords: Integer-valued polynomial, matrix, triangular matrix, integral closure, pullback, polynomially dense set. accepted for publication in the volume "Commutative rings, integer-valued polynomials and polynomial functions", M. Fontana, S. Frisch and S. Glaz (editors), Springer 201

On the molecules of numerical semigroups, Puiseux monoids, and Puiseux algebras

A molecule is a nonzero non-unit element of an integral domain (resp., commutative cancellative monoid) having a unique factorization into irreducibles (resp., atoms). Here we study the molecules of Puiseux monoids as well as the molecules of their corresponding semigroup algebras, which we call Puiseux algebras. We begin by presenting, in the context of numerical semigroups, some results on the possible cardinalities of the sets of molecules and the sets of reducible molecules (i.e., molecules that are not irreducibles/atoms). Then we study the molecules in the more general context of Puiseux monoids. We construct infinitely many non-isomorphic atomic Puiseux monoids all whose molecules are atoms. In addition, we characterize the molecules of Puiseux monoids generated by rationals with prime denominators. Finally, we turn to investigate the molecules of Puiseux algebras. We provide a characterization of the molecules of the Puiseux algebras corresponding to root-closed Puiseux monoids. Then we use such a characterization to find an infinite class of Puiseux algebras with infinitely many non-associated reducible molecules.Comment: 21 pages, 2 figure

Crossover effects in the Wolf-Villain model of epitaxial growth in 1+1 and 2+1 dimensions

A simple model of epitaxial growth proposed by Wolf and Villain is investigated using extensive computer simulations. We find an unexpectedly complex crossover behavior of the original model in both 1+1 and 2+1 dimensions. A crossover from the effective growth exponent $\beta_{\rm eff}\!\approx\!0.37$ to $\beta_{\rm eff}\!\approx\!0.33$ is observed in 1+1 dimensions, whereas additional crossovers, which we believe are to the scaling behavior of an Edwards--Wilkinson type, are observed in both 1+1 and 2+1 dimensions. Anomalous scaling due to power--law growth of the average step height is found in 1+1 D, and also at short time and length scales in 2+1~D. The roughness exponents $\zeta_{\rm eff}^{\rm c}$ obtained from the height--height correlation functions in 1+1~D ($\approx\!3/4$) and 2+1~D ($\approx\!2/3$) cannot be simultaneously explained by any of the continuum equations proposed so far to describe epitaxial growth.Comment: 11 pages, REVTeX 3.0, IC-DDV-93-00

A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres

We construct an invariant J_M of integral homology spheres M with values in a completion \hat{Z[q]} of the polynomial ring Z[q] such that the evaluation at each root of unity \zeta gives the the SU(2) Witten-Reshetikhin-Turaev invariant \tau_\zeta(M) of M at \zeta. Thus J_M unifies all the SU(2) Witten-Reshetikhin-Turaev invariants of M. As a consequence, \tau_\zeta(M) is an algebraic integer. Moreover, it follows that \tau_\zeta(M) as a function on \zeta behaves like an ``analytic function'' defined on the set of roots of unity. That is, the \tau_\zeta(M) for all roots of unity are determined by a "Taylor expansion" at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. In particular, \tau_\zeta(M) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q=1.Comment: 66 pages, 8 figure

Fast evolving pair-instability supernova models: evolution, explosion, light curves

With an increasing number of superluminous supernovae (SLSNe) discovered, the question of their origin remains open and causes heated debates in the supernova community. Currently, there are three proposed mechanisms for SLSNe: (1) pair-instability supernovae (PISNe), (2) magnetar-driven supernovae and (3) models in which the supernova ejecta interacts with a circumstellar material ejected before the explosion. Based on current observations of SLSNe, the PISN origin has been disfavoured for a number of reasons. Many PISN models provide overly broad light curves and too reddened spectra, because of massive ejecta and a high amount of nickel. In the current study, we re-examine PISN properties using progenitor models computed with the GENEC code. We calculate supernova explosions with FLASH and light-curve evolution with the radiation hydrodynamics code STELLA. We find that high-mass models (200 and 250 M⊙) at relatively high metallicity (Z = 0.001) do not retain hydrogen in the outer layers and produce relatively fast evolving PISNe Type I and might be suitable to explain some SLSNe. We also investigate uncertainties in light-curve modelling due to codes, opacities, the nickel-bubble effect and progenitor structure and composition

Monte Carlo with Absorbing Markov Chains: Fast Local Algorithms for Slow Dynamics

A class of Monte Carlo algorithms which incorporate absorbing Markov chains is presented. In a particular limit, the lowest-order of these algorithms reduces to the $n$-fold way algorithm. These algorithms are applied to study the escape from the metastable state in the two-dimensional square-lattice nearest-neighbor Ising ferromagnet in an unfavorable applied field, and the agreement with theoretical predictions is very good. It is demonstrated that the higher-order algorithms can be many orders of magnitude faster than either the traditional Monte Carlo or $n$-fold way algorithms.Comment: ReVTeX, Request 3 figures from [email protected]

Some closure operations in Zariski-Riemann spaces of valuation domains: a survey

In this survey we present several results concerning various topologies that were introduced in recent years on spaces of valuation domains

Balancing influence between actors in healthcare decision making

<p>Abstract</p> <p>Background</p> <p>Healthcare costs in most developed countries are not clearly linked to better patient and public health outcomes, but are rather associated with service delivery orientation. In the U.S. this has resulted in large variation in healthcare availability and use, increased cost, reduced employer participation in health insurance programs, and reduced overall population health outcomes. Recent U.S. healthcare reform legislation addresses only some of these issues. Other countries face similar healthcare issues.</p> <p>Discussion</p> <p>A major goal of healthcare is to enhance patient health outcomes. This objective is not realized in many countries because incentives and structures are currently not aligned for maximizing population health. The misalignment occurs because of the competing interests between "actors" in healthcare. In a simplified model these are individuals motivated to enhance their own health; enterprises (including a mix of nonprofit, for profit and government providers, payers, and suppliers, etc.) motivated by profit, political, organizational and other forces; and government which often acts in the conflicting roles of a healthcare payer and provider in addition to its role as the representative and protector of the people. An imbalance exists between the actors, due to the resources and information control of the enterprise and government actors relative to the individual and the public. Failure to use effective preventive interventions is perhaps the best example of the misalignment of incentives. We consider the current Pareto efficient balance between the actors in relation to the Pareto frontier, and show that a significant change in the healthcare market requires major changes in the utilities of the enterprise and government actors.</p> <p>Summary</p> <p>A variety of actions are necessary for maximizing population health within the constraints of available resources and the current balance between the actors. These actions include improved transparency of all aspects of medical decision making, greater involvement of patients in shared medical decision making, greater oversight of guideline development and coverage decisions, limitations on direct to consumer advertising, and the need for an enhanced role of the government as the public advocate.</p