Finite generation of powers of ideals

Suppose is a maximal ideal of a commutative integral domain and that some power of is finitely generated. We show that is finitely generated in each of the following cases: (i) is of height one, (ii) is integrally closed and , (iii) is a monoid domain over a field , where is a cancellative torsion-free monoid such that , and is the maximal ideal . We extend the above results to ideals of a reduced ring such that is Noetherian. We prove that a reduced ring is Noetherian if each prime ideal of has a power that is finitely generated. For each with , we establish existence of a -dimensional integral domain having a nonfinitely generated maximal ideal of height such that is -generated.

     

A generalized Dedekind-Mertens lemma and its converse

We study content ideals of polynomials and their behavior under multiplication. We give a generalization of the Lemma of Dedekind-Mertens and prove the converse under suitable dimensionality restrictions.

     

EOL and ETOL array languages

In this paper we study the families of ETOL and EOL array languages. Standard forms for ETOL and EOL array systems are defined and closure properties of the families are studied. Relations of these families with other developmental array languages and other array languages are studied.     

Some concepts of positive dependence for bivariate interchangeable distributions

In this work we consider some familiar and some new concepts of positive dependence for interchangeable bivariate distributions. By characterizing distributions which are positively dependent according to some of these concepts, we indicate real situations in which these concepts arise naturally. For the various families of positively dependent distributions we prove some closure properties and demonstrate all the possible logical relations. Some inequalities are shown and applied to determine whether under- (or over-) estimates, of various probabilistic quantities, occur when a positively dependent distribution is assumed (falsely) to be the product of its marginals (that is, when two positively dependent random variables are assumed, falsely, to be independent). Specific applications in reliability theory, statistical mechanics and reversible Markov processes are discussed. This work was partially supported by National Science Foundation GP-30707X1. It is part of the author's Ph.D. dissertation prepared at the University of Rochester and supervised by A. W. Marshall. Now at Indiana University.     

Lower closure and existence theorems for optimal control problems with infinite horizon

Lower closure theorems are proved for optimal control problems governed by ordinary differential equations for which the interval of definition may be unbounded. One theorem assumes that Cesari's property (Q) holds. Two theorems are proved which do not require property (Q), but assume either a generalized Lipschitz condition or a bound on the controls in an appropriateL _p-space. An example shows that these hypotheses can hold without property (Q) holding.     

A half-moment model for radiative transfer in a 3D gray medium and its reduction to a moment model for hot, opaque sources

We present in this paper a new 3D half-moment model for radiative transfer in a gray medium, called the model, which uses maximum entropy closure. This model is a generalization to 3D of the 1D version recently proposed in (J. Comp. Phys. 180 (2002) 584). The direction space Ω is divided into two pieces, Ω⁺ and Ω^-, in a dynamical way by the plane perpendicular to the total radiative flux, and the half moments are defined from these subspaces. The model closure and the integrations of the radiative transfer equation performed on the moving Ω^± spaces are detailed. 1D planar results, which have motivated the extension of the model of (J. Comp. Phys. 180 (2002) 584) to multi-dimensions, are shown. These results are very good. The model is thereafter derived for 3D spherically symmetric geometry, where the correctness of the non-trivial border terms can be checked. Two 3D spherically symmetric problems are numerically solved in order to show the accuracy of the closure and the role of the border terms. Once again, compared to the solution obtained with a ray tracing solver, results are very good. From the 3D half-moment model, a new moment model, called , is derived for the particular case of a 3D hot and opaque source radiating into a cold medium, for applications such as simulations of stellar atmospheres and fires. Two-dimensional numerical results are presented and compared to those obtained solving the RTE and with other moment models. They demonstrate the very good accuracy of the model, its good convergence properties, and better prediction compared to all other existing moment models in its domain of applicability.     

The uniform closure of non-dense rational spaces on the unit interval

Let P_n denote the set of all algebraic polynomials of degree at most n with real coefficients. Associated with a set of poles a₁,a₂,…,a_n R[-1,1] we define the rational function spaces Associated with a set of poles a₁,a₂,… R[-1,1], we define the rational function spacesIt is an interesting problem to characterize sets a₁,a₂,… R[-1,1] for which P(a₁,a₂,…) is not dense in C[-1,1], where C[-1,1] denotes the space of all continuous functions equipped with the uniform norm on [-1,1]. Akhieser showed that the density of P(a₁,a₂,…) is characterized by the divergence of the series .In this paper, we show that the so-called Clarkson–Erdős–Schwartz phenomenon occurs in the non-dense case. Namely, if P(a₁,a₂,…) is not dense in C[-1,1], then it is “very much not so”. More precisely, we prove the following result.Theorem Let a₁,a₂,… R[-1,1]. Suppose P(a₁,a₂,…) is not dense in C[-1,1], that is,Then every function in the uniform closure of P(a₁,a₂,…) in C[-1,1] can be extended analytically throughout the set C -1,1,a₁,a₂,… .     

Ideals in a perfect closure, linear growth of primary decompositions, and tight closure

This paper is concerned with tight closure in a commutative Noetherian ring of prime characteristic , and is motivated by an argument of K. E. Smith and I. Swanson that shows that, if the sequence of Frobenius powers of a proper ideal of has linear growth of primary decompositions, then tight closure (of ) `commutes with localization at the powers of a single element'. It is shown in this paper that, provided has a weak test element, linear growth of primary decompositions for other sequences of ideals of that approximate, in a certain sense, the sequence of Frobenius powers of would not only be just as good in this context, but, in the presence of a certain additional finiteness property, would actually imply that tight closure (of ) commutes with localization at an arbitrary multiplicatively closed subset of .

Work of M. Katzman on the localization problem for tight closure raised the question as to whether the union of the associated primes of the tight closures of the Frobenius powers of has only finitely many maximal members. This paper develops, through a careful analysis of the ideal theory of the perfect closure of , strategies for showing that tight closure (of a specified ideal of ) commutes with localization at an arbitrary multiplicatively closed subset of and for showing that the union of the associated primes of the tight closures of the Frobenius powers of is actually a finite set. Several applications of the strategies are presented; in most of them it was already known that tight closure commutes with localization, but the resulting affirmative answers to Katzman's question in the various situations considered are believed to be new.