The Skolem property in rings of integer-valued polynomials

Let be an integral domain with quotient field and . We investigate the relationship between the Skolem and completely integrally closed properties in the ring of integer-valued polynomials

Among other things, we show for the case and that the following are equivalent: (1) is strongly Skolem, (2) is completely integrally closed, and (3) for every .

     

Gorenstein Prüfer整环的一些新刻画

设R是整环.众所周知,R是Prüfer整环当且仅当每个可除模是FP-内射模当且仅当每个h-可除模是FP-内射模.本文引进了一种新的Gorenstein FP-内射模,并且证明了R是Gorenstein Prüfer整环当且仅当每个可除模是Gorenstein FP-内射模,当且仅当每个h-可除模是Gorenstein FP-内射模.     

Eigenvalue ratios for the regular Sturm-Liouville system

Following the method of Ashbaugh-Benguria in Comm. Math. Phys. 124 (1989), 403--415; J. Differential Equations 103 (1993), 205--219, we prove an upper estimate of the arbitrary eigenvalue ratio for the regular Sturm-Liouville system. This upper estimate is sharp for Neumann boundary conditions. We also discuss the sign of and include an elementary proof of a useful trigonometric inequality first given in the aforementioned articles.

     

Localization of injective modules over valuation rings

It is proved that is injective if is an injective module over a valuation ring , for each prime ideal . Moreover, if or is flat, then is injective, too. It follows that localizations of injective modules over h-local Prüfer domains are injective, too.

     

The -spectrum and spanning trees of tensor products of bipartite graphs

Recently, Knuth and Ciucu independently proved the surprising fact, conjectured by Stanley, that one connected component of the tensor product of a path with itself (the so-called ``Aztec diamond graph') has four times as many spanning trees as the other connected component, independent of the length of the path. We show here that much more is true: the connected components of the tensor product of any connected bipartite multigraphs all have essentially the same -spectrum. It follows at once that there is a simple formula relating their numbers of spanning trees.

     

A generalized enumeration of labeled trees and reverse Prüfer algorithm

A leader of a tree T on [n] is a vertex which has no smaller descendants in T. Gessel and Seo showed that

     

Generalizations of Müntz's Theorem via a Remez-type inequality for Müntz spaces

The principal result of this paper is a Remez-type inequality for Müntz polynomials:

or equivalently for Dirichlet sums:

where . The most useful form of this inequality states that for every sequence satisfying , there is a constant depending only on and (and not on , , or ) so that

for every Müntz polynomial , as above, associated with , and for every set of Lebesgue measure at least . Here denotes the supremum norm on . This Remez-type inequality allows us to resolve two reasonably long-standing conjectures.

The first conjecture it lets us resolve is due to D. J. Newman and dates from 1978. It asserts that if , then the set of products is not dense in .

The second is a complete extension of Müntz's classical theorem on the denseness of Müntz spaces in to denseness in , where
is an arbitrary compact set with positive Lebesgue measure. That is, for an
arbitrary compact set with positive Lebesgue measure,
is dense in if and only if .

Several other interesting consequences are also presented.

     

Self-pure-generators over Dedekind domains

We prove that all pure submodules of a finite rank torsion-free module A over a Dedekind domain are A-generated (i.e. A is a self-pure-generator) if and only if A has a rank 1 direct summand B such that $\mathbf{type}(B)$ is the inner type of A. This result is applied to describe the direct products of torsion-free groups of finite rank which are self-pure-generators.     

Low dimensional modules over quantum complete intersections in two variables

We classify all the indecomposable modules of dimension ≤ 5 over the quantum exterior algebra k(x, y)/(x^2, y^2, xy + qyx) in two variables, and all the indecomposable modules of dimension ≤3 over the quantum complete intersection k(x,y)/(x^m,y^n,xy + qyx) in two variables, where m or n ≥3, by giving explicitly their diagram presentations.     

Zero distribution of Müntz extremal polynomials in

Let be a sequence of distinct positive numbers. Let and let denote the extremal Müntz polynomial in with exponents . We investigate the zero distribution of . In particular, we show that if

then the normalized zero counting measure of converges weakly as to

while if or , the limiting measure is a Dirac delta at or , respectively.

     

On finitely injective modules and locally pure-injective modules over Prüfer domains

Over Matlis valuation domains there exist finitely injective modules which are not direct sums of injective modules, as well as complete locally pure-injective modules which are not the completion of a direct sum of pure-injective modules. Over Prüfer domains which are either almost maximal, or -local Matlis, finitely injective torsion modules and complete torsion-free locally pure-injective modules correspond to each other under the Matlis equivalence. Almost maximal Prüfer domains are characterized by the property that every torsion-free complete module is locally pure-injective. It is derived that semi-Dedekind domains are Dedekind.

     

Closed geodesics and non-differentiability of the metric in infinite-dimensional Teichmüller spaces

In this paper we construct a closed geodesic in any infinite-
dimensional Teichmüller space. The construction also leads to a proof of non-differentiability of the metric in infinite-dimensional Teichmüller spaces, which provides a negative answer to a problem of Goldberg.

     

Tilting modules over path algebras of Dynkin type and complete slice modules

Let T be a tilting A-module over a path algebra of Dynkin type.We prove thatif the indecomposable direct summands of T are in the different (?)-orbits of the AR-quiverof A,then T is a complete slice module.     

Applications of phase plane analysis of a Liénard system to positive solutions of Schrödinger equations

This paper deals with semilinear elliptic equations in an exterior domain of with . Sufficient conditions are obtained for the equation to have a positive solution which decays at infinity. The main result is proved by means of a supersolution-subsolution method presented by Noussair and Swanson. By using phase plane analysis of a system of Liénard type, a suitable positive supersolution is found out. Asymptotic decay estimation on a solution of the Liénard system gains a positive subsolution. Examples are given to illustrate the main result.

     

Geometry of regular modules over canonical algebras

We classify canonical algebras such that for every dimension vector of a regular module the corresponding module variety is normal (respectively, a complete intersection). We also prove that for the dimension vectors of regular modules normality is equivalent to irreducibility.

     

On nonsingular modules over infinite-dimensional simple novikov algebra of characteristic 0

In this paper we show that an A-module is a direct sum of a trivial module and a nosingular module, and we also give the structure of nonsingular module over infinite dimensional simple Novikov algebras of types (II) and (III) over a field F of Characteristic 0.     

Classification of irreducible non-zero level quasifinite modules over twisted affine Nappi-Witten algebra

We obtain that every irreducible quasifinite module with non-zero level over the twisted affine Nappi-Witten algebra is either a highest weight module or a lowest one.