Existence of unliftable modules

Let be a commutative Noetherian local ring, and let where is a non-zerodivisor of contained in . Then a finitely generated -module is said to lift to if there exists a finitely generated -module such that is -regular and . In this paper we give a general construction of finitely generated -modules of finite projective dimension over which fail to lift to provided and the depth of is at least 2.

     

On the excess of sets of complex exponentials

For let be complex numbers such that is bounded. For define , where . Then the excesses in the sense of Paley and Wiener satisfy .

     

Cyclic torsion of elliptic curves

Let be an elliptic curve over a number field such that
and let denote the number of roots of unity in . Ross proposed a question: Is isogenous over to an elliptic curve such that is cyclic of order dividing ? A counter-example of this question is given. We show that is isogenous to such that . In case has complex multiplication and , we obtain certain criteria whether or not is isogenous to such that .

     

Extremal points of a functional on the set of convex functions

We investigate the extremal points of a functional , for a convex or concave function . The admissible functions are convex themselves and satisfy a condition . We show that the extremal points are exactly and if these functions are convex and coincide on the boundary . No explicit regularity condition is imposed on , , or . Subsequently we discuss a number of extensions, such as the case when or are non-convex or do not coincide on the boundary, when the function also depends on , etc.

     

On a theorem of Scott and Swarup

Let be an exact sequence of hyperbolic groups induced by an automorphism of the free group . Let be a finitely generated distorted subgroup of . Then there exist and a free factor of such that the conjugacy class of is preserved by and contains a finite index subgroup of a conjugate of . This is an analog of a theorem of Scott and Swarup for surfaces in hyperbolic 3-manifolds.

     

The Weierstrass approximation theorem and a characterization of the unit circle

We study real algebraic morphisms from nonsingular real algebraic varieties with into nonsingular real algebraic curves . We show, among other things, that the set of real algebraic morphisms from into is never dense in the space of all maps from into , unless is biregularly isomorphic to a Zariski open subset of the unit circle.

     

The Furuta inequality with negative powers

Let be bounded linear operators on a Hilbert space satisfying . Furuta showed the operator inequality
as long as positive real numbers satisfy and . In this paper, we show this inequality is valid if negative real numbers satisfy a certain condition. Also, we investigate the optimality of that condition.

     

Function algebras and the lattice of compactifications

We provide some conditions as to when for two locally compact spaces and (where is the lattice of all Hausdorff compactifications of ). More specifically, we prove that if and only if . Using this result, we prove several extensions to the case where is embedded as a sub-lattice of and to where and are not locally compact.

One major contribution is in the use of function algebra techniques. The use of these techniques makes the extensions simple and clean and brings new tools to the subject.

     

On symmetry of discrete polynomial hypergroups

Let be a discrete polynomial hypergoup on with Plancherel measure If the hypergroup is symmetric, the set of characters can be identified with a compact subset of the real line which contains the support of We show that the lower and upper bounds of and coincide. In particular, the trivial character belongs to the support of the Plancherel measure.

     

On the asymptotic stability in functional differential equations

Consider a system of functional differential equations where is the vector-valued functional. The classical asymptotic stability result for such a system calls for a positive definite functional and negative definite functional . In applications one can construct a positive definite functional , whose derivative is not negative definite but is less than or equal to zero. Exactly for such cases J. Hale created the effective asymptotic stability criterion if the functional in functional differential equations is autonomous ( does not depend on ), and N. N. Krasovskii created such criterion for the case where the functional is periodic in . For the general case of the non-autonomous functional V. M. Matrosov proved that this criterion is not right even for ordinary differential equations. The goal of this paper is to prove this criterion for the case when is almost periodic in . This case is a particular case of the class of non-autonomous functionals.

     

Sharper changes in topologies

Let be a Polish group, a Polish topology on a space , acting continuously on , with -invariant and in the Borel algebra generated by . Then there is a larger Polish topology on so that is open with respect to , still acts continuously on , and has a basis consisting of sets that are of the same Borel rank as relative to .

     

Bousfield localizations of classifying spaces of nilpotent groups

Let be a finitely generated nilpotent group. The object of this paper is to identify the Bousfield localization of the classifying space with respect to a multiplicative complex oriented homology theory . We show that is the same as the localization of with respect to the ordinary homology theory determined by the ring .

     

Relative Brauer groups of discrete valued fields

Let be a non-trivial finite Galois extension of a field . In this paper we investigate the role that valuation-theoretic properties of play in determining the non-triviality of the relative Brauer group, , of over . In particular, we show that when is finitely generated of transcendence degree 1 over a -adic field and is a prime dividing , then the following conditions are equivalent: (i) the -primary component, , is non-trivial, (ii) is infinite, and (iii) there exists a valuation of trivial on such that divides the order of the decomposition group of at .

     

On a problem of Dynkin

Consider an -superdiffusion on , where is an uniformly elliptic differential operator in , and . The -polar sets for are subsets of which have no intersection with the graph of , and they are related to the removable singularities for a corresponding nonlinear parabolic partial differential equation. Dynkin characterized the -polarity of a general analytic set in term of the Bessel capacity of , and Sheu in term of the restricted Hausdorff dimension. In this paper we study in particular the -polarity of sets of the form , where and are two Borel subsets of and respectively. We establish a relationship between the restricted Hausdorff dimension of and the usual Hausdorff dimensions of and . As an application, we obtain a criterion for -polarity of in terms of the Hausdorff dimensions of and , which also gives an answer to a problem proposed by Dynkin in the 1991 Wald Memorial Lectures.

     

A Dauns-Hofmann theorem for TAF-algebras

Let be a TAF-algebra, the centre of the ideal lattice of , and the space of meet-irreducible elements of , equipped with the hull-kernel topology. It is shown that is a compact, locally compact, second countable, -space, that is an algebraic lattice isomorphic to the lattice of open subsets of , and that is isomorphic to the algebra of continuous, complex functions on . If is semisimple, then is isomorphic to the algebra of continuous, complex functions on , the primitive ideal space of . If is strongly maximal, then the sum of two closed ideals of is closed.

     

On covering multiplicity

Let be a system of arithmetic sequences which forms an -cover of (i.e. every integer belongs at least to members of ). In this paper we show the following surprising properties of : (a) For each there exist at least subsets of with such that . (b) If forms a minimal -cover of , then for any there is an such that for every there exists an for which and

     

Helly-type theorems for hollow axis-aligned boxes

A hollow axis-aligned box is the boundary of the cartesian product of compact intervals in . We show that for , if any of a collection of hollow axis-aligned boxes have non-empty intersection, then the whole collection has non-empty intersection; and if any of a collection of hollow axis-aligned rectangles in have non-empty intersection, then the whole collection has non-empty intersection. The values for and for are the best possible in general. We also characterize the collections of hollow boxes which would be counterexamples if were lowered to , and to , respectively.

     

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.

     

An application of the regularized Siegel-Weil formula on unitary groups to a theta lifting problem

Let and be the pair of unitary groups over a global field and an irreducible cuspidal representation of which satisfies a certain -function condition. By using a regularized Siegel-Weil formula, we can show that the global theta lifting of in is non-trivial if every local factor of has a local theta lifting (Howe lifting) in .

     

On the product of two generalized derivations

Two elements and in a ring determine a generalized derivation on by setting for any in . We characterize when the product is a generalized derivation in the cases when the ring is the algebra of all bounded operators on a Banach space , and when is a -algebra . We use these characterizations to compute the commutant of the range of .