LCM-splitting sets in some ring extensions

Let be a saturated multiplicative set of an integral domain . Call an lcm splitting set if and are principal ideals for every and . We show that if is an -stable overring of (that is, if whenever and is principal, it follows that and if is an lcm splitting set of , then the saturation of in is an lcm splitting set in . Consequently, if is Noetherian and is a (nonzero) prime element, then is also a prime element of the integral closure of . Also, if is Noetherian, is generated by prime elements of and if the integral closure of is a UFD, then so is the integral closure of .

     

Index of B-Fredholm operators and generalization of a Weyl theorem

The aim of this paper is to show that if and are commuting B-Fredholm operators acting on a Banach space , then is a B-Fredholm operator and , where means the index. Moreover if is a B-Fredholm operator and is a finite rank operator, then is a B-Fredholm operator and We also show that if is isolated in the spectrum of , then is a B-Fredholm operator of index if and only if is Drazin invertible. In the case of a normal bounded linear operator acting on a Hilbert space , we obtain a generalization of a classical Weyl theorem.     

On one problem of uniqueness of meromorphic functions concerning small functions

In this paper, we show that if two non-constant meromorphic functions and satisfy for , where are five distinct small functions with respect to and , and is a positive integer or with , then . As a special case this also answers the long-standing problem on uniqueness of meromorphic functions concerning small functions.

     

Linearity of dimension functions for semilinear -spheres

In this paper, we show that the dimension function of every semilinear -sphere is equal to that of a linear -sphere for finite nilpotent groups of order , where , are primes. We also show that there exists a semilinear -sphere whose dimension function is not virtually linear for an arbitrary nonsolvable compact Lie group .

     

Normal bases for Hopf-Galois algebras

Let be a Hopf algebra over a commutative ring such that is a finitely generated, projective module over , let be a right -comodule algebra, and let be the subalgebra of -coinvariant elements of . If is a Galois extension of and is a local subalgebra of the center of , then is a cleft right -comodule algebra or, equivalently, there is a normal basis for over .

     

An automatic adjoint theorem and its applications

In this paper, we prove the following automatic adjoint theorem: For any sequence spaces and , if has the signed-weak gliding hump property and is an infinite matrix which transforms into , then the transpose matrix of transforms into , and for any and , . That is, the adjoint operator of automatically exists and is just the transpose matrix of . From the theorem we obtain a class of infinite matrix topological algebras , and prove also a -multiplier convergence theorem of Orlicz-Pettis type. The theorem improves substantially the famous Stiles' Orlicz-Pettis theorem.

     

On the class number of certain imaginary quadratic fields

Theorem. Let 2$"> denote an integer, the square-free part of and the class number of the field . Then except for the case , divides .

     

Existence of multiwavelets in

For a -regular Multiresolution Analysis of multiplicity with arbitrary dilation matrix for a general lattice in , we give necessary and sufficient conditions in terms of the mask and the symbol of the vector scaling function in order that an associated wavelet basis exists. We also show that if where is the absolute value of the determinant of , then these conditions are always met, and therefore an associated wavelet basis of -regular functions always exists. This extends known results to the case of multiwavelets in several variables with an arbitrary dilation matrix for a lattice .

     

The strong radical and finite-dimensional ideals

Let be a semi-prime Banach algebra with strong radical (intersection of its two-sided modular maximal ideals). A minimal left or right ideal of is infinite-dimensional if and only if . Thus all minimal one-sided ideals in are finite-dimensional if is strongly semi-simple.

     

Invariant ideals of abelian group algebras under the multiplicative action of a field. II

Let be a division ring and let be a finite-dimensional right -vector space, viewed multiplicatively. If is the multiplicative group of , then acts on and hence on any group algebra . In this paper, we completely describe the semiprime -stable ideals of , and conclude that these ideals satisfy the ascending chain condition. As it turns out, this result follows fairly easily from the corresponding results for the field of rational numbers (due to Brookes and Evans) and for infinite locally-finite fields (handled in Part I).

     

A global pinching theorem for surfaces with constant mean curvature in

Let be a compact immersed surface in the unit sphere with constant mean curvature . Denote by the linear map from into , , where is the linear map associated to the second fundamental form and is the identity map. Let denote the square of the length of . We prove that if , then is either totally umbilical or an -torus, where is a constant depending only on the mean curvature .

     

Primary decomposition: Compatibility, independence and linear growth

For finitely generated modules over a Noetherian ring , we study the following properties about primary decomposition: (1) The Compatibility property, which says that if and is a -primary component of for each , then ; (2) For a given subset , is an open subset of if and only if the intersections for all possible -primary components and of ; (3) A new proof of the `Linear Growth' property, which says that for any fixed ideals of there exists a such that for any there exists a primary decomposition of such that every -primary component of that primary decomposition contains .

     

On the irreducibility of the iterates of

Let be a field and suppose that is irreducible in . We discuss the following question: under what conditions are all iterates of irreducible over ?

     

Radicals and Plotkin's problem concerning geometrically equivalent groups

If and are groups and is a normal subgroup of , then the -closure of in is the normal subgroup of . In particular, is the -radical of . Plotkin calls two groups and geometrically equivalent, written , if for any free group of finite rank and any normal subgroup of the -closure and the -closure of in are the same. Quasi-identities are formulas of the form for any words in a free group. Generally geometrically equivalent groups satisfy the same quasi-identities. Plotkin showed that nilpotent groups and satisfy the same quasi-identities if and only if and are geometrically equivalent. Hence he conjectured that this might hold for any pair of groups. We provide a counterexample.

     

Van der Waerden spaces

A topological space is van der Waerden if for every sequence in there exists a converging subsequence so that contains arbitrarily long finite arithmetic progressions. Not every sequentially compact space is van der Waerden. The product of two van der Waerden spaces is van der Waerden.

The following condition on a Hausdorff space is sufficent for to be van der Waerden:

The closure of every countable set in is compact and first-countable.

A Hausdorff space that satisfies satisfies, in fact, a stronger property: for every sequence in :

There exists so that is converging, and contains arbitrarily long finite arithmetic progressions and sets of the form for arbitrarily large finite sets .

There are nonmetrizable and noncompact spaces which satisfy . In particular, every sequence of ordinal numbers and every bounded sequence of real monotone functions on satisfy .

     

Signed sums of polynomial values

We give a generalization of Bleicher's result on signed sums of th powers. Let be an integral-valued polynomial of degree satisfying the necessary condition that there exists no integer 1$"> dividing the values for all integers . Then, for every positive integer and every integer , there are infinitely many integers and choices of such that

     

Covering maps that are not compositions of covering maps of lesser order

In 1995 J.W. Heath asked which exactly -to-one maps are compositions of exactly -to-one maps with . This paper deals with compositions of covering maps. Exactly -to-one covering maps on locally arcwise connected continua that are not factorable into covering maps of order are constructed for all 's, and characterized in algebraic terms (fundamental groups). They are not proper compositions of exactly -to-one maps, open maps, or locally one-to-one maps.     

The homogeneous spectrum of a graded commutative ring

Suppose is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a -graded commutative ring is Noetherian if its homogeneous spectrum is Noetherian, thus answering a question of David Rush. Suppose is a commutative ring having Noetherian spectrum. We determine conditions in order that the monoid ring have Noetherian spectrum. If , we show that has Noetherian spectrum, while for each we establish existence of an example where the homogeneous spectrum of is not Noetherian.

     

On the absolute continuity of a class of invariant measures

Let be a compact connected subset of , let , be contractive self-conformal maps on a neighborhood of , and let be a family of positive continuous functions on . We consider the probability measure that satisfies the eigen-equation

for some 0$">. We prove that if the attractor is an -set and is absolutely continuous with respect to , the Hausdorff -dimensional measure restricted on the attractor , then is absolutely continuous with respect to (i.e., they are equivalent). A special case of the result was considered by Mauldin and Simon (1998). In another direction, we also consider the -property of the Radon-Nikodym derivative of and give a condition for which is unbounded.

     

Exactness of one relator groups

A discrete group is -exact if the reduced crossed product with converts a short exact sequence of --algebras into a short exact sequence of -algebras. A one relator group is a discrete group admitting a presentation where is a countable set and is a single word over . In this short paper we prove that all one relator discrete groups are -exact. Using the Bass-Serre theory we also prove that a countable discrete group acting without inversion on a tree is -exact if the vertex stabilizers of the action are -exact.