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 .

     

Simple algebras of Weyl type, II

Over a field of any characteristic, for a commutative associative algebra , and for a commutative subalgebra of , the vector space which consists of polynomials of elements in with coefficients in and which is regarded as operators on forms naturally an associative algebra. It is proved that, as an associative algebra, is simple if and only if is -simple. Suppose is -simple. Then, (a) is a free left -module; (b) as a Lie algebra, the subquotient is simple (except for one case), where is the center of . The structure of this subquotient is explicitly described. This extends the results obtained by Su and Zhao.

     

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 .

     

On the predictability of discrete dynamical systems

Let be a metric space. A function is said to be non-sensitive at a point if for every 0$"> there is a 0$"> such that for any choice of points , , , we have that for every . Let be the set of all homeomorphisms from onto endowed with the topology of uniform convergence. The main goal of the present paper is to prove that for certain spaces , ``most' functions in are non-sensitive at ``most' points of .

     

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.

     

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.     

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.

     

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 ?

     

Equivariant cohomology with local coefficients

We show that for a discrete group , the equivariant cohomology of a -space with -local coefficients is isomorphic to the Bredon-Illman cohomology of with equivariant local coefficients .

     

The f-depth of an ideal on a module

Let be an ideal of a Noetherian local ring and a finitely generated -module. The f-depth of on is the least integer such that the local cohomology module is not Artinian. This paper presents some part of the theory of f-depth including characterizations of f-depth and a relation between f-depth and f-modules.

     

On representable linearly compact modules

For a flat module we prove that is a functor from the category of linearly compact modules to itself and is exact. Moreover, is representable when is linearly compact and representable. This gives an affirmative answer to a question of L. Melkersson (1995) for linearly compact modules without the condition of finite Goldie dimension. The set of attached prime ideals of the co-localization of a linearly compact representable module with respect to a multiplicative set in is described.

     

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 .

     

Isomorphism of commutative group algebras of closed -local algebraically compact groups

Let be an abelian group and let be a field of 0$">. It is shown via a universal algorithm that if the modified Direct-Factor Problem holds, then the -isomorphism for some group yields provided is a closed -group or a -local algebraically compact group. In particular, this is the case when is closed -primary of arbitrary power, or is -local algebraically compact with cardinality at most and is in cardinality not exceeding . The last claim completely settles a question raised by W. May in Proc. Amer. Math. Soc. (1979) and partially extends our results published in Rend. Sem. Mat. Univ. Padova (1999) and Southeast Asian Bull. Math. (2001).

     

The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank

We prove that the Euler characteristic of an even-dimensional compact manifold with positive (nonnegative) sectional curvature is positive (nonnegative) provided that the manifold admits an isometric action of a compact Lie group with principal isotropy group and cohomogeneity such that . Moreover, we prove that the Euler characteristic of a compact Riemannian manifold or with positive sectional curvature is positive if admits an effective isometric action of a torus , i.e., if the symmetry rank of is .

     

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 .

     

Proof of the prime power conjecture for projective planes of order

Let be an abelian collineation group of order of a projective plane of order . We show that must be a prime power, and that the -rank of is at least if for an odd prime .

     

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 sumset phenomenon

Answering a problem posed by Keisler and Leth, we prove a theorem in non-standard analysis to reveal a phenomenon about sumsets, which says that if two sets and are large in terms of ``measure', then the sum is not small in terms of ``order-topology'. The theorem has several corollaries about sumset phenomenon in the standard world; these are described in sections 2-4. One of these is a new result in additive number theory; it says that if two sets and of non-negative integers have positive upper or upper Banach density, then is piecewise syndetic.

     

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.

     

-Weyl's theorem for operator matrices

If is a upper triangular matrix on the Hilbert space , then -Weyl's theorem for and need not imply -Weyl's theorem for , even when . In this note we explore how -Weyl's theorem and -Browder's theorem survive for operator matrices on the Hilbert space.