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.

     

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 .

     

Strongly representable atom structures of relation algebras

A relation algebra atom structure is said to be strongly representable if all atomic relation algebras with that atom structure are representable. This is equivalent to saying that the complex algebra is a representable relation algebra. We show that the class of all strongly representable relation algebra atom structures is not closed under ultraproducts and is therefore not elementary. This answers a question of Maddux (1982).

Our proof is based on the following construction. From an arbitrary undirected, loop-free graph , we construct a relation algebra atom structure and prove, for infinite , that is strongly representable if and only if the chromatic number of is infinite. A construction of Erdös shows that there are graphs () with infinite chromatic number, with a non-principal ultraproduct whose chromatic number is just two. It follows that is strongly representable (each ) but is not.

     

A uniqueness result for harmonic functions

Let , , and suppose is harmonic in and on the closure of . If the gradient of vanishes continuously on a subset of of positive -dimensional Lebesgue measure and satisfies certain regularity conditions, then must be identically constant.

     

On the invariant translation approximation property for discrete groups

Recently J.Roe considered the question of whether for a discrete group the reduced group -algebra is the fixed point algebra of Ad acting on the uniform Roe algebra . is said to have the invariant translation approximation property in this case. We consider a slight generalization of this property which, for exact , is equivalent to the operator space approximation property of . We also give a new characterization of exactness and a short proof of the equivalence of exactness of and exactness of for discrete groups.

     

Two characterizations of pure injective modules

Let be a commutative ring with identity and an -module. It is shown that if is pure injective, then is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows that if is Noetherian, then is pure injective if and only if is isomorphic to a direct summand of the direct product of a family of Artinian modules. Moreover, it is proved that is pure injective if and only if there is a family of -algebras which are finitely presented as -modules, such that is isomorphic to a direct summand of a module of the form , where for each , is an injective -module.

     

Equivalence of domains with isomorphic semigroups of endomorphisms

For two bounded domains in whose semigroups of analytic endomorphisms are isomorphic with an isomorphism , Eremenko proved in 1993 that there exists a conformal or anticonformal map such that for all .

In the present paper we prove an analogue of this result for the case of bounded domains in .

     

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.

     

Character degree graphs that are complete graphs

Let be a finite group, and write for the set of degrees of irreducible characters of . We define to be the graph whose vertex set is , and there is an edge between and if . We prove that if is a complete graph, then is a solvable group.

     

Automatic continuity of -derivations on -algebras

Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.

     

Parallel tangent hyperplanes

Let be a smooth strictly convex closed hypersurface in and let be any oriented smooth connected manifold immersed in Suppose that is a continuous function from to Then there is at least one point such that the hyperplane tangent to at is parallel to the hyperplane tangent to the immersed manifold at the point corresponding to If there did not exist at least two such points, would have to be compact and the Hurewicz homomorphism of into would have to be surjective. If in addition our immersion was an embedding, the Euler characteristic of would have to be equal to For any and any immersed we could always get maps for which the number of points satisfying the conditions of our theorem exactly equaled two. An example can be given in which both and are the unit sphere about the origin in and there is only one such point .

     

Convex curves, Radon transforms and convolution operators defined by singular measures

Let be a convex curve in the plane and let be the arc-length measure of Let us rotate by an angle and let be the corresponding measure. Let . Then This is optimal for an arbitrary . Depending on the curvature of , this estimate can be improved by introducing mixed-norm estimates of the form where and are conjugate exponents.     

Metacompact subspaces of products of ordinals

Let be a subspace of the product of finitely many ordinals. is countably metacompact, and is metacompact iff has no closed subset homeomorphic to a stationary subset of a regular uncountable cardinal. A theorem generalizing these two results is: is -metacompact iff has no closed subset homeomorphic to a -stationary set where .

     

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.

     

A question of B. Plotkin about the semigroup of endomorphisms of a free group

Let be a free group of finite rank , let be the semigroup of endomorphisms of , and let be the group of automorphisms of .

Theorem. If is an automorphism of , then there is an such that for all .

     

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 .

     

-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.

     

On the finiteness properties of extension and torsion functors of local cohomology modules

Let be a commutative Noetherian ring with non-zero identity, and ideals of with , and a finitely generated -module. In this paper, for fixed integers and , we study the finiteness of and in several cases.

     

Specker's theorem for Nöbeling's group

Specker proved that the group of integer-valued sequences is far from free; all its homomorphisms to factor through finite subproducts. Nöbeling proved that the subgroup consisting of the bounded sequences is free and therefore has many homomorphisms to . We prove that all ``reasonable' homomorphisms factor through finite subproducts. Among the reasonable homomorphisms are all those that are Borel with respect to a natural topology on . In the absence of the axiom of choice, it is consistent that all homomorphisms are reasonable and therefore that Specker's theorem applies to as well as to .

     

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.