Group homomorphisms inducing cohomology monomorphisms

Let be a homomorphism of -groups such that
is injective, for . We prove that the non-bijectivity of implies the existence of a quotient of containing as a proper direct factor. This gives a refined proof of a result of Evens, which asserts that is bijective if is.

     

Beals-Cordes-type characterizations of pseudodifferential operators

We show that, if is the representation of on given by (2.11), and is a bounded operator on , then belongs to if and only if

is a function on with values in the Banach space .

     

When is a -block?

Let and be distinct prime numbers and let be a finite group. If is a -block of and is a -block, we study when the set of ordinary irreducible characters in the blocks and coincide.

     

Rigidity of compact manifolds with boundary and nonnegative Ricci curvature

Let be an ()-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary . Assume that the principal curvatures of are bounded from below by a positive constant . In this paper, we prove that the first nonzero eigenvalue of the Laplacian of acting on functions on satisfies with equality holding if and only if is isometric to an -dimensional Euclidean ball of radius . Some related rigidity theorems for are also proved.

     

Upper bounds for the number of facets of a simplicial complex

Here we study the maximal dimension of the annihilator ideals
of artinian graded rings with a given Hilbert function, where is the polynomial ring in the variables over a field with each , is a graded ideal of , and is the graded maximal ideal of . As an application to combinatorics, we introduce the notion of -facets and obtain some informations on the number of -facets of simplicial complexes with a given -vector.

     

On simultaneous extension of continuous partial functions

For a metric space let (that is, the set of all graphs of real-valued continuous functions with a compact domain in ) be equipped with the Hausdorff metric induced by the hyperspace of nonempty closed subsets of It is shown that there exists a continuous mapping satisfying the following conditions: (i) for all partial functions (ii) For every nonempty compact subset of is a linear positive operator such that .

     

All maps of type are boundary maps

Let be a continuous map of an interval into itself having periodic points of period for all and no other periods. It is shown that every neighborhood of contains a map such that the set of periods of the periodic points of is finite. This answers a question posed by L. S. Block and W. A. Coppel.

     

Isomorphisms of row and column finite matrix rings

This paper investigates the ring-theoretic similarities and the categorical dissimilarities between the ring of row finite matrices and the ring of row and column finite matrices. For example, we prove that two rings and are Morita equivalent if and only if the rings and are isomorphic. This resembles the result of V. P. Camillo (1984) for . We also show that the Picard groups of and are isomorphic, even though the rings and are never Morita equivalent.

     

-holomorphic curves in almost complex surfaces do not always minimize the genus

The adjunction formula computes the genus of an almost complex curve embedded in an almost complex surface in terms of the homology class of . If is Kähler (or at least symplectic) and the self-intersection of is non-negative then the genus of any other surface embedded in and homologous to is not less then the genus of (the proof of this statement (which is a generalization of the Thom conjecture for ) was recently given by the Seiberg-Witten theory). This paper shows that the extra assumptions on are essential for the genus-minimizing properties of embedded almost complex curves.

     

A remark on commuting operator exponentials

In a previous paper the author proved that for square matrices with algebraic entries expexpexpexp if and only if . This result is extended here to bounded operators on an arbitrary Banach space.

     

Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption

In the present paper, the following result is shown: Let be a real Banach space with a uniformly convex dual , and let be a nonempty closed convex and bounded subset of . Assume that is a continuous strong pseudocontraction. Let and be two real sequences satisfying (i) for all ; (ii) ; and (iii) as Then the Ishikawa iterative sequence generated by

converges strongly to the unique fixed point of .

     

On Bernstein-Sato polynomials

We show that for fixed and the set of Bernstein-Sato polynomials of all the polynomials in at most variables of degrees at most is finite. As a corollary, we show that there exists an integer depending only on and such that generates as a module over the ring of the -linear differential operators of , where is an arbitrary field of characteristic 0, is the ring of polynomials in variables over and is an arbitrary non-zero polynomial of degree at most .

     

Extension and convergence theorems for families of normal maps in several complex variables

Let represent the family of holomorphic (continuous) maps from a complex (topological) space to a complex (topological) space , and let be the Alexandroff one-point compactification of if is not compact, if is compact. We say that is uniformly normal if , is relatively compact in (with the compact-open topology) for each complex manifold . We show that normal maps as defined and studied by authors in various settings are, as singleton sets, uniformly normal families, and prove extension and convergence theorems for uniformly normal families. These theorems include (1) extension theorems of big Picard type for such families - defined on complex manifolds having divisors with normal crossings - which encompass results of Järvi, Kiernan, Kobayashi, and Kwack as special cases, and (2) generalizations to such families of an extension-convergence theorem due to Noguchi.

     

A Strict Version of the Non-commutative Urysohn Lemma

Given a pair , of -commuting, hereditary -subalgebras of a unital -algebra , such that is -unital and , there is an element in , with , such that is strictly positive in and is strictly positive in in . Moreover, is strictly positive in in .

     

Inner derivations on ultraprime normed algebras

We show that, for every ultraprime Banach algebra , there exists a positive number satisfying for all in , where denotes the centre of and denotes the inner derivation on induced by . Moreover, the number depends only on the ``constant of ultraprimeness' of .

     

Commensurators of parabolic subgroups of Coxeter groups

Let be a Coxeter system, and let be a subset of . The subgroup of generated by is denoted by and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of in is the subgroup of in such that has finite index in both and . The subgroup can be decomposed in the form where is finite and all the irreducible components of are infinite. Let be the set of in such that for all . We prove that the commensurator of is . In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and is its own commensurator if and only if .

     

Commutative group algebras of -summable abelian groups

In this note we study the commutative modular and semisimple group rings of -summable abelian -groups, which group class was introduced by R. Linton and Ch. Megibben. It is proved that is -summable if and only if is -summable, provided is an abelian group and is a commutative ring with 1 of prime characteristic , having a trivial nilradical. If is a -summable -group and the group algebras and over a field of characteristic are -isomorphic, then is a -summable -group, too. In particular provided is totally projective of a countable length.

Moreover, when is a first kind field with respect to and is -torsion, is -summable if and only if is a direct sum of cyclic groups.

     

Invariant subspaces for polynomially hyponormal operators

We show that if is a bounded operator on a Hilbert space such that for every polynomial , then has a nontrivial invariant subspace.

     

Bilocal derivations of standard operator algebras

In this paper, we shall show the following two results: (1) Let be a standard operator algebra with , if is a linear mapping on which satisfies that maps into for all , then is of the form for some in . (2) Let be a Hilbert space, if is a norm-continuous linear mapping on which satisfies that maps into for all self-adjoint projection in , then is of the form for some in .

     

Semi-free actions of zero-dimensional compact groups on Menger compacta

Let be the -dimensional universal Menger compactum, a -set in and a metrizable zero-dimensional compact group with the unit. It is proved that there exists a semi-free -action on such that is the fixed point set of every . As a corollary, it follows that each compactum with can be embedded in as the fixed point set of some semi-free -action on .