Catalan paths and quasi-symmetric functions

We investigate the quotient ring of the ring of formal power series over the closure of the ideal generated by non-constant quasi-symmetric functions. We show that a Hilbert basis of the quotient is naturally indexed by Catalan paths (infinite Dyck paths). We also give a filtration of ideals related to Catalan paths from and above the line . We investigate as well the quotient ring of polynomial ring in variables over the ideal generated by non-constant quasi-symmetric polynomials. We show that the dimension of is bounded above by the th Catalan number.

     

On functions whose graph is a Hamel basis

We say that a function is a Hamel function ( ) if , considered as a subset of , is a Hamel basis for . We prove that every function from into can be represented as a pointwise sum of two Hamel functions. The latter is equivalent to the statement: for all there is a such that . We show that this fails for infinitely many functions.

     

Ordered group invariants for nonorientable one-dimensional generalized solenoids

Let be an edge-wrapping rule which presents a one-dimensional generalized solenoid , and let be the adjacency matrix of . When is a wedge of circles and leaves the unique branch point fixed, we show that the stationary dimension group of is an invariant of homeomorphism of even if is not orientable.

     

On a theorem of R. Steinberg on rings of coinvariants

Let be a representation of a finite group over the field . Denote by the algebra of polynomial functions on the vector space . The group acts on and hence also on . The algebra of coinvariants is , where is the ideal generated by all the homogeneous -invariant forms of strictly positive degree. If the field has characteristic zero, then R. Steinberg has shown (this is the formulation of R. Kane) that is a Poincaré duality algebra if and only if is a pseudoreflection group. In this note we explore the situation for fields of nonzero characteristic. We prove an analogue of Steinberg's theorem for the case and give a counterexample in the modular case when .

     

Fuchs' problem 34 for mixed Abelian groups

This paper investigates the extent to which an Abelian group is determined by the homomorphism groups . A class of Abelian groups is a Fuchs 34 class if and in are isomorphic if and only if for all . Two -groups and satisfy for all groups if and only if they have the same -Ulm-Kaplansky-invariants and the same final rank. The mixed groups considered in this context are the adjusted cotorsion groups and the class introduced by Glaz and Wickless. While is a Fuchs 34 class, the class of (adjusted) cotorsion groups is not.     

Approximation of measurable mappings by sequences of continuous functions

Let be a completely regular Hausdorff space, a positive, finite Baire measure on , and a separable metrizable locally convex space. Suppose is a measurable mapping. Then there exists a sequence of functions in which converges to a.e. . If the function is assumed to be weakly continuous and the measure is assumed to be -smooth, then a separability condition is not needed.

     

Farrell sets for harmonic functions

Let denote a relatively closed subset of the unit ball of . The purpose of this paper is to characterize those sets which have the following property: any harmonic function on which satisfies on (where 0$">) can be locally uniformly approximated on by a sequence of harmonic polynomials which satisfy the same inequality on . This answers a question posed by Stray, who had earlier solved the corresponding problem for holomorphic functions on the unit disc.

     

The alternative Dunford-Pettis property in -algebras and von Neumann preduals

A Banach space is said to have the alternative Dunford-Pettis property if, whenever a sequence weakly in with , we have for each weakly null sequence in X. We show that a -algebra has the alternative Dunford-Pettis property if and only if every one of its irreducible representations is finite dimensional so that, for -algebras, the alternative and the usual Dunford-Pettis properties coincide as was conjectured by Freedman. We further show that the predual of a von Neumann algebra has the alternative Dunford-Pettis property if and only if the von Neumann algebra is of type I.

     

Borel subrings of the reals

A Borel (or even analytic) subring of either has Hausdorff dimension or is all of . Extensions of the method of proof yield (among other things) that any analytic subring of having positive Hausdorff dimension is equal to either or .

     

A characterization of quasiconvex vector-valued functions

The aim of this paper is to characterize in terms of scalar quasiconvexity the vector-valued functions which are -quasiconvex with respect to a closed convex cone in a Banach space. Our main result extends a well-known characterization of -quasiconvexity by means of extreme directions of the polar cone of , obtained by Dinh The Luc in the particular case when is a polyhedral cone generated by exactly linearly independent vectors in the Euclidean space .

     

Reduction numbers and initial ideals

The reduction number of a standard graded algebra is the least integer such that there exists a minimal reduction of the homogeneous maximal ideal of such that . Vasconcelos conjectured that where is the initial ideal of an ideal in a polynomial ring with respect to a term order. The goal of this note is to prove the conjecture.

     

A remark on quasi-isometries

We show that if is an quasi-isometry, with , defined on the unit ball of , then there is an affine isometry with where is a universal constant. This result is sharp.

     

On approximately convex functions

A real valued function defined on a real interval is called -convex if it satisfies

The main results of the paper offer various characterizations for -convexity. One of the main results states that is -convex for some positive and if and only if can be decomposed into the sum of a convex function, a function with bounded supremum norm, and a function with bounded Lipschitz-modulus. In the special case , the results reduce to that of Hyers, Ulam, and Green obtained in 1952 concerning the so-called -convexity.

     

Existence of local solutions of the complex Monge-Ampère equation

We prove the local solvability of the -dimensional complex Monge-Ampère equation , 0$">, in a neighborhood of any point where but .

     

Vanishing of cohomology over Gorenstein rings of small codimension

We prove that if , are finite modules over a Gorenstein local ring of codimension at most , then the vanishing of for is equivalent to the vanishing of for . Furthermore, if has no embedded deformation, then such vanishing occurs if and only if or has finite projective dimension.

     

Characterizations of paracompactness and Lindelöfness by the separation property

The separation property in our title is that, for two spaces and , any two disjoint closed copies of in are separated by open sets in . It is proved that a Tychonoff space is paracompact if and only if this separation property holds for the space and every Tychonoff space which is a perfect image of (where denotes the Stone-Cech compactification of ). Moreover, we give a characterization of Lindelöfness in a similar way under the assumption of paracompactness.

     

On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces

In this paper, we prove the following strong convergence theorem: Let be a closed convex subset of a Hilbert space . Let be a strongly continuous semigroup of nonexpansive mappings on such that . Let and be sequences of real numbers satisfying , 0$"> and . Fix and define a sequence in by for . Then converges strongly to the element of nearest to .

     

Some finiteness conditions on the set of overrings of an integral domain

Let be an integral domain with quotient field and integral closure . An overring of is a subring of containing , and denotes the set of overrings of . We consider primarily two finiteness conditions on : (FO), which states that is finite, and (FC), the condition that each chain of distinct elements of is finite. (FO) is strictly stronger than (FC), but if , each of (FO) and (FC) is equivalent to the condition that is a Prüfer domain with finite prime spectrum. In general satisfies (FC) iff satisfies (FC) and all chains of subrings of containing have finite length. The corresponding statement for (FO) is also valid.

     

Extremal metrics for the first eigenvalue of the Laplacian in a conformal class

Let be a compact manifold. First, we give necessary and sufficient conditions for a Riemannian metric on to be extremal for with respect to conformal deformations of fixed volume. In particular, these conditions show that for any lattice of , the flat metric induced on from the standard metric of is extremal (in the previous sense). In the second part, we give, for any , an upper bound of on the conformal class of and exhibit a class of lattices for which the metric maximizes on its conformal class.

     

On quasi-affine transforms of Read's operator

We show that C. J. Read's example of an operator on which does not have any non-trivial invariant subspaces is not the adjoint of an operator on a predual of . Furthermore, we present a bounded diagonal operator such that even though is unbounded, the operator is a bounded operator on with invariant subspaces, and is adjoint to an operator on .