Sufficient conditions for a linear functional to be multiplicative

A commutative Banach algebra is said to have the property if the following holds: Let be a closed subspace of finite codimension such that, for every , the Gelfand transform has at least distinct zeros in , the maximal ideal space of . Then there exists a subset of of cardinality such that vanishes on , the set of common zeros of . In this paper we show that if is compact and nowhere dense, then , the uniform closure of the space of rational functions with poles off , has the property for all . We also investigate the property for the algebra of real continuous functions on a compact Hausdorff space.

     

Iterative approximation of fixed points of Lipschitz pseudocontractive maps

Let be a -uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g., ). Let be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset of and let be arbitrary. Then the iteration sequence defined by , converges strongly to a fixed point of , provided that and have certain properties. If is a Hilbert space, then converges strongly to the unique fixed point of closest to .

     

Invariant subspaces for bounded operators with large localizable spectrum

Suppose is a complex Hilbert space and is a bounded operator. For each closed set let denote the corresponding spectral manifold. Let denote the set of all points with the property that for any open neighborhood of In this paper we show that if is dominating in some bounded open set, then has a nontrivial invariant subspace. As a corollary, every Hilbert space operator which is a quasiaffine transform of a subdecomposable operator with large spectrum has a nontrivial invariant subspace.

     

The depth of an ideal with a given Hilbert function

Let denote the polynomial ring in variables over a field with each . Let be a homogeneous ideal of with and the Hilbert function of the quotient algebra . Given a numerical function satisfying for some homogeneous ideal of , we write for the set of those integers such that there exists a homogeneous ideal of with and with . It will be proved that one has either for some or .

     

A new construction of semi-free actions on Menger manifolds

A new construction of semi-free actions on Menger manifolds is presented. As an application we prove a theorem about simultaneous coexistence of countably many semi-free actions of compact metric zero-dimensional groups with the prescribed fixed-point sets: Let be a compact metric zero-dimensional group, represented as the direct product of subgroups , a -manifold and (resp., ) its pseudo-interior (resp., pseudo-boundary). Then, given closed subsets of , there exists a -action on such that (1) and are invariant subsets of ; and (2) each is the fixed point set of any element .

     

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.

     

Criteria of Pólya type for radial positive definite functions

This article presents sufficient conditions for the positive definiteness of radial functions , , in terms of the derivatives of . The criterion extends and unifies the previous analogues of Pólya's theorem and applies to arbitrarily smooth functions. In particular, it provides upper bounds on the Kuttner-Golubov function which gives the minimal value of such that the truncated power function , , is positive definite. Analogous problems and criteria of Pólya type for -dependent functions, 0$">, are also considered.

     

A splitting theorem for degrees

We prove that, for any , and with _{T}A\oplus U$"> and r.e., in , there are pairs and such that ; ; and, for any and from and any set , if and , then . We then deduce that for any degrees , , and such that and are recursive in , , and is into , can be split over avoiding . This shows that the Main Theorem of Cooper (Bull. Amer. Math. Soc. 23 (1990), 151-158) is false.

     

Dynamics of the function and the Green-Tao theorem on arithmetic progressions in the primes

Let be the set of all positive integers , where are primes and possibly two, but not all three of them are equal. For any , define a function by where is the largest prime factor of . It is clear that if , then . For any , define , for . An element is semi-periodic if there exists a nonnegative integer and a positive integer such that . We use ind to denote the least such nonnegative integer . Wushi Goldring [Dynamics of the function and primes, J. Number Theory 119(2006), 86-98] proved that any element is semi-periodic. He showed that there exists such that , ind, and conjectured that ind can be arbitrarily large.

In this paper, it is proved that for any we have ind , and the Green-Tao Theorem on arithmetic progressions in the primes is employed to confirm Goldring's above conjecture.

     

A rigidity theorem for holomorphic generators on the Hilbert ball

We present a rigidity property of holomorphic generators on the open unit ball of a Hilbert space . Namely, if is the generator of a one-parameter continuous semigroup on such that for some boundary point , the admissible limit - , then vanishes identically on .

     

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.

     

Collapsibility of and some related CW complexes

Let denote the order complex of the partition lattice. The natural -action on the set induces an -action on . We show that the regular CW complex is collapsible. Even more, we show that is collapsible, where is a suitable type selection of the partition lattice. This allows us to generalize and reprove in a conceptual way several previous results regarding the multiplicity of the trivial character in the -representation on .

     

On zeros of Eisenstein series for genus zero Fuchsian groups

Let SL be a genus zero Fuchsian group of the first kind with as a cusp, and let be the holomorphic Eisenstein series of weight on that is nonvanishing at and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on and on a choice of a fundamental domain , we prove that all but possibly of the nontrivial zeros of lie on a certain subset of . Here is a constant that does not depend on the weight, is the upper half-plane, and is the canonical hauptmodul for

     

Symmetric numerical semigroups with arbitrary multiplicity and embedding dimension

We construct symmetric numerical semigroups for every minimal number of generators and multiplicity , . Furthermore we show that the set of their defining congruence is minimally generated by elements.

     

Characterizing indecomposable plane continua from their complements

We show that a plane continuum is indecomposable iff has a sequence of not necessarily distinct complementary domains satisfying the double-pass condition: for any sequence of open arcs, with and , there is a sequence of shadows , where each is a shadow of , such that . Such an open arc divides into disjoint subdomains and , and a shadow (of ) is one of the sets .

     

Some results related to the Corach-Porta-Recht inequality

Let be the algebra of all bounded operators on a complex Hilbert space and let be an invertible self-adjoint (or skew-symmetric) operator of . Corach-Porta-Recht proved that

The problem considered here is that of finding (i) some consequences of the Corach-Porta-Recht Inequality; (ii) a necessary condition (resp. necessary and sufficient condition, when for the invertible positive operators to satisfy the operator-norm inequality for all in ; (iii) a necessary and sufficient condition for the invertible operator in to satisfy

     

On the interpolation constant for Orlicz spaces

In this paper we deal with the interpolation from Lebesgue spaces and , into an Orlicz space , where and for some concave function , with special attention to the interpolation constant . For a bounded linear operator in and , we prove modular inequalities, which allow us to get the estimate for both the Orlicz norm and the Luxemburg norm,

where the interpolation constant depends only on and . We give estimates for , which imply . Moreover, if either or , then . If , then , and, in particular, for the case this gives the classical Orlicz interpolation theorem with the constant .

     

Metric geodesics of isometries in a Hilbert space and the extension problem

We consider the problem of finding short smooth curves of isometries in a Hilbert space . The length of a smooth curve , , is measured by means of , where denotes the usual norm of operators. The initial value problem is solved: for any isometry and each tangent vector at (which is an operator of the form with ) with norm less than or equal to , there exist curves of the form , with initial velocity , which are short along their path. These curves, which we call metric geodesics, need not be unique, and correspond to the so called extension problem considered by M.G. Krein and others: in our context, given a symmetric operator

find all possible extending to all , with . We also consider the problem of finding metric geodesics joining two given isometries and . It is well known that if there exists a continuous path joining and , then both ranges have the same codimension. We show that if this number is finite, then there exist metric geodesics joining and .

     

On the - boundedness of operators

We prove that if is in , is a Banach space, and is a linear operator defined on the space of finite linear combinations of -atoms in with the property that

then admits a (unique) continuous extension to a bounded linear operator from to . We show that the same is true if we replace -atoms by continuous -atoms. This is known to be false for -atoms.

     

Coefficient ideals and the Cohen-Macaulay property of Rees algebras

Let be a local ring and let be an ideal of positive height. If is a reduction of , then the coefficient ideal is by definition the largest ideal such that . In this article we study the ideal when the Rees algebra is Cohen-Macaulay.