A function space -compact space

Assuming that the minimal cardinality of a dominating family in is equal to , we construct a subset of a real line such that the space of continuous real-valued functions on does not admit any continuous bijection onto a -compact space. This gives a consistent answer to a question of Arhangel'skii.

     

On bifurcation points of a complex polynomial

Let be a polynomial of degree . Assume that the set there is a sequence s.t. and is finite. We prove that the set of generalized critical values of (hence in particular the set of bifurcation points of ) has at most points. Moreover, We also compute the set effectively.

     

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 .

     

Fully transitive -groups with finite first Ulm subgroup

An abelian -group is called (fully) transitive if for all with ( ) there exists an automorphism (endomorphism) of which maps onto . It is a long-standing problem of A. L. S. Corner whether there exist non-transitive but fully transitive -groups with finite first Ulm subgroup. In this paper we restrict ourselves to -groups of type , this is to say -groups satisfying . We show that the answer to Corner's question is no if is finite and is of type .

     

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.

     

Bernstein-Walsh inequalities and the exponential curve in

It is shown that for the pluripolar set in there is a global Bernstein-Walsh inequality: If is a polynomial of degree on and on , this inequality gives an upper bound for which grows like . The result is used to obtain sharp estimates for .

     

Van der Waerden spaces and Hindman spaces are not the same

A Hausdorff topological space is van der Waerden if for every sequence in there is a converging subsequence where contains arithmetic progressions of all finite lengths. A Hausdorff topological space is Hindman if for every sequence in there is an IP-converging subsequence for some infinite .

We show that the continuum hypothesis implies the existence of a van der Waerden space which is not Hindman.

     

The Nevanlinna counting functions for Rudin's orthogonal functions

and denote the Hardy spaces on the open unit disc . Let be a function in and . If is an inner function and , then is orthogonal in . W.Rudin asked if the converse is true and C. Sundberg and C. Bishop showed that the converse is not true. Therefore there exists a function such that is not an inner function and is orthogonal in . In this paper, the following is shown: is orthogonal in if and only if there exists a unique probability measure on [0,1] with supp such that for nearly all in where is the Nevanlinna counting function of . If is an inner function, then is a Dirac measure at .

     

Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures

Let and denote the Gaussian and Poisson measures on , respectively. We show that there exists a unique measure on such that under the Segal-Bargmann transform the space is isomorphic to the space of analytic -functions on with respect to . We also introduce the Segal-Bargmann transform for the Poisson measure and prove the corresponding result. As a consequence, when and have the same variance, and are isomorphic to the same space under the - and -transforms, respectively. However, we show that the multiplication operators by on and on act quite differently on .

     

Every set of finite Hausdorff measure is a countable union of sets whose Hausdorff measure and content coincide

A set is -straight if has finite Hausdorff -measure equal to its Hausdorff -content, where is continuous and non-decreasing with . Here, if satisfies the standard doubling condition, then every set of finite Hausdorff -measure in is shown to be a countable union of -straight sets. This also settles a conjecture of Foran that when , every set of finite -measure is a countable union of -straight sets.

     

An inverse problem for an inhomogeneous conformal Killing field equation

Let be a Riemannian metric defined on a bounded domain with boundary and let be a vector field on satisfying . We show that if is a gradient field of a solution to the equation on , then both inner products and are uniquely determined by the restriction of the tensor to the gradient field , where is the Lie derivative of the metric tensor under the vector field and . This work solves a problem related to an inverse boundary value problem for nonlinear elliptic equations.

     

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.

     

Extended Cesàro operators on mixed norm spaces

We define an extended Cesàro operator with holomorphic symbol in the unit ball of as

where is the radial derivative of . In this paper we characterize those for which is bounded (or compact) on the mixed norm space .

     

The real powers of the convolution of a negative binomial distribution and a Bernoulli distribution

For 0,$"> this note computes essentially the set of in such that the entire series in defined by has all its coefficients non-negative. If and are independent random variables which have respectively Bernoulli and negative binomial distributions, denote by the distribution of . The above problem is equivalent to finding the set of 0$"> such that exists; this set is a finite union of intervals and may be the first example of this type in the literature. This gives the final touch to the classification of the natural exponential families with variance functions of Babel type, i.e. of the form , where is a polynomial with degree

     

On a proposed characterization of Schatten-class composition operators

For an analytic function which maps the open unit disc to itself, let be the operator of composition with on the Bergman space . It has been a longstanding problem to determine whether or not the membership of in the Schatten class , , is equivalent to the condition that the function has a finite integral with respect to the Möbius-invariant measure on . We show that the answer is negative when .

     

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.

     

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.

     

Reflecting point-countable families

It is shown that if every -sized subspace of a (regular) space of density has a point-countable base, then so does . Similar results hold for meta-Lindelöfness. Dow's reflection theorem and a number of other results are deduced as corollaries and applications.

     

Nonvanishing of Fourier coefficients of modular forms

Let be a cusp form with integer weight that is not a linear combination of forms with complex multiplication. For , let

Improving on work of Balog, Ono, and Serre we show that for almost all , where is any good function (e.g. such as ) monotonically tending to infinity with . Using a result of Fouvry and Iwaniec, if is a weight 2 cusp form for an elliptic curve without complex multiplication, then we show for all that . We also obtain conditional results depending on the Generalized Riemann Hypothesis and the Lang-Trotter Conjecture.

     

Nilpotency degree of cohomology rings in characteristic

Let be an odd prime number. The purpose of this paper is to provide a -group whose mod- cohomology ring has a nilpotent element satisfying .