Some remarks on totally imperfect sets

We prove the following two theorems.

Theorem 1. Let be a strongly meager subset of . Then it is dual Ramsey null.

We will say that a -ideal of subsets of satisfies the condition iff for every , if

then .

Theorem 2. The -ideals of perfectly meager sets, universally meager sets and perfectly meager sets in the transitive sense satisfy the condition .

     

Cofinality changes required for a large set of unapproachable ordinals below

In , assume that is a strong limit cardinal and . Let be the set of approachable ordinals less than . An open question of M. Foreman is whether can be non-stationary in some and preserving extension of . It is shown here that if is such an outer model, then is infinite, for each positive integer .

     

On a conjecture about MRA Riesz wavelet bases

Let be a compactly supported refinable function in such that the shifts of are stable and for a -periodic trigonometric polynomial . A wavelet function can be derived from by . If is an orthogonal refinable function, then it is well known that generates an orthonormal wavelet basis in . Recently, it has been shown in the literature that if is a -spline or pseudo-spline refinable function, then always generates a Riesz wavelet basis in . It was an open problem whether can always generate a Riesz wavelet basis in for any compactly supported refinable function in with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function does not generate a Riesz wavelet basis in . Our proof is based on some necessary and sufficient conditions on the -periodic functions and in such that the wavelet function , defined by , generates a Riesz wavelet basis in .

     

A new proof and generalizations of Gearhart's theorem

Let be a Hilbert space, let be the space of almost periodic functions from to , and let be a closed densely defined linear operator on . For a closed subset , let be the subspace of consisting of functions with spectrum contained in . We prove that the following properties are equivalent: (i) for every function there exists a unique mild solution of equation ; (ii) and . The case yields a new proof of the well-known Gearhart's spectral mapping theorem.

     

Counting cusps of subgroups of

Let be a number field with real places and complex places, and let be the ring of integers of . The quotient has cusps, where is the class number of . We show that under the assumption of the generalized Riemann hypothesis that if is not or an imaginary quadratic field and if , then has infinitely many maximal subgroups with cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.

     

On the characteristic polynomial of the almost Mathieu operator

Let be the rotation C*-algebra for angle . For with and relatively prime, is the sub-C*-algebra of generated by a pair of unitaries and satisfying . Let

be the almost Mathieu operator. By proving an identity of rational functions we show that for even, the constant term in the characteristic polynomial of is .

     

Some results on the Hochschild cohomology of group algebras

In this paper we investigate structure of the second cohomology of a discrete group . First, for a -set we show that an isomorphism of vector spaces from onto exists, where is the set of orbits of . Next we define the notion of pseudoderivation and apply it for the calculation of .

     

Nonstandard topologies with bases that consist only of standard sets

Let be an infinite set, a set of pseudo-metrics on and If is limited (finite) for every and every then, for each we can define a pseudo-metric on by writing st We investigate the conditions under which the topology induced on by has a basis consisting only of standard sets. This investigation produces a theory with a variety of applications in functional analysis. For example, a specialization of some of our general results will yield such classical compactness theorems as Schauder's theorem, Mazur's theorem, and Gelfand-Philips's theorem.

     

Fixed points of nonexpansive mappings in spaces of continuous functions

Let be a compact metrizable space and let be the Banach space of all real continuous functions defined on with the maximum norm. It is known that fails to have the weak fixed point property for nonexpansive mappings (w-FPP) when contains a perfect set. However the space , where and is the first infinite ordinal number, enjoys the w-FPP, and so also satisfies this property if . It is unknown if has the w-FPP when is a scattered set such that . In this paper we prove that certain subspaces of , with , satisfy the w-FPP. To prove this result we introduce the notion of -almost weak orthogonality and we prove that an -almost weakly orthogonal closed subspace of enjoys the w-FPP. We show an example of an -almost weakly orthogonal subspace of which is not contained in for any .

     

Lipscomb's space

It is known that Lipscomb's space can be imbedded in Hilbert's space . Let be the imbedded version of endowed with the -induced topology. We show how to construct as the attractor of an iterated function system containing an infinite number of affine transformations of . In this way we answer an open question of J.C. Perry.

     

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

     

planes in

We establish a homeomorphism between the moduli space of ordered -tuples of 2-dimensional linear subspaces (mod ) and the quotient by simultaneous conjugation of a certain open subset . For , this leads to an explicit computation of the moduli space of central 2-arrangements in mod and its subspace of those classes that contain a complex hyperplane arrangement.

     

Hypercyclicity in omega

A sequence of operators is said to be hypercyclic if there exists a vector , called hypercyclic for , such that is dense. A hypercyclic subspace for is a closed infinite-dimensional subspace of, except for zero, hypercyclic vectors. We prove that if is a sequence of operators on that has a hypercyclic subspace, then there exist (i) a sequence of one variable polynomials such that is hypercyclic for every fixed and (ii) an operator that maps nonzero vectors onto hypercyclic vectors for .

We complement earlier work of several authors.

     

On the local Hölder continuity of the inverse of the -Laplace operator

We prove an interpolation type inequality between , and spaces and use it to establish the local Hölder continuity of the inverse of the -Laplace operator: , for any and in a bounded set in .

     

Tauberian type theorem for operators with interpolation spectrum for Hölder classes

We consider an invertible operator on a Banach space whose spectrum is an interpolating set for Hölder classes. We show that if , , with and , then for all , assuming that satisfies suitable regularity conditions. When is a Hilbert space and (i.e. is a contraction), we show that under the same assumptions, is unitary and this is sharp.

     

Uniqueness of dilation invariant norms

Let be a nontrivial dilation. We show that every complete norm on that makes from into itself continuous is equivalent to . also determines the norm of both and with in a weaker sense. Furthermore, we show that even all the dilations do not determine the norm on .

     

Principal values for the Cauchy integral and rectifiability

We give a geometric characterization of those positive finite measures on with the upper density finite at -almost every , such that the principal value of the Cauchy integral of ,

{\varepsilon}} \frac{1}{\xi-z}\, d\mu(\xi),\end{displaymath}">

exists for -almost all . This characterization is given in terms of the curvature of the measure . In particular, we get that for , -measurable (where is the Hausdorff -dimensional measure) with , if the principal value of the Cauchy integral of exists -almost everywhere in , then is rectifiable.

     

Weak properties of weighted convolution algebras

Suppose that is a weighted convolution algebra on with the weight normalized so that the corresponding space of measures is the dual space of the space of continuous functions. Suppose that is a continuous nonzero homomorphism, where is also a convolution algebra. If is norm dense in , we show that is (relatively) weak dense in , and we identify the norm closure of with the convergence set for a particular semigroup. When is weak continuous it is enough for to be weak dense in . We also give sufficient conditions and characterizations of weak continuity of . In addition, we show that, for all nonzero in , the sequence converges weak to 0. When is regulated, converges to 0 in norm.

     

Countable dense homogeneity of definable spaces

We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zero-dimensional Borel CDH spaces. We also show that for a Borel the following are equivalent: (1) is in , (2) is CDH and (3) is homeomorphic to or to . Assuming the Axiom of Projective Determinacy the results extend to all projective sets and under the Axiom of Determinacy to all separable metric spaces. In particular, modulo a large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Stepr ns and Zhou, by showing that is not CDH.