Exponents of class groups of real quadratic function fields

We show that there are polynomials with such that the ideal class group of the real quadratic extensions has an element of order .

     

Vector measure Banach spaces containing a complemented copy of

Let a Banach space and a -algebra of subsets of a set . We say that a vector measure Banach space has the bounded Vitaly-Hahn-Sacks Property if it satisfies the following condition: Every vector measure , for which there exists a bounded sequence in verifying for all , must belong to . Among other results, we prove that, if is a vector measure Banach space with the bounded V-H-S Property and containing a complemented copy of , then contains a copy of .

     

The s-elementary frame wavelets are path connected

An s-elementary frame wavelet is a function which is a frame wavelet and is defined by a Lebesgue measurable set such that . In this paper we prove that the family of s-elementary frame wavelets is a path-connected set in the -norm. This result also holds for s-elementary -dilation frame wavelets in in general. On the other hand, we prove that the path-connectedness of s-elementary frame wavelets cannot be strengthened to uniform path-connectedness. In fact, the sets of normalized tight frame wavelets and frame wavelets are not uniformly path-connected either.

     

Control of radii of convergence and extension of subanalytic functions

Let : denote a real analytic function on an open subset of , and let denote the points where does not admit a local analytic extension. We show that if is semialgebraic (respectively, globally subanalytic), then is semialgebraic (respectively, subanalytic) and extends to a semialgebraic (respectively, subanalytic) neighbourhood of . (In the general subanalytic case, is not necessarily subanalytic.) Our proof depends on controlling the radii of convergence of power series centred at points in the image of an analytic mapping , in terms of the radii of convergence of at points , where denotes the Taylor expansion of at .

     

Notes on the largest irreducible character degree of a finite group

Let be a finite group and the largest irreducible character degree of . In this note, we show the following results: if , then ; if and, in addition, is -solvable with abelian Sylow -subgroup, then .

     

Strong comparison principle for solutions of quasilinear equations

Let , , be a bounded smooth connected open set and be a map satisfying the hypotheses (H1)-(H4) below. Let with , in and with be two weak solutions of

Suppose that in . Then we show that u_1$"> in under the following assumptions: either u_1$"> on , or on and in . We also show a measure-theoretic version of the Strong Comparison Principle.

     

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 .

     

On two problems of Erdos and Hechler: New methods in singular madness

For an infinite cardinal , denotes the set of all cardinalities of nontrivial maximal almost disjoint families over .

Erdos and Hechler proved in 1973 the consistency of for a singular cardinal and asked if it was ever possible for a singular that , and also whether for every singular cardinal .

We introduce a new method for controlling for a singular and, among other new results about the structure of for singular , settle both problems affirmatively.

     

On commutators of fractional integrals

Let be the infinitesimal generator of an analytic semigroup on with Gaussian kernel bounds, and let be the fractional integrals of for . For a BMO function on , we show boundedness of the commutators from to , where . Our result of this boundedness still holds when is replaced by a Lipschitz domain of with infinite measure. We give applications to large classes of differential operators such as the magnetic Schrödinger operators and second-order elliptic operators of divergence form.

     

A note on commutativity up to a factor of bounded operators

In this note, we explore commutativity up to a factor for bounded operators and in a complex Hilbert space. Conditions on possible values of the factor are formulated and shown to depend on spectral properties of the operators. Commutativity up to a unitary factor is considered. In some cases, we obtain some properties of the solution space of the operator equation and explore the structures of and that satisfy for some A quantum effect is an operator on a complex Hilbert space that satisfies The sequential product of quantum effects and is defined by We also obtain properties of the sequential product.

     

Conservativeness of diffusion processes with drift

We show the conservativeness of the Girsanov transformed diffusion process by drift with or 4d/(d+2)$">, or if is of the Hardy class with sufficiently small coefficient of energy . Here 0$"> is the lower bound of the symmetric measurable matrix-valued function appearing in the given Dirichlet form. In particular, our result improves the conservativeness of the transformed process by when .

     

Approximation of holomorphic maps with a lower bound on the rank

Let be a closed polydisc or ball in , and let be a quasi-projective algebraic manifold which is Zariski locally equivalent to , or a complement of an algebraic subvariety of codimension in such a manifold. If is an integer satisfying , then every holomorphic map from a neighborhood of to with rank at every point of can be approximated uniformly on by entire maps with rank at every point of .

     

Distinguished representations and poles of twisted tensor -functions

Let be a quadratic extension of -adic fields. If is an admissible representation of that is parabolically induced from discrete series representations, then we prove that the space of -invariant linear functionals on has dimension one, where is the mirabolic subgroup. As a corollary, it is deduced that if is distinguished by , then the twisted tensor -function associated to has a pole at . It follows that if is a discrete series representation, then at most one of the representations and is distinguished, where is an extension of the local class field theory character associated to . This is in agreement with a conjecture of Flicker and Rallis that relates the set of distinguished representations with the image of base change from a suitable unitary group.

     

On Korenblum's maximum principle

Let be the Bergman space over the open unit disk in the complex plane. Korenblum's maximum principle states that there is an absolute constant , such that whenever ( ) in the annulus , then . In this paper we prove that Korenblum's maximum principle holds with .

     

Bounds of Gauss sums in finite fields

We consider Gauss sums of the form

with a nontrivial additive character of a finite field of elements of characteristic . The classical bound becomes trivial for . We show that, combining some recent bounds of Heath-Brown and Konyagin with several bounds due to Deligne, Katz, and Li, one can obtain the bound on which is nontrivial for the values of of order up to . We also show that for almost all primes one can obtain a bound which is nontrivial for the values of of order up to .

     

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

     

Mapping properties of analytic functions on the disk

There is a universal constant with the following property. Suppose that is an analytic function on the unit disk , and suppose that there exists a constant so that the Euclidean area, counting multiplicity, of the portion of which lies over the disk , centered at and of radius , is strictly less than the area of . Then must send into . This answers a conjecture of Don Marshall.

     

The positivity of linear functionals on Cuntz algebras associated to unit vectors

We study the linear functional on the Cuntz algebra associated to a sequence of unit vectors in that is a generalization of the Cuntz state. We prove that is positive if and only if is a constant sequence.

     

Algebraic characterizations of measure algebras

We present necessary and sufficient conditions for the existence of a countably additive measure on a Boolean -algebra. For instance, a Boolean -algebra is a measure algebra if and only if is the union of a chain of sets such that for every ,

(i)

every antichain in has at most elements (for some integer ),

(ii)

if is a sequence with for each , then , and

(iii)

for every , if is a sequence with , then for eventually all , .

The chain is essentially unique.

     

Closed similarity Lorentzian affine manifolds

A affine manifold is an -dimensional affine manifold whose linear holonomy lies in the similarity Lorentzian group but not in the Lorentzian group. In this paper, we show that a compact affine manifold is incomplete. Let be the Lorentz form, and the map on defined by . We show that for a compact radiant affine manifold , if a connected component of intersects the image of the universal cover of by the developing map, then either or a connected component of , where is a hyperplane, is contained in this image.