A bifurcation result for harmonic maps from an annulus to with not symmetric boundary data

We consider the problem of minimizing the energy of the maps from the annulus to such that is equal to for , and to , for , where is a fixed angle.

We prove that the minimum is attained at a unique harmonic map which is a planar map if , while it is not planar in the case \pi^2$">.

Moreover, we show that tends to as , where minimizes the energy of the maps from to , with the boundary condition , .

     

The number of Hall -subgroups of a -separable group

We observe a simple formula to compute the number of Hall -subgroups of a -separable finite group in terms of only the action of a fixed Hall -subgroup of on a set of normal -sections of . As a consequence, we obtain that divides whenever is a subgroup of a finite -separable group . This generalizes a recent result of Navarro. In addition, our method gives an alternative proof of Navarro's result.

     

Covering by graphs of -ary functions and long linear orderings of Turing degrees

A point is covered by a function iff there is a permutation of such that .

By a theorem of Kuratowski, for every infinite cardinal exactly -ary functions are needed to cover all of . We show that for arbitrarily large uncountable it is consistent that the size of the continuum is and is covered by -ary continuous functions.

We study other cardinal invariants of the -ideal on generated by continuous -ary functions and finally relate the question of how many continuous functions are necessary to cover to the least size of a set of parameters such that the Turing degrees relative to this set of parameters are linearly ordered.

     

Cellular generators

The aim of this paper is twofold. On the one hand, we show that the kernel of the Bousfield periodization functor is cellularly generated by a space , i.e., we construct a space such that the smallest closed class containing is exactly . On the other hand, we show that the partial order is a complete lattice, where if . Finally, as a corollary we obtain Bousfield's theorem, which states that )$"> is a complete lattice, where A$"> if .

     

On the evaluation of generalized Watson integrals

The triple integrals

and

where and are complex variables in suitably defined cut planes, were first evaluated by Watson in 1939 for the special cases and , respectively. In the present paper simple direct methods are used to prove that can be expressed in terms of squares of complete elliptic integrals of the first kind for general values of and . It is also shown that and are related by the transformation formula

where

Thus both of Watson's results for are contained within a single formula for .

     

The numerical range of a nilpotent operator on a Hilbert space

We prove that the numerical range of an arbitrary nilpotent operator on a complex Hilbert space is a circle (open or closed) with center at and radius not exceeding where is the power of nilpotency of

     

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 .

     

The duality theory of a finite dimensional discrete quantum group

Suppose that is a finite dimensional discrete quantum group and is a Hilbert space. This paper shows that if there exists an action of on so that is a modular algebra and the inner product on is -invariant, then there is a unique C*-representation of on supplemented by the The commutant of in is exactly the -invariant subalgebra of . As an application, a new proof of the classical Schur-Weyl duality theory of type A is given.

     

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 .

     

Characterizations of elements with compact support in the dual spaces of -modules of

For a locally compact group and , let be the Figà-Talamanca-Herz algebra and let be its dual Banach space. For a Banach -module of , we denote the norm closure of the subspace of the elements in with compact support by . We prove that an element of is in if and only if for any 0$">, there exists a compact subset of such that for all with and . In particular, we have that an element of is in if and only if for any 0$">, there exists a compact subset of such that for all with . If has an orthogonal complement in , we characterize by the following condition: is in if and only if for any 0$"> and any compact subset of , there exists some with and such that \Vert u\Vert - \epsilon $">. Some results of Flory (1971) and Miao (1999) can be obtained from our main theorems by taking and as some -subalgebras of .

     

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.

     

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 .

     

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.

     

The Russo-Dye Theorem in a weakly closed -module

Suppose that is admissible. It is shown that the convex hull of unitary elements of a weakly closed -module contains the whole unit ball of if and only if and for any 0$">, 0$">.

     

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.

     

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.

     

On embeddings of full amalgamated free product C-algebras

We examine the question of when the -homomorphism of full amalgamated free product C-algebras, arising from compatible inclusions of C-algebras , and , is an embedding. Results giving sufficient conditions for to be injective, as well as classes of examples where fails to be injective, are obtained. As an application, we give necessary and sufficient conditions for the full amalgamated free product of finite-dimensional C-algebras to be residually finite dimensional.

     

Dense subsets of the boundary of a Coxeter system

In this paper, we investigate dense subsets of the boundary of a Coxeter system. We show that for a Coxeter system , if is quasi-dense in and the order for some , then there exists a point in the boundary of the Coxeter system such that the orbit is dense in . Here . We also show that if the set is quasi-dense in , then is dense in .

     

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 .

     

Generalized frames and their redundancy

Let be a generalized frame in a separable Hilbert space indexed by a measure space , and assume its analysing operator is surjective. It is shown that is essentially discrete; that is, the corresponding index measure space can be decomposed into atoms such that is isometrically isomorphic to the weighted space of all sequences of complex numbers with , where This provides a new proof for the redundancy of the windowed Fourier transform as well as any wavelet family in .