Transference for amenable actions

Suppose acts amenably on a measure space with quasi-invariant -finite measure . Let be an isometric representation of on and a finite Radon measure on . We show that the operator has -operator norm not exceeding the -operator norm of the convolution operator defined by . We shall also prove an analogous result for the maximal function associated to a countable family of Radon measures .

     

Multiplicity results for a class of superlinear elliptic problems

We study a class of superlinear elliptic problems under the Dirichlet boundary condition on a bounded smooth domain in . Assuming that the nonlinearity is superlinear in a neighborhood of , we study the dependence of the number of signed and sign-changing solutions on the parameter .

     

Tietze extension theorem for Hilbert -modules

We prove the following generalization of the noncommutative Tietze extension theorem: if is a countably generated Hilbert -module over a -unital -algebra, then the canonical extension of a surjective morphism of Hilbert -modules to extended (multiplier) modules, , is also surjective.

     

-function of an operator: A white noise approach

Let be the canonical framework of white noise analysis over the Gel'fand triple and be the space of continuous linear operators from to . Let be a self-adjoint operator in with spectral representation . In this paper, it is proved that under appropriate conditions upon , there exists a unique linear mapping such that for each . The mapping is then naturally used to define as , where is the Dirac -function. Finally, properties of the mapping are investigated and several results are obtained.

     

Remarks concerning linear characters of reflection groups

Let be a finite group generated by unitary reflections in a Hermitian space , and let be a root of unity. Let be a subspace of , maximal with respect to the property of being a -eigenspace of an element of , and let be the parabolic subgroup of elements fixing pointwise. If is any linear character of , we give a condition for the restriction of to to be trivial in terms of the invariant theory of , and give a formula for the polynomial , where is the dimension of the -eigenspace of . Applications include criteria for regularity, and new connections between the invariant theory and the structure of .

     

On the projective-injective modules over cellular algebras

We show that the projective module over a cellular algebra is injective if and only if the socle of coincides with the top of , and this is also equivalent to the condition that the th socle layer of is isomorphic to the th radical layer of for each positive integer . This eases the process of determining the Loewy series of the projective-injective modules over cellular algebras.

     

Associated primes of local cohomology modules

Let be an ideal of a commutative Noetherian ring and a finitely generated -module. Let be a natural integer. It is shown that there is a finite subset of , such that is contained in union with the union of the sets , where and . As an immediate consequence, we deduce that the first non- -cofinite local cohomology module of with respect to has only finitely many associated prime ideals.

     

Perturbed smooth Lipschitz extensions of uniformly continuous functions on Banach spaces

We show that if is a separable subspace of a Banach space such that both and the quotient have -smooth Lipschitz bump functions, and is a bounded open subset of , then, for every uniformly continuous function and every 0$">, there exists a -smooth Lipschitz function such that for every .

     

Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert space

Let be a real Hilbert space. Let , be bounded monotone mappings with , where and are closed convex subsets of satisfying certain conditions. Suppose the equation has a solution in . Then explicit iterative methods are constructed that converge strongly to such a solution. No invertibility assumption is imposed on , and the operators and need not be defined on compact subsets of .

     

Witt kernels of bilinear forms for algebraic extensions in characteristic

Let be a field of characteristic and let be a purely inseparable extension of exponent . We determine the kernel of the natural restriction map between the Witt rings of bilinear forms of and , respectively. This complements a result by Laghribi who computed the kernel for the Witt groups of quadratic forms for such an extension . Based on this result, we will determine for a wide class of finite extensions which are not necessarily purely inseparable.

     

On stable equivalences of Morita type for finite dimensional algebras

In this paper, we assume that algebras are finite dimensional algebras with 1 over a fixed field and modules over an algebra are finitely generated left unitary modules. Let and be two algebras (where is a splitting field for and ) with no semisimple summands. If two bimodules and induce a stable equivalence of Morita type between and , and if maps any simple -module to a simple -module, then is a Morita equivalence. This conclusion generalizes Linckelmann's result for selfinjective algebras. Our proof here is based on the construction of almost split sequences.

     

-additive families and the invariance of Borel classes

We prove that any -additive family of sets in an absolutely Souslin metric space has a -discrete refinement provided every partial selector set for is -discrete. As a corollary we obtain that every mapping of a metric space onto an absolutely Souslin metric space, which maps -sets to -sets and has complete fibers, admits a section of the first class. The invariance of Borel and Souslin sets under mappings with complete fibers, which preserves -sets, is shown as an application of the previous result.

     

On -supplemented maximal and minimal subgroups of Sylow subgroups of finite groups

This paper proves: Let be a saturated formation containing . Suppose that is a group with a normal subgroup such that .

(1) If all maximal subgroups of any Sylow subgroup of are -supple- mented in , then ;

(2) If all minimal subgroups and all cyclic subgroups with order 4 of are -supplemented in , then .

     

Linear series over real and -adic fields

We note that the degeneration arguments given by the author in 2003 to derive a formula for the number of maps from a general curve of genus to with prescribed ramification also yields weaker results when working over the real numbers or -adic fields. Specifically, let be such a field: we see that given , , , and satisfying , there exists smooth curves of genus together with points such that all maps from to can, up to automorphism of the image, be defined over . We also note that the analagous result will follow from maps to higher-dimensional projective spaces if it is proven in the case , , and that thanks to work of Sottile, unconditional results may be obtained for special ramification conditions.

     

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.

     

The Hausmann-Weinberger 4-manifold invariant of abelian groups

The Hausmann-Weinberger invariant of a group is the minimal Euler characteristic of a closed orientable 4-manifold with fundamental group . We compute this invariant for finitely generated free abelian groups and estimate the invariant for all finitely generated abelian groups.

     

A simple closure condition for the normal cone intersection formula

In this paper it is shown that if and are two closed convex subsets of a Banach space and , then whenever the convex cone, , is weak* closed, where and are the support function and the normal cone of the set respectively. This closure condition is shown to be weaker than the standard interior-point-like conditions and the bounded linear regularity condition.

     

Characterization of scaling functions in a multiresolution analysis

We characterize the scaling functions of a multiresolution analysis in a general context, where instead of the dyadic dilation one considers the dilation given by a fixed linear map such that and all (complex) eigenvalues of have absolute value greater than In the general case the conditions depend on the map We identify some maps for which the obtained condition is equivalent to the dyadic case, i.e., when is a diagonal matrix with all numbers in the diagonal equal to There are also easy examples of expanding maps for which the obtained condition is not compatible with the dyadic case. The complete characterization of the maps for which the obtained conditions are equivalent is out of the scope of the present note.

     

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.

     

-Weyl's theorem for operator matrices

If is a upper triangular matrix on the Hilbert space , then -Weyl's theorem for and need not imply -Weyl's theorem for , even when . In this note we explore how -Weyl's theorem and -Browder's theorem survive for operator matrices on the Hilbert space.