Existence and uniqueness theorems for singular anisotropic quasilinear elliptic boundary value problems

On bounded domains we consider the anisotropic problems in with 1$"> and on and in with and on . Moreover, we generalize these boundary value problems to space-dimensions 2$">. Under geometric conditions on and monotonicity assumption on we prove existence and uniqueness of positive solutions.     

Normalizers of the congruence subgroups of the Hecke group II

Let . Let be an ideal of and let be the maximal ideal of such that . Then . In particular, if is square free, then is self-normalized in .

     

Uniqueness of the least-energy solution for a semilinear Neumann problem

We prove that the least-energy solution of the problem

where is a ball, and if , if , is unique (up to rotation) if is small enough.

     

The Postnikov Tower and the Steenrod problem

The Steenrod problem asks: given a -module, when does there exist a Moore space realizing the module? By using the equivariant Postnikov Tower, it is shown that a -module is -realizable if and only if it is -realizable for all -Sylow subgroups , for all primes .

     

A two-dimensional Hahn-Banach theorem

Let , where and is a Banach space. Let be an extension of to all of (i.e., ) such that has minimal (operator) norm. In this paper we show in particular that, in the case and the field is R, there exists a rank- such that for all if and only if the unit ball of is either not smooth or not strictly convex. In this case we show, furthermore, that, for some , there exists a choice of basis such that ; i.e., each is a Hahn-Banach extension of .

     

The Dedekind-Mertens lemma and the contents of polynomials

Let be a commutative ring, let be an indeterminate, and let . There has been much recent work concerned with determining the Dedekind-Mertens number =min , especially on determining when = . In this note we introduce a universal Dedekind-Mertens number , which takes into account the fact that deg() + for any ring containing as a subring, and show that behaves more predictably than .

     

Dense subsets of maximally almost periodic groups

A (discrete) group is said to be maximally almost periodic if the points of are distinguished by homomorphisms into compact Hausdorff groups. A Hausdorff topology on a group is totally bounded if whenever there is such that . For purposes of this abstract, a family with a totally bounded topological group is a strongly extraresolvable family if (a)  \vert G\vert$">, (b) each is dense in , and (c) distinct satisfy ; a totally bounded topological group with such a family is a strongly extraresolvable topological group.

We give two theorems, the second generalizing the first.

Theorem 1. Every infinite totally bounded group contains a dense strongly extraresolvable subgroup.

Corollary. In its largest totally bounded group topology, every infinite Abelian group is strongly extraresolvable.

Theorem 2. Let be maximally almost periodic. Then there are a subgroup of and a family such that

(i) is dense in every totally bounded group topology on ;

(ii) the family is a strongly extraresolvable family for every totally bounded group topology on such that ; and

(iii) admits a totally bounded group topology as in (ii).

Remark. In certain cases, for example when is Abelian, one must in Theorem 2 choose . In certain other cases, for example when the largest totally bounded group topology on is compact, the choice is impossible.

     

A classification of prime segments in simple artinian rings

Let be a simple artinian ring. A valuation ring of is a Bézout order of so that is simple artinian, a Goldie prime is a prime ideal of so that is Goldie, and a prime segment of is a pair of neighbouring Goldie primes of A prime segment is archimedean if is equal to it is simple if and it is exceptional if In this last case, is a prime ideal of so that is not Goldie. Using the group of divisorial ideals, these results are applied to classify rank one valuation rings according to the structure of their ideal lattices. The exceptional case splits further into infinitely many cases depending on the minimal so that is not divisorial for

     

A topological property of integrable systems

If we are given real-valued smooth functions on which are in involution, then, under some mild hypotheses, the subset of where these functions are linearly independent is not simply connected.

     

Purely infinite, simple -algebras arising from free product constructions. III

In the reduced free product of C-algebras, with respect to faithful states and , is purely infinite and simple if is a reduced crossed product for an infinite group, if is well behaved with respect to this crossed product decomposition, if and if is not a trace.     

Are covering (enveloping) morphisms minimal?

We prove that for certain classes of modules such that direct sums of -covers ( -envelopes) are -covers ( -envelopes), -covering ( -enveloping) homomorphisms are always right (left) minimal. As a particular case we see that over noetherian rings, essential monomorphisms are left minimal. The same type of results are given when direct products of -covers are -covers. Finally we prove that over commutative noetherian rings, any direct product of flat covers of modules of finite length is a flat cover.     

A refinement of the toral rank conjecture for 2-step nilpotent Lie algebras

It is known that the total (co)-homoloy of a 2-step nilpotent Lie algebra is at least , where is the center of . We improve this result by showing that a better lower bound is , where and is a complement of in . Furthermore, we provide evidence that this is the best possible bound of the form .

     

Sufficient conditions for one domain to contain another in a space of constant curvature

As an application of the analogue of C-S. Chen's kinematic formula in the 3-dimensional space of constant curvature , that is, Euclidean space , -sphere , hyperbolic space (, respectively), we obtain sufficient conditions for one domain to contain another domain in either an Euclidean space , or a -sphere or a hyperbolic space .

     

Linear functionals on the Cuntz algebra

For a pure state on , which is an extension of a pure state on with the property that if is a corresponding representation, then , induces a unital shift of of the Powers index . We describe states on by using sequences of unit vectors in . We study the linear functionals on the Cuntz algebra whose restrictions are the product pure state on . We find conditions on the sequence of unit vectors for which the corresponding linear functionals on become states under these conditions.

     

Inequalities for the gamma function

We prove the following two theorems:

(i) Let be the th power mean of and . The inequality

holds for all if and only if , where denotes Euler's constant. This refines results established by W. Gautschi (1974) and the author (1997).

(ii) The inequalities

are valid for all if and only if and , while holds for all if and only if and . These bounds for improve those given by G. D. Anderson an S.-L. Qiu (1997).

     

A note on branching theorems

Let be a complex, simply connected semisimple analytic group with a closed connected reductive subgroup. Suppose is an irreducible holomorphic -module and an irreducible holomorphic -module. We prove that Hom possesses the structure of an irreducible -module whenever is . Moreover, for all and if and only if is commutative.

     

Multiplicative bijections of

Let be a compact Hausdorff space which satisfies the first axiom of countability, let and let , be the set of all continuous functions from to If , ,is a bijective multiplicative map, then there exist a homeomorphism and a continuous map such that for all and for all

     

The Hausdorff operator is bounded on the real Hardy space

We prove that the Hausdorff operator generated by a function is bounded on the real Hardy space . The proof is based on the closed graph theorem and on the fact that if a function in is such that its Fourier transform equals for (or for ), then .

     

A continuous decomposition of the Menger curve into pseudo-arcs

It is proved that the Menger universal curve admits a continuous decomposition into pseudo-arcs with the quotient space homeomorphic to .