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.

     

Convergence theorem for zeros of generalized Lipschitz generalized phi-quasi-accretive operators

Let be a uniformly smooth real Banach space and let be a mapping with . Suppose is a generalized Lipschitz generalized -quasi-accretive mapping. Let and be real sequences in [0,1] satisfying the following conditions: (i) ; (ii) ; (iii) ; (iv) Let be generated iteratively from arbitrary by

where is defined by and is an arbitrary bounded sequence in . Then, there exists such that if the sequence converges strongly to the unique solution of the equation . A related result deals with approximation of the unique fixed point of a generalized Lipschitz and generalized -hemi-contractive mapping.

     

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.     

Blowup for

In this note we consider the global regularity of smooth solutions to the vector-valued Cauchy problem

We show that if , the gradient-blowup phenomenon occurs in finite time for suitably chosen vanishing at infinity. We also present a simple example of the -blowup solutions for for any 0$">, if .

     

Enriched Reedy categories

We define the notion of an enriched Reedy category and show that if is a -Reedy category for some symmetric monoidal model category and is a -model category, the category of -functors and -natural transformations from to is again a model category.

     

On the Diophantine equation

Let be an odd prime. In this paper, using some theorems of Adachi and the author, we prove that if and , then the equation , and the equation , have no integral solutions respectively. Here is th Bernoulli number.

     

On functions whose graph is a Hamel basis

We say that a function is a Hamel function ( ) if , considered as a subset of , is a Hamel basis for . We prove that every function from into can be represented as a pointwise sum of two Hamel functions. The latter is equivalent to the statement: for all there is a such that . We show that this fails for infinitely many functions.

     

The -invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Let be a regular local ring and let be a filtration of ideals in such that is a Noetherian ring with . Let and let be the -invariant of . Then the theorem says that is a principal ideal and for all if and only if is a Gorenstein ring and . Hence , if is a Gorenstein ring, but the ideal is not principal.

     

Codes over rings of size four, Hermitian lattices, and corresponding theta functions

Let be an imaginary quadratic field with ring of integers , where is a square free integer such that , and let is a linear code defined over . The level theta function of is defined on the lattice , where is the natural projection. In this paper, we prove that:

i) for any such that , and have the same coefficients up to ,

ii) for , determines the code uniquely,

iii) for , there is a positive dimensional family of symmetrized weight enumerator polynomials corresponding to .

     

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

     

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.

     

The structure of quantum spheres

We show that the C*-algebra of a quantum sphere , 1$">, consists of continuous fields of operators in a C*-algebra , which contains the algebra of compact operators with , such that is a constant function of , where is the quotient map and is the unit circle.

     

Sur une question de capitulation

Let and be prime numbers such that and . Let , , and let be the 2-Hilbert class field of , the 2-Hilbert class field of and the Galois group of . The 2-part of the class group of is of type , so contains three extensions . Our goal is to study the problem of capitulation of the 2-classes of in , and to determine the structure of .

RSESUM´E. Soient et deux nombres premiers tels que et , , , , le 2-corps de classes de Hilbert de , le 2-corps de classes de Hilbert de et le groupe de Galois de . La 2-partie du groupe de classes de est de type , par suite contient trois extensions . On s'intéresse au problème de capitulation des 2-classes de dans , et à déterminer la structure de .

     

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.     

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 .

     

The real rank zero property of crossed product

Let be a unital -algebra, and let be a -dynamical system with abelian and discrete. In this paper, we introduce the continuous affine map from the trace state space of the crossed product to the -invariant trace state space of . If is of real rank zero and is connected, we have proved that is homeomorphic. Conversely, if is homeomorphic, we also get some properties and real rank zero characterization of . In particular, in that case, is of real rank zero if and only if each unitary element in with the form can be approximated by the unitary elements in with finite spectrum, where , , and if moreover is a unital inductive limit of the direct sums of non-elementary simple -algebras of real rank zero, then the above can be cancelled.

     

On the triple jump of the set of atoms of a Boolean algebra

We prove the following result concerning the degree spectrum of the atom relation on a computable Boolean algebra. Let be a computable Boolean algebra with infinitely many atoms and be the Turing degree of the atom relation of . If is a c.e. degree such that , then there is a computable copy of where the atom relation has degree . In particular, for every c.e. degree , any computable Boolean algebra with infinitely many atoms has a computable copy where the atom relation has degree .

     

Sobolev spaces and the Cayley transform

The generalised Cayley transform  from an Iwasawa -group into the corresponding real unit sphere  induces isomorphisms between suitable Sobolev spaces and . We study the differential of  , and we obtain a criterion for a function to be 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.

     

Obstructions to deformations of d.g. modules

Let be a field and . There exist a differential graded -module and various approximations to a differential on one of which gives a non-trivial deformation, another is obstructed, and another is unobstructed at order . The analogous problem in the category of -algebras in characteristic zero remains a long-standing open question.