The Resolvent Problem and $${{H^{\infty}}}$$ -calculus of the Stokes Operator in Unbounded Cylinders with Several Exits to Infinity

It is proved that the Stokes operator in L^q -space on an infinite cylindrical domain of , , with several exits to infinity generates a bounded and exponentially decaying analytic semigroup and admits a bounded -calculus. For the resolvent estimates, the Stokes resolvent system with a prescribed divergence in an infinite straight cylinder with bounded cross-section is studied in L ^q where and is an arbitrary Muckenhoupt weight. The proofs use cut-off techniques and the theory of Schauder decomposition of UMD spaces based on -boundedness of operator families and on square function estimates involving Muckenhoupt weights.     

Number of Complete N-ary Subtrees on Galton-Watson Family Trees

We associate with a Bienaymé-Galton-Watson branching process a family tree rooted at the ancestor. For a positive integer , define a complete -ary tree to be the family tree of a deterministic branching process with offspring generating function . We study the random variables and counting the number of disjoint complete -ary subtrees, rooted at the ancestor, and having height and , respectively. Dekking (1991) and Pakes and Dekking (1991) find recursive relations for and involving the offspring probability generation function (pgf) and its derivatives. We extend their results determining the probability distributions of and . It turns out that they can be expressed in terms of the offspring pgf, its derivatives, and the above probabilities. We show how the general results simplify in case of fractional linear, geometric, Poisson, and one-or-many offspring laws.      

On the Structure and Symmetry Properties of Almost $$\cal{S}$$-manifolds

We prove that any simply connected -manifold of CR-codimension s 2 is noncompact by showing that the complete, simply connected -manifolds are all the CR products N × ^{s-1} with N Sasakian, endowed with a suitable product metric. N is a Sasakian -symmetric space if and only if M is CR-symmetric. The locally CR-symmetric -manifolds are characterized by

The Invariance Principle for p-θ^→ Chain

There are two parts in this paper. In the first part we construct the Markov chain in random environment（MCRE）, the skew product Markov chain and p-θ^→ chain from a random transition matrix and a two-dimensional probability distribution, and in the second part we prove that the invarianee principle for p-θ^→ chain, a more complex non-homogeneous Markov chain, is true under some reasonable conditions. This result is more powerful.     

On the Schwarz reflection principle for monogenic functions

Let ∑ be either an oriented hyperplane or the unit sphere in , let be open and connected and let be an open and connected domain in such that . If in is a null solution of the Dirac operator (also called a monogenic function in ) which is continuously extendable to , then conditions upon are given enabling the monogenic extension of across . In such a way Schwarz reflection type principles for monogenic functions are established in the Spin (1) and Spin cases. The Spin (1) case includes the classical Schwarz reflection principle for holomorphic functions in the plane. The Spin case deals with so-called “half boundary value problems” for the Dirac operator. Received: 2 February 2006     

Fixed points in {\mathcal{M}}-fuzzy metric spaces

In this paper, we give some common fixed point theorems for five mappings satisfying some conditions in -fuzzy metric spaces.     

Lectures on the Orbital Stability of Standing Waves and Application to the Nonlinear Schrödinger Equation

In the first part of these notes, we deal with first order Hamiltonian systems in the form where the phase space X may be infinite dimensional so as to accommodate some partial differential equations. The Hamiltonian is required to be invariant with respect to the action of a group of isometries where is skew-symmetric and JA  = AJ. A standing wave is a solution having the form for some and such that . Given a solution of this type, it is natural to investigate its stability with respect to perturbations of the initial condition. In this context, the appropriate notion of stability is orbital stability in the usual sense for a dynamical system. We present some of the important criteria for establishing orbital stability of standing waves. In the second part we consider the nonlinear Schr?dinger equation which provides an interesting example of this situation where standing waves appear as time-harmonic solutions. We show how the general theory applies to this case and review what is known about stability. Received: January 2008     

Permutable subnormal subgroups of finite groups

The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called -groups. In particular, it is shown that the finite solvable -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugate-permutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not -groups but all subnormal subgroups of defect two are permutable. Received: 13 August 2008     

On microfunctions at the boundary along CR manifolds

Let X be a complex analytic manifold, a C ² submanifold, an openset with C ² boundary .Denote by (resp. ) the microlocalization along M (resp. ) of the sheaf of holomorphic functions.In the literature (cf. [A-G], [K-S 1,2])one encounters two classical results concerning the vanishing of the cohomology groups .The most general gives the vanishing outside a range of indices j whose length is equal to (with being the number of respectively positive, negative and null eigenvalues for themicrolocal Levi form ).The sharpest result gives the concentration in a single degree, provided that the difference is locally constant for near p (with for z the base point of p).The first result was restated for the complex in [D'A-Z 2], in the case codim We extend it here to any codimension and moreover we also restate for the second vanishing theorem.We also point out that the principle of our proof, related to a criterion for constancy of sheaves due to [K-S 1], is a quite new one.     

On Complete Arcs Arising from Plane Curves

We point out an interplay between -Frobenius non-classical plane curves and complete -arcs in . A typical example that shows how this works is the one concerning an Hermitian curve. We present some other examples here which give rise to the existence of new complete -arcs with parameters and being a power of the characteristic. In addition, for q a square, new complete -arcs with either and or and are constructed by using certain reducible plane curves.     

Hilbert-Schmidt Hankel Operators on the Bergman Space of Planar Domains

In this paper we study the problem of the membership of H _ϕ in the Hilbert-Schmidt class, when and Ω is a planar domain. We find a necessary and sufficient condition.We apply this result to the problem of joint membership of H _φ and in the Hilbert-Schmidt class. Using the notion of Berezin Transform and a result of K. Zhu we are able to give a necessary and sufficient condition. Finally, we recover a result of Arazy, Fisher and Peetre on the case with φ holomorphic.     

A Topological View of Ramsey Families of Finite Subsets of Positive Integers

If is an initially hereditary family of finite subsets of positive integers (i.e., if and G is initial segment of F then ) and M an infinite subset of positive integers then we define an ordinal index . We prove that if is a family of finite subsets of positive integers such that for every the characteristic function χ_F is isolated point of the subspace

Passive Systems with a Normal Main Operator and Quasi-selfadjoint Systems

Passive systems with and as an input and output space and as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system with is said to be quasi-selfadjoint if ran . The subclass of the Schur class is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass and the Q-function of T is given. Received: December 16, 2007., Accepted: March 4, 2008.     

The Unfolding of Equivariant Bifurcation Problems with Two Types of State Variables in the Presence of Parameter Symmetry

The unfolding of equivariant bifurcation problems with two types of state variables under the action of group K（Г, △） is discussed by using DA-algebraic tools. One of the main results is the equivariant versal unfolding theorem.     

Redefined fuzzy B-algebras

By two relations belonging to and quasi-coincidence (q) between fuzzy points and fuzzy sets, we define the concept of (α, β)-fuzzy subalgebras where α, β are any two of with . We state and prove some theorems in (α, β)-fuzzy B-algebras.     

An Application of U(g)-bimodules to Representation Theory of Affine Lie Algebras

Let be the affine Lie algebra associated to the simple finite-dimensional Lie algebra . We consider the tensor product of the loop -module associated to the irreducible finite-dimensional -module V() and the irreducible highest weight -module L _k,. Then L _k, can be viewed as an irreducible module for the vertex operator algebra M _k,0. Let A(L _k,) be the corresponding -bimodule. We prove that if the -module is zero, then the -module is irreducible. As an example, we apply this result on integrable representations for affine Lie algebras.     

Schur–Weyl duality for higher levels

We extend Schur–Weyl duality to an arbitrary level l ≥ 1, level one recovering the classical duality between the symmetric and general linear groups. In general, the symmetric group is replaced by the degenerate cyclotomic Hecke algebra over parametrized by a dominant weight of level l for the root system of type A_∞. As an application, we prove that the degenerate analogue of the quasi-hereditary cover of the cyclotomic Hecke algebra constructed by Dipper, James and Mathas is Morita equivalent to certain blocks of parabolic category for the general linear Lie algebra.      

Sur la cohomologie à supports compacts des variétés de Shimura pour GSp(4)_\mathbb{Q}

In this paper we compute the cohomology with compact supports of a Siegelthreefold as a virtual module over the product of the Galois group of over and the Hecke algebra. We use a method which has been developed by Ihara, Langlands and Kottwitz: comparison of the Grothendieck--Lefschetz formula and the Arthur--Selberg trace formula.     

Hyperbolic Function Theory in the Clifford Algebra {\mathcal {C}}\ell_{n+1, 0}

The aim of this paper is to give the basic principles of hyperbolic function theory on the Clifford algebra . The structure of the theory is quite similar to the case of Clifford algebras with negative generators, but the proofs are not obvious. The (real) Clifford algebra is generated by unit vectors with positive squares e²_i = + 1. The hyperbolic Dirac operator is of the form where Q₀f is represented by the composition . If is a solution of H_kf = 0, then f is called k-hypergenic in Ω, where is an open set. We introduce some basic results of hyperbolic function theory and give some representation theorems on . Received: October, 2007. Accepted: February, 2008.