An Extension of the Admissibility-Type Conditions for the Exponential Dichotomy of C 0-Semigroups

In the present paper we obtain a sufficient condition for the exponential dichotomy of a strongly continuous, one-parameter semigroup , in terms of the admissibility of the pair . It is already known the equivalence between the -admissibility condition and and the hyperbolicity of a C ₀-semigroup , when we assume a priori that the kernel of the dichotomic projector (denoted here by X ₂) is T(t)-invariant and is an invertible operator. We succeed to prove in this paper that the admissibility of the pair still implies the existence of an exponential dichotomy for a C ₀-semigroup even in the general case where the kernel of the dichotomic projector, X ₂, is not assumed to be T(t)-invariant.      

Invariance of Fock spaces under the action of the Heisenberg group

We show that there is only one non-trivial Hilbert space of entire functions that is invariant under the action of a certain unitary representation of the Heisenberg group.     

An algebraic slice in the coadjoint space of the Borel and the Coxeter element

Let g be a complex simple Lie algebra and b a Borel subalgebra. The algebra Y of polynomial semi-invariants on the dual b^? of b is a polynomial algebra on rank g generators (Grothendieck and Dieudonné (1965–1967)) [16]. The analogy with the semisimple case suggests there exists an algebraic slice to coadjoint action, that is an affine translate y+V of a vector subspace of b^? such that the restriction map induces an isomorphism of Y onto the algebra R[y+V] of regular functions on y+V. This holds in type A and even extends to all biparabolic subalgebras (Joseph (2007)) [20]; but the construction fails in general even with respect to the Borel. Moreover already in type C(2) no algebraic slice exists.Very surprisingly the exception of type C(2) is itself an exception. Indeed an algebraic slice for the coadjoint action of the Borel subalgebra is constructed for all simple Lie algebras except those of types B(2m), C(n) and F(4).Outside type A, the slice obtained meets an open dense subset of regular orbits, even though the special point y of the slice is not itself regular. This explains the failure of our previous construction.     

Weighted integral inequalities for solutions of the A-harmonic equation

We prove the parametric versions of -weighted integral inequalities for differential forms satisfying the A-harmonic equation. These results can be considered as extensions of the classical inequalities for Sobolev functions.     

Some characterizations of the Besov space and the α-Bloch space

In this paper we obtain some characterizations of Besov spaces and α-Bloch spaces on the unit ball of Cn, which extend some results in the settings of the unit disk.     

On the Wl q-Regularity of Solutions to Systems of Differential Equations in the Case When the Equations Are Constructed from Discontinuous Functions

Some solution, final in a sense from the standpoint of the theory of Sobolev spaces, is obtained to the problem of regularity of solutions to a system of (generally) nonlinear partial differential equations in the case when the system is locally close to elliptic systems of linear equations with constant coefficients. The main consequences of this result are Theorems 5 and 8. According to the first of them, the higher derivatives of an elliptic C ^l -smooth solution to a system of lth-order nonlinear partial differential equations constructed from C ^l -smooth functions meet the local Hoelder condition with every exponent , 0<<1. Theorem 8 claims that if a system of linear partial differential equations of order l with measurable coefficients and right-hand sides is uniformly elliptic then, under the hypothesis of a (sufficiently) slow variation of its leading coefficients, the degree of local integrability of lth-order partial derivatives of every W ^l _q,loc-solution, q>1, to the system coincides with the degree of local integrability of lower coefficients and right-hand sides.     

On the Homotopy Type of the Unitary Group and the Grassmann Space of Purely Infinite Simple C*-Algebras

We completely determine the homotopy groups _n (.) of the unitary group and the space of projections of purely infinite simple C ^*-algebras in terms of K-theory. We also prove that the unitary group of a purely infinite simple C ^*-algebra A is a contractible topological space if and only if K0(A) = K1(A) = {0}, and again if and only if the unitary group of the associated generalized Calkin algebra L(HA) / K(HA) is contractible. The well-known Kuiper's theorem is extended to a new class of C ^*-algebras.     

An identity of Andrews and the Askey-Wilson integral

Andrews (Adv. Math. 41:137–172, 1981) derived a four-variable q-series identity, which is an extension of the Ramanujan ₁ ψ ₁ summation. In this paper, we shall give a simple evaluation of the Askey-Wilson integral by using this identity. The author was supported by the National Science Foundation of China, PCSIRT and Innovation Program of Shanghai Municipal Education Commission.     

On the number of groups in clustering

Clustering is the problem of partitioning data into a finite number k of homogeneous and separate groups, called clusters. A good choice of k is essential for building meaningful clusters. In this paper, this task is addressed from the point of view of model selection via penalization. We design an appropriate penalty shape and derive an associated oracle-type inequality. The method is illustrated on both simulated and real-life data sets.     

On the irrationality of factorial series III

In this paper we derive some irrationality and linear independence results for series of the form where is either a non-negative integer sequence with υ_n = o(log n/log log n) or a non-decreasing integer sequence with .     

Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold

The paper studies the first homology of finite regular branched coverings of a universal Borromean orbifold called B _4,4,4ℍ³. We investigate the irreducible components of the first homology as a representation space of the finite covering transformation group G. This gives information on the first betti number of finite coverings of general 3-manifolds by the universality of B _4,4,4. The main result of the paper is a criterion in terms of the irreducible character whether a given irreducible representation of G is an irreducible component of the first homology when G admits certain symmetries. As a special case of the motivating argument the criterion is applied to principal congruence subgroups of B _4,4,4. The group theoretic computation shows that most of the, possibly nonprincipal, congruence subgroups are of positive first Betti number. This work is partially supported by the Sonderforschungsbereich 288.     

Approximation in the mean by rational functions

In this paper we proved that for a large class of compact subsetsK in the complex plane,R(K) is dense inL ^q () if and only if the set of analytic bounded point evaluations forR ^q (K, ) is empty. As a consequence, we showed that this result is true for allK ifR(K) is replaced byA(K). Our main result includes the corresponding result of James Thomson for polynomials approximation as such a special case thatK is a disk.     

The free spectrum of the variety of bands

In this short note we give an asymptotic formula for the p _n sequence of the variety of bands, namely, for some constant K. This yields a formula for the free spectrum of this variety.     

Nonlinear stability of traveling wave solutions for the compressible fluid models of Korteweg type

This paper is concerned with the existence and time-asymptotic nonlinear stability of traveling wave solutions to the Cauchy problem of the one-dimensional compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. The existence of traveling wave solutions is obtained by the phase plane analysis, then the traveling wave solution is shown to be asymptotically stable by the elementary L²$L^2$-energy method.     

On the structure of the K2 of the ring of integers in a number field

For any prime p, the p-primary part of the tame and the wild kernel of a number field F is described in terms of ideal class groups of p-power cyclotomic extensions of F.     

Some remarks on the asymptotic profiles of solutions for strongly damped wave equations on the 1-D half space

We consider a mixed problem with the Dirichlet null boundary conditions on the 1-dimensional half space (0,+∞)$(0,+\infty )$. We will derive an asymptotic profiles of the corresponding solutions in the case when the initial data belong to a weighted L^1,2(0,∞)$L^{1,2}(0,\infty )$ space by employing a method recently introduced in [7] or [8].