Strong and uniform mean stability of cosine and sine operator functions

It is first observed that a uniformly bounded cosine operator function C() and the associated sine function S() are totally non-stable. Then, using a zero-one law for the Abel limit of a closed linear operator, we prove some results concerning strong mean stability and uniform mean stability of C(). Among them are: (1) C() is strongly (C,1)-mean stable (or (C,2)-mean stable, or Abel-mean stable) if and only if 0ρ(A)σ_c(A); (2) C() is uniformly (C,2)-mean stable if and only if S() is uniformly (C,1)-mean stable, if and only if , if and only if , if and only if C() is uniformly Abel-mean stable, if and only if S() is uniformly Abel-mean stable, if and only if 0ρ(A).     

Two canonical passive state/signal shift realizations of passive discrete time behaviors

A discrete time invariant linear state/signal system Σ with a Hilbert state space and a Kren signal space has trajectories (x(),w()) that are solutions of the equation , where F is a bounded linear operator from into with a closed domain whose projection onto is all of . This system is passive if the graph of F is a maximal nonnegative subspace of the Kren space . The future behavior of a passive system Σ is the set of all signal components w() of trajectories (x(),w()) of Σ on with x(0)=0 and . This is always a maximal nonnegative shift-invariant subspace of the Kren space , i.e., the space endowed with the indefinite inner product inherited from . Subspaces of with this property are called passive future behaviors. In this work we study passive state/signal systems and passive behaviors (future, full, and past). In particular, we define and study the input and output maps of a passive state/signal system, and the past/future map of a passive behavior. We then turn to the inverse problem, and construct two passive state/signal realizations of a given passive future behavior , one of which is observable and backward conservative, and the other controllable and forward conservative. Both of these are canonical in the sense that they are uniquely determined by the given data , in contrast earlier realizations that depend not only on , but also on some arbitrarily chosen fundamental decomposition of the signal space . From our canonical realizations we are able to recover the two standard de Branges–Rovnyak input/state/output shift realizations of a given operator-valued Schur function in the unit disk.     

Existence of vector minimizers for nonconvex 1-dim integrals with almost convex Lagrangian

This paper proves new results of existence of minimizers for the nonconvex integral , among the AC functions with x(a)=A, x(b)=B. Our Lagrangian L() is e.g. lsc with superlinear growth, assuming +∞ values freely. We replace convexity by almost convexity, a hypothesis which in the radial superlinear case L(s,ξ)=f(s,|ξ|) is automatically satisfied provided f(s,) is convex at zero.     

Projective change between two classes of -metrics

In this paper, we find equations to characterize projective change between (α,β)-metric and Randers metric on a manifold with dimension n3, where α and are two Riemannian metrics, β and are two nonzero one forms. Moreover, we consider this projective change when F has some special curvature properties.     

Propagating sharp group homology decompositions

A collection of subgroups of a finite group G can give rise to three different standard formulas for the cohomology of G in terms of either the subgroups in or their centralizers or their normalizers. We give a short but systematic study of the relationship among such formulas for nine standard collections of p-subgroups, obtaining some new formulas in the process. To do this, we exhibit some sufficient conditions on the poset which imply comparison results.     

On the dual of a Coulter–Matthews bent function

Coulter–Matthews (CM) bent functions are from to defined by , where and (α,2n)=1. It is not known if these bent functions are weakly regular in general. In this paper, we show that when n is even and α=n+1 (or n−1), the CM bent function is weakly regular. Moreover, we explicitly determine the dual of the CM bent function in this case. The dual is a bent function not reported previously.     

A supplement to the Davis–Gut law

Let {X,X_n;n1} be a sequence of i.i.d. real-valued random variables and set , n1. Let h() be a positive nondecreasing function such that . Define Lt=log_emax{e,t} for t0. In this note we prove that

if and only if E(X)=0 and E(X²)=1, where , t1. When h(t)≡1, this result yields what is called the Davis–Gut law. Specializing our result to h(t)=(Lt)^r, 0<r1, we obtain an analog of the Davis–Gut law.     

RELATIONS BETWEEN PACKING PREMEASURE AND MEASURE ON METRIC SPACE

Let X be a metric space andμa finite Borel measure on X. Let pμq,t and pμq,t be the packing premeasure and the packing measure on X, respectively, defined by the gauge (μB(x,r))q(2r)t, where q, t∈R. For any compact set E of finite packing premeasure the authors prove: (1) if q≤0 then pμq,t(E)=pμq,t(E);(2)if q>0 andμis doubling on E then pμq,t(E) and pμq,t(E) are both zero or neither.     

Relative rearrangement and Lebesgue spaces with variable exponent

We apply the techniques of monotone and relative rearrangements to the nonrearrangement invariant spaces L^p()(Ω) with variable exponent. In particular, we show that the maps uL^p()(Ω)→k(t)u_*L^p_*()(0,measΩ) and uL^p()(Ω)→u_*L^p*()(0,measΩ) are locally -Hölderian (u_* (resp. p^*) is the decreasing (resp. increasing) rearrangement of u (resp. p)). The pointwise relations for the relative rearrangement are applied to derive the Sobolev embedding with eventually discontinuous exponents.     

Sequential definitions of compactness

A subset F of a topological space is sequentially compact if any sequence of points in F has a convergent subsequence whose limit is in F. We say that a subset F of a topological group X is G-sequentially compact if any sequence of points in F has a convergent subsequence such that where G is an additive function from a subgroup of the group of all sequences of points in X. We investigate the impact of changing the definition of convergence of sequences on the structure of sequentially compactness of sets in the sense of G-sequential compactness. Sequential compactness is a special case of this generalization when G=lim.     

Gruenhage compacta and strictly convex dual norms

We prove that if K is a Gruenhage compact space then admits an equivalent, strictly convex dual norm. As a corollary, we show that if X is a Banach space and , where K is a Gruenhage compact in the w^*-topology and |||||| is equivalent to a coarser, w^*-lower semicontinuous norm on X^*, then X^* admits an equivalent, strictly convex dual norm. We give a partial converse to the first result by showing that if is a tree, then admits an equivalent, strictly convex dual norm if and only if is a Gruenhage space. Finally, we present some stability properties satisfied by Gruenhage spaces; in particular, Gruenhage spaces are stable under perfect images.     

Norms of some operators on the Bergman and the Hardy space in the unit polydisk and the unit ball

Let H(X) be the class of all holomorphic functions on the set and uH(X). We calculate operator norms of the multiplication operators M_u(f)=uf, on the weighted Bergman space , as well as on the Hardy space H^p(X), where X is the unit polydisk or the unit ball in . We also calculate the norm of the weighted composition operator from the weighted Bergman space , and the Hardy space , to a weighted-type space on the unit polydisk.     

Inverse eigenproblem for -symmetric matrices and their approximation

Let be a nontrivial involution, i.e., R=R⁻¹≠±I_n. We say that is R-symmetric if RGR=G. The set of all -symmetric matrices is denoted by . In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors in and a set of complex numbers , find a matrix such that and are, respectively, the eigenvalues and eigenvectors of A. We then consider the following approximation problem: Given an n×n matrix , find such that , where is the solution set of IEP and is the Frobenius norm. We provide an explicit formula for the best approximation solution by means of the canonical correlation decomposition.     

Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source

In this paper we investigate the global existence and finite time blow-up of solutions to the system of nonlinear viscoelastic wave equations

in Ω×(0,T) with initial and Dirichlet boundary conditions, where Ω is a bounded domain in . Under suitable assumptions on the functions g_i(), , the initial data and the parameters in the equations, we establish several results concerning local existence, global existence, uniqueness and finite time blow-up property.     

Renorming of and the fixed point property

For any , let P_k denote the natural projections on ℓ₁. Let |||||| be an equivalent norm of ℓ₁ that satisfies all of the following four conditions:

(1) There are α>4 and a positive (decreasing) sequence (α_n) in (0,1) such that for any normalized block basis {f_n} of (ℓ₁,||||||) and xℓ₁ with P_k−1(x)=x and |||x|||<α_k,

(2) There are two strictly decreasing sequences {β_k} and {γ_k} with

such that for any normalized block basis {f_n} of (ℓ₁,||||||) and x with (I−P_k)(x)=x,

(3) For any , I−P_k=1.

(4) The unit ball of (ℓ₁,||||||) is σ(ℓ₁,c₀)-closed.

In this article, we prove that the space (ℓ₁,||||||) has the fixed point property for the nonexpansive mapping. This improves a previous result of the author.

Nonwandering set of points of skew-product maps with base having closed set of periodic points

The aim of the present paper is to study the structure of the nonwandering set of points Ω() for the skew-product maps of the unit square , (x,y)→(f(x),g(x,y)), with base f having closed set of periodic points. For every and every point (x,y) with x periodic of period p_x by f and y not chain recurrent of F^p_x|_Ix, where , we prove that (x,y)Ω(F). On the other hand we construct a map with an isolated fixed point x₀ of f and y₀Ω(F|_Ix0) such that (x₀,y₀)Ω(F₀).     

On rank-one perturbations of normal operators

This paper is concerned with operators on Hilbert space of the form T=D+uv where D is a diagonalizable normal operator and uv is a rank-one operator. It is shown that if and the vectors u and v have Fourier coefficients and with respect to an orthonormal basis that diagonalizes D that satisfy , then T has a nontrivial hyperinvariant subspace. This partially answers an open question of at least 30 years duration.     

All-derivable points in continuous nest algebras

Let be an operator algebra on a Hilbert space. We say that an element is an all-derivable point of for the strong operator topology if every strong operator topology continuous derivable linear mapping φ at G (i.e. φ(ST)=φ(S)T+Sφ(T) for any with ST=G) is a derivation. Let be a continuous nest on a complex and separable Hilbert space H. We show in this paper that every orthogonal projection operator P(M) () is an all-derivable point of for the strong operator topology.     

Theorems about the attractor for incompressible non-Newtonian flow driven by external forces that are rapidly oscillating in time but have a smooth average

This paper discusses the incompressible non-Newtonian fluid with rapidly oscillating external forces g(x,t)=g(x,t,t/) possessing the average g⁰(x,t) as →0⁺, where 0<₀<1. Firstly, with assumptions (A₁)–(A₅) on the functions g(x,t,ξ) and g⁰(x,t), we prove that the Hausdorff distance between the uniform attractors and in space H, corresponding to the oscillating equations and the averaged equation, respectively, is less than O() as →0⁺. Then we establish that the Hausdorff distance between the uniform attractors and in space V is also less than O() as →0⁺. Finally, we show for each [0,₀].