The continuity of superposition operators on some sequence spaces defined by moduli

Let λ and μ be solid sequence spaces. For a sequence of modulus functions Φ = (ϕ k) let λ(Φ) = {x = (x _k ): (ϕk(|x _k |)) ∈ λ}. Given another sequence of modulus functions Ψ = (ψ_k), we characterize the continuity of the superposition operators P _f from λ(Φ) into μ (Ψ) for some Banach sequence spaces λ and μ under the assumptions that the moduli ϕk (k ∈ ℕ) are unbounded and the topologies on the sequence spaces λ(Φ) and μ(Ψ) are given by certain F-norms. As applications we consider superposition operators on some multiplier sequence spaces of Maddox type. This research was supported by Estonian Science Foundation Grant 5376.     

Multidimensional version of M. A. Krasnoseľskii’s generalized contraction principle

Let M be a complete K-metric space with n-dimensional metric ρ(x, y): M × M → R ⁿ , where K is the cone of nonnegative vectors in R ⁿ . A mapping F: M → M is called a Q-contraction if ρ (Fx,Fy) ⩽ Qρ (x,y), where Q: K → K is a semi-additive absolutely stable mapping. A Q-contraction always has a unique fixed point x* in M, and ρ(x*,a) ⩽ (I - Q)^-1 ρ(Fa, a) for every point a in M. The point x* can be obtained by the successive approximation method x _k = Fx _k-1, k = 1, 2,..., starting from an arbitrary point x ₀ in M, and the following error estimates hold: ρ (x*, x _k ) ⩽ Q ^k (I - Q)^-1ρ(x ₁, _x ⁰) ⩽ (I - Q)^-1 Q ^k ρ(x ₁, x ₀), k = 1, 2,.... Generally the mappings (I - Q)^-1 and Q ^k do not commute. For n = 1, the result is close to M. A. Krasnosel’skii’s generalized contraction principle.     

On tense modules for extended algebras over rational fields

Let k(x) be the field of fractions of the polynomial algebra k[x] over the field k. We prove that, for an arbitrary finite dimensional k-algebra Λ, any finitely generated Λ ⊗_k k(x)-module M such that its minimal projective presentation admits no non-trivial selfextension is of the form M ≅ N^k(x), for some finitely generated Λ-module N. Some consequences are derived for tilting modules over the rational algebra Λ ⊗_k k(x) and for some generic modules for Λ. Received: 24 November 2003; revised: 11 February 2005     

Strongly almost convergent generalized difference sequences associated with multiplier sequences

Let Λ = (λ _k ) be a sequence of non-zero complex numbers. In this paper we introduce the strongly almost convergent generalized difference sequence spaces associated with multiplier sequences i.e. w ₀[A,Δ ^m ,Λ,p], w ₁[A,Λ ^m ,Λ,p], w _∞[A,Δ ^m ,Λ,p] and study their different properties. We also introduce Δ _Λ ^m -statistically convergent sequences and give some inclusion relations between w ₁[Δ ^m ,λ,p] convergence and Δ _Λ ^m -statistical convergence. Communicated by Pavel Kostyrko     

On convergence of gradient-dependent integrands

We study convergence properties of {υ(∇u _k )}k∈ℕ if υ ∈ C(ℝ ^m×m ), |υ(s)| ⩽ C(1+|s| ^p ), 1 < p < + ∞, has a finite quasiconvex envelope, u _k → u weakly in W ^1,p (Ω; ℝ ^m ) and for some g ∈ C(Ω) it holds that ∫_Ω g(x)υ(∇u _k (x))dx → ∫_Ω g(x)Qυ(∇u(x))dx as k → ∞. In particular, we give necessary and sufficient conditions for L ¹-weak convergence of {det ∇u _k } _k∈ℕ to det ∇u if m = n = p. Dedicated to Jiří V. Outrata on the occasion of his 60th birthday This work was supported by the grants IAA 1075402 (GA AV ČR) and VZ6840770021 (MŠMT ČR).     

Higher order Riesz transforms associated with Bessel operators

In this paper we investigate Riesz transforms R _μ ^(k) of order k≥1 related to the Bessel operator Δ_μ f(x)=-f”(x)-((2μ+1)/x)f’(x) and extend the results of Muckenhoupt and Stein for the conjugate Hankel transform (a Riesz transform of order one). We obtain that for every k≥1, R _μ ^(k) is a principal value operator of strong type (p,p), p∈(1,∞), and weak type (1,1) with respect to the measure dλ(x)=x ^2μ+1  dx in (0,∞). We also characterize the class of weights ω on (0,∞) for which R _μ ^(k) maps L ^p (ω) into itself and L ¹(ω) into L ^1,∞(ω) boundedly. This class of weights is wider than the Muckenhoupt class of weights for the doubling measure dλ. These weighted results extend the ones obtained by Andersen and Kerman.     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

On number system constructions

We investigate various number system constructions. After summarizing earlier results we prove that for a given lattice Λ and expansive matrix M: Λ → Λ if ρ(M ⁻¹) < 1/2 then there always exists a suitable digit set D for which (Λ, M, D) is a number system. Here ρ means the spectral radius of M ⁻¹. We shall prove further that if the polynomial f(x) = c ₀ + c ₁ x + ··· + c _k x ^k ∈ Z[x], c _k = 1 satisfies the condition |c ₀| > 2 Σ _i=1 ^k |c _i | then there is a suitable digit set D for which (Z ^k , M, D) is a number system, where M is the companion matrix of f(x). The research was supported by OTKA-T043657 and Bolyai Fellowship Committee.     

On a Class of Vector-Valued Sequences Associated with Multiplier Sequences

In this article we introduce the vector valued sequence space m(E_k,φ,∧),associated with themultiplier sequence ∧=(λ_k) of non-zero complex numbers,and the terms of the sequence are chosen from theseminormed spaces E_k,seminormed by f_k for all k∈N.This generalizes the sequence space m(φ) introducedand studied by Sargent.We study some of its properties like solidity,completeness,and obtain some inclusionresults.We also characterize the multiplier problem and obtain the corresponding spaces dual to m(E_k,φ,∧).We prove some general results too.     

Representation of measurable functions by series in Walsh subsystems

For any sequence {ω(n)} _n∈ℕ tending to infinity we construct a “quasiquadratic” representation spectrum Λ = {n ² + o(ω(n))} _n∈ℕ: for any almost everywhere (a. e.) finite measurable function f(x) there exists a series in the form $ \mathop \sum \limits_{k \in \Lambda } $ \mathop \sum \limits_{k \in \Lambda } α _k ω _k (x) that converges a. e. to this function, where {w _k (x)} _k∈ℕ is the Walsh system. We find representation spectra in the form {n ^l + o(n ^l )} _n∈ℕ, where l ∈ {2 ^k } _k∈ℕ.     

(<Emphasis Type="Italic">p, λ</Emphasis>)-Koszul algebras and modules

In this paper, the notions of (p, λ)-Koszul algebra and (p, λ)-Koszul module are introduced. Some criteria theorems for a positively graded algebra A to be (p, λ)-Koszul are given. The notion of weakly (p, λ)-Koszul module is defined as well and let WK _λ ^p (A) denote the category of weakly (p, λ)-Koszul modules. We show that M ∈ WK _λ ^p (A) if and only if it can be approximated by (p, λ)-Koszul submodules, which is equivalent to that G(M) is a (p, λ)-Koszul module, where G(M) denotes the associated graded module of M. As applications, the relationships of the minimal graded projective resolutions of M, G(M) and (p, λ)-Koszul submodules are established. In particular, for a module M ∈ WK _λ ^p (A) we prove that ⊕ _i≥0 Ext _A ⁱ (M,A ₀) ∈ gr ₀(E(A)), we also get as a consequence that the finitistic dimension conjecture is valid in WK _λ ^p (A) under certain conditions.     

非对称Cantor集的Hausdorff中心测度

§ 1　 IntroductionThe class of Cantor sets is a typical one of sets in fractal geometry.Mathematicianshave paid their attentions to such sets for a long time.Itis well known that the Hausdorffmeasure of the Cantor middle- third set is1(see[1]) .Recently,Feng[3] obtained the exactvalues of the packing measure for a class of linear Cantor sets.Using Feng s method,Zhuand Zhou[5] obtained the exactvalue of Hausdorff centred measure of the symmetry Cantorsets.In this papar,we consider the Ha…     

About the Blowup of Quasimodes on Riemannian Manifolds

On any compact Riemannian manifold (M,g) of dimension n, the L ²-normalized eigenfunctions φ _λ satisfy ||f_l||_￥ ￡ Cl^\fracn-12\|\phi_{\lambda}\|_{\infty}\leq C\lambda^{\frac{n-1}{2}} where −Δφ _λ =λ ² φ _λ . The bound is sharp in the class of all (M,g) since it is obtained by zonal spherical harmonics on the standard n-sphere S ⁿ . But of course, it is not sharp for many Riemannian manifolds, e.g., flat tori ℝ ⁿ /Γ. We say that S ⁿ , but not ℝ ⁿ /Γ, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the (M,g) with maximal eigenfunction growth. In an earlier work, two of us showed that such an (M,g) must have a point x where the set ℒ _x of geodesic loops at x has positive measure in S^*_xMS^{*}_{x}M. We strengthen this result here by showing that such a manifold must have a point where the set ℛ _x of recurrent directions for the geodesic flow through x satisfies |{ℛ} _x |>0. We also show that if there are no such points, L ²-normalized quasimodes have sup-norms that are o(λ ^(n−1)/2), and, in the other extreme, we show that if there is a point blow-down x at which the first return map for the flow is the identity, then there is a sequence of quasimodes with L ^∞-norms that are Ω(λ ^(n−1)/2).     

The convergence rates of empirical Bayes estimation in a multiple linear regression model

Empirical Bayes (EB) estimation of the parameter vector =(,²) in a multiple linear regression modelY=X+ is considered, where is the vector of regression coefficient, N(0,² I) and ² is unknown. In this paper, we have constructed the EB estimators of by using the kernel estimation of multivariate density function and its partial derivatives. Under suitable conditions it is shown that the convergence rates of the EB estimators areO(n ^{-(k-1)(k-2)/k(2k+p+1)}), where the natural numberk3, 1/3<<1, andp is the dimension of vector .The project is supported by the National Natural Science Foundation of China.     

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

Recurrence relations for the coefficients of the Fourier series expansions with respect to q-classical orthogonal polynomials

We propose an algorithm to construct recurrence relations for the coefficients of the Fourier series expansions with respect to the q-classical orthogonal polynomials p_k(x;q). Examples dealing with inversion problems, connection between any two sequences of q-classical polynomials, linearization of ϑ_m(x) p_n(x;q), where ϑ_m(x) is x^mor (x;q)_m, and the expansion of the Hahn-Exton q-Bessel function in the little q-Jacobi polynomials are discussed in detail. This revised version was published online in June 2006 with corrections to the Cover Date.     

Weakly equicompact sets of operators defined on Banach spaces

Let X and Y be Banach spaces. We say that a set (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x_n) in X, there exists a subsequence (x_k(n)) so that (Tx_k(n)) is weakly uniformly convergent for T ∈ M. We study some properties of weakly equicompact sets and, among other results, we prove: 1) if is collectively weakly compact, then M^* is weakly equicompact iff M^** x^**={T^** x^** : T ∈ M} is relatively compact in Y for every x^** ∈X^**; 2) weakly equicompact sets are precompact in for the topology of uniform convergence on the weakly null sequences in X. Received: 14 February 2005; revised: 1 June 2005     

Extremal probabilities for Gaussian quadratic forms

 Denote by Q an arbitrary positive semidefinite quadratic form in centered Gaussian random variables such that E(Q)=1. We prove that for an arbitrary x>0, inf _Q P(Q≤x)=P(χ² _n /n≤x), where χ _n ² is a chi-square distributed rv with n=n(x) degrees of freedom, n(x) is a non-increasing function of x, n=1 iff x>x(1)=1.5364…, n=2 iff x[x(2),x(1)], where x(2)=1.2989…, etc., n(x)≤rank(Q). A similar statement is not true for the supremum: if 1<x<2 and Z ₁ ,Z ₂ are independent standard Gaussian rv's, then sup_0≤λ≤1/2 P{λZ ₁ ² +(1−λ)Z ₂ ² ≤x} is taken not at λ=0 or at λ=1/2 but at 0<λ=λ(x)<1/2, where λ(x) is a continuous, increasing function from λ(1)=0 to λ(2)=1/2, e.g. λ(1.5)=.15…. Applications of our theorems include asymptotic quantiles of U and V-statistics, signal detection, and stochastic orderings of integrals of squared Gaussian processes. Received: 24 June 2002 / Revised version: 26 January 2003 Published online: 15 April 2003 Research supported by NSA Grant MDA904-02-1-0091 Mathematics Subject Classification (2000): Primary 60E15, 60G15; Secondary 62G10