<Emphasis Type="Italic">q</Emphasis>-Besselian Frames in Banach Spaces

In this paper, we introduce the concepts of q-Besselian frame and （p, σ）-near Riesz basis in a Banach space, where a is a finite subset of positive integers and 1/p＋1/q = 1 with p 〉 1, q 〉 1, and determine the relations among q-frame, p-Riesz basis, q-Besselian frame and （p, σ）-near Riesz basis in a Banach space. We also give some sufficient and necessary conditions on a q-Besselian frame for a Banach space. In particular, we prove reconstruction formulas for Banach spaces X and X^＊ that if {xn}n=1^∞ C X is a q-Besselian frame for X, then there exists a p-Besselian frame {y＆＊}n=1^∞ belong to X^＊ for X^＊ such that x = ∑n=1^∞ yn^＊（x）xn for all x ∈ X, and x^＊ =∑n=1^∞ x^＊（xn）yn^＊ for all x^＊ ∈ X^＊. Lastly, we consider the stability of a q-Besselian frame for the Banach space X under perturbation. Some results of J. R. Holub, P. G. Casazza, O. Christensen and others in Hilbert spaces are extended to Banach spaces.     

<Emphasis Type="Italic">K</Emphasis><Subscript>0</Subscript>-group and approximate similarity invariants of strongly irreducible operators

Let H olenote a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator T ∈ L(H) is said to be strongly irreducible if T does not commute with any nontrivial idempotent. Herrero and Jiang showed that the norm-closure of the class of all strongly irreducible operators is the class of all operators with connected spectrum. This result can be considered as an approximate inverse of the Riesz decomposition theorem. In the paper, we give a more precise charact...     

Positive Solutions for Semipositone <Emphasis Type="Italic">m</Emphasis>-point Boundary-value Problems

Abstract   Let ξ _i ∈ (0, 1) with 0 < ξ₁ < ξ₂ < ··· < ξ _m−2 < 1, a _i , b _i ∈ [0,∞) with and . We consider the m-point boundary-value problem

一类m-点边值问题的正解

§ 1　 IntroductionIn[1 ] ,Karakostas and Tsamatos studied the existence of positive solutions for two-pointboundary value problemx″+ sign( 1 -c) q( t) f( x,x′) x′=0 ,( 1 .1 )x( 0 ) =0 ,x′( 1 ) =cx′( 0 ) ,( 1 .2 )where c∈ ( 0 ,1 )∪ ( 1 ,∞ ) .By using indices ofconvergence ofthe nonlinearity at0 and +∞and fixed point theorem in cones,they provided a priori upper and lower bounds for theslope of the solutions.The“c∈ ( 0 ,1 ) part”of their results has been extended to the fol-lowing …     

On <Emphasis Type="Italic">t</Emphasis><Superscript>2</Superscript>-like solutions of certain second order differential equations

Via an integral transformation, we establish two embedding results between the Emden-Fowler type equation , t ≥ t ₀ > 0, with solutions x such that as , , and the equation , u > 0, with solutions y such that for given k > 0. The conclusions of our investigation are used to derive conditions for the existence of radial solutions to the elliptic equation , , that blow up as in the two dimensional case.      

Unilateral <Emphasis Type="Italic">o</Emphasis>-Groups

For a big number of varieties of groups close to Engelian, it is proved that a variety of lattice-ordered groups generated by all linearly ordered groups in the class does not coincide with the variety of all o-approximable lattice-ordered groups. Supported by FP “Universities of Russia” grant No. UR.04.01.001. __________ Translated from Algebra i Logika, Vol. 45, No. 1, pp. 20–27, January–February, 2006.     

Approximately Linear Mappings in Banach Modules over a C^＊-algebra

Let X and Y be vector spaces. The authors show that a mapping f ： X →Y satisfies the functional equation 2d f（∑^2d j=1（-1）^j＋1xj/2d）=∑^2dj=1（-1）^j＋1f（xj） with f（0） = 0 if and only if the mapping f ： X→ Y is Cauchy additive, and prove the stability of the functional equation （≠） in Banach modules over a unital C^＊-algebra, and in Poisson Banach modules over a unital Poisson C＊-algebra. Let A and B be unital C^＊-algebras, Poisson C^＊-algebras or Poisson JC^＊- algebras. As an application, the authors show that every almost homomorphism h ： A →B of A into is a homomorphism when h（（2d-1）^nuy） =- h（（2d-1）^nu）h（y） or h（（2d-1）^nuoy） = h（（2d-1）^nu）oh（y） for all unitaries u ∈A, all y ∈ A, n = 0, 1, 2,.... Moreover, the authors prove the stability of homomorphisms in C^＊-algebras, Poisson C^＊-algebras or Poisson JC^＊-algebras.     

The Dual Spaces of Sets of Difference Sequences of Order <Emphasis Type="Italic">m</Emphasis> and Matrix Transformations

Let p = （pk）k=0^∞ be a bounded sequence of positive reals, m C N and u be s sequence of nonzero terms. If x = （xk）k=0^∞ is any sequence of complex numbers we write Δ（m）x for the sequence of the m th order differences of x and Δu^（m）X = {x=（x）k=0^∞ uΔ（m）x ∈ X} for any set X of sequences. We determine the α-, β- and γ-duals of the sets Δμ^（m）X for X=co（p）,c（p）,l∞（p） and characterize some matrix transformations between these spaces Δ^（m）X.     

A note on <Emphasis Type="Italic">q</Emphasis>-multiplicative functions

We prove the following statement. Let , and let . Suppose that, for all and , the sequence satisfies the relation

On representation theorem of <Emphasis Type="Italic">G</Emphasis>-expectations and paths of <Emphasis Type="Italic">G</Emphasis>-Brownian motion

We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ ： θ∈θ} representing an important sublinear expectation- G-expectation E[·]. We also give a concrete approximation of a bounded continuous function X（ω） by an increasing sequence of cylinder functions Lip（Ω） in order to prove that Cb（Ω） belongs to the completion of Lip（Ω） under the natural norm E[｜·｜].     

Inequalities for derivatives of functions in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

  We obtain a new sharp inequality for the local norms of functions x ∈ L _{∞, ∞} ^r (R), namely,

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> approximation capability of RBF neural networks

L ^p approximation capability of radial basis function (RBF) neural networks is investigated. If g: R ₊¹ → R ¹ and ∈ L _loc ^p (R ⁿ ) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in L ^p (K) with any accuracy for any compact set K in R ⁿ , if and only if g(x) is not an even polynomial. Partly supported by the National Natural Science Foundation of China (10471017)     

Ground state alternative for <Emphasis Type="Italic">p</Emphasis>-Laplacian with potential term

Let Ω be a domain in , d ≥ 2, and 1 < p < ∞. Fix . Consider the functional Q and its Gateaux derivative Q′ given by If Q ≥ 0 on, then either there is a positive continuous function W such that for all, or there is a sequence and a function v > 0 satisfying Q′ (v) = 0, such that Q(u _k ) → 0, and in . In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every satisfying there exists a constant C > 0 such that . As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.     

Multivariate Refinement Equations and Convergence of Cascade Algorithms in Lp（0〈p〈1）Spaces

We consider the solutions of refinement equations written in the form

Functoriality of automorphic <Emphasis Type="Italic">L</Emphasis>-functions through their zeros

Let E be a Galois extension of ℚ of degree l, not necessarily solvable. In this paper we first prove that the L-function L(s, π) attached to an automorphic cuspidal representation π of cannot be factored nontrivially into a product of L-functions over E. Next, we compare the n-level correlation of normalized nontrivial zeros of L(s, π₁)…L(s, π _k ), where π _j , j = 1,…, k, are automorphic cuspidal representations of , with that of L(s,π). We prove a necessary condition for L(s, π) having a factorization into a product of L-functions attached to automorphic cuspidal representations of specific , j = 1,…,k. In particular, if π is not invariant under the action of any nontrivial σ ∈ Gal _E/ℚ, then L(s, π) must equal a single L-function attached to a cuspidal representation of and π has an automorphic induction, provided L(s, π) can factored into a product of L-functions over ℚ. As E is not assumed to be solvable over ℚ, our results are beyond the scope of the current theory of base change and automorphic induction. Our results are unconditional when m,m ₁,…,m _k are small, but are under Hypothesis H and a bound toward the Ramanujan conjecture in other cases. The first author was supported by the National Basic Research Program of China, the National Natural Science Foundation of China (Grant No. 10531060), and Ministry of Education of China (Grant No. 305009). The second author was supported by the National Security Agency (Grant No. H98230-06-1-0075). The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein     

On the rate of convergence of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> norms in the CLT for poisson and gamma random variables

In the paper, we present upper bounds of L _p norms of order ( X)^-1/2 for all 1 ≤ p ≤ ∞ in the central limit theorem for a standardized random variable (X− X)/ √ X, where a random variable X is distributed by the Poisson distribution with parameter λ > 0 or by the standard gamma distribution Γ(α, 0, 1) with parameter α > 0. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09.     

Compact corigid objects in triangulated categories and co-<Emphasis Type="Italic">t</Emphasis>-structures

In the work of Hoshino, Kato and Miyachi, [11], the authors look at t-structures induced by a compact object, , of a triangulated category, , which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on whose heart is equivalent to Mod(End()^op). Rigid objects in a triangulated category can the thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave like cochain DGAs naturally gives the dual notion of a corigid object. Here, we see that a compact corigid object, , of a triangulated category, , induces a structure similar to a t-structure which we shall call a co-t-structure. We also show that the coheart of this non-degenerate co-t-structure is equivalent to Mod(End()^op), and hence an abelian subcategory of .      

On the <Emphasis Type="Italic">C</Emphasis>*-valued triangle equality and inequality in Hilbert <Emphasis Type="Italic">C</Emphasis>*-modules

We prove that for two elements x, y in a Hilbert C*-module V over a C*-algebra the C*-valued triangle equality |x + y| = |x| + |y| holds if and only if 〈x, y〉 = |x| |y|. In addition, if has a unit e, then for every x, y ∊ V and every ɛ > 0 there are contractions u, υ ∊ such that |x + y| ≦ u|x|u* + υ|y|υ* + ɛe.      

Riesz <Emphasis Type="Italic">s</Emphasis>-Equilibrium Measures on <Emphasis Type="Italic">d</Emphasis>-Rectifiable Sets as <Emphasis Type="Italic">s</Emphasis> Approaches <Emphasis Type="Italic">d</Emphasis>

Let A be a compact set in of Hausdorff dimension d. For s ∈ (0,d) the Riesz s-equilibrium measure μ ^s is the unique Borel probability measure with support in A that minimizes

Norm of Berezin transformon <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> space

It is shown that the Berezin transform B on L ^p (D), where D is the unit disc, has norm . Furthermore, the norms of a family of operators (on L ^p (D)) whose kernels are moduli of Bergman type kernels are also calculated. Partially supported by MNZZS, Grant ON144010