A New Class of Banach Spaces with Uniform Normal Structure

Let X be a Banach space, S(X) be the unit sphere of X, φ be a function: S(X)→ S(X *) such that φ(x)∈ x, and v φ(ε) =inf 1-12x+y: x,y∈S(X), and 〈φ(x), x-y 〉≥ε, 0≤ε≤2, where x is the set of norm 1 supporting functionals of S(X) at x. A geometric concept, modulus of V convexity V(ε)= sup {V φ(ε), for all φ: S(X)→S(X *)}, is introduced; the properties of V(ε) and the relationship between V(ε) and other geometric concepts are discussed. The main result is that V12>0 implies normal structure.

NORMAL STRUCTURE AND SOME GEOMETRIC PARAMETERS RELATED TO THE MODULUS OF U-CONVEXITY IN BANACH SPACES

We present a short and simple proof of the recent result of Yang and Wang [12]. Stimulated by their idea, two geometric parameters Uax（ε） and βx （ε）, both related to Gao＇s modulus of U-convexity of a Banach space X, are introduced. Their properties and the relationships with normal structure are studied. Some existing results involving normal structure and fixed points for non-expansive mappings in Banach spaces are improved.

UNIFORM NORMAL STRUCTURE AND SOLUTIONS OF REICH’S OPEN QUESTION

The open question raised by Reich is studied in a Banach space with uniform normal structure, whose norm is uniformly Gateaux differentiable. Under more suitable assumptions imposed on an asymptotically nonexpansive mapping, an affirmative answer to Reich＇ s open question is given. The results presented extend and improve Zhang Shisheng＇ s recent ones in the following aspects ： （i） Zhang＇ s stronger condition that the sequence of iterative parameters converges to zero is removed; （ii） Zhang＇ s stronger assumption that the asymptotically nonexpansive mapping has a fixed point is removed; （iii） Zhang＇ s stronger condition that the sequence generated by the Banach Contraction Principle is strongly convergent is also removed. Moreover, these also extend and improve the corresponding ones obtained previously by several authors including Reich, Shioji, Takahashi,Ueda and Wittmann.

ON A CLASS OF KTHE SEQUENCE SPACES WITH NORMAL STRUCTURE

In this note, we investigate the generalized modulus of convexity δ（λ） and the generalized modulus smoothness p（λ）. We obtain some estimates of these moduli for X=lp. We obtain inequalities between WCS coefficient of a KSthe sequence space X and δX（λ）. We show that, for a wide class of class the sequence spaces X, if for some ε∈（0, 9/10] holds δx（e） 〉 1/3（1- √3/2）ε, then X has normal structure.

L^0-CONVEX COMPACTNESS AND RANDOM NORMAL STRUCTURE IN L^0(F,B)

Let(B,||·||)be a Banach space,(?,F,P)a probability space,and L^0(F,B)the set of equivalence classes of strong random elements(or strongly measurable functions)from(?,F,P)to(B,||·||).It is well known that L^0(F,B)becomes a complete random normed module,which has played an important role in the process of applications of random normed modules to the theory of Lebesgue-Bochner function spaces and random operator theory.Let V be a closed convex subset of B and L^0(F,V)the set of equivalence classes of strong random elements from(?,F,P)to V.The central purpose of this article is to prove the following two results:(1)L^0(F,V)is L^0-convexly compact if and only if V is weakly compact;(2)L^0(F,V)has random normal structure if V is weakly compact and has normal structure.As an application,a general random fixed point theorem for a strong random nonexpansive operator is given,which generalizes and improves several well known results.We hope that our new method,namely skillfully combining measurable selection theorems,the theory of random normed modules,and Banach space techniques,can be applied in the other related aspects.

ON A GENERALIZED GEOMETRIC CONSTANT AND SUFFICIENT CONDITIONS FOR NORMAL STRUCTURE IN BANACH SPACES

In this paper, we introduce a new geometric constant C;（a, X） of a Banach space X, which is closely related to the generalized von Neumann-Jordan constant and analyze some properties of the constant. Subsequently, we present several sufficient conditions for normal structure of a Banach space in terms of this new constant, the generalized James constant, the generalized Garc′?a-Falset coefficient and the coefficient of weak orthogonality of Sims. Our main results of the paper generalize some known results in the recent literature.

ON A GENERALIZED MODULUS OF CONVEXITY AND UNIFORM NORMAL STRUCTURE

In this article, the authors study a generalized modulus of convexity, δ（α） （ε）. Certain related geometrical properties of this modulus are analyzed. Their main result is that Banach space X has uniform normal structure if there exists ε, 0 ≤e≤ 1, such that δ^（α）（1＋ε ） 〉 （1- α）ε.

Weak Uniform Normal Structure and Fixed Points of Asymptotically Regular Semigroups

Let X be a Banach space with a weak uniform normal structure and C a non–empty convex weakly compact subset of X. Under some suitable restriction, we prove that every asymptotically regular semigroup T = {T(t) : t ∈? S} of selfmappings on C satisfying

A FIXED POINT THEOREM FOR MULTIVALUED NONEXPANSIVE MAPPINGS IN BANACH SPACES OF NORMAL STRUCTURE

Since the appearance of T. C. Lim’s fixed point theorem for multivalued nonexpansive mappings in uniformly convex spaces in 1974, various generalizations and modifications have been obtained (e.g. [2, 3] and other references in [4, 5]). However, the corresponding fixed point problem for Banach spaces of normal structure remains open. The present report shall give a positive answer to it.

Coexistence of ferromagnetism and superconductivity in normal mental/superconductor/ferromagnet structures

We extend the Blonder, Tinkham and Klapwijk theory to the study of the inverse proximity effects in the normal mental/superconductor/ferromagnet structures. In the superconducting film, there are the gapless superconductivity and the spin-dependent density of states both within and without the energy gap. It indicates an appearance of the inverse-proximity-effect-induced ferromagnetism and a coexistence of ferromagnetism and superconductivity near the interface. The influence of exchange energy in the ferromagnet and barrier strength at the superconductor/ferromagnet interface on the inverse proximity effects is discussed.

Synthesis and Structure of Ag_3PSe_4

The title compound Ag3PSe4 was synthesized by the reaction of Ag powder, P2Se5 and Se in a molar ratio of 1:1:1 at 500 C and structurally characterized by X-ray crystallography. The crystal belongs to orthorhombic, space group Pmn21 with cell parameters: a = 7.689(4), b = 6.660(3), c = 6.379(4) , V = 326.7(3) 3, Z = 2, Dc = 6.816 g/cm3, Mr = 670.42, F(000) = 584, m = 31.302 mm-1, R = 0.0606, wR = 0.1289 and S = 1.012. The 3-D structure can be regarded as constructed from the stacking of puckered AgPSe honeycomb-like sheets along the c direction, in which the Ag, P and Se atoms are bonded to each other to form a chair-like six-membered ring, and the rings then build the sheets by sharing edges.

FIXED POINTS OF NONEXPANSIVE MAPPINGS ON STAR-SHAPED SUBSETS OF A CONVEX METRIC SPACE

In this paper, we give some characteristic properties of star-shaped sets which include a subset of a convex metric space. Using the characteristic properties, we discuss the existence problems of fixed points of nonexpansive type mappings on star-shaped subsets of convex metric spaces, which generalize the recent results obtained by Ding Xie-ping, Beg and Azam. Finally, we give an example which shows that our generalizations are essential.

Synthesis and Re-refinement of Cu_3PSe_4

The title compound Cu3PSe4 was synthesized by the reaction of CuCl, P2Se5 and Se in a molar ratio of 1:1:1 at 500 C and structurally characterized by X-ray crystallography. The crystal belongs to orthorhombic, space group Pmn21 with cell parameters: a = 7.685(2), b = 6.656(1), c = 6.377(1) , V = 326.2(1) 3, Z = 2, Dc = 5.472 g/cm3, Mr = 537.43, F(000) = 476, m = 32.12 mm-1, R = 0.0642, wR = 0.1481 and S = 1.037. The 3-D structure can be regarded as constructed from the alternately stacking of [Cu(2)Se4] tetrahedral layers and Cu(1)PSe tetrahedral layers along the b direction, in which the Cu(2)Se layer is comprised of corner-sharing [Cu(2)Se4] tetrahedra along the a and c directions, and the Cu(1)PSe layer is consisted of alternately corner-sharing [Cu(1)Se4] tetrahedra and [PSe4] tetrahedra along the a and c directions.

Structure normal velocity reconstruction withsparse measurement points

To achieve normal velocity reconstruction of a vibrating surface with sparse mea- surement points, a reconstruction method is proposed by exploiting of acoustic radiation modes as expansion functions, which are capable of describing the geometric shape of a vibrating surface. Firstly, acoustic radiation modes of the vibrating surface are calculated and the rela- tionship between normal velocity and acoustic radiation modes is built. Then actual measured normal velocity values are expressed by corresponding acoustic radiation modes and the expan- sion coefficients are calculated. Subsequently, all normal velocity values can be reconstructed by the obtained expansion coefficients. Experimental validations have been performed by a double-layer steel cylindrical shell with enclosed ends in an anechoic water tank. Two cases with different wavenumber components distribution were designed by a vibration shaker and a rotor device respectively. Two experimental results both show that actual vibration distribution cannot be revealed exactly by the sparse measurement points, which corresponds to severe loss of vibration related wavenumber components. On the other hand, normal velocity and corresponding wavenumber components can be restored accurately in both two wavenumber components distribution cases according to the proposed method, which demonstrates obvious effectiveness of the proposed method.

Fibrous Structure of Connective Tissue of Normal Human Lamina Cribrosa.

Fibrous components and structural morphology of the connective tissue of the lamina cribrosa obtained from 35 normal human autopsy eyes were examined by histochemical staining, transmission electron microscopic and computer-

RF-thermal-structural-RF coupled analysis on a travelling wave disk-loaded accelerating structure

The travelling wave （TW） disk-loaded accelerating structure is one of the key components in normal conducting （NC） linear accelerators, and has been studied for many years. In the design process, usually after the dimensions of each cell and the two couplers are finalized, the structure is fabricated and tuned, and then the whole structure RF characteristics are measured by using a vector network analyzer. Before fabrication, the whole structure characteristics （including RF, thermal and structural ones） are less simulated due to the limited capability of currently available computers. In this paper, we described a method for performing RF-thermal-structural-RF coupled analysis on a TW disk-loaded structure using only one PC. In order to validate our method, we first analyzed and compared our RF simulation results on the 3 m long BEPC Ⅱ structure with the corresponding experimental results, which shows very good consistency. Finally, the RF-thermal-structure-RF coupled analysis results on the 1.35 m long NSC KIPT linac accelerating structure are presented.

The Effect of Particle Shape on the Structure and Rheological Properties of Carbon-based Particle Suspensions

The structure and rheological properties of carbon-based particle suspensions, i.e., carbon black（CB）, multi-wall carbon nanotube（MWNT）, graphene and hollow carbon sphere（HCS） suspended in polydimethylsiloxane（PDMS）, are investigated. In order to study the effect of particle shape on the structure and rheological properties of suspensions, the content of surface oxygen-containing functional groups of carbon-based particles is controlled to be similar. Original spherical-like CB（fractal filler）, rod-like MWNT and sheet-like graphene form large agglomerates in PDMS, while spherical HCS particles disperse relatively well in PDMS. The dispersion state of carbon-based particles affects the critical concentration of forming a rheological percolation network. Under weak shear, negative normal stress differences（ΔN） are observed in CB, MWNT and graphene suspensions, while ΔN is nearly zero for HCS suspensions. It is concluded that the vorticity alignment of CB, MWNT and graphene agglomerates under shear results in the negative ΔN. However, no obvious structural change is observed in HCS suspension under weak shear, and accordingly, the ΔN is almost zero.

ON THE EXISTENCE OF FIXED POINTS FOR MAPPINGS OF ASYMPTOTICALLY NONEXPANSIVE TYPE

Let C be a nonempty weakly compact convex subset of a Banach space X, and T : C →C a mapping of asymptotically nonexpansive type. Then there hold the following conclusions: (i) if X has uniform normal structure and limsup |||TjN||| < N(X)~1/(N(X)) , where|||TjN||| is the exact Lipschitz constant of TjN , N is some positive integer, and N(X) is the normal structure coefficient of X, then T has a fixed point; (ii) if X is uniformly convex in every direction and has weak uniform normal structure, then T has a fixed point.

On Characterization of Iterative Approximation for Asymptotically Pseudocontractive Mappings

Let C be a nonempty bounded closed convex subset of a Banach space X, and T : C → C be uniformly L-Lipschitzian with L ≥ 1 and asymptotically pseudocontractive with a sequence {kn}(?)[1, ∞), limn→∞ kn = 1. Fix u ∈ C. For each n ≥ 1, xn is a unique fixed point of the contraction Sn(x) = (1 - (tn)/(Lkn))u + (tn)/(Lkn)Tnx(?)x ∈ C, where {tn}(?)[0,1). Under suitable conditions, the strong convergence of the sequence{xn}to a fixed point of T is characterized.

ON THE EXISTENCE OF FIXED POINTS FOR LIPSCHITZIAN SEMIGROUPS IN BANACH SPACES

Let C be a nonempty bounded subset of a p-uniformly convex Banach space X, and T = {T(t): t S} be a Lipschitzian semigroup on C with lim inf |||T(t)||| < Np, where Np is n→ t s the normal structure coefficient of X. Suppose also there exists a nonempty bounded closed convex subset E of C with the following properties: (P1)x: E implies ωω(χ) C E; (P2)T is asymptotically regular on E. The authors prove that there exists a z E such that T(s)z = z for all s S. Fruther, under the similar condition, the existence of fixed points of Lipschitzian semigroups in a uniformly convex Banach space is discussed.