The James constant, the Jordan-von Neumann constant, weak orthogonality, and fixed points for multivalued mappings

We give some sufficient conditions for the Domínguez-Lorenzo condition in terms of the James constant, the Jordan-von Neumann constant, and the coefficient of weak orthogonality. As a consequence, we obtain fixed point theorems for multivalued nonexpansive mappings.     

Lim's theorems for multivalued mappings in CAT(0) spaces

Let X be a complete CAT(0) space. We prove that, if E is a nonempty bounded closed convex subset of X and a nonexpansive mapping satisfying the weakly inward condition, i.e., there exists p∈E such that ∀x∈E, ∀α∈[0,1], then T has a fixed point. In Banach spaces, this is a result of Lim [On asymptotic centers and fixed points of nonexpansive mappings, Canad. J. Math. 32 (1980) 421-430]. The related result for unbounded R-trees is given.     

The Domínguez–Lorenzo condition and multivalued nonexpansive mappings

Let E be a nonempty bounded closed convex separable subset of a reflexive Banach space X   which satisfies the Domínguez–Lorenzo condition, i.e., an inequality concerning the asymptotic radius of a sequence and the Chebyshev radius of its asymptotic center. We prove that a multivalued nonexpansive mapping T:E→2^X$T:E\to 2^X$ which is compact convex valued and such that T(E)$T(E)$ is bounded and satisfies an inwardness condition has a fixed point. As a consequence, we obtain a fixed-point theorem for multivalued nonexpansive mappings in uniformly nonsquare Banach spaces which satisfy the property WORTH, extending a known result for the case of nonexpansive single-valued mappings. We also prove a common fixed point theorem for two nonexpansive commuting mappings t:E→E$t:E\to E$ and T:E→KC(E)$T:E\to \mathit{KC}(E)$ (where KC(E)$\mathit{KC}(E)$ denotes the class of all compact convex subsets of E) when X is a uniformly convex Banach space.     

Determination of effective atomic numbers and effective electron densities for Cu/Zn alloy

The mass attenuation coefficients, total interaction cross-sections, effective atomic numbers, effective electron densities and photon mean free paths of the Cu/Zn alloy were determined on the basis of the mixture rule at 356, 511, 662, 835 and 1275 keV gamma-ray energies. The gamma-rays were detected by using an ordinary NaI(Tl) scintillation detection system with a resolution of 10.2% at 662 keV.It was observed that the mixture rule is a suitable method for determination of these parameters. The effective atomic numbers and effective electron densities tend to be almost constant as a function of energy. There is good agreement between experiment and theory, calculated by WinXCom.     

On Kirk’s strong convergence theorem for multivalued nonexpansive mappings on CAT(0) spaces

We prove a strong convergence theorem for multivalued nonexpansive mappings which includes Kirk’s convergence theorem on CAT(0) spaces. The theorem properly contains a result of Jung for Hilbert spaces. We then apply the result to approximate a common fixed point of a countable family of single-valued nonexpansive mappings and a compact valued nonexpansive mapping.     

Optical and electronic polarizability investigation of Nd-doped soda-lime silicate glasses

This paper reports on different physical and optical properties of Nd³⁺-doped soda-lime silicate glass. The glasses containing Nd³⁺ in (65−x)SiO₂:25Na₂O:10CaO:xNd₂O₃ (where x=0.0-5.0 mol%) have been prepared by the melt-quenching method. In order to understand the role of Nd₂O₃ in these glasses the density, molar volume, refractive index and optical absorption were investigated. The results show that the density and molar volume of the glasses increase with an increase in Nd₂O₃ concentration and consequently generate more non-bridging oxygen (NBOs) into glass matrix. The optical absorption spectra were measured in the wavelength range from 300 to 700 nm and the optical band gaps were determined. It was found that the optical band gap decreases with an increase in Nd₂O₃ concentration. On the basis of the measured values of density and refractive index, the Nd³⁺ ion concentration in glasses, the polarizability of oxide ions and optical basicity were theoretically determined.     

Synthesis of the Northern hemisphere of the briaranes

A highly functionalized fragment corresponding to the Northern hemisphere of the briaranes has been synthesized employing a sulfonyl diene Diels-Alder reaction, oxidative desulfonylation, Eschenmoser rearrangement and Keck radical allylation.     

On a generalized James constant

We introduce a generalized James constant J(a,X) for a Banach space X, and prove that, if J(a,X)<(3+a)/2 for some a∈[0,1], then X has uniform normal structure. The class of spaces X with J(1,X)<2 is proved to contain all u-spaces and their generalizations. For the James constant J(X) itself, we show that X has uniform normal structure provided that , improving the previous known upper bound at 3/2. Finally, we establish the stability of uniform normal structure of Banach spaces.