A Note on Kantorovich's Majorant Method

We prove that in certain function spaces Kantorovich majorants are also majorants for the Hausdorff measure of noncompactness.     

A Note on Existence and Convergence of Best Proximity Points for Pointwise Cyclic Contractions

We introduce a notion of pointwise cyclic contraction T satisfying TA ? B and TB ? A to obtain the existence of a point x ∈ A, such that d(x, Tx) = dist(A, B), known as a best proximity point for such a map. We also prove that for any x ∈ A, the Picard iteration {T²ⁿx} converges to a best proximity point.     

Coherent X‐ray diffractive imaging of protein crystals

The technique of coherent X‐ray diffraction imaging (CXDI) has recently shown great promise for the study of inorganic nanocrystals. In this work the CXDI method has been applied to the study of micrometer‐size protein crystals. Finely sampled diffraction patterns of single crystals were measured and iterative phase‐retrieval algorithms were used to reconstruct the two‐dimensional shape of the crystal. The density maps have limited reproducibility because of radiation damage, but show clear evidence for crystal facets. Qualitative analysis of a number of single‐crystal diffraction peaks indicates the presence of inward surface contraction on 2 µm size crystals. A survey of several hundred diffraction patterns yielded a number of examples with dramatic single‐sided streaks, for which a plausible model is constructed.     

Probing the Contraction and Association of Polystyrene Chains in Semidilute Solution by Non‐Radiative Energy Transfer

The interchain contraction and association of polystyrene in semidilute solution near the Θ temperature are studies using NET. The results reveal that during a drop of temperature, polystyrene chains pass through different states: a smooth change of conformation at temperatures above the Θ temperature, an accelerated contraction and association, aggregation of polymer chains, and then further continuous contraction below the Θ temperature. A pronounced hysteresis is observed during repeated heating. A viscosity study of the same solution was compared with the NET results. It indicates that NET can effectively be used to study conformational changes like contraction and association of polymer chains in a semidilute solution.

     

A new idea to construct the fractal interpolation function

The aim of this paper is to present a new idea to construct the nonlinear fractal interpolation function, in which we exploit the Matkowski and the Rakotch fixed point theorems. Our technique is different from the methods presented in the previous literatures.     

Contraction of Exterior Powers in Characteristic 2 and the Spin Module

Let K be a field of characteristic 2 and letV be a vector space of dimension 2m over K. Let f be a non-degenerate alternating bilinear form defined on V × V. The symplectic group Sp(2m, K) acts on the exterior powers ^k V for 0 k. 2m There is a contraction map defined on the exterior algebra , which commutes with the Sp(2m, K) action and satisfies ² = 0 and ( ^k V) ^k–1 V We prove that ( ^k V)= ker ^k–1 V except when k=m+2. In the exceptional case, ( ^m+2 V) has codimension 2^m in ker ^m V and we show that the quotient module ker ^m V/ ^m+2 V is a spin module for Sp(2m,K). When K is algebraically closed, we show that this spin module occurs with multiplicity 1 in ^m V and multiplicity 0 in all other components of V.     

A remark on normal derivations

Given a Hilbert space , let be operators on . Anderson has proved that if is normal and , then for all operators . Using this inequality, Du Hong-Ke has recently shown that if (instead) , then for all operators . In this note we improve the Du Hong-Ke inequality to for all operators . Indeed, we prove the equivalence of Du Hong-Ke and Anderson inequalities, and show that the Du Hong-Ke inequality holds for unitarily invariant norms.

     

An efficient one-step method for the synthesis of 2-(indolizin-2-yl)benzimidazoles from quinoxalinones and α-picoline via a novel rearrangement

A highly efficient one-step and versatile method for the synthesis of 2-(indolizin-2-yl)benzimidazoles has been developed on the basis of the novel ring contraction of 3-arylchloromethyl- and alkylchloromethylquinoxalin-2-ones with α-picoline.