Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space whose dual space E^∗ satisfies the Kadec-Klee property, K be a closed convex nonempty subset of E. Let be asymptotically nonexpansive mappings of K into E with sequences (respectively) satisfying kin→1 as n→∞, i=1,2,…,m, and . For arbitrary ?∈(0,1), let be a sequence in [?,1−?], for each i∈{1,2,…,m} (respectively). Let {xn} be a sequence generated for m?2 by

     

Positivity and boundedness of trigonometric sums

We give a systematic account of results which assure positivity and boundedness of partial sums of cosine or sine series. New proofs of recent results are sketched.     

Shellability of noncrossing partition lattices

We give a case-free proof that the lattice of noncrossing partitions associated to any finite real reflection group is EL-shellable. Shellability of these lattices was open for the groups of type and those of exceptional type and rank at least three.

     

Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator

We give a complete classification of -invariant real hypersurfaces in complex two-plane Grassmannians G ₂(C ^m+2) with commuting normal Jacobi operator . The first author was supported by MCYT-FEDER grant BFM 2001-2871-C04-01, the second author by grant Proj. No. KRF-2006-351-C00004 from Korea Research Foundation and the third author by grant Proj. No. R14-2002-003-01001-0 from Korea Research Foundation, Korea 2006 and Proj. No. R17-2007-006-01000-0 from KOSEF.     

Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis

^** Email: saito{at}infsup.jp Finite-element approximation for a non-linear parabolic–ellipticsystem is considered. The system describes the aggregation ofslime moulds resulting from their chemotactic features and iscalled a simplified Keller–Segel system. Applying an upwindtechnique, first we present a finite-element scheme that satisfiesboth positivity and mass conservation properties. Consequently,if the triangulation is of acute type, our finite-element approximationpreserves the L¹ norm, which is an important property of theoriginal system. Then, under some assumptions on the regularityof a solution and on the triangulation, we establish error estimatesin L^p x W^1, with a suitable p > d, where d is the dimensionof a spatial domain. Our scheme is well suited for practicalcomputations. Some numerical examples that validate our theoreticalresults are also presented.     

Construction of nonstandard finite difference schemes for $1{1\over 2}$ space‐dimension‐coupled PDEs

Two coupled PDEs, where one has a diffusion term and the other does not, are defined to be space‐dimension systems. We show how to construct nonstandard finite difference schemes for such systems and demonstrate that they are positivity‐preserving. These methods also allow the calculation of an explicit functional relationships between the time and space step‐sizes. The case of water flowing through fractured bedrock is used to illustrate our procedure. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Structure–dynamics relationships of the α‐relaxation in flexible copolyesters during crystallization as revealed by real‐time methods

The evolution of the α‐relaxation during an isothermal crystallization process of a series of flexible copolyesters of hydroxybutyrate (HB) and hydroxyvalerate (HV) has been followed in real‐time by wide‐angle X‐ray scattering and dielectric complex permittivity measurements. The change of the dielectric parameters with crystallization time can be phenomenologically described in terms of the Havriliak‐Negami equation. The dielectric strength follows a sigmoidal‐shaped pattern similar to that shown by the crystallinity. A reduction of the overall mobility with crystallization time of the polymeric chains in the amorphous phase has been observed. This slowing down effect depends on the HV molar content. The influence of the chain flexibility on the crystalline‐induced restriction has been discussed in the light of similar studies carried out with more rigid polymers. Dielectric experiments suggest that the progressive immobilization of polymer segments as crystallization proceeds cannot be exclusively associated with the amount of crystalline material. Differences in microstructure, depending on the HV molar content, seem to be responsible for the observed behavior. The progressive broadening and symmetrization of the α‐relaxation with increasing crystallization time has been explained as due to a restriction of the large‐scale motions of the polymeric chains, as the material is being filled in with crystals. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 37–49, 1999     

Real‐Time Monitoring of Phosphorylation Kinetics with Self‐Assembled Nano‐oscillators

Phosphorylation is a post‐translational modification that is involved in many basic cellular processes and diseases, but is difficult to detect in real time with existing technologies. A label‐free detection of phosphorylation is reported in real time with self‐assembled nano‐oscillators. Each nano‐oscillator consists of a gold nanoparticle tethered to a gold surface with a molecular linker. When the nanoparticle is charged, the nano‐oscillator can be driven into oscillation with an electric field and detected with a plasmonic imaging approach. The nano‐oscillators measure charge change associated with phosphorylation of peptides attached onto a single nanoparticle, allowing us to study the dynamic process of phosphorylation in real time without antibodies down to a few molecules, from which Michaelis and catalytic rate constants are determined.     

Identification and control of multiple leaky plasmon modes in silver nanowires

Leaky plasmon modes (LPMs) in metal nanowires (NWs), which combine the physical characteristic of both “plasmonics” and “leaky radiation”, present distinguished performances in terms of guiding and radiating light. In contrast to traditional light‐guiding in metal NWs with one single LPM, multiple LPMs are crucial for advanced uses such as augmenting data transmission channels, enhancing sensing performance, manipulating polarization and converting mode. Here, we demonstrate experimentally the control over multiple LPMs in pentagonal silver NWs. By combining far‐field real‐space imaging and leakage radiation microscopy, the three typical LPMs with fields mainly concentrating in corners surrounded by air are specifically identified. By manipulating excitation wavelengths and NW diameters, the number of the excited LPMs can be controlled. These findings reveal the physics of LPMs in silver NWs, thereby paving the way towards applying the high‐order leaky modes in silver NWs for photonic integrated circuits, nanoscale confinement, plasmonic sensing, QD‐nanowire coupling, etc.

     

Decomposer and associative functional equations

In the previous researches [2,3] b-integer and b-decimal parts of real numbers were introduced and studied by M.H. Hooshmand. The b-parts real functions have many interesting number theoretic explanations, analytic and algebraic properties, and satisfy the functional equation f (f(x) + y - f(y)) = f(x). These functions have led him to a more general topic in semigroups and groups (even in an arbitrary set with a binary operation [4] and the following functional equations have been introduced: Associative equations:

f(xf(yz))=f(f(xy)z),f(xf(yz))=f(f(xy)z)=f(xyz)