Generalized Hardy-Sobolev Inequalities and Exponential Decay of the Entropy of $g(x)\dot{u}=\Delta u$

Provided the non-negative function allows for a generalized Hardy-Sobolev inequality, existence and uniqueness of global weak solutions of the possibly degenerate parabolic PDE , subject to homogeneous Dirichlet boundary conditions, is proved. The maximum/minimum principle holds. The associated entropy decays exponentially as t with a rate not exceeding 2/C, where C is the constant arising in the generalized Hardy-Sobolev inequality.A.U. acknowledges support from the DFG Forschungszentrum Mathematics for Key Technologies, project D10 (Berlin) and from the EU Research Network HYKE.M.R. acknowledges the hospitality of the mathematical department, Universität Kaiserslautern, where this work was carried out.     

Valency Changes of Iron and Tin in Framework-Substituted Molecular Sieves Investigated by in Situ Mössbauer Spectroscopy

Valency changes of iron and tin in framework-substituted microporous and mesoporous molecular sieves (Fe-ETS-10, Fe-ZSM-5, Fe-AlPO-11, Sn-ZSM-12 and Sn-MCM-41) have been studied under reducing and oxidizing conditions by in situ Mössbauer spectroscopy. It is demonstrated that the structure of the host matrices strongly influences the stabilization and the oxidation state of the ions. Debye temperature, _D, data also are estimated, they support the suggested structural assignment of various species.     

Lithium ion conductivity of Li5+xBaxLa3?xTa2O12 (x?=?0–2) with garnet-related structure in dependence of the barium content

The stoichiometry range and lithium ion conductivity of Li_5+x Ba _x La_3−x Ta₂O₁₂ (x = 0, 0.25, 0.50, 1.00, 1.25, 1.50, 1.75, 2.00) with garnet-like structure were studied. The powder X-ray diffraction data of Li_5+x Ba _x La_3−x Ta₂O₁₂ indicated that single phase oxides with garnet-like structure exist over the compositional range 0 ≤ x ≤ 1.25; while for x = 1.5, 1.75 and 2.00, the presence of second phase in addition to the major garnet like phase was observed. The cubic lattice parameter increases with increasing x and reaches a maximum at x = 1.25 then decreases slightly with further increase in x in Li_5+x Ba _x La_3−x Ta₂O₁₂. The impedance plots of Li_5+x Ba _x La_3−x Ta₂O₁₂ samples obtained at 33 °C indicated a minimum grain-boundary resistance (R _gb) contribution to the total resistance (R _b + R _gb) at x = 1.0. The total (bulk + grain boundary) ionic conductivity increases with increasing lithium and barium content and reaches a maximum at x = 1.25 and then decreases with further increase in x in Li_5+x Ba _x La_3−x Ta₂O₁₂. Scanning electron microscope investigations revealed that Li_6.25Ba_1.25La_1.75Ta₂O₁₂ is much more dense, and the grains are more regular in shape. Among the investigated compounds, Li_6.25Ba_1.25La_1.75Ta₂O₁₂ exhibits the highest total (bulk + grain boundary) and bulk ionic conductivity of 5.0 × 10⁻⁵ and 7.4 × 10⁻⁵ S/cm at 33 °C, respectively.     

Rheology of active-particle suspensions

We study the interplay of activity, order, and flow through a set of coarse-grained equations governing the hydrodynamic velocity, concentration, and stress fields in a suspension of active, energy-dissipating particles. We make several predictions for the rheology of such systems, which can be tested on bacterial suspensions, cell extracts with motors and filaments, or artificial machines in a fluid. The phenomena of cytoplasmic streaming, elastotaxis, and active mechanosensing find natural explanations within our model.     

Effect of Cation/Anion Co‐doping on the Photocatalytic Performance of Na3SbO(PO4)2

Silver/copper‐nitrogen co‐doped Na₃SbO(PO₄)₂ were prepared by solid state/ion exchange methods. The materials were characterized by powder XRD, SEM‐EDS, TG analysis, as well as FT‐IR and UV/Vis diffuse reflectance spectroscopy. All the doped materials crystallize in orthorhombic lattice with space group P2₁2₁2₁. The infrared spectra of the samples gave characteristic bands corresponding to PO₄ and SbO₆ groups. The bandgap energy of all the samples was deduced from their UV/Vis DRS profiles. The bandgap energy of doped materials decreased substantially. Visible‐light‐induced photocatalytic oxidation of the methyl violet (MV) and methylene blue (MB) was examined. The influence of cation, anion, and co‐doping on the photoactivity of Na₃SbO(PO₄)₂ was studied. The Ag⁺ and nitrogen co‐doped Na₃SbO(PO₄)₂ exhibited the highest photocatalytic activity among the materials investigated.     

Interior local null controllability of one‐dimensional compressible flow near a constant steady state

We consider the one‐dimensional compressible Navier–Stokes equations with periodic boundary conditions, with initial conditions in a small neighborhood of a state of uniform density and uniform nonzero velocity. We prove that, with a control given only by a body force localized in a subinterval, we can steer the system to uniform density and velocity. Copyright © 2016 John Wiley & Sons, Ltd.     

Fourier transform infrared imaging and MR microscopy studies detect compositional and structural changes in cartilage in a rabbit model of osteoarthritis

Assessment of subtle changes in proteoglycan (PG) and collagen, the primary macromolecular components of cartilage, which is critical for diagnosis of the early stages of osteoarthritis (OA), has so far remained a challenge. In this study we induced osteoarthritic cartilage changes in a rabbit model by ligament transection and medial meniscectomy and monitored disease progression by infrared fiber optic probe (IFOP) spectroscopy, Fourier transform infrared imaging spectroscopy (FT-IRIS), and magnetic resonance imaging (MRI) microscopy. IFOP studies combined with chemometric partial least-squares analysis enabled us to monitor progressive cartilage surface changes from two to twelve weeks post-surgery. FT-IRIS studies of histological sections of femoral condyle cartilage revealed that compared with control cartilage the OA cartilage had significantly reduced PG content 2 and 4 weeks post-surgery, collagen fibril orientation changes 2 and 4 weeks post-surgery, and changes in collagen integrity 2 and 10 weeks post-surgery, but no significant changes in collagen content at any time. MR microscopy studies revealed reduced fixed charge density (FCD), indicative of reduced PG content, in the OA cartilage, compared with controls, 4 weeks post-surgery. A non-significant trend toward higher apparent MT exchange rate, k_m, was also found in the OA cartilage at this time point, suggesting changes in collagen structural features. These two MR findings for FCD and k_m parallel the FT-IRIS findings of reduced PG content and altered collagen integrity, respectively. MR microscopy studies of the cartilage at the 12-week time point also found a trend toward longer T ₂ values and reduced anisotropy in the deep zone of the OA cartilage, consistent with increased hydration and less ordered collagen. These studies reveal that FT-IRIS and MR microscopy provide complementary data on compositional changes in articular cartilage in the early stages of osteoarthritic degradation.     

Monovalent Group 13 Organometallic Compounds: Weak Association to Monomeric,Versatile Two-Electron Donors

A weakly associated hexamer is formed for [GaCp*] (Cp*=C₅Me₅) in the solid state (see picture). The recent X-ray crystal structure analyses of [GaCp*] as well as the monomeric In^I and Tl^I compounds [M(2,4,6-Trip₃C₆H₂)] (Trip=2,4,6-iPr₃C₆H₂) throw new light on the association and aggregation of monovalent Group 13 elements in the solid state. The synthesis of [Ni⁰{In[C(SiMe₃)₃]}₄], a complex with terminally bonded In^IR ligands, offers alternative σ-donor/π-acceptor ligands to organometallic chemists. The newest results in this area are likely to open up new and intriguing possibilities in the preparation of main group–transition metal clusters.     

An extended mover—stayer model for diagnosing the dynamics of trial and repeat for a new brand

This paper proposes an approach that extends the mover—stayer model to examine the impact of a new brand on brand choice patterns in a mature, frequently purchased product category. The objective is to obtain insights into the dynamics of brand choice following trial and repeat purchase of the new brand. The analysis uses consumer panel data on brand choices over a limited number of consecutive purchase occasions spanning the new brand introduction. The estimated parameters of the model provide diagnostic information for possible managerial action. An illustrative empirical application of the model demonstrates some of the insights into consumer buying patterns following a new brand introduction and potential managerial implications.     

Symmetry of Ground States of p‐Laplace Equations via the Moving Plane Method

. In this paper we use the moving plane method to get the radial symmetry about a point of the positive ground state solutions of the equation in , in the case . We assume f to be locally Lipschitz continuous in and nonincreasing near zero but we do not require any hypothesis on the critical set of the solution. To apply the moving plane method we first prove a weak comparison theorem for solutions of differential inequalities in unbounded domains. (Accepted September 21, 1998)