Geometric dilation of closed planar curves: New lower bounds

Given two points on a closed planar curve, C, we can divide the length of a shortest connecting path in C by their Euclidean distance. The supremum of these ratios, taken over all pairs of points on the curve, is called the geometric dilation of C. We provide lower bounds for the dilation of closed curves in terms of their geometric properties, and prove that the circle is the only closed curve achieving a dilation of π/2, which is the smallest dilation possible. Our main tool is a new geometric transformation technique based on the perimeter halving pairs of C.     

Positive Solutions to Singular Second and Third Order Differential Equations for Quantum Fluids

We analyze a quantum trajectory model given by a steady-state hydrodynamic system for quantum fluids with positive constant temperature in bounded domains for arbitrary large data. The momentum equation can be written as a dispersive third-order equation for the particle density where viscous effects are incorporated. The phenomena that admit positivity of the solutions are studied. The cases, one space dimensional dispersive or non-dispersive, viscous or non-viscous, are thoroughly analyzed with respect to positivity and existence or non-existence of solutions, all depending on the constitutive relation for the pressure law. We distinguish between isothermal (linear) and isentropic (power law) pressure functions of the density. It is proved that in the dispersive, non-viscous model, a classical positive solution only exists for “small” (positive) particle current densities, both for the isentropic and isothermal case. Uniqueness is also shown in the isentropic subsonic case, when the pressure law is strictly convex. However, we prove that no weak isentropic solution can exist for “large” current densities. The dispersive, viscous problem admits a classical positive solution for all current densities, both for the isentropic and isothermal case, with an “ultra-diffusion” condition. The proofs are based on a reformulation of the equations as a singular elliptic second-order problem and on a variant of the Stampacchia truncation technique. Some of the results are extended to general third-order equations in any space dimension. Accepted July 1, 2000?Published online February 14, 2001     

Analysis of a bipolar energy-transport model for a metal-oxide-semiconductor diode

A simplified bipolar energy-transport model for a metal-oxide-semiconductor diode (MOS) with nonconstant lattice temperature is considered. The electron and hole current densities vanish in the diode but the particle temperature may be large. The existence of weak solutions to the system of quasilinear elliptic equations with nonlinear boundary conditions is proved using a Stampacchia trunction technique and maximum principle arguments. Further, an asymptotic analysis for the one-dimensional MOS diode is presented, which shows that only the boundary temperature influences the capacitance-voltage characteristics of the device. The analytical results are underlined by numerical experiments.     

Structurally diverse polyamides obtained from monomers derived via the ugi multicomponent reaction

The combination of the Ugi four-component reaction (Ugi-4CR) with acyclic diene metathesis (ADMET) or thiol-ene polymerization led to the formation of poly-1-(alkylcarbamoyl) carboxamides, a new class of substituted polyamides with amide moieties in the polymer backbone, as well as its side chains. 10-Undecenoic acid, obtained by pyrolysis of ricinoleic acid, the main fatty acid of castor oil, was used as the key renewable building block. The use of different primary amines, as well as isonitriles (isocyanides) for the described Ugi reactions provided monomers with high structural diversity. Furthermore, the possibility of versatile post-modification of functional groups in the side chains of the corresponding polymers should be of considerable interest in materials science. The obtained monomers were polymerized by ADMET, as well as thiol-ene, chemistry and all polymers were fully characterized. Finally, ortho-nitrobenzylamide-containing polyamides obtained by this route were shown to be photoresponsive and exhibited a dramatic change of their properties upon irradiation with light.     

Achieving Portable High Performance for Iterative Solvers on Accelerators

We propose performance enhancements for the implementation of the conjugate gradient method and the generalized minimum residual method for accelerators such as graphics processing units. Through a minimization of memory transfers from global memory via pipelining as well as a reduction of the number of compute kernels through kernel fusion, the performance is improved by up to two-fold when compared to standard implementations based on vendor-tuned routines. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ugi Reactions with CO2: Access to Functionalized Polyurethanes,Polycarbonates, Polyamides,and Polyhydantoins

A novel strategy for the incorporation of carbon dioxide into polymers is introduced. For this purpose, the Ugi five‐component condensation (Ugi‐5CC) of an alcohol, CO₂, an amine, an aldehyde, and an isocyanide is used to obtain step‐growth monomers. Polymerization via thiol‐ene reaction or polycondensation with diphenyl carbonate gives diversely substituted polyurethanes or alternating polyurethane‐polycarbonates, respectively. Furthermore, the application of 1,12‐diaminododecane and 1,6‐diisocyanohexane as bifunctional components in the Ugi‐5CC directly results in the corresponding polyamide bearing methyl carbamate side chains ( = 19 850 g mol⁻¹). The latter polymer is further converted into the corresponding polyhydantoin in a highly straightforward fashion.

     

An Asymptotic Limit of a Navier–Stokes System with Capillary Effects

A combined incompressible and vanishing capillarity limit in the barotropic compressible Navier–Stokes equations for smooth solutions is proved. The equations are considered on the two-dimensional torus with well prepared initial data. The momentum equation contains a rotational term originating from a Coriolis force, a general Korteweg-type tensor modeling capillary effects, and a density-dependent viscosity. The limiting model is the viscous quasi-geostrophic equation for the “rotated” velocity potential. The proof of the singular limit is based on the modulated energy method with a careful choice of the correction terms.     

Stoichiometry and morphology in terbium nitrate-poly(ethylene oxide) macromolecular complex

The Tb (NO₃)₃-PEO system was characterized to have a solubility limit equivalent to n ? 6.5 and a coordination number of 12.4 ± 0.1, compared to 5.0 and 10.9 ± 0.1 for the Nd(NO₃)₃-PEO system. The greater coordination involving Tb³⁺ appears to be responsible for the lower asymptotic glass transition temperature at high doping levels of the Tb(NO₃)₃-PEO complex. In both systems, anhydrous amorphous complexes were obtained below the respective solubility limits, over which apparent solubilities were found to be enhanced by the presence of tightly bound moisture while retaining the amorphous character of the complexes. Whereas partial precipitation was observed in the Tb(NO₃)₃-PEO system upon exposure to ambient moisture, both complexation and amorphous character were found to persist in the Nd(NO₃)₃-PEO system in the presence of absorbed moisture. © 1994 John Wiley & Sons, Inc.