On Abrikosov Lattice Solutions of the Ginzburg-Landau Equations

We prove existence of Abrikosov vortex lattice solutions of the Ginzburg-Landau equations of superconductivity, with multiple magnetic flux quanta per fundamental cell. We also revisit the existence proof for the Abrikosov vortex lattices, streamlining some arguments and providing some essential details missing in earlier proofs for a single magnetic flux quantum per a fundamental cell.     

Caccioppoli’s inequalities on constant mean curvature hypersurfaces in Riemannian manifolds

We prove some Caccioppoli’s inequalities for the traceless part of the second fundamental form of a complete, noncompact, finite index, constant mean curvature hypersurface of a Riemannian manifold, satisfying some curvature conditions. This allows us to unify and clarify many results scattered in the literature and to obtain some new results. For example, we prove that there is no stable, complete, noncompact hypersurface in ${{\mathbb R}^{n+1}, n \leq 5}$ , with constant mean curvature ${H \not=0}$ , provided that, for suitable p, the L ^p norm of the traceless part of second fundamental form satisfies some growth condition.     

Percentage problems in bridging courses

Research on teaching high school mathematics shows that the topic of percentages often causes learning difficulties. This article describes a method of teaching percentages that the authors used in university bridging courses. In this method, the information from a word problem about percentages is presented in a two-way table. Such a table gives a logical structure to the problem and provides an algorithm for finding a simple equation for the unknown value of interest. The use of this procedure is illustrated by several examples of different levels of difficulty. The method can be applied to many types of percentage problems, so it is quite universal.     

Applying change of variable to calculus problems

This article describes the technique of introducing a new variable in some calculus problems to help students master the skills of integration and evaluation of limits. This technique is algorithmic and easy to apply.     

Sorption and Diffusion of Propylene and Ethylene in Heterophasic Polypropylene Copolymers

Summary: Sorption experiments of ethylene and propylene in different polypropylene powder samples, both homopolymer and heterophasic copolymers with different rubber content, have been carried out in a high-pressure magnetic suspension balance at 10 bars pressure and 70 °C. The gross solubilities measured can be well correlated with the rubber content of the polymer samples. Solubility of ethylene and propylene in the rubber phase differ from solubility in the amorphous fraction of the homopolymer, especially the concentration ratio of propylene to ethylene differs significantly between rubber phase and amorphous fraction of the homopolymer. From the slope of monomer uptake, information on kinetics of mass-transfer can be gained. No significant differences were observed in terms of mass-transfer for ethylene and propylene. With increasing rubber content, effective diffusion coefficients increased slightly. By combined sorption studies with powder samples and compressed films, information about both effective diffusion coefficients and the effective length scale of diffusion could be gained. It could be shown, that the particle radius is not the characteristic length of diffusion in the studied powder samples. Mass transfer of nearly all samples could be described by a constant diffusion length of 120 to 130 µm, independently on particle size. This indicates that the effective scale of diffusion in polymer particles is in between microparticle and macroparticle scale used in classical particle modeling.     

A speculative transmission scheme for scheduling-latency reduction

Low latency is a critical requirement in some switching applications, specifically in parallel computer interconnection networks. The minimum latency in switches with centralized scheduling comprises two components, namely, the control-path latency and the data-path latency, which in a practical high-capacity, distributed switch implementation can be far greater than the cell duration. We introduce a speculative transmission scheme to significantly reduce the average control-path latency by allowing cells to proceed without waiting for a grant, under certain conditions. It operates in conjunction with any centralized matching algorithm to achieve a high maximum utilization and incorporates a reliable delivery mechanism to deal with failed speculations. The speculative transmission scheme is employed in a non-blocking N ×N R input-queued crossbar switch with R receivers per output. The control-path latency can be almost entirely eliminated for loads up to 50%. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)