Sinc-Galerkin method for solving linear sixth-order boundary-value problems

There are few techniques available to numerically solve sixth-order boundary-value problems with two-point boundary conditions. In this paper we show that the Sinc-Galerkin method is a very effective tool in numerically solving such problems. The method is then tested on examples with homogeneous and nonhomogeneous boundary conditions and a comparison with the modified decomposition method is made. It is shown that the Sinc-Galerkin method yields better results.

     

The role of emitter quasi-bound state and scattering on intrinsic bistability in resonant tunneling diodes

Usually, numerical self-consistent calculations predict a much larger intrinsic bistability region than actually is measured in resonant tunneling diodes (RTDs). In addition, numerical calculations have shown that scattering in the well reduces bistability. We used a unified treatment of current flowing from continuum states and emitter quasi-bound states to show numerically and analytically that not only the scattering in the quantum well but also the scattering in the emitter reduces bistability. Moreover, within the Hartree approximation, bistability occurs by tunneling resonantly between emitter quasi-bound state and well quasi-bound state as a pitchfork bifurcation.     

CONSTRUCTION OF SOME KIESSWETTER-LIKE FUNCTIONS-THE CONTINUOUS BUT NON-DIFFERENTIABLE FUNCTION DEFINED BY QUINARY DECIMAL

In this paper, we construct some continuous but non-differentiable functions defined by quinary decimal, that are Kiesswetter-like functions. We discuss their properties, then investigate the Hausdorff dimensions of graphs of these functions and give a detailed proof.     

Multi-point Taylor expansions of analytic functions

Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.

     

Maximum norms of random sums and transient pattern formation

Many interesting and complicated patterns in the applied sciences are formed through transient pattern formation processes. In this paper we concentrate on the phenomenon of spinodal decomposition in metal alloys as described by the Cahn-Hilliard equation. This model depends on a small parameter, and one is generally interested in establishing sharp lower bounds on the amplitudes of the patterns as the parameter approaches zero. Recent results on spinodal decomposition have produced such lower bounds. Unfortunately, for higher-dimensional base domains these bounds are orders of magnitude smaller than what one would expect from simulations and experiments. The bounds exhibit a dependence on the dimension of the domain, which from a theoretical point of view seemed unavoidable, but which could not be observed in practice.

In this paper we resolve this apparent paradox. By employing probabilistic methods, we can improve the lower bounds for certain domains and remove the dimension dependence. We thereby obtain optimal results which close the gap between analytical methods and numerical observations, and provide more insight into the nature of the decomposition process. We also indicate how our results can be adapted to other situations.

     

Partially Classical States of a Boson Field

Employing positive-definiteness arguments we analyse Boson field states, which combine classical and quantum mechanical features (signal and noise), in a constructive manner. Mathematically, they constitute Bauer simplexes within the convex, weak-*-compact state space of the C*-Weyl algebra, defined by a presymplectic test function space (smooth one-Boson wave functions) and are affinely homeomorphic to a state space of a classical field. The regular elements are expressed in terms of weak distributions (probability premeasures) on the dual test function space. The Bauer simplex arising from the bare vacuum is shown to generalize the quantum optical photon field states with positive P-functions.     

FeCoB-SiO2磁性纳米颗粒膜的微波电磁特性

采用交替沉积磁控溅射工艺制备了超薄多层的FeCoB SiO2 磁性纳米颗粒膜 .利用x射线衍射仪、扫描探针显微镜、透射电子显微镜分析了薄膜的微结构和形貌特征 .采用振动样品磁强计、四探针法、微波矢量分析仪及谐振腔法测量薄膜试样的磁电性能和微波复磁导率 .重点对SiO2 介质相含量、薄膜微结构对电磁性能产生重要影响的机理做了分析和探讨 .结果表明 :这类FeCoB SiO2 磁性纳米颗粒膜具有良好的软磁性能和高频电磁性能 ,2GHz时的磁导率 μ′高于 70 ,可以应用于高频微磁器件或微波吸收材料的设计     

Directed Motion of a Molecular Motor Based on the Four-State Model with Unequal Substeps

A periodic one-dimensional four-state hopping model is proposed. In the model, the substeps betweenarbitrary adjacent states are unequal, and an explicit solution of the master equation is first obtained for the probabilitydistribution as a function of the time and position for any initial distribution with all the transients included. Next, thetransient behaviors in the initial period of time and the characteristic time to reach the steady state for the molecularmotor are discussed. Finally, we compare the steady state results to experiments and illustrate qualitatively the kineticbehaviors of a molecular motor under external load F.     

Comparison of higher-order accurate schemes for solving the two-dimensional unsteady Burgers' equation

This paper is devoted to the testing and comparison of numerical solutions obtained from higher-order accurate finite difference schemes for the two-dimensional Burgers' equation having moderate to severe internal gradients. The fourth-order accurate two-point compact scheme, and the fourth-order accurate Du Fort Frankel scheme are derived. The numerical stability and convergence are presented. The cases of shock waves of severe gradient are solved and checked with the fourth-order accurate Du Fort Frankel scheme solutions. The present study shows that the fourth-order two-point compact scheme is highly stable and efficient in comparison with the fourth-order accurate Du Fort Frankel scheme.     

Formula 1 – A Mathematical Microworld with CAS: Analysis of Learning Opportunities and Experiences with Students

In this article we describe a mathematical microworld for investigating car motion on a racing course and its use with a group of grade 12 students. The microworld is concerned with the mathematical construction of courses and functions which describe car motion. It is implemented in the computer algebra system, Maple^®, which provides the means to represent courses and functions symbolically and graphically. We describe the learning opportunities offered by the microworld in relation to the research literature on functions. Various facets and layers of the function concept are addressed in the microworld, and we suggest how work in the microworld might help in overcoming some well-known misconceptions.This revised version was published online in September 2005 with corrections to the Cover Date.