代数课程教学改革的一些探索

本文结合代数教学中的实际情况．针对代数教学过程中出现的问题。提出了在代数教学过程中改进教学方法的几条措施．     

The Hopf Property for Subgroups of Hyperbolic Groups

A group is said to be Hopfian if every surjective endomorphism of the group is injective. We show that finitely generated subgroups of torsion-free hyperbolic groups are Hopfian. Our proof generalizes a theorem of Sela (Topology 35 (2) 1999, 301–321).     

Upper and lower bounds for the energy of bipartite graphs

Using Lagrange's multiplier rule, we find upper and lower bounds of the energy of a bipartite graph G, in terms of the number of vertices, edges and the spectral moment of fourth order. Moreover, the upper bound is attained in a graph G if and only if G is the graph of a symmetric balanced incomplete block design (BIBD). Also, we determine the graphs for which the lower bound is sharp.     

Global Solution of Optimization Problems with Parameter-Embedded Linear Dynamic Systems

This paper develops a theory for the global solution of nonconvex optimization problems with parameter-embedded linear dynamic systems. A quite general problem formulation is introduced and a solution is shown to exists. A convexity theory for integrals is then developed to construct convex relaxations for utilization in a branch-and-bound framework to calculate a global minimum. Interval analysis is employed to generate bounds on the state variables implied by the bounds on the embedded parameters. These bounds, along with basic integration theory, are used to prove convergence of the branch-and-bound algorithm to the global minimum of the optimization problem. The implementation of the algorithm is then considered and several numerical case studies are examined thoroughly     

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.     

S[a,b]类中边界Nevanlinna-Pick插值(Ⅱ)

继续作者最近的研究,用所谓的Hankel向量方法建立S[a,b]函数类中带边界插值数据的Nevanlinna-Pick插值问题与[a,b]上的某种带约束条件的Hausdorff矩量问题之间解集之间明确的一一对应关系.通过N[a,b]函数类与S[a,b]函数类之间的联系,从而由BNP(N[a,b])问题的可解性准则和解的参数化描述获得BNP(S[a,b)问题的可解性准则和解的参数化表示.     

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.