Adaptive boundary element methods for strongly elliptic integral equations

Summary Integral operators are nonlocal operators. The operators defined in boundary integral equations to elliptic boundary value problems, however, are pseudo-differential operators on the boundary and, therefore, provide additional pseudolocal properties. These allow the successful application of adaptive procedures to some boundary element methods. In this paper we analyze these methods for general strongly elliptic integral equations and obtain a-posteriori error estimates for boundary element solutions. We also apply these methods to nodal collocation with odd degree splines. Some numerical examples show that these adaptive procedures are reliable and effective.This work was carried out while Dr. De-hao Yu was an Alexander-von-Humboldt-Stiftung research fellow at the University of Stuttgart in 1987, 1988     

On the solution of singular linear systems of algebraic equations by semiiterative methods

Summary For a square matrixT ^n,n , where (I–T) is possibly singular, we investigate the solution of the linear fixed point problemx=T x+c by applying semiiterative methods (SIM's) to the basic iterationx ₀ ⁿ ,x _k T c _k–1+c(k1). Such problems arise if one splits the coefficient matrix of a linear systemA x=b of algebraic equations according toA=M–N (M nonsingular) which leads tox=M ^–1 N x+M ^–1 bT x+c. Even ifx=T x+c is consistent there are cases where the basic iteration fails to converge, namely ifT possesses eigenvalues 1 with ||1, or if =1 is an eigenvalue ofT with nonlinear elementary divisors. In these cases — and also ifx=T x+c is incompatible — we derive necessary and sufficient conditions implying that a SIM tends to a vector which can be described in terms of the Drazin inverse of (I–T). We further give conditions under which is a solution or a least squares solution of (I–T)x=c.Research supported in part by the Alexander von Humboldt-Stiftung     

Differential form methods in the theory of variational systems and Lagrangian field theories

This paper will attempt to unify diverse material from physics and engineering in terms of differential forms on manifolds. A variational system will be defined by means of a scalar-valued differential form on a manifold and an ideal in the Grassmann algebra of differential forms on that manifold to serve as constraints. Two types of extremal submanifolds will be defined. The first-called the Euler-Lagrange extremals-will be defined by a method that is the generalization of the classical methods in the calculus of variations. The second—a generalization of a method used by Cartan in his treatise Leçons sur les invariants intégraux-will define extremals as integral submanifolds of an exterior differential system invariently attached to the variational system. As examples, the variational systems attached to string theories in Riemannian manifolds and Yang-Mills fields will be discussed from this differential form point of view. Finally, as application, the differential geometric properties and definition of energy will be presented from the differential form point of view.This work was supported by a grant from the Applied Mathematics program of the National Science Foundation.     

On the width of an orientation of a tree

There are 2 ^n-1 ways in which a tree on n vertices can be oriented. Each of these can be regarded as the (Hasse) diagram of a partially ordered set. The maximal and minimal widths of these posets are determined. The maximal width depends on the bipartition of the tree as a bipartite graph and it can be determined in time O(n). The minimal width is one of [/2] or [/2]+1, where is the number of leaves of the tree. An algorithm of execution time O(n + ² log ) to construct the minimal width orientation is given.This research was partially funded by the National Science and Engineering Research Council of Canada under Grant Number A4219.     

A short proof that N 3 is not a circle containment order

A partially ordered set P is called a circle containment order provided one can assign to each xP a circle C _x so that . We show that the infinite three-dimensional poset N ³ is not a circle containment order and note that it is still unknown whether or not [n]³ is such an order for arbitrarily large n.     

N,C10-Überbrückte Morphinalkaloide, 5H-10,13-Iminoethano-phenanthro[4,5-bcd]furan, 2. Mitt.

Summary Following our concept of rather new structural variations in the nitrogen region of morphine alkaloids we describe the first synthesis of N/C-10-bridged tertiary bases.

     

大环二萜类化合物的研究(Ⅳ)——Isosarcophytol—A前体化合物的合成

Isosarcophytol-A(1)是1982年首次从澳大利亚软珊瑚(Nephthea brassica)中分离鉴定的西松烷型(Cembrane)大环二萜类化合物,其结构为6,10,14-三甲基-3-异丙基-3E,5E,9E,13E-环十四碳四烯-1-醇,是Sarcophytol—A(2)的异构体,但有关1的生物活性试验和全合成研究尚未见报道.我们在前文报道了以低价钛诱导的分子内二羰基偶联为环化方法,完成了天然大环二萜类化合物Cembrene—C的全合成和Sarcophytol—A(2)苄醚衍生物(3)的合成.本文报道以天然法呢醇4为起始原料,经区域选择性氧化、羟醛缩合等六步反应,合成了1的前体化合物11.合成路线如下:     

反相高效液相色谱法半制备分离[T(4-MOP)PS4]卟啉的研究

本文研究了新水溶性５，１０，１５，２０－四（４－甲氧基－３－磺酸苯基）卟啉［Ｔ（４－ＭＯＰ）ＰＳ４］的反相高效液相色谱（ＨＰＬＣ）分离条件。采用ＳｈｉｍｐａｃｋＰＲＥＰ－ＯＤＳ半制备色谱柱，用含有１０ｍｍｏｌ／Ｌ四乙基碘化铵的乙腈－水（体积比２５：７５）为流动相，流速１８ｍＬ／ｍｉｎ，于４１８ｎｍ波长下检测。［Ｔ（４－ＭＯＰ）ＰＳ４］与合成中生成的杂质组分完全分离。经此制备的卟啉纯度高，已成功地应用于自来水样中微量钴、锌、铜离子的ＨＰＬＣ测定中。     

On a characterization of the normal distribution by means of identically distributed linear forms

Let X₁, X₂,…, be independent, identically distributed random variables. Suppose that the linear forms L₁ = Σ_j=1^∞a_jX_j and L₂ = Σ_j=1^∞b_jX_j exist with probability one and are identically distributed; necessary and sufficient conditions assuring that X₁ is normally distributed are presented. The result is an extension of a theorem of Linnik (Ukrainian Math. J.5 (1953), 207–243, 247–290) concerning the case that the linear forms L₁ and L₂ have a finite number of nonvanishing components. This proof only makes use of elementary properties of characteristic functions and of meromorphic functions.