Regressions and monotone chains II: The poset of integer intervals

A regressive function (also called a regression or contractive mapping) on a partial order P is a function mapping P to itself such that (x)x. A monotone k-chain for is a k-chain on which is order-preserving; i.e., a chain x ₁<...ksuch that (x ₁)...(x_k). Let P _nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P _nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x ^t(x,j–1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t( + _k, k) , where _k 0 as k. Alternatively, the largest k such that every regression on P _nis guaranteed to have a monotone k-chain lies between lg^*(n) and lg^*(n)–2, inclusive, where lg^*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element.     

A new C21-steroidal glycoside from Cynanchum stauntonii

A new C21-steroidal glycoside with two known compounds were isolated from the root of Cynanchum Stauntonii.Based on thespectral analysis,including MS,1H NMR,13C NMR,DEPT,1H-1H COSY,13C-1H COSY,HMQC and HMBC,their chemicalstructures were determinated as glaucogenin-C 3-O-a-L-cymaropyranosyl-(1→4)-b-D-digitoxopyranosyl-(1→4)-β-D-canaropyranoside(1),stigmasterol(2)and ursolic acid(3).     

Why is space three-dimensional and how may groups be seen?

First we propose a model of visual perception essentially based on the Keldysh-Chernavsky-Sossinsky three-channel theorem, from which three-dimensionality of space follows. Second, we associate with a system of subgroups H ₁, ..., H_s of a given group G a geometric object, called a group crystal, in order to visualize G. How this notion works is illustrated via the Burnside problem.     

Mixed finite element approximation of the vector potential

Summary We consider a mixed finite element approximation of the three dimensional vector potential, which plays an important rôle in the simulation of perfect fluids and in the calculation of rotational corrections to transonic potential flows. The central point of our approach is a saddlepoint formulation of the essential boundary conditions. In particular, this avoids the wellknown Babuka paradox when approximating smooth domains by polyhedrons. Using piecewise linear/piecewise constant elements for the vector potential/the boundary terms, we obtain optimal error estimates under minimal regularity assumptions for the solution of the continuous problem.     

An asymptotic finite element method for improvement of solutions of boundary layer problems

Summary A scheme that uses singular perturbation theory to improve the performance of existing finite element methods is presented. The proposed scheme improves the error bounds of the standard Galerkin finite element scheme by a factor of O(ⁿ⁺¹) (where is the small parameter andn is the order of the asymptotic approximation). Numerical results for linear second order O.D.E.'s are given and are compared with several other schemes.     

Finite element approximation on incompressible Navier-Stokes equations with slip boundary condition

Summary We consider a mixed finite element approximation of the stationary, incompressible Navier-Stokes equations with slip boundary condition, which plays an important rôle in the simulation of flows with free surfaces and incompressible viscous flows at high angles of attack and high Reynold's numbers. The central point is a saddle-point formulation of the boundary conditions which avoids the well-known Babuka paradox when approximating smooth domains by polyhedrons. We prove that for the new formulation one can use any stable mixed finite element for the Navier-Stokes equations with no-slip boundary condition provided suitable bubble functions on the boundary are added to the velocity space. We obtain optimal error estimates under minimal regularity assumptions for the solution of the continous problem. The techniques apply as well to the more general Navier boundary condition.     

A priori-Schranken für diskrete Lösungen monotoner Operatorgleichungen und Anwendungen

Summary Operator equationsTu=f are approximated by Galerkin's method, whereT is a monotone operator in the sense of Browder and Minty, so that existence results are available in a reflexive Banach spaceX. In a normed spaceY error estimates are established, which require a priori bounds for the discrete solutionsu _h in the norm of a suitable space . Sufficient conditions for the uniform boundedness u _{h Z} =O(1) ash0 are proved. Well-known error estimates in [3] for the special caseX=Y=Z are generalized by this. The theory is applied to quasilinear elliptic boundary value problems of order 2m in a bounded domain . The approximating subspaces are finite element spaces. Especially the caseX=W ₀ ^{m, p} (), 1<p<,Y=W ₀ ^{m. 2} (),Z=W ₀ ^{m. max (2,p)} ()W^m, () is treated. Some examples for 1<p<2 are considered. Forp2 a refined technique is introduced in the author's paper [7].

     

相变温度低且宽的含有胆甾烯基和反式-4-烷基环己基苯氧基的二介晶化合物

Unsymmetrical dimesogenic compounds with cholesteryl and 4-（trans-4-n-alkylcyclohexyl）phenoxy parts were synthesized by condensation of cholester-3-yl 6-bromohexanoate with appropriate 4-（trans-4-n-alkylcyclohexyl）- phenols. Structures and thermal phase behaviour of these dimesogenic compounds have been confirmed by IR, IH NMR, elemental analysis, DSC, polarity microscopy and XRD measurements. Their thermal phase behaviour was significantly different with that of other cholesterol-based dimesogens while they exhibited low and wide phase transition temperature.     

不饱和类锗烯H2C=GeNaBr的密度泛函理论研究

应用密度泛函理论DFT方法,在B3LYP/6-311G(d,p)水平上研究了不饱和类锗烯H2C=GeNaBr的结构及异构化反应.结果表明,不饱和类锗烯H2C=GeNaBr有3种平衡构型,其中非平面的p-配合物型构型能量最低,是其存在的主要构型.并对平衡构型间异构化反应的过渡态进行了计算,求得了转化势垒,预言了最稳定构型的振动频率和红外强度.     

基团的电子效应与单取代苯对位^1H,^13C的化学位移

基团的电子效应与单取代苯对位１Ｈ、１３Ｃ的化学位移＊韩长日冯娇杨钟照平（海南师范学院化学系海口５７１１５８）关键词１Ｈ的化学位移１３Ｃ的化学位移基团电负性共轭效应引言在１Ｈ、１３Ｃ核磁共振谱中，化学位移值的大小主要取决于屏蔽作用的大小，而屏蔽作用的大...