Extendible elements of the alternating groups

An element σ of A_n, the Alternating group of degree n, is extendible in S_n, the Symmetric group of degree n, if there exists a subgroup H of S_n but not A_n whose intersection with A_n is the cyclic group generated by σ. A simple number-theoretic criterion, in terms of the cycle-decomposition, for an element of A_n to be extendible in S_n is given here.     

酚醛型吸附树脂对水体系中吡啶和N,N-二甲基苯胺的吸附性能研究

研究了酚醛型吸附树脂在水体系中对吡啶和N，N-二甲基苯胺的静态和动态吸附行为．结果表明，在水中树脂对吡啶和N，N-二甲基苯胺的吸附主要以疏水吸附机理进行；吸附吡啶和N．N-二甲基苯胺的初始阶段，即达到38．3～48．9％平衡吸附时，吸附速率数据和半经验速率方程很吻合：酚醛型吸附树脂等温吸附吡啶和N，N-二甲基苯胺的平衡吸附数据符合Langmuir方程，相关系数在0．99以上，酚醛型吸附树脂吸附吡啶和N，N-二甲基苯胺属单分子层吸附：用80％的乙醇溶液作洗脱剂来洗脱吸附吡啶已达饱和的JDW-2树脂，效果是很理想的．在3．6个床体积内洗脱率达91．52％，4．8个床体积内洗脱率达到94．85％。表明酚醛型吸附树脂具有优良的洗脱性能．     

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.     

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].

     

Conductance studies in amide-water mixtures. II. Sodium salts inN,N-dimethylformamide-water mixtures at 35°C

Conductance data for sodium nitrite, chloride, and acetate in water andN,N-dimethylformamide (DMF)-water mixtures (74.82D42.48) for the concentration range 0.001–0.04N, as well as the densities, viscosities, and dielectric constants of the solvent mixtures at 35°C, are reported. The data have been analyzed by the Fuoss (1975) equation. The existence of a maximum in the viscosity at a 13 mole ratio of DMF and water is indicated. The Walden products for all the three salts pass through a maximum while the equivalent conductances show a minimum with increasing DMF content. The maxima in the Walden product are attributed to the dehydration of ions by the cosolvent (DMF).Part I:Indian J. Chem. 14A, 1015 (1976).Deceased.     

Growth functions for semi-regular tilings of the hyperbolic plane

This paper studies the growth function, with respect to the generating set of edge identifications, of a surface group with fundamental domainD in the hyperbolic plane ann-gon whose angles alternate between /p and /q. The possibilities ofn,p andq for which a torsion-free surface group can have such a fundamental polygon are classified, and the growth functions are computed. Conditions are given for which the denominator of the growth function is a product of cyclotomic polynomials and a Salem polynomial.This work was supported in part by NSF Research Grants.     

Incomplete idempotent Schröder quasigroups and related packing designs

Summary In this paper it is proved that, for any positive integern 2, 3 (mod 4),n 7, there exists an incomplete idempotent Schröder quasigroup with one hole of size two IISQ(n, 2) except forn = 10. It is also proved that for any positive integern 0, 1 (mod 4), there exists an idempotent Schröder quasigroup ISQ(n) except forn = 5 and 9. These results completely determine the spectrum of ISQ(n) and provide an application to the packing of a class of edge-coloured block designs.Research supported by NSERC grant A-5320.Research supported by NSFC grant 19231060-2.     

Combinatorial invariance of Stanley-Reisner rings

In this short note we show that Stanley—Reisner rings of simplicial complexes, which have had a dramatic application in combinatorics [2, p. 41], possess a rigidity property in the sense that they determine their underlying simplicial complexes.     

Homotopy types for tamely approximable Sobolev maps between manifolds

We define a class ^p (M,N) of Sobolev maps from a manifoldM into a manifoldN, in such a way that each mapu ^p (M, N) has a well defined [p]-homotopy type, providedN satisfies a topological hypothesis. Using this, we prove the existence of minimizers in [p]-homotopy classes for some polyconvex variational problems.     

Variable step size destabilizes the Störmer/leapfrog/verlet method

For fixed step-sizeh the Störmer method is stable for the standard test equationÿ= ² y,>0, if and only ifh<2. We show that for variable step sizeh _n there does not exist a (positive) limit onh which ensures stability. Nor can we guarantee stability if, in addition, we limit the step size ratioh _n/h _n–1.This work was supported in part by National Science Foundation Grant DMS 90 15533.