Describing Neighborhoods of 5-Vertices in a Class of 3-Polytopes with Minimum Degree 5

Lebesgue proved in 1940 that each 3-polytope with minimum degree 5 contains a 5-vertex for which the set of degrees of its neighbors is majorized by one of the following sequences(6, 6, 7, 7, 7), (6, 6, 6, 7, 9), (6, 6, 6, 6, 11)(5, 6, 7, 7, 8), (5, 6, 6, 7, 12), (5, 6, 6, 8, 10), (5, 6, 6, 6, 17)(5, 5, 7, 7, 13), (5, 5, 7, 8, 10), (5, 5, 6, 7, 27), (5, 5, 6, 6,∞), (5, 5, 6, 8, 15), (5, 5, 6, 9, 11)(5, 5, 5, 7, 41), (5, 5, 5, 8, 23), (5, 5, 5, 9, 17), (5, 5, 5, 10, 14), (5, 5, 5, 11, 13).We prove that each 3-polytope with minimum degree 5 without vertices of degree from 7 to 10 contains a 5-vertex whose set of degrees of its neighbors is majorized by one of the following sequences: (5, 6, 6, 5, ∞), (5, 6, 6, 6, 15), and (6, 6, 6, 6, 6), where all parameters are tight.     

Book reviews

L.N. TREFETHEN and D. BAU, III,Numerical Linear Algebra,SIAM, Philadelphia, 1997G.-C. ROTA,Indiscrete Thoughts,Birkhäuser, Boston, 1997D.E. KEYES, A. SAMEH and V. VENKATAKRISHNAN, eds.Parallel Numerical Algorithms,Kluwer, Dordrecht, 1997A. KIRSCH,An Introduction to the Mathematical Theory of Inverse Problems,Springer, New York, 1996L.F. SHAMPINE, R.C. ALLEN, Jr. and S. PRUESS,Fundamentals of Numerical Computing,Wiley, New York, 1997C.W. UEBERHUBERNumerical Computation, 2 vols.Springer, Berlin, 1997W.G. McCALLUM et al.Multivariate Calculus,Wiley, New York, 1997ZHI-QUAN LUO, JONG-SHI PANG and D. RALPH,Mathematical Programs with Equilibrium Constraints,Cambridge University Press, Cambridge, 1996P.R. POPIVANOV and D.K. PALAGACHEV,The Degenerate Oblique Derivative Problem for Elliptic and Parabolic Equations,Akademie Verlag, Berlin, 1997     

Regularisierte Faltung von Distributionen. Teil 2: Eine Tabelle von Fundamentallösungen

Zusammenfassung In der folgenden Liste sollen einige häufig vorkommende Differentialoperatoren und ihre Fundamentallösungen in der Art einer Integraltafel zusammengefaßt werden. Dabei ist zu beachten, daß Fundamentallösungen im allgemeinen (d. h. ohne Zusatzbedingungen) nicht eindeutig bestimmt sind. In den meisten Fällen sind Berechnungsverfahren zur Herleitung der angegebenen Fundamentallösungen skizziert. DAbei wurden sowohl neue Methoden (z. B. Regularisierung, Penalisierung, regularisierte Faltung: 3, 6, 8, 10, 15, 22, 23, 24, 28, 31, 38, 40, 44, 48, 50, 51, 52, 64, 65, 68, 69, 72, 73) als auch neue Fundamentallösungen (15, 16, 20, 22, 23, 24, 31, 68, 69, 70) angegeben, deren Berechnung im 1. Teil (Abschnitt 3) aus Platzgründen nicht aufgenommen wurde. Die Berechnung von Fundamentallösungen mittelspartieller Fourier- oder Laplacetrans-formation, oder durch Spezialisierung in gewissen allgemeinen, von F. J. Bureau oder G. Herglotz angegebenen Integralen wurde außer Betracht gelassen und wird nicht zitiert. Für die Operatoren 26, 66, 67, 71, 74 und 77 wurden keine Fundamentallösungen angeschrieben, da entweder die entsprechende Literatur nur auszugsweise zur Verfügung stand oder die Formeln zu umfangreich sind.

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Summary Similar to a table of integrals the following table contains some frequently occurring differential operators and their fundamental solutions. However one has to pay attention to the fact, that fundamental solutions are not given uniquely (i.e. without assuming further conditions). In most cases you find methods to derive the listed fundamental solutions. We give new methods (e.g. regularization, penalization, regularized convolution: 3, 6, 8, 10, 15, 22, 23, 24, 28, 31, 38, 40, 44, 48, 50, 51, 52, 64, 65, 68, 69, 72, 73) as well as new fundamental solutions (15, 16, 20, 22, 23, 24, 31, 68, 69, 70), whose computation is not contained in part one (section 3) for lack of space. The derivation of fundamental solutions bypartial Fourier- or Laplace Transformation, or by specializing certain general integrals given by F. J. Bureau or G. Herglotz is not taken into account. For the operators 26, 66, 67, 71, 74 and 77 no fundamental solutions are given either for the corresponding reference being only partially available or the formulas being too long.

     

某一类色唯一的K_4-同胚图

令K_4(i,j,k,l,m,n)表示两两三度点间的路长分别为i,j,k,l,m,n的K_4-同胚图.对6条路的长均大于1且有4条路的长相等其余两条路互不相等的K_4-同胚图的着色进行了研究,得到了一类色唯一的K_4-同胚图.     

Book reviews

Devroye, L.; L. Györfi: Nonparametric Density Estimation: the L1 View. John Wiley & Sons New York, Chichester, Brisbane, Toronto, Singapore 1985, XI, 356 p., £ 45.90, ISBN 0-471816469

Gericke, H.: Mathematik in Antike und Orient. Springer-Verlag Berlin, Heidelberg, New York, Tokyo 1984, 140.Abb., 4 Kartenskizzen, XII, 292 S., DNI 98,-, ISBN 3-540-11647-8

K. ITö (ed.) StochasticAnalysis. Proceedings of the Traniguchi Internationaloal Symposium on Stochastic Analysis, Katata and Kyoto, 1982. North-Holland Mathematical Library, Vol. 32, North-Holland Publishing Company, Amsterdam 1984, 488 pages, Dfl. 250.- ISBN 0-444-87588-3

Weizenbaum, J.: Die Macht der Computer und dic Ohnmaeht der Vernuutt, Suhrkamp Taschenbuch Wissenschaf't 274. Suhrkamp Verlag Frankfurt/M. 1978, 368 S., DM 16,-, ISBN 3-518-27874-6

Krylov, N.V.; R. Sh. Lipiser; A.A Novikov (eds.): Statistics and Control of Stochastic Processes. Steklov Seminar, 1984. Springer-Verlag Berlin, Heidelberg, New York, Tokyo 1985, XIII, 507 S., DM 198, -, ISBN 3.540-96101-1

Sachs, H. (ed.): Graphs, Hypergraphs and Applications. Proceedings of the Conference on Graph Theory held in Eyba, Oct. 1984.. 'I'eubner-Texte zur Mathematik 73 Teubner Verlagsgesellschaft Leipzig 1985, 224 S., 23,-M, ISBN 666275-5

Bhat, U. Narayan; Elements of Applied Stochastic Processes. 2. Ed. Jonoh Wilet & Sons New York, Chichester, Brisbane, Toronto, Singapore 1984, IX, 685 p., £ 52.00, ISBN 0-471-87826-X

Methods of Operations Research. Bd, 66. Verlag Anton Hain Meisenheim, Königstein 1985, 410 S., DM 120,-, ISBN 3..445..02400..6

Maier, T.: Intervall-Input-Ontput-Rechnung. Mathernat.ical Systems in Economics. 101. Verlag Anton Rain, Meisenheim GmbH InigiteinTs. 1985, X, 338 S., DM 72,-, ISBN 3-445-02399-9.

Sawaragi, Y.; H. Nakayama; T. Tanino: Theory of Multi-objective Optimization. Mathematics in Science and Engineering. 176. Academic Press Orlando, San Diego, New York, London, Toronto 1985, 320 pp., $ 48.00, ISBN 0-12-620370-9.

Krämer, W.: Trend in ökonometrischen Modellen. Mathematical Systems in Economics. 93. Athenaum/Hain Konigstein 1985, 172 S., DM 48,-, ISBN 3-445-02386-7.     

抽象空间中非时齐马氏过程的分析理论(Ⅰ)——连续性、可微性及柯氏方程式

<正> 马氏过程的分析理论(这是主要指马氏过程的转移函数及其对应的半群的分析理论),是马氏过程基本理论的一个重要组成部分,对时齐的马氏过程的分析理论,不论其状态空间是可数的或一般的,国内外作者都进行过大量研究,得到过一系列结果.然而,对     

半质环的交换性条件

本文对满足可变恒等式的半质环在某种有界条件下给出了一个判断环Ｒ交换性的简便准则，使文献［２－２７］中所有相应结果均成为其直接推论．此外，对不限有界的情况，也得到较为广泛的结论．     

Uniform meyer solution to the three dimensional cauchy problem for laplace equation

We consider the three dimensional Cauchy problem for the Laplace equation uxx(x,y,z)+ uyy(x,y,z)+ uzz(x,y,z) = 0, x ∈ R,y ∈ R,0 z ≤ 1, u(x,y,0) = g(x,y), x ∈ R,y ∈ R, uz(x,y,0) = 0, x ∈ R,y ∈ R, where the data is given at z = 0 and a solution is sought in the region x,y ∈ R,0 z 1. The problem is ill-posed, the solution (if it exists) doesn't depend continuously on the initial data. Using Galerkin method and Meyer wavelets, we get the uniform stable wavelet approximate solution. Furthermore, we shall give a recipe for choosing the coarse level resolution.     

Toader型平均与几种经典平均的确界

给出了最佳参数α_1,α_2,α_3,β_1,β_2,β_3∈R,使得双向不等式α_1Q(a,b)+(1-α_1)G(a,b)0且a≠b成立.其中A(a,b)=(a+b)/2,H(a,b)=2ab/(a+b),G(a,b)=(ab)~(1/2),Q(a,b)=((a~2+b~2)/2)~(1/2),C(a,b)=(a~2+b~2)/(a+b),T(a,b)=2/π∫_0~(π/2)(a~2cos~2t+b~2sin~2)~(1/2)tdt分别是两个正数a和b的算术平均,调和平均,几何平均,二次平均,反调和平均和Toader平均.     

On Stability and Convergence of the Finite Difference Methods for the Nonlinear Pseudo-Parabolic System

In this paper, we deal with the finite difference method for the initial boundary value problem of the nonlinear pseudo-parabolic system $(-1)^Mu_t+A(x,t,u,u_x,\cdots,u_x 2M-1)u_x2M_t=F(x,t,u,u_x,\cdots,u_x 2M)$,$u_xk(o,t)=\psi_{0k}(t), u_xk(L,t)=\psi_{1k}(t),k=0,1,\cdots,M-1,u(x,0)=\phi (x)$ in the rectangular domain $D=[0\leq X\leq L,0\leq t\leq T]$, where $u(x,t)=(u_1(x,t),u_2(x,t),\cdots,u_m(x,t)),\phi (x),\psi_{0k}(t),\psi_{1k}(t),F(x,t,u,u_x,\cdots,u_x 2M)$ are $m$-dimensional vector functions, and $A(x,t,u,u_x,\cdots,u_x2M-1)$ is an $m\times m$ positive definite matrix. The existence and uniqueness of solution for the finite difference system are proved by fixed-point theory. Stability, convergence and error estimates are derived.     

BOOK REVIEWS

Book reviewed in this article: The Origin of the Carolina Bays , by Douglas Johnson. A Guide to Naval Aviation , by Lt. Wallace W. Elton, USNR; Lt. Alfred H. Driscoll, USNR; Lt. Robert N. Burchmore, USNR; and Lt. Gray B. Larkum, USNR. Mathematics for Navigators , by Delwyn Hyatt, Commander, USN, (Ret.) and Bennett M. Dodson, Commander, USNR. The Text -Activity Books . Elementary Topography and Map Reading , by Samuel L. Greitzer. Earth Science , A Physiography, by Gustave L. Fletcher. Shopwork , by Edward C. Wicks, Instructor of Industrial Arts, Morristown High School, New Jersey, John Poljacik, Jr., Instructor of Industrial Arts, Morrislown High School, and John Ellberg, author of Tales of A Rambler. Shop Practice , by William J. Kennedy. Plane Trigonometry and Statics , by Norman Miller. The Story of Flying , by Archibald Black. Electricity for Everyone , by Joseph R. Lunt, late Professor of Science at Rhode Island College of Education, and William T. Wyman. Air Navigation for Beginners , A Ground School Primer for Aerial Navigators , by Scott G. Lamb, B.S., M.S. Essentials of Algebra (Complete Second Year Course ), by Walter W. Hart. Solid Geometry , by F. Eugene Seymour, Supervisor of Mathematics, New York State Department of Education, and Paul James Smith, Head of Department of Mathematics, East High School, Rochester, New York. Modern Science Series , by Charles E. Dull. Organic Chemistry , by C. W. Porter and T. D. Stewart. Man's Physical Universe , by Arthur Talbot Bawden. Man and His Physical Universe , by Frank Covert, Jean Ezra Clarence Harrah, and Fred Louis Herman. Elementary Electricity , by Edgar P. Slack.     

一类不定方程组的解的个数

本文将证明以下结论:设m为正整数,当(a,c,δ)取(m,m+1,-1),(m,m+ 2,-2),(m,m+4,-4),或者(m+2,m,2)时,联立的不定方程组■的正整数解(x,y,z)的个数不超过1。     

关于阶乘的Erds-Stewart猜想

摘要:设n是正整数;P_0=1,P_i(i=1,2,…)是第i个素数.本文证明了:方程 n!+1=P_k~aP_(k+1)~b,P_(k-1)0,b>0,仅有解(n,P_k,P_(k+1),a,b)=(1,1,2,1,0),(2,3,5,1,0),(3,5,7,0,1),(4,5,7,2,0),(5,7,11,0,2).上述结果证实了Erds和Stewart提出的一个猜想.     

(3,2,1)-共轭r-正交拉丁方的存在性

如果两个v阶拉丁方L和M的重叠产生恰好r个不同的有序对,则称L和M是r-正交的.如果L还是M的(i,j,k)-共轭,则称L是(i,j,k)-共轭r-正交的,简记为(i,j,k)-r-COLS(v)((i,j,k)-r-conjugate orthogonal Latin square of order v),其中{i,j,k}={1,2,3}.本文研究(3,2,1)-r-COLS(v)的存在性问题.对于v 23,除去少数几个可能的例外值,本文给出关于(3,2,1)-r-COLS(v)的几乎完整的解.对于v23,如果r∈[v,v2]\{v+1,v+2,v+3,v+5,v+7,v2 1},除去可能的例外r=v2 3,都存在(3,2,1)-r-COLS(v).由于(3,2,1)-r-COLS(v)的存在性与(1,3,2)-r-COLS(v)的存在性是等价的,本文得到关于(1,3,2)-r-COLS(v)的同样结论.     

On codes from cyclotomic fields for multidimensional signals

This paper derives an algorithm for constructing codes overfinite fields of the algebraic integer ring of the cyclotomicfield Q(e^2i/n) modulo irreducible elements with the norm p²,where n {3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20,21, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 48, 60, 84},where p x n and p is a prime number.     

L—Fuzzy命题逻辑系统的语构分层方案

建立一种L－Fuzzy命题逻辑的语构分层构造,引进诸如λ－信息,λ－公理系,λ－语构,λ－证明,λ－定理,λ－语构结论,λ－语构闭包,λ－紧性,,上L－Fuzzy语构闭包,下Fuzzy语构闭包,λ－语构分层和谐,语构分层和谐,强语构分层和谐等新的观念和方法,从而L－Fuzzy命题逻辑语构和语义分层次的程度化的和谐性度量提供了一种方案。     

Hybrid unified theory

We shall investigate, in this paper, theories related to mind, consciousness, life, evolution, existence, non-existence, cosmology, quantum fields, quantum jump, Big Bang, and higher reality. A hybrid unification between ancient and modern discoveries is discussed with a view to shedding some light on the existing theories and strengthening them.     

一般对角同余式的解的个数

设Ｐ为素数,ｎ,ｒ以及１ｌ２…,ｌｒ均为正整数,ａ１,ａ２,…,ａｒ及ｂ均为整数,诸ａｉ,ｌｉ均与ｐ互素．本文给出一般对角同余式ａ１ｘ１ｌ１＋ａ２ｘ２ｌ２＋…＋ａｒｘｒｌｒ＝ｂ（ｍｏｄｐｎ）的解数公式．