Errata for“The Approximate Valuesof Certain SeriesInvolving the Prowersof Generalized Fibonacciand Lucas Numbers

On page1 5 8,( 6) ,( 7) ,( 8) ,and( 9) should be,respectively:∑∞n=1α4nU22 n U22 n+ 2≈ ΔV2p2 α212 4 - 18lnα+ π296( lnα) 2 - 3α4p4,∑∞n=0α4nV22 n V22 n+ 2≈ V2Δα2 p218+ 18lnα+ π24 ( lnα) 2 ( eπ2 / ( 2 lnα) - 2 ) - 14Δα4p2 - 1Δα2 Δ p3 ,∑∞n=0α4nU22 n+ 1U22 n+ 3≈ ΔV2p2 α418lnα- π24 ( eπ2 / ( 2 lnα) + 2 ) ( lnα) 2 - 1α6p2 - 2α5p3 ,∑∞n=0α4nV22 n+ 1V22 n+ 3≈ V2Δα4p2π24 ( lnα) 2 ( 2 eπ2 / lnα + 1 ) + π232 ( lnα) 2 - 18lnα - 1Δα6p4- 2Δα5p4Δ.In t…     

三角变换的若干技巧

一、化归为条件角例1 已知cos(a π/4)=3/5,π/2≤a<3π/2,求cos(2a π/4)的值. (2002年全国新课程卷高考题) 简析a π/4是题设条件涉及的角,可简称为条件角.2a π/4是题目所求涉及的角,可叫目标角.将目标角转化为条件角,问题就解决了.     

一道课本习题的再探究

人教版新课标教材《数学4》（必修）第151页有如下问题： 观察以下各等式： sin^2 π/6＋cos^2 π/3＋sin π/6 cosπ/3=3/4，     

一个有极限的单调增加数列

证明了{n(16n^2+4n+3)/16n^2-4~n+3^(1/2) integral from 0 to π/2 sin^nxdx}为严格单调增加数列,且极限为π/2^(1/2),因而得π(16n^2+36n+23)/2(n+1)(16n^2+28n+15)^(1/2)相似文献   

数学问题解答

20 0 3年 1月号问题解答(解答由问题提供人给出 )1 41 1　求大于 1的整数ｋ,使ｆ(ｘ) =ｓｉｎｋｘ·ｓｉｎｋｘ+ｃｏｓｋｘ·ｃｏｓｋｘ-ｃｏｓｋ2ｘ为常值函数 .(湖北省襄樊市一中　王必廷　 441 0 0 0 )解　取ｘ=0 ,得ｆ(0 ) =0 ,故ｆ(ｘ) =0取ｘ =π/ｋ ,则ｓｉｎπ·ｓｉｎｋ πｋ +ｃｏｓπ·ｃｏｓｋ πｋ -ｃｏｓｋ2πｋ =0所以 -ｃｏｓｋ πｋ =ｃｏｓｋ2πｋ所以ｋ为奇数 ,且 -ｃｏｓ πｋ =ｃｏｓ2πｋ所以ｃｏｓ2πｋ =ｃｏｓπ - πｋ所以π- πｋ =2ｎπ±2πｋ所以 1ｋ =2ｎ - 1或3ｋ =1 - 2ｎ　　ｎ∈ｚ所以ｋ=1或 3经检验知…     

数学问题解答

20 0 1年 9月号问题解答(解答由问题提供人给出 )1 331　解方程 :8ｘ3- 6ｘ 1 =0(山东省新泰一中九九级 1 0班南区学生　田茂江　 2 71 2 0 0 )解 :将方程变形为 :12 =3ｘ - 4ｘ3ｓｉｎ( π6 2ｋπ) =3ｘ - 4ｘ3 ①由三倍角公式得ｓｉｎ( π6 2ｋπ) =3ｓｉｎ( π1 8 2ｋπ3)- 4ｓｉｎ3( π1 8 2ｋπ3)②由①②得ｘ =ｓｉｎ( π1 8 2ｋπ3)即ｘ1 =ｓｉｎ π1 8,ｘ2 =ｓｉｎ1 3π1 8,ｘ3=ｓｉｎ2 5π1 8又∵三次方程最多有三个根 ,∴以上即为原方程的全部根1 332　函数ｆ(ｘ) ,ｘ∈ [0 , ∞ ) ,ｆ(ｘ)不恒等于0 ,对任意ｘ ,ｙ∈ [0 , …     

A Relation in the Stable Homotopy Groups of Spheres

Let p≥7 be an odd prime. Based on the Toda bracket α1βp-11, α1 β1, p, γs,the authors show that the relation α1βp-11h2,0 γs= βp/p-1γs holds. As a result, they can obtain α1βp1h2,0 γs = 0 ∈π*(S0) for 2≤s≤p- 2, even though α1h2,0γs and β1α1h2,0 γs are not trivial. They also prove that βp-11α1 h2,0 γ3 is nontrivial in π*(S0) and conjecture that βp-11α1 h2,0 γs is nontrivial in π*(S0) for 3≤s≤p- 2. Moreover, it is known thatβp/p-1γ3 = 0 ∈ Ext5,*BP*BP(BP*, BP*), but βp/p-1γ3 is nontrivial in π*(S0) and represents the element βp-11α1 h2,0 γ3.     

Radius of locally convex subsets in Alexandrov spaces with curvature ≥1 and radius 〉 π/2

Let X be a complete Alexandrov space with curvature ≥1 and radius 〉 π/2. We prove that any connected, complete, and locally convex subset without boundary in X also has the radius 〉 π/2.     

评2003年高中联赛

与往年的试题相比 ,2 0 0 3年的试题计算量较小 ,而思维的程度有所增加 ,更有利于培养人才 .选择题的 3是过抛物线 y2 =8(x + 2 )的焦点F作倾斜角为 6 0°的弦AB ,AB的中垂线交x轴于P ,求PF ,本题焦点F为原点 ,直线AB方程为y =3x ,所以A ,B横坐标适合方程 :3x2 - 8x - 1 6 =0 .由韦达定理 ,AB中点E的横坐标为12 ×83=43.由于AB倾斜角为 6 0° ,所以FE =2×43,PF =2FE =4×43=1 63.选择题的 4是x∈ [- 5π1 2 ,- π3],求 y =tan(x + 2π3) -tan(x + π6 ) +cos(x + π6 )的最大值 .本题可先化 y为同角的三角函数的代数和 .y =-cot(…     

对新题征展(31)第3题的订正

题　已知复数 z满足条件 | z| =1 ,求| z - i| .| z - 12 32 - i|的最大值 .解法 1　设 z =cosθ isinθ,其中θ∈[0 ,2π) ,| z - i| =| cosθ i( sinθ - 1 ) |= cos2 θ ( sinθ - 1 ) 2 =2 ( 1 - sinθ)= 2 [1 - cos( π2 -θ) ]=2 | sin( π4 - θ2 ) || z - 12 32 i|= | ( cosθ - 12 ) i( sinθ 32 ) |= ( cosθ - 12 ) 2 ( sinθ 32 ) 2= 2 2 sin(θ - π6 )=2 [1 cos( 2π3-θ) ]=2 .2 cos2 ( π3- θ2 )=2 | cos( π3- θ2 ) | .则　 | z - i| .| z - 12 32 i|=4 | sin( π4 - θ2 ) .cos( π3- θ2 ) |=…     

Comparison of two norms of matrices

Any complex n × n matrix A satisfies the inequality

A ₁ ≤ n 1/2 A _d

where .₁ is the trace norm and ._d is the norm defined by

Full-size image

,

where B is the set of orthonormal bases in the space of n × 1 matrices. The present work is devoted to the study of matrices A satisfying the identity:

A₁ = n1/2 A _d

This paper is a first step towards a characterization of matrices satisfying this identity. Actually, a workable characterization of matrices subject to this condition is obtained only for n = 2. For n = 3, a partial result on nilpotent matrices is presented. Like our previous study (J. Dazord, Linear Algebra Appl. 254 (1997) 67), this study is a continuation of the work of M. Marcus and M. Sandy (M. Marcus and M. Sandy, Linear and Multilinear Algebra 29 (1991) 283). Also this study is related to the work of R. Gabriel on classification of matrices with respect to unitary similarity (see R. Gabriel, J. Riene Angew, Math. 307/308 (1979) 31; R. Gabriel, Math. Z. 200 (1989) 591).     

Atoms of set systems with a fixed number of pairwise unions

For a finite set system with ground set X, we let

Full-size image

. An atom of H is a nonempty maximal subset C of X such that for all A H, either C A or C ∩ A = 0. We obtain a best possible upper bound for the number of atoms determined by a set system H with H = k and H H = u for all integers k and u. This answers a problem posed by Sós.     

Strong continuity of the relative rearrangement maps and application to a Galerkin approach for nonlocal problems

We study the strong continuity of the map u   (b_*u, b_*u(| > u(·)|)). Here,

Full-size image

for σ]0 means Ω[, u_* (respectively, (b|_{u=u*(σ)})*) denotes the decreasing rearrangement of u (respectively b restricted to the set {u = u_*(σ)}) and |E| denotes the Lebesgue measure of a set E included in a domain Ω. The results are useful for solving plasmas physics equations or any nonlocal problems involving the monotone rearrangement, its inverse or its derivatives.     

A single-element extension of antimatroids

An antimatroid is a family of sets which is accessible, closed under union, and includes an empty set. A number of examples of antimatroids arise from various kinds of shellings and searches on combinatorial objects, such as, edge/node shelling of trees, poset shelling, node-search on graphs, etc. (Discrete Math. 78 (1989) 223; Geom. Dedicata 19 (1985) 247; Greedoids, Springer, Berlin, 1980) [1, 2 and 3]. We introduce a one-element extension of antimatroids, called a lifting, and the converse operation, called a reduction. It is shown that a family of sets is an antimatroid if and only if it is constructed by applying lifting repeatedly to a trivial lattice. Furthermore, we introduce two specific types of liftings, 1-lifting and 2-lifting, and show that a family of sets is an antimatroid of poset shelling if and only if it is constructed from a trivial lattice by repeating 1-lifting. Similarly, an antimatroid of edge-shelling of a tree is shown to be constructed by repeating 2-lifting, and vice versa.     

On approximation of solutions of multidimensional SDE's with reflecting boundary conditions

Let D be either a convex domain in ^d or a domain satisfying the conditions (A) and (B) considered by Lions and Sznitman [7] and Saisho [11]. We estimate the rate of L^p convergence for Euler and Euler–Peano schemes for stochastic differential equations in D with normal reflection at the boundary of the form

Full-size image

, where W is a d-dimensional Wiener process. As a consequence we give the rate of almost sure convergence for these schemes.     

Bounds on the number of isolates in sum graph labeling

A simple undirected graph H is called a sum graph if there is a labeling L of the vertices of H into distinct positive integers such that any two vertices u and v of H are adjacent if and only if there is a vertex w with label L(w)=L(u)+L(v). The sum number σ(G) of a graph G=(V,E) is the least integer r such that the graph H consisting of G and r isolated vertices is a sum graph. It is clear that σ(G)|E|. In this paper, we discuss general upper and lower bounds on the sum number. In particular, we prove that, over all graphs G=(V,E) with fixed |V|3 and |E|, the average of σ(G) is at least

Full-size image

. In other words, for most graphs, σ(G)Ω(|E|).     

Solution of bordered singular systems in numerical continuation and bifurcation

In numerical continuation and bifurcation problems linear systems with coefficient matrices in the block form arise naturally. Here

Full-size image

and n may be large but m is small. A usually has a special structure (banded, block banded, sparse,…) and B, C, D are dense, so that it is advisable to use a specialized solver for A and to solve with M by some block method. Unfortunately, A is often also a nearly singular matrix (in fact, made nonsingular only by roundoff and truncation errors). On the other hand, M is usually nonsingular but can be ill-conditioned and in certain situations will degenerate to singularity as well. We describe numerical tests for this problem using the mixed block elimination method of Govaerts and Pryce (1993) for solving bordered linear systems with possibly nearly singular blocks A. To this end, we compute by Newton's method a triple-point bifurcation point in a parameterized reaction—diffusion equation (the Brusselator). The numerical tests show that the linear systems are solved in a stable way, in spite of the use of a black-box solver (SGBTRS from LAPACK) for a nearly singular matrix.     

Chromatic-index-critical graphs of orders 13 and 14

A graph is chromatic-index-critical if it cannot be edge-coloured with Δ colours (with Δ the maximal degree of the graph), and if the removal of any edge decreases its chromatic index. The Critical Graph Conjecture stated that any such graph has odd order. It has been proved false and the smallest known counterexample has order [[18] A.J.W. Hilton, R.J. Wilson, Edge-colorings of graphs: a progress report, in: M.F. Cabobianco, et al. (Eds.), Graph Theory and its Applications: East and West, New York, 1989, pp. 241–249; [31] H.P. Yap, Some topics in graph theory, London Mathematical Society, Lecture Note Series, vol. 108, Cambridge University Press, Cambridge, 1986].

In this paper we show that there are no chromatic-index-critical graphs of order 14. Our result extends that of [[5] G. Brinkmann, E. Steffen, Chromatic-index-critical graphs of orders 11 and 12, European J. Combin. 19 (1998) 889–900] and leaves order 16 as the only case to be checked in order to decide on the minimality of the counterexamples given by Chetwynd and Fiol. In addition we list all nontrivial critical graphs of order 13.     

The practice of operational research in Taiwan

This paper presents the results of a questionnaire survey of operational research/management science (OR/MS) practices in Taiwanese companies. The survey was addressed to the current status and future perspectives of OR/MS in Taiwan. During the spring of 2001, 2000 questionnaires were mailed to the top 1000 domestic manufacturing firms and top 1000 domestic service organizations. These asked questions concerning specific OR/MS techniques being used, application areas of OR/MS, reasons for using OR/MS, reasons for not using OR/MS, difficulties encountered practically, and future perspectives. Results from 262 valid respondents provided several informative findings. Compared with the 1995 survey of Kao et al, it found that the percentage of companies that have used OR/MS techniques increased from 62.7% in 1995 to 76.7% in 2001. Also, the percentage of companies that have never heard of OR/MS decreased from 16.1% in 1995 to 11.1% in 2001. Because of the Taiwan government recently developing educational and training programs in OR/MS, it found that the importance of OR/MS has been recognized by Taiwan companies as an efficient management tool for economic development, and the usage of OR/MS will continuously increase.