数学问题解答及解答者名单

265．把直径是一寸的钢球放到一尺见方、一寸高的铁盒内，你能放多少进去？为什么？（何思明提）．答案是IO6个．今将其摆法与证明叙述于下：因为铁盒的高是一寸，故盒内只能摆一层球．因此问题变为考虑在一尺见方的正方形内可以摆多少个直径是1寸的圆的问题．首先沿正方形的底边摆IO个圆作为第一行，其次在第一行的每个圆的上面摆下IO个圆作为第二行，然后再在第二行的上面摆下IO个圆作为第三行，显然第三行的IO个圆的每一个圆心至底边的距离是25寸，于是第三行的IO个圆的圆心在～条直线上，这条直线与底边平行，距离是2．5寸．紧靠着…     

拉格朗日与"米制"

1 度量衡改革——大革命后的一项重要改革。在历史上，长度计量单位是比较混乱的，古罗马帝国查理一世规定以他的脚长为1尺．我国古代有过以夏禹身高为1丈的记载．汉代也有“中妇女手掌八寸”之说，即长短适中的妇女，手掌之长定为八寸；又有“度人之两臂为寻，八尺也”的记载，即以张开人的两臂之长定为八尺．这些规定现在看来十分荒唐可笑．直到18世纪，欧洲资本主义兴起，商业贸易频繁，改变传统的、不科学的度量衡制，逐步过度到统一的计量单位势在必行．1799年，米制诞生于巴黎．     

换比萨

星期天．精灵鼠小弟和爸爸、妈妈去必胜客吃饭，精灵鼠小弟点了一个12寸的比萨，谁知道服务员说没有了，建议给他们一个9寸的外加一个6寸的比萨。爸爸妈妈觉得不错，可精灵鼠小弟不乐意了，坚持叫服务员拿一个9寸的，外加两个6寸的!     

两个近似方法

数学通报1958年12月号20頁和11月号12頁介紹了两个来自农民羣众的近似方法。一个是用圓周长C表圆面积A的公式:A=C~2/12,一个是正五边形的作图法:“一尺头頂六,八五两边分”。意思是要作一个正五边形,可先作线段GFA,使GF=1尺,FA=6寸;再作EFB⊥GA,使EF=FB=8寸,又DGC⊥AG,使DG=GC     

双曲线抛物线切线的尺规作法

文[1]介绍了椭圆切线的尺规作图方法，作为补充，本文介绍双曲线、抛物线切线的尺规作法．     

珠算算理浅论

四、远古时代数学发展的可能性 （一）度量单位的发展用尺量长度实际早已有记载。1976年上海人民出版社出版《中国古代的发明创造》一书揭示，公元前2698年以后，已有尺，其尺折24.88厘米，该尺以九为进制，足见我国在4700年前对长度就有了研究。关于长度的研究是不是更远一些，有待进一步考证，若伏羲研究的矩有数量关系，那么可以把历史推进7000年。尺的出现不是小事。它说明数轴已经出现。黄帝时有尺，说明中国有了历史可载的“数”学，数学理性化已经开始，“隶首作数”决不是虚言。这个数学经过大禹时代可能因治水的需要，得到了很大的发展，到了商代周代数学已经达到理性化的一个顶峰。     

巧构妙用简化解几运算

在解决圆锥曲线的问题中，大部分学生觉得“计算量太大，太复杂，没信心继续算下去”．其实，学好圆锥曲线的关键是过好两个关：方法关与运算关．而计算量大往往与选择的方法有很大关系．笔者就如何构建函数、方程等手段，巧妙利用好韦达定理，把繁复的计算变得简洁流畅，进行探究．、1构造函数，运用韦达定理     

HPM视角下“三等分角”习题课教学与思考

一、数学史话将数学史与数学融合在一起共同促进学生的发展是HPM(Historyand Pedagogyof Mathematics)研究的一个主要涵义.我们知道,三等分角是古希腊三大几何问题之一.即在尺规作图(即用没有刻度的直尺和圆规作图)的前提下,无法将一个给定的角三等分.若将条件放宽,却可以引发三等分的作法.比如历史上关于"三等分角"的故事的版本很多,其中有一段是与阿基米德有关.这里简要概述这段故事.     

一些事物的有限性

1把一张2寸照片等分成（比如）1百万个小的正方或长方格，让每个格只取黑白2色，则只有a=21000000张这种精确度（指：等分成1百万个小格）的照片．到若干年之后，地球上的人数（包括已死的，健在的和未生的）会超过a．那时，就至少有2人的2寸照片完全相同（按上述精确度）．因而，此2人的面貌（从照相机的那个侧面看）也就（按上述精确度）完全相同了。     

米勒问题帮你解题

米勒（Johannes Miiller1436·1476），德国数学家，对三角学做出了巨大贡献，是斐波那契以来欧洲最有影响的数学家．米勒1533年发表的名著《三角全书》是使三角学在欧洲取得独立地位的第一部系统性著作．该书共分五册，前两册讲平面三角，后三册讲球面三角．此外，他还讨论到一个新颖的极值问题：天花板挂一垂直的杆，长10尺，下端离地面4尺。     

Diversity and Popularity in Organizations and Communities

This paper examines social groupings whose structure depends only on the distribution of ability to attract attention. When people‘s attention is a scarce resource, those individuals who are rated highest by a large number of other individuals will have to ration their attention, resulting in constraints on the social structure of the group. The incidence of popular individuals by that definition reflects the extent to which individuals agree on who their highest-rated colleague is. We propose three basic distributions or ways to generate the matrix of perceived ability so as to yield popularity profiles that can be parametrically adjusted to match observations. We demonstrate that each of these assumption sets leads to a slightly different correlation between the value of the assumption‘s parameter and the set of observable popularity patterns. Since popularity, in real life, often determines such things as power, centrality, over-utilization and perhaps reduced accessibility, having more realistic ways of representing it is important for modeling and understanding virtual organizations and communities.     

Ritz and harmonic Ritz values and the convergence of FOM and GMRES

The Ritz and harmonic Ritz values are approximate eigenvalues, which can be computed cheaply within the FOM and GMRES Krylov subspace iterative methods for solving non‐symmetric linear systems. They are also the zeros of the residual polynomials of FOM and GMRES, respectively. In this paper we show that the Walker–Zhou interpretation of GMRES enables us to formulate the relation between the harmonic Ritz values and GMRES in the same way as the relation between the Ritz values and FOM. We present an upper bound for the norm of the difference between the matrices from which the Ritz and harmonic Ritz values are computed. The differences between the Ritz and harmonic Ritz values enable us to describe the breakdown of FOM and stagnation of GMRES. Copyright © 1999 John Wiley & Sons, Ltd.     

Polytopes and arrangements: Diameter and curvature

We introduce a continuous analogue of the Hirsch conjecture and a discrete analogue of the result of Dedieu, Malajovich and Shub. We prove a continuous analogue of the result of Holt and Klee, namely, we construct a family of polytopes which attain the conjectured order of the largest total curvature.     

Three- and Four-Dimensional Einstein-like Manifolds and Homogeneity

The aim of this paper is to study three- and four-dimensional Einstein-like Riemannian manifolds which are Ricci-curvature homogeneous, that is, have constant Ricci eigenvalues. In the three-dimensional case, we present the complete classification of these spaces while, in the four-dimensional case, this classification is obtained in the special case where the manifold is locally homogeneous. We also present explicit examples of four-dimensional locally homogeneous Riemannian manifolds whose Ricci tensor is cyclic-parallel (that is, are of type A) and has distinct eigenvalues. These examples are invalidating an expectation stated by F. Podestá and A. Spiro, and illustrating a striking contrast with the three-dimensional case (where this situation cannot occur). Finally, we also investigate the relation between three- and four-dimensional Einstein-like manifolds of type A and D'Atri spaces, that is, Riemannian manifolds whose geodesic symmetries are volume-preserving (up to sign).     

E—matrices and Several Necessary and Sufficient Conditions

AER-x.,Aiscalledanonnegative(nonPositive)matrixifau>O(a,jO(aiirtO),markedA>O(AO(i#j),auO,wecallp(A)=max{IA,l,lA,l'.-.,lAnl},(whereA,,A2,..',A"areeigenva1ueofA)asspectralradiusofA.LemmalIfAe7'.,thenareanonnegativesquarematrxBandapositivenumberksuchthatA=B-kI(whereIistheidentitymatrixofordern).ProofLetbti=…     

Singular quasilinear and Hessian equations and inequalities

We solve the existence problem in the renormalized, or viscosity sense, and obtain global pointwise estimates of solutions for quasilinear and Hessian equations with measure coefficients and data, including the following model problems:

     

Vertex- and edge-minimal and locally minimal graphs

Given a family of graphs , a graph is called edge-minimal (vertex-minimal) if for every subgraph (induced subgraph) G^′ of G; furthermore, G is called locally edge-minimal (locally vertex-minimal) if whenever G^′ is obtained from G by deleting an edge (a vertex). Similarly, the concepts of minimality and local minimality are introduced for directed graphs (digraphs) and, more generally, for partially ordered sets.For example, by the Strong Perfect Graph Theorem, the only vertex-minimal graphs with χ>ω are odd holes and anti-holes. In contrast, the only locally vertex-minimal graphs with χ>ω are partitionable graphs. Somewhat surprisingly, there are infinitely many non-trivial perfect graphs that are locally edge-minimal and -maximal simultaneously. In other words, such a graph is perfect but it becomes imperfect after any edge is deleted from or added to it.In this paper we consider vertex- and edge-minimal and locally minimal graphs in the following families: (i) perfect and imperfect graphs, (ii) graphs with χ=ω and with χ>ω, (iii) digraphs that have a kernel and kernel-free digraphs, and finally, (iv) vertex-minimal complementary connected d-graphs.     

On exact and inexact differentials and applications

Considering the important role played by mathematical derivatives in the study of physical–chemical processes, this paper discusses the different possibilities and formulations of this concept and its application. In particular, in Chemical Thermodynamics, we study exact differentials associated with the so-called state functions and inexact differentials associated with path functions. As an application, we discuss the equation of state associated with a gas.     

Multidimensional analogs of direct and converse Jackson and Bernshtein theorems and their generalizations

We obtain an instance of the multidimensional analogs of the classical direct and converse Jackson and Bernshtein-Vallée-Poussin theorems (a version stronger than those obtained by S. M. Nikol’skii) and significant generalizations of these theorems to polynomial approximations of functions of real variables from Nikol’skii and Besov spaces on bounded domains with Lipschitzian boundary. The direct theorem concerning the Nikol’skii spaces was previously obtained by Yu. A. Brudnyi and generalized to joint polynomial approximations of functions and their derivatives by B. N. Konovalov. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 608–615, April, 2000.     

面积比较与上调和函数及其应用

本文研究了无穷远处具非负截曲率的完备非紧流形的几何和解析性质,并将 它应用于调和映照.