关于丢番图方程x^2±2^m=y^n

本文证明了：方程ｘ２＋２ｍ＝ｙｎ，ｘ，ｙ，ｍ，ｎ∈Ｎ，ｇｃｄ（ｘ，ｙ）＝１，ｎ＞２仅有有限多组解（ｘ，ｙ，ｍ，ｎ），而且当（ｘ，ｙ，ｍ，ｎ）≠（５，３，１，３），（１１，５，２，３），（７，３，５，４）时，ｎ是适合ｎ≡７（ｍｏｄ８）以及２３≤ｎ＜８．５·１０６的奇素数，ｍａｘ（ｘ，ｙ，ｍ）＜Ｃ１；方程ｘ２－２ｍ＝ｙｎ，ｘ，ｙ，ｍ，ｎ∈Ｎ，ｇｃｄ（ｘ，ｙ）＝１，ｙ＞１；ｎ＞２仅有有限多组解（ｘ，ｙ，ｍ，ｎ），而且这些解都满足ｎ＜２·１０９炉以及ｍａｘ（ｘ，ｙ，ｍ）＜Ｃ２，这里Ｃ１，Ｃ２是可有效计算的绝对常数．     

齐民友先生谈数学教育改革──在PMN全国会议上的讲话(摘要)

这个会非常重要，我很高兴，主动来参加这个会．近年在国内，关于教育方面的种种问题都在考虑，不仅在中国，在美国也一样，例如在加州围绕教学改革，打了一场"数学之战"，最近这一年我与华中师大路刚同志谈的较多，无锡的徐沥泉老师，试着把数学方法论用于数学教学，做了很好的工作，数学教学问题的研究，郁很大的发展，英藉匈牙利人拉卡托斯，通过研究多面体的欧拉公式，以此为素材，为案例，搞助探论（heuristic，康宏逵先生的译法是准确的，有人译为探索法，亦可，译成"启发法"就在些勉强了），起初在一个杂志登出来，后来他的学生把它…     

读者来信

多年来，本刊录用的每一篇文章，都是先致信作者，通知稿件录用，并要求回信说明文章是否他投，大多数作者回信都会如实说明情况，但是，最近发现有些作者并未说真话，导致文章在多家杂志先后登出，造成了恶劣影响．这种影响是多方面的，首先，这是对作者自身诚信的一种否定；其二，这是对审稿人劳动的不尊重；其三，杂志的版面有限，这种行为直接剥夺了其他作者展现自己成果的机会；其四，这种行为严重影响了杂志的声誉，一些热心的读者通过电话，电子邮件和信件等形式指出这种现象，并列出了这些作者的名单和发表文章的题目，强烈谴责这种行为，在此，我们向这些关心和支持杂志的读者表示感谢，此前，为了保护作者的积极性，并未进行公开批评，针对当前这种现象呈增多之势，本着保护其他作者积极性、维护杂志声誉的想法，对于今后出现一稿多投的作者，我们将公开批评，列出作者姓名和发表文章题目，以正视听，下面摘登了朱永厂老师的一封来信．     

关于折线的一个猜想的证明

文[1]中，王方汉老师有如下猜想：当n为形如4m－1（mEN）的素数时，生成数列凡一l＝（1，2，…，牛／，M，…，2，l）所确定的数列B，，具有遍历性．也就是说，设数列凡1＝（r，12，…，r。－l），h为正整数，B。＝（hi，bZ，…，b。），1＝hi＜bZ＜…＜b。且均为整数，设凡；与B。之间有如下映射关系：。。。。，；；＝（1，2，…，宁，宁，…，2，1）且n为形如4m－1的素数时，数列B。各项被n除的余数恰是0，1，2，…，n－1的一个全排列．显然，数列B。＝（hi，bZ，…，b。）各项被n除的余数恰是0，1，2，…，n一且的一个全排列等价于…     

一个随机偏差定理

设{Xn，n≥0}是可列非齐次马尔可夫链，Sn（i0，i1，…，im-1，ω）表示m元状态序组（i0，i1，…，im-1）在序列（X0，X1，…，Xm-1），（X1，X2，…，Xm），…，（Xn-1，Xn，…，Xn＋m-2）中出现的次数.本文给出了关于Sn（i0，i1，…，im-1，ω）的一个对任意可列非齐次马尔可夫链普遍成立的随机偏差定理，即用不等式表示的一个强极限定理.     

关于方程xy+yz+zx=n的正整数解

本文在广义Ｒｉｅｍａｎｎ猜想成立的条件下证明了：当且仅当正整数ｎ＝１，２，４，６，１０，１８，２２，３０，４２，５８，７０，７８，１０２，１３０，１９０，２１０，３３０，４６２时，方程ｘｙ＋ｙｚ＋ｚｘ＝ｎ无正整数解（ｘ，ｙ，ｚ）．     

基于巴斯卡数据成败型元件串联系统可靠性经典精确置信限

设Ａ为由Ｋ个相互独立的成败型元件组成的串联系统，第ｉ个元件的可靠性ｐｉ，ｐｉ未知，ｉ＝１，２，…，Ｋ．设对第ｉ个元件，对于给定的ｍｉ，有ｎｉ个巴斯卡试验数据：Ｘｉ１，Ｘｉ２，…，Ｘｉｎｉ，其中Ｘｉｊ表示对第ｉ个元件进行试验，试验进行到ｍｉ次成功时所需要的试验次数ｊ＝１，２，…，ｎｉ，ｉ＝１，２，…，Ｋ．记Ｔｉ＝Ｘｉ１＋Ｘｉ２＋…＋Ｘｉｎｉ，ｉ＝１，２，…，ｋ．本文研究基于统计量（Ｔ１，Ｔ２，…，Ｔｋ）求串联系统Ａ的可靠性经典精确最优置信下限．     

两向量组与其对应的齐次线性方程组之间关系的讨论

两向量组与其对应的齐次线性方程组之间关系的讨论宋福民邱根胜（南昌航空工业学院，南昌３３００３４）设有两个ｎ维向量组：Ａ：αｊ＝（ａｊ１，ａｊ２，…，ａｊｎ），ｊ＝１，２，…，ｓ１，Ｂ：βｊ＝（ｂｊ１，ｂｊ２，…，ｂｊｎ），ｊ＝１，２，…，ｓ２．记Ａ＝...     

关于Shannon－McMillan定理的若干研究

设｛Ｘｎ，ｎ≥１｝是字母集为Ｓ＝｛１，２，…，Ｎ｝上的任意信源，其分布为ｐ｛ｘ１，…，ｘｎ｝，（ｐｋ（１），ｐｋ（２），…，ｐｋ（Ｎ），ｋ＝１，２，…，是Ｓ上的一列分布　称为｛Ｘｋ，１≤ｋ≤ｎ｝相对于乘积分布　的熵密度偏差，本文利用这个概念研究任意信源的极限性质，其中包括在没有平稳性和遍历性假设情况下对ＳｈａｎｎｏｎＭｃＭｉｌｌａｎ定理的某些讨论．     

成功数不确定但已知权分布成败型元件可靠性点估计及置信限

设成败型元件可靠性为Ｐ，Ｐ未知，今对此元件进行ｎ次独立试验，确切的成功数并不知道，但只知成功数在ｓ，ｓ＋１，…，ｓ＋ｍ之间，且设对成功数为ｓ，ｓ＋１，…，ｓ＋ｍ的相信程度分别为β０，β１，…，βｍ（βｉ≥０，Σβ１＝１），本文研究基于数组（ｎ，ｓ，ｍ，β０，β１，…，βｍ）求此元件可靠性后的点估计及置信限的问题。     

Dual divergence estimators and tests: Robustness results

The class of dual ?-divergence estimators (introduced in Broniatowski and Keziou (2009) [5]) is explored with respect to robustness through the influence function approach. For scale and location models, this class is investigated in terms of robustness and asymptotic relative efficiency. Some hypothesis tests based on dual divergence criteria are proposed and their robustness properties are studied. The empirical performances of these estimators and tests are illustrated by Monte Carlo simulation for both non-contaminated and contaminated data.     

Invariant distributions on then-fold metaplectic covers ofGL(r, F), F p-adic

We describe the unitary and tempered dual of the n-fold metaplectic covers of SL(2, F), where F is a p-adic field with p not dividing 2n. We show that any tempered distribution on the n-fold metaplectic covers of SL(2, F) or of GL(r, F) (satisfying the assumptions of Section 1.1 below) may be expressed as a distributional integral over the tempered dual. We also show that any invariant distribution on the n-fold metaplectic covers of SL(2, F) or of GL(r, F) is supported on the tempered dual. Dedicated to John Benedetto on his 60th birthday.     

A modification of Hochbaum and Shmoys' algorithm for scheduling problems

The dual approximation algorithm of Hochbaum and Shmoys for solving the minimum makespan problem on a set of identical machines is modified in such a way that it turns into an optimal algorithm when the relative error is chosen properly.     

A Dual-Primal FETI method for incompressible Stokes equations

In this paper, a dual-primal FETI method is developed for incompressible Stokes equations approximated by mixed finite elements with discontinuous pressures. The domain of the problem is decomposed into nonoverlapping subdomains, and the continuity of the velocity across the subdomain interface is enforced by introducing Lagrange multipliers. By a Schur complement procedure, the solution of an indefinite Stokes problem is reduced to solving a symmetric positive definite problem for the dual variables, i.e., the Lagrange multipliers. This dual problem is solved by the conjugate gradient method with a Dirichlet preconditioner. In each iteration step, both subdomain problems and a coarse level problem are solved by a direct method. It is proved that the condition number of this preconditioned dual problem is independent of the number of subdomains and bounded from above by the square of the product of the inverse of the inf-sup constant of the discrete problem and the logarithm of the number of unknowns in the individual subdomains. Numerical experiments demonstrate the scalability of this new method. This work is based on a doctoral dissertation completed at Courant Institute of Mathematical Sciences, New York University. This work was supported in part by the National Science Foundation under Grants NSF-CCR-9732208, and in part by the U.S. Department of Energy under contract DE-FG02-92ER25127.     

Congruences for Hecke eigenvalues in higher level cases

In Guerzhoy (2008) [6], Guerzhoy defined certain quotient space dual to the space of cusp forms of given weight, developed its properties and applied them to the congruences for Hecke eigenvalues in level one case. We extend his result to higher level cases.     

多目标变分问题的混合对偶性

本文给出了一类多目标变分问题的混合对偶 ,使得 Wolfe型对偶和 Mond-Weir型对偶是其特殊情况 ,并在函数 (F ,ρ) -凸性的条件下建立了多目标变分问题关于有效解的混合对偶理论 .     

The structured sensitivity of Vandermonde-like systems

Summary We consider a general class of structured matrices that includes (possibly confluent) Vandermonde and Vandermonde-like matrices. Here the entries in the matrix depend nonlinearly upon a vector of parameters. We define, condition numbers that measure the componentwise sensitivity of the associated primal and dual solutions to small componentwise perturbations in the parameters and in the right-hand side. Convenient expressions are derived for the infinity norm based condition numbers, and order-of-magnitude estimates are given for condition numbers defined in terms of a general vector norm. We then discuss the computation of the corresponding backward errors. After linearising the constraints, we derive an exact expression for the infinity norm dual backward error and show that the corresponding primal backward error is given by the minimum infinity-norm solution of an underdetermined linear system. Exact componentwise condition numbers are also derived for matrix inversion and the least squares problem, and the linearised least squares backward error is characterised.     

A combinatorial theory of Grünbaum's new regular polyhedra,Part II: Complete enumeration

The new regular polyhedra as defined by Branko Grünbaum in 1977 (cf. [5]) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. [3], [2] or [1]) the standard crystallographic restrictions are established for the isometry groups of the non finite (Grünbaum-)polyhedra. Then, using an appropriate classification scheme which—compared with the similar, geometrically motivated scheme, used originally by Grünbaum—is suggested rather by the group theoretical investigations in [4], it turns out that the list of examples given in [5] is essentially complete except for one additional polyhedron.So altogether—up to similarity—there are two classes of planar polyhedra, each consisting of 3 individuals and each class consisting of the Petrie duals of the other class, and there are ten classes of non planar polyhedra: two mutually Petrie dual classes of finite polyhedra, each consisting of 9 individuals, two mutually Petrie dual classes of infinite polyhedra which are contained between two parallel planes with each of those two classes consisting of three one-parameter families of polyhedra, two further mutually Petrie dual classes each of which consists of three one parameter families of polyhedra whose convex span is the whole 3-space, two further mutually Petrie dual classes consisting of three individuals each of which spanE ³ and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE ³, two of which are Petrie dual to each other, the remaining one being Petrie dual to itself.In addition, a new classification scheme for regular polygons inE ⁿ is worked out in §9.     

Robust tests based on dual divergence estimators and saddlepoint approximations

This paper is devoted to robust hypothesis testing based on saddlepoint approximations in the framework of general parametric models. As is known, two main problems can arise when using classical tests. First, the models are approximations of reality and slight deviations from them can lead to unreliable results when using classical tests based on these models. Then, even if a model is correctly chosen, the classical tests are based on first order asymptotic theory. This can lead to inaccurate p-values when the sample size is moderate or small. To overcome these problems, robust tests based on dual divergence estimators and saddlepoint approximations, with good performances in small samples, are proposed.