2500年研究探寻相亲数

设σ(n)为n的所有正因子(包括1和n本身在内)之和.正整数对(m,n)被称之为相亲数(或双亲数,因为这种数总是成双成对出现的)如果他们满足 σ(m)=σ(n) = m + n.如果n=n, σ(m)=2m,则m被称之为完全数(或单亲数,因为这种数总是单独出现的).更一般的,如果κ个(κ＞2)正整数(m1,m2,…mmk)满足下列条件σ(m1)=m1+m2,σ(m2)=m2+m3,σ(mk)=mκ+m1.则这κ个正整数被称之为多亲数.第一对相亲数(220,284)是在2500年前的古希腊数学家毕达哥拉斯发现的.不过迄今为止,人们对相亲数的情况、尤其对相亲数的分布情况仍然知之甚少.与相亲数有关的难题、尤其是悬而未决千百年的难题还很多就是在今夭,我们仍然不知道是不是有无穷多对相亲数,我们甚至连一个生成相亲数的充分必要条件(定义除外)都没有.在这篇文章中,我们试图给出人类在2500年的漫长历史长河中研究、探寻相亲数的大致情况与重要结果,并着重介绍从古至今生成相亲数的各种数值方法与代数方法.完全数的研究探寻史几乎与相亲数的研究探寻史是一样长的.比如2350年前的古希腊数学家欧几理德就在其数学名著<几何原本>中列出了前四个完全数,不过迄今为止,人们总共也只找到39个完全数,并且这些完全数还都是偶完全数.至于有没有奇完全数的存在,则是一个悬而未决两千多年的著名数学难题.最早的两串多亲数(一串为5个.另一串为28个),则是由法国数学家Poulet于1918年发现的.多亲数的研究探寻史虽然比相亲数的研究探寻史要短得多,但目前人们对它们的注意力与日俱增.由于相亲数与完全数及多亲数密切相关、紧密相连(我们可以将其统一称之为亲和数,因为它们都与相关数的因子和有关),因此在本文中,我们除了要讨论介绍相亲数外,也将顺便介绍完全数与多亲数的研究与探寻简史、以及人们在研究探寻这些数时所获得的一些重要结果.附注截止2004年3月25日作者校勘清样时,人们已经发现了共40个完全数和6262871对相亲数.     

关于亲和数的一个注记

本文研究了一类整数序列(2n)2n 1的某些性质,利用费玛数和数论函数的某些性质,获得了验证此类整数是否是亲和数和完全数的方法,既不与其他正整数构成亲和数对也不是完全数.     

On the amicability of orthogonal designs

Although it is known that the maximum number of variables in two amicable orthogonal designs of order 2ⁿp, where p is an odd integer, never exceeds 2n+2, not much is known about the existence of amicable orthogonal designs lacking zero entries that have 2n+2 variables in total. In this paper we develop two methods to construct amicable orthogonal designs of order 2ⁿp where p odd, with no zero entries and with the total number of variables equal or nearly equal to 2n+2. In doing so, we make a surprising connection between the two concepts of amicable sets of matrices and an amicable pair of matrices. With the recent discovery of a link between the theory of amicable orthogonal designs and space‐time codes, this paper may have applications in space‐time codes. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 240‐252, 2009     

素数的立方都是孤立数

本文运用数论函数的基本性质讨论了相亲数的存在性,证明了素数的立方都是孤立数.     

New amicable pairs of type

A UBASIC computer program was developed to implement a method of te Riele for finding amicable pairs of type . Hundreds of new pairs were found, including a new largest pair and several ``daughter', ``granddaughter', and ``great granddaughter' pairs.

     

The first known type amicable pair

Amicable pairs of type have been the object of special search partly because they breed other amicable pairs. Examples exist for and 6. In this paper we exhibit the first known type pair.

     

New orthogonal designs of order 56

Amicable sets of eight matrices are very useful in the construction of orthogonal designs using the Kharaghani array. In this article we use a simple procedure to construct many new amicable sets of eight matrices of order 7 and then new orthogonal designs of order 56. Some of these are restricted to be short amicable sets of two or four circulant matrices.. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 387–393, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10030     

New amicable four-cycles

Fifty new amicable four-cycles are discovered by the constructive method invented in 1969 by the second author.

     

关于数论函数$\sigma(n)$的一个注记

对于两个不相同的正整数$m$和$n$, 如果满足$\sigma(m)=\sigma(n)=m+n$, 则称之为一对亲和数, 这里$\sigma(n)=\sum_{d|n}d$.本文给出了$f(x,y)=x^{2^{x}}+y^{2^{x}}(x>y\geq{1},(x,y)=1)$不与任何正整数构成亲和数对的结论, 这里$x$,$y$具有不同的奇偶性, 即, 关于$z$的方程$\sigma(f(x,y))=\sigma(z)=f(x,y)+z$不存在正整数解.     

Calculating approximation guarantees for partial set cover of pairs

As a part of a heuristic for the fast detection of new word combinations in text streams, we consider the NP-hard Partial Set Cover of Pairs problem. There we wish to cover a maximum number of pairs of elements by a prescribed number of sets from a given set family. While the approximation ratio of the greedy algorithm for the classic Partial Set Cover problem is completely understood, the same question for covering of pairs is intrinsically more complicated, since the pairs insert some graph-theoretic structure. The best approximation guarantee for the first greedy step can be rephrased as a problem in extremal combinatorics: Assume that we may place a fixed number of subsets of fixed and equal size in a set, how many different pairs of elements can we cover? In this paper we introduce a method to calculate optimal approximation guarantees, and we demonstrate its use on the smallest set families.     

The radar method: an effective line search for piecewise linear concave functions

The maximization of one-dimensional piecewise linear concave (OPLC) functions arises in the line search associated with the maximization of piecewise linear concave functions (e.g. Kelley cutting plane method). The OPLC line search is usually done by the next-break-point method, where one goes from break point to break point up to the optimum. If the number of break points is large this method will be computationally expensive. One can also use some classical derivative-free line search method as for example the golden section method. Such methods do not take advantage of the OPLC geometry. As an alternative, we propose an improved version of the so-called radar method, which maximizes an OPLC function by maximizing successive outer approximations. We prove superlinear and finite convergence of the radar method. Furthermore, our computational test shows that the radar method is highly effective independently from the number of break points.     

A basic variable neighborhood search heuristic for the uncapacitated multiple allocation <Emphasis Type="Italic">p</Emphasis>-hub center problem

The uncapacitated multiple allocation p-hub center problem (UMApHCP) consists of choosing p hub locations from a set of nodes with pairwise traffic demands in order to route the traffic between the origin-destination pairs such that the maximum cost between origin-destination pairs is minimum. It is assumed that transportation between non-hub nodes is possible only via chosen hub nodes. In this paper we propose a basic variable neighborhood search (VNS) heuristic for solving this NP hard problem. In addition we apply two mathematical formulations of the UMApHCP in order to detect limitations of the current state-of-the-art solver used for this problem. The heuristics are tested on benchmark instances for p-hub problems. The obtained results reveal the superiority of the proposed basic VNS over the state-of-the-art as well as over a multi-start local search heuristic developed by us in this paper.     

Multi-coloring and job-scheduling with assignment and incompatibility costs

Consider a scheduling problem (P) which consists of a set of jobs to be performed within a limited number of time periods. For each job, we know its duration as an integer number of time periods, and preemptions are allowed. The goal is to assign the required number of time periods to each job while minimizing the assignment and incompatibility costs. When a job is performed within a time period, an assignment cost is encountered, which depends on the involved job and on the considered time period. In addition, for some pairs of jobs, incompatibility costs are encountered if they are performed within common time periods. (P) can be seen as an extension of the multi-coloring problem. We propose various solution methods for (P) (namely a greedy algorithm, a descent method, a tabu search and a genetic local search), as well as an exact approach. All these methods are compared on different types of instances.     

Optimal Multipliers for Linear Congruential Pseudo-Random Number Generators with Prime Moduli: Parallel Computation and Properties

Two systematic search methods are employed to find multipliers for linear congruential pseudo-random number generation which are optimal with respect to an upper bound for the discrepancy of pairs of successive pseudo-random numbers. The efficiency of these search procedures when executed on parallel systems is assessed by experimental results of a MIMD parallel implementation on a Meiko CS-2 and a workstation cluster. Furthermore the quality of the computed multipliers is evaluated by using the spectral test in dimensions 2–8 and by calculating the actual discrepancy of pairs of the resulting full-period sequences.     

An analytic approach for the analysis of rotations in fringe-balanced binary search trees

This paper presents an analytic approach to the construction cost of fringe-balanced binary search trees. In [7], Mahmoud used a bottom-up approach and an urn model of Pólya. The present method is top-down and uses differential equations and Hwang's quasi-power theorem to derive the asymptotic normality of the number of rotations needed to construct such afringe balanced search tree. We also obtain the exact expectation and variance with this method. Although Pólya's urn model is no longer needed, we also present an elegant analysis of it based on an operator calculus as in [4].This research was supported by the Austrian Research Society (FWF) under the project number P12599-MAT.     

Notes on Th bit ibn Qurra and his rule for amicable numbers

On the basis of the Arabic text, we investigate how Th bit ibn Qurra (ninth century) could have found his rule for amicable numbers.     

A hybrid method for quantum global optimization

This paper gives a quantum algorithm for global optimization. The heart of such approaches employ Grover’s database search (1996; Phys Rev Lett 79(23):4709–4712, 1997a; 79(2):325–328, 1997b). Chi and Kim (1998) show that when the phases of the generalized Grover database search operator are optimally chosen, it is capable of finding a solution by a single query. To apply this method to global optimization requires knowledge of the number of marked points m to calculate the optimal phases, but this value is seldom known. This paper focuses on overcoming this hurdle by showing that an estimate of the optimal phases can be found and used to replace the optimal phases while maintaining a high probability of finding a solution. Merging this finding with a recently discovered dynamic quantum global optimization algorithm (BBW2D) that reduces the problem to finding successively improving regions using Grover’s search, we present a hybrid method that improves the efficiency and reduces the variance of the search algorithm when empirically compared to other existing quantum search algorithms.     

A new direct search method based on separable fractional interpolation model

In this paper, we propose a new separable fractional interpolation model which can be established by 2n interpolation points where n is the number of variables. Based on this model, a new direct search method is presented. In this method, a new iterate is determined by solving the fractional interpolation model in trust region. Under mild assumptions, the convergence results of this method are given and proved. Numerical experiments show that the new method is promising.     

A max-conflicts based heuristic search for the stable marriage problem with ties and incomplete lists

In this paper, we propose a heuristic search algorithm based on maximum conflicts to find a weakly stable matching of maximum size for the stable marriage problem with ties and incomplete lists. The key idea of our approach is to define a heuristic function based on the information extracted from undominated blocking pairs from the men’s point of view. By choosing a man corresponding to the maximum value of the heuristic function, we aim to not only remove all the blocking pairs formed by the man but also reject as many blocking pairs as possible for an unstable matching from the women’s point of view to obtain a solution of the problem as quickly as possible. Experiments show that our algorithm is efficient in terms of both execution time and solution quality for solving the problem.