面尺法求解哥古猜想解答

哥德巴赫猜想与古叶猜想合成哥古猜想（Goldbach-Guye’s Conjecture）：任何一个非零偶数（z）,都可表达为两个素数（y及x）之＂和＂或＂差＂;即有z＋=y＋x,或有z-=y-x。当知z＋或z-时,如何求解y及x配偶成副呢？文章介绍了求解哥古猜想解答的一种面尺法（利用‘定尺’与‘动尺’间的相对位移）,并按参考文献[8]的‘提示’提出了改进措施,使缩尺法更趋完善。     

选尺法求解哥古猜想解答(续)

哥德巴赫猜想与古叶猜想合成哥古猜想( Goldbach -Guye's Conjecture):任何一个非零偶数(z),都可表达为两个素数(y及x)之‘和’或‘差’;即有z+=y+x,或有z=y-x.当知z+或z-时,如何求解y及x配偶成副呢?文章介绍了求解哥古猜想解答的一种选尺法(利用‘y尺’与‘x尺’间的相对位移)...     

一类整函数系数线性微分方程解的增长性

本文主要研究某类整函数系数高阶线性微分方程解的增长性,这类方程有一个系数为满足Denjoy猜想极值情况的整函数．运用亚纯函数值分布理论和整函数的渐近值理论,通过比较方程中每一项的模的大小,得到这类方程解的增长级的估计．对只有一个系数起控制作用的方程,当其存在一个系数为二阶微分方程的解时,得到上述方程的非零解都为无穷级．对系数具有相同增长级的方程,当其系数具指数函数形式时,得到上述方程的非零解也为无穷级．文中所得结果是对线性微分方程相关结果的推广和补充．     

关于Diophantine方程(an-1)(bn-1)=x2的Cohn猜想

设a和b是两个不相等的正整数.针对Cohn猜想,即方程(an-1)(bn-1)=x2没有正整数解(x,n),其中n＞4.利用初等数论方法和指数Diophantine万程的性质,得到了如果a和b具有相反的奇偶性,那么方程没有满足n＞4和2| n的正整数解(x,n).     

一个极小值问题的证明

将一个与莫朗猜想等价的极小值问题转化为非线性规划问题，并利用Ｋ－Ｔ条件求解，从而证明了莫朗猜想．     

Complementarity and Additivity for Covariant Channels

This paper contains several new results concerning covariant quantum channels in d ≥ 2 dimensions. The first part, Sec. 3, based on [4], is devoted to unitarily covariant channels, namely depolarizing and transpose-depolarizing channels. The second part, Sec. 4, based on [10], studies Weyl-covariant channels. These results are preceded by Sec. 2 in which we discuss various representations of general completely positive maps and channels. In the first part of the paper we compute complementary channels for depolarizing and transpose-depolarizing channels. This method easily yields minimal Kraus representations from non-minimal ones. We also study properties of the output purity of the tensor product of a channel and its complementary. In the second part, the formalism of discrete noncommutative Fourier transform is developed and applied to the study of Weyl-covariant maps and channels. We then extend a result in [16] concerning a bound for the maximal output 2-norm of a Weyl-covariant channel. A class of maps which attain the bound is introduced, for which the multiplicativity of the maximal output 2-norm is proven. The complementary channels are described which have the same multiplicativity properties as the Weyl-covariant channels.     

An introduction to Wu's method for mechanical theorem proving in geometry

Wu's algebraic method for mechanically proving geometry theorems is presented at a level as elementary as possible with sufficient examples for further understanding the complete method.The work reported here was supported by NSF Grant DCR-8503498.     

Graham's problem on shortest networks for points on a circle

Suppose a configurationX consists ofn points lying on a circle of radiusr. If at most one of the edges joining neighboring points has length strictly greater thanr, then the Steiner treeS consists of all these edges with a longest edge removed. In order to showS is, in fact, just the minimal spanning treeT, a variational approach is used to show the Steiner ratio for this configuration is at least one and equals one only ifS andT coincide. The variational approach greatly reduces the number of possible Steiner trees that need to be considered.     

Partition arguments in multiparty communication complexity

Consider the “Number in Hand” multiparty communication complexity model, where k players holding inputs x₁,…,x_k∈{0,1}ⁿ communicate to compute the value f(x₁,…,x_k) of a function f known to all of them. The main lower bound technique for the communication complexity of such problems is that of partition arguments: partition the k players into two disjoint sets of players and find a lower bound for the induced two-party communication complexity problem.In this paper, we study the power of partition arguments. Our two main results are very different in nature:

(i)

For randomized communication complexity, we show that partition arguments may yield bounds that are exponentially far from the true communication complexity. Specifically, we prove that there exists a 3-argument function f whose communication complexity is Ω(n), while partition arguments can only yield an Ω(logn) lower bound. The same holds for nondeterministiccommunication complexity.

(ii)

For deterministic communication complexity, we prove that finding significant gaps between the true communication complexity and the best lower bound that can be obtained via partition arguments, would imply progress on a generalized version of the “log-rank conjecture” in communication complexity. We also observe that, in the case of computing relations (search problems), very large gaps do exist.

We conclude with two results on the multiparty “fooling set technique”, another method for obtaining communication complexity lower bounds.