有限时—频平移的线性无关性

本文介绍时—频分析中空间L~2(■)上有限时—频平移的线性无关性猜想.首先说明提出这个猜想的背景.然后综述近二十五年来对这个猜想研究的主要进展.最后简略说明该猜想高维情形和其他情形的部分结果.     

基于光的波粒二象性猜想的数学建模与仿真

光具有波粒二象性,诸多学者都对此提出了不同猜想,用以合理解释光的直线传播、衍射和干涉等问题.对三种基于光的波粒二象性的猜想进行了数学建模与仿真验证.以Matlab为工具对光的衍射等现象的进行了数学仿真,并与《光学原理》中的相关结果作了对比,最终验证了猜想的正确性.     

图的中间图2-pebbling性质和Graham猜想

本文研究了图的2-pebbling性质和Graham猜想.利用图的pebbling数的一些结果,我们研究了路和圈的中间图具有2-pebbling性质,从而也证明了路的中间图满足Graham猜想.     

具有二阶细焦点的二次系统极限环的唯一性

人们猜想,平面二次系统二阶细焦点外围至多存在一个极限环,但迄今未能证实.文[1,2]在某些参数取特定值之下证明了这一猜想.最近文[5]在具有零特征根奇点之下也证明了这一猜想.本文则在较一般的情况下证明了这一猜想,并使文[5]的结果成为本文的特例.此外,本文还给出了若干有环无环的条件.     

图的中间图2-pebbling性质和Graham猜想（英文）

本文研究了图的2-pebbling性质和Graham猜想.利用图的pebbling数的一些结果,我们研究了路和圈的中间图具有2-pebbling性质,从而也证明了路的中间图满足Graham猜想.     

K(1,4)-自由的模k泛圈图(英文)

设G是2-连通的K1,4自由图.本文证明了当δ(G)≥k 1时,G是模k泛圈图.这一结果肯定了猜想2,继而也肯定了Thomassen猜想在2-连通图中的正确性.     

自由作业稠密时间表的性质研究

稠密时间表作为自由作业问题的近似解,其加工总长与最优值之比具有上界2-1/m(m为机器数),是一个尚未证明的猜想.利用组合方法证明了稠密时间表性能比猜想成立的一个充分条件.利用该条件及有关文献的结果,给出了机器数不超过7的自由作业稠密时间表性能比猜想的证明.     

Lin-Bose问题的进一步研究

基于Lin-Bose猜想,研究l×m阶矩阵MLP分解问题,利用l×m阶矩阵的l-1阶子式、l-2阶子式来刻画矩阵MLP分解的条件,得到的一些结果推广了Lin-Bose猜想中的某些结果.     

一个不等式猜想的证明

文[1]末提出了四个不等式猜想,文[2],文[3]均给出了猜想1的详细证明,文[2]还对猜想1作了更深入的讨论.事实上,只要取a=-1,b=-2,c=32,便可知:abc>1,因而猜想2并非十分准确,同样猜想3,4亦有漏洞,本文对猜想4作一细小的修正,并给予证明.作为特例的猜想2,3也就一并解决了.猜想4若ni     

齐性动力系统的有界轨道

齐性动力系统是由李群给出的一类特殊动力系统,与Diophantine逼近等数论方向有着紧密的联系.本文概述关于齐性动力系统有界轨道的主要问题和结果,并介绍它们与Oppenheim猜想、Littlewood猜想和Schmidt猜想等Diophantine逼近问题的关系.     

Vizing's conjecture: a survey and recent results

Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw‐free graphs with arbitrary graphs. Open problems, questions and related conjectures are discussed throughout the paper. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 46–76, 2012     

A game with divisors and absolute differences of exponents

In this work, we discuss a number game that develops in a manner similar to that on which Gilbreath's conjecture on iterated absolute differences between consecutive primes is formulated. In our case the action occurs at the exponent level and there, the evolution is reminiscent of that in a final Ducci game. We present features of the whole field of the game created by the successive generations, prove an analogue of Gilbreath's conjecture and raise some open questions.     

Mean value conjectures for rational maps

Let p be a polynomial in one complex variable. Smale's mean value conjecture estimates |p′(z)| in terms of the gradient of a chord from (z,?p(z)) to some stationary point on the graph of p. The conjecture does not immediately generalize to rational maps since its formulation is invariant under the group of affine maps, not the full Möbius group. Here we give two possible generalizations to rational maps, both of which are Möbius invariant. In both cases we prove a version with a weaker constant, in parallel to the situation for Smale's mean value conjecture. Finally, we discuss some candidate extremal rational maps, namely rational maps all of whose critical points are fixed points.     

Alternating sums over π-subgroups

Dade's conjecture predicts that if p is a prime, then the number of irreducible characters of a finite group of a given p-defect is determined by local subgroups. In this paper we replace p by a set of primes π and prove a π-version of Dade's conjecture for π-separable groups. This extends the (known) p-solvable case of the original conjecture and relates to a π-version of Alperin's weight conjecture previously established by the authors.     

The Arc‐Weighted Version of the Second Neighborhood Conjecture

Seymour conjectured that every oriented simple graph contains a vertex whose second neighborhood is at least as large as its first. Seymour's conjecture has been verified in several special cases, most notably for tournaments by Fisher  6 . One extension of the conjecture that has been used by several researchers is to consider vertex‐weighted digraphs. In this article we introduce a version of the conjecture for arc‐weighted digraphs. We prove the conjecture in the special case of arc‐weighted tournaments, strengthening Fisher's theorem. Our proof does not rely on Fisher's result, and thus can be seen as an alternate proof of said theorem.     

Fair reception and Vizing's conjecture

In this paper we introduce the concept of fair reception of a graph which is related to its domination number. We prove that all graphs G with a fair reception of size γ(G) satisfy Vizing's conjecture on the domination number of Cartesian product graphs, by which we extend the well‐known result of Barcalkin and German concerning decomposable graphs. Combining our concept with a result of Aharoni, Berger and Ziv, we obtain an alternative proof of the theorem of Aharoni and Szabó that chordal graphs satisfy Vizing's conjecture. A new infinite family of graphs that satisfy Vizing's conjecture is also presented. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 45‐54, 2009     

A generalization of Mühlbach's extension principle for determinantal identities

Mühlbach's extension principle for determinantal identities generalizes Muir's law of extensible minors. Some particular issues with Mühlbach–Beckermann's identity [A general determinantal identity of Sylvester type and some applications, Linear Algebra Appl. 197, 198 (1994), pp. 93–112] led to the conjecture of a more general extension method than Mühlbach's. However, no confirmation seems to have been reported so far. In this note, we present a generalization of Mühlbach's extension principle which confirms that conjecture. The whole identity of Mühlbach–Beckermann is put in a simpler form from which a new interpretation as an extension of Leibniz's definition of a determinant.     

Irreducible values of polynomials

Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove a quantitative arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields via model theory. A main tool in the proof is an irreducibility theorem à la Hilbert.     

Matching theory and Barnette's conjecture

Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian.A graph, other than the path of length three, is a brace if it is bipartite and any two disjoint edges are part of a perfect matching. Our perspective allows us to observe that Barnette's Conjecture can be reduced to cubic, planar braces. We show a similar reduction to braces for cubic, 3-connected, bipartite graphs regarding four stronger versions of Hamiltonicity. Note that in these cases we do not need planarity.As a practical application of these results, we provide some supplements to a generation procedure for cubic, 3-connected, planar, bipartite graphs discovered by Holton et al. (1985) [14]. These allow us to check whether a graph we generated is a brace.