关于Mycielski图循环色数的猜想

通过引进Mycielski图点集的一类特殊划分,利用该划分在Mycielski图循环着色中的特点改进了如下猜想:完全图的Mycielski图的循环色数等于它的点色数.     

Log‐concavity of genus distributions for circular ladders

A well‐known conjecture in topological graph theory says that the genus distribution of every graph is log‐concave. In this paper, the genus distribution of the circular ladder is re‐derived, using overlap matrices and Chebyshev polynomials, which facilitates proof that this genus distribution is log‐concave.     

Variation of Heegner points in Hida families

Given a weight two modular form f with associated p-adic Galois representation V _f , for certain quadratic imaginary fields K one can construct canonical classes in the Galois cohomology of V _f by taking the Kummer images of Heegner points on the modular abelian variety attached to f. We show that these classes can be interpolated as f varies in a Hida family and construct an Euler system of big Heegner points for Hida’s universal ordinary deformation of V _f . We show that the specialization of this big Euler system to any form in the Hida family is nontrivial, extending results of Cornut and Vatsal from modular forms of weight two and trivial character to all ordinary modular forms, and propose a horizontal nonvanishing conjecture for these cohomology classes. The horizontal nonvanishing conjecture implies, via the theory of Euler systems, a conjecture of Greenberg on the generic ranks of Selmer groups in Hida families.     

Fractional chromatic number and circular chromatic number for distance graphs with large clique size

An Erratum has been published for this article in Journal of Graph Theory 48: 329–330, 2005 . Let M be a set of positive integers. The distance graph generated by M, denoted by G(Z, M), has the set Z of all integers as the vertex set, and edges ij whenever |i?j| ∈ M. We investigate the fractional chromatic number and the circular chromatic number for distance graphs, and discuss their close connections with some number theory problems. In particular, we determine the fractional chromatic number and the circular chromatic number for all distance graphs G(Z, M) with clique size at least |M|, except for one case of such graphs. For the exceptional case, a lower bound for the fractional chromatic number and an upper bound for the circular chromatic number are presented; these bounds are sharp enough to determine the chromatic number for such graphs. Our results confirm a conjecture of Rabinowitz and Proulx 22 on the density of integral sets with missing differences, and generalize some known results on the circular chromatic number of distance graphs and the parameter involved in the Wills' conjecture 26 (also known as the “lonely runner conjecture” 1 ). © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 129–146, 2004     

Euler systems in global function fields

In this paper, the construction of Euler systems of cyclotomic units in a general global function fields is explained. As an application, an analogue of Gras’ conjecture in a global function field is proved.     

Asymptotic behaviour and bifurcation in competitive Lotka-Volterra Systems

A conjecture about global attraction in autonomous competitive Lotka-Volterra systems is clarified by investigating a special system with a circular matrix. Under suitable assumptions, this system meets the condition of the conjecture but Hopf bifurcation occurs in a particular instance. This shows that the conjecture is not true in general and the condition of the conjecture is too weak to guarantee global attraction of an equilibrium. Sufficient conditions for global attraction are also obtained for this system.     

Fixed Energy Universality for Generalized Wigner Matrices

We prove the Wigner‐Dyson‐Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics.© 2016 Wiley Periodicals, Inc.     

On the strong circular 5‐flow conjecture

The Strong Circular 5‐flow Conjecture of Mohar claims that each snark—with the sole exception of the Petersen graph—has circular flow number smaller than 5. We disprove this conjecture by constructing an infinite family of cyclically 4‐edge connected snarks whose circular flow number equals 5. © 2006 Wiley Periodicals, Inc. J Graph Theory     

On the maximum number of pairwise compatible euler cycles

B. Jackson [4] made the following conjecture: If G is an Eulerian graph with δ(G) ≥ 2k, then G has a set of 2k - 2 pairwise compatible Euler cycles (i.e., every pair of adjacent edges appears in at most one of these Euler cycles as a pair of consecutive edges). We verify this conjecture in the case where every circuit of G is a block of G.     

A universality theorem for projectively unique polytopes and a conjecture of Shephard

We prove that every polytope described by algebraic coordinates is the face of a projectively unique polytope. This provides a universality property for projectively unique polytopes. Using a closely related result of Below, we construct a combinatorial type of 5-dimensional polytope that is not realizable as a subpolytope of any stacked polytope. This disproves a classical conjecture in polytope theory, first formulated by Shephard in the seventies.     

Coarsening in one dimension: invariant and asymptotic states

We study a coarsening process of one-dimensional cell complexes. We show that if cell boundaries move with velocities proportional to the difference in size of neighboring cells, then the average cell size grows at a prescribed exponential rate and the Poisson distribution is precisely invariant for the distribution of the whole process, rescaled in space by its average growth rate. We present numerical evidence toward the following universality conjecture: starting from any finite mean stationary renewal process, the system when rescaled by e ^?2t converges to a Poisson point process. For a limited case, this makes precise what has been observed previously in experiments and simulations, and lays the foundation for a theory of universal asymptotic states of dynamical cell complexes.     

关于Euler数一个猜想的证明

研究Euler数的猜想:对于任何素数p≡1 mod4,E(p-1)／20 mod p．根据Euler数本身的性质和Fourier级数对于Gauss和的应用,证明了这个猜想是成立的．     

基于光子结构猜想的波动性现象建模仿真研究

综合龚祖同院士的光子类氢原子结构论和北大俎栋林教授的光子电磁场结构论,提出了两种结构相统一的理论猜想.基于猜想对单缝衍射、双缝干涉以及多缝干涉中光强与光子尺度、细缝宽度等参数的关系进行建模,并以双缝干涉为例进行了仿真实验.通过实验结果与经典波动光学中光强分布的对比,验证了模型的合理性.     

圆盘形和圆环形压膜轴承的气动力分析

本文按照气体动力学理论,对圆盘形和圆环形压膜轴承的气动力作了详细的分析;给出了这两种轴承的压力分布的解析表达式,并用该方法修正了圆盘形压膜轴承的部分计算公式;还给出了这两种轴承的压力分布和承载能力的计算结果,可供设计者参考.     

The Christoffel Function for Orthogonal Polynomials on a Circular Arc

For the special type of weight functions on circular arc we study the asymptotic behavior of the Christoffel kernel off the arc and of the Christoffel function inside the arc. We prove Totik's conjecture for the Christoffel function corresponding to such weight functions.     

Partitions with fixed largest hook length

Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub’s analogue of Euler’s Odd-Distinct partition theorem, derive a generalization in the spirit of Alder’s conjecture, as well as a curious analogue of the first Rogers–Ramanujan identity. Moreover, we obtain a partition theorem that is the counterpart of Euler’s pentagonal number theorem in this setting, and connect it with the Rogers–Fine identity. We conclude with some congruence properties.     

THE EULER EQUATION AND ONSAGER CONJECTURE

In this paper, we introduce the progress of the Euler equation and Onsager conjecture. We also introduce the Euler''s life, the researches about the incompressible Euler equation, and the Onsager conjecture.     

Special values of multiple polylogarithms

Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier.

     

Le Système d'Euler de Kato en Famillie(Ⅱ)

This article is the second article on the generalization of Kato's Euler system. The main subject of this article is to construct a family of Kato's Euler systems over the cuspidal eigencurve, which interpolate the Kato's Euler systems associated to the modular forms parametrized by the cuspidal eigencurve. We also explain how to use this family of Kato's Euler system to construct a family of distributions on Z_p over the cuspidal eigencurve; this distribution gives us a two-variable p-adic L function which interpolate the p-adic L function of modular forms.     

不用克希霍夫—拉夫假设的弹性圆板理论再探

本文在不用克希霍夫一拉夫假设的弹性板一般理论的基础上,建立了不用克希霍夫一拉夫假设的弹性圆板的一级近似理论,对圆板在四周固定和均布载荷的条件下,得到了具体的轴对称分析解,并和经典的圆薄板解进行了比较,证明本文新解更加接近实验结果,本文也具体地讨论了理论结果中厚度增大时的影响。