A characterization for a sequence to be potentially K r+1 − e-graphic

Let n 〉 r, let lr --- （dl,d2,-,dn） be a non-increasing sequence of nonnegative integers and let Kr＋l - e be the graph obtained from Kr＋l by deleting one edge. If zr has a realization G containing Kr＋l - e as a subgraph, then r is said to be potentially Kr＋l - e-graphic. In this paper, we give a characterization for a sequence π to be potentially Kr＋l - e-graphic.     

2-Arc-transitive cyclic covers of K n,n −nK 2

Du et al. (in J. Comb. Theory B 74:276–290, 1998 and J. Comb. Theory B 93:73–93, 2005), classified regular covers of complete graph whose fiber-preserving automorphism group acts 2-arc-transitively, and whose covering transformation group is either cyclic or isomorphic to $\mathbb{Z}_{p}^{2}$ or $\mathbb{Z}_{p}^{3}$ with p a prime. In this paper, a complete classification is achieved of all the regular covers of bipartite complete graphs minus a matching K _n,n ?nK ₂ with cyclic covering transformation groups, whose fiber-preserving automorphism groups act 2-arc-transitively.     

On the chromatic uniqueness of the graphW(n,n − 2, k)

This paper shows that the graphW(n, n – 2, k) is chromatically unique for any even integern 6 and any integerk 1.     

The local rank of an ergodic symmetric power T ⊙n does not exceed n!n −n

The infinity of the rank of ergodic symmetric powers of automorphisms of the Lebesgue space is proved, and sharp upper bounds for their local rank are found.     

Zolotarev's first problem — the best approximation by polynomials of degree ≤n − 2 tox n − nσx n−1 in [−1, 1]

Chebyshev determined $$\mathop {\min }\limits_{(a)} \mathop {\max }\limits_{ - 1 \le x \le 1} |x^n + a_1 x^{n - 1} + \cdots + a_n |$$ as 2^1?n , which is attained when the polynomial is 2^1?n T _n(x), whereT _n(x) = cos(n arc cosx). Zolotarev's First Problem is to determine $$\mathop {\min }\limits_{(a)} \mathop {\max }\limits_{ - 1 \le x \le 1} |x^n - n\sigma x^{n - 1} + a_2 x^{n - 2} + \cdots + a_n |$$ as a function ofn and the parameter σ and to find the extremal polynomials. He solved this in 1878. Another discussion was given by Achieser in 1928, and another by Erdös and Szegö in 1942. The case when 0≤|σ|≤ tan²(π/2n) is quite simple, but that for |σ|> tan²(π/2n) is quite different and very complicated. We give two new versions of the proof and discuss the change in character of the solution. Both make use of the Equal Ripple Theorem.     

（n,n2,…nk）型（r1,r2,…rk）—循环矩阵非异性的一个判定

当|r1|=|r2|=…=|rk|=1时,本文给出了（n1,n2,…,nk）型k重（r1,r2,…rk)－循环矩阵非异性的一个判定方法。     

（n1,n2）型二重（r1,r2）—循环矩阵逆矩阵的插值求法

本文用插值法给出n1n2阶（n1，n2）型二重（r1，r2）－循环矩阵逆矩阵计算公式．     

On the collection of spectral multiplicities {2, 4, …, 2 n } for totally ergodic ℤ2-actions

The paper is devoted to the realization of collections of spectral multiplicities for ergodic ?²-actions. Sufficient conditions ensuring the realizability of multiplicities of the form {2, 4, …, 2 ⁿ } are given.     

On the Diophantine Equations (2 n − 1)(6 n − 1) = x 2 and (a n − 1)(a kn − 1) = x 2

In this paper we prove that the equation (2 ⁿ – 1)(6 ⁿ – 1) = x ² has no solutions in positive integers n and x. Furthermore, the equation (a ⁿ – 1) (a ^kn – 1) = x ² in positive integers a > 1, n, k > 1 (kn > 2) and x is also considered. We show that this equation has the only solutions (a,n,k,x) = (2,3,2,21), (3,1,5,22) and (7,1,4,120).     

Sections of U(2n)/Sp(s) over S4n−1

We describe the sections of U(2n)/Sp(s) over S⁴ⁿ⁻¹ in terms of the sections of the symplectic Stiefel manifold Sp(n)/Sp(s) and we express the orders of obstructions to sectioning U(2n)/Sp(s) in terms of orders of obstructions to sectioning Sp(n)/Sp(s). In certain cases we give the exact values of these orders.