Intermediate rings between D+I And K [y1,…,yt]

Let R = k [y₁,…,y_t] be an affine domain (where k is a field) having krull dimension =n>0. Let I be a nonzero proper ideal of R and D be a subring of K. In section 1 we determine necessary and sufficient conditions in order that (S,R) is a 'lying over pair' where S = D+I. In section 2 we chaaracterize when S is a Maximal non-Noetherian subring of R. Further we determine when S is a maximal subring of R.     

关于公式1+2+…+n=(n(n+1))/2

<正>九年义务教育三年制初中《代数》第一册(上)第38页,B组第二题为:正整数从1开始,逐个相加,一直加到n,它们的和记作s,即s=1+2+3+…+n(n表示一个正整数),写出计算s的公式.这道题目中含有字母且设问富有思考性,解题方法体现了数学方法,更重要的是结果能作公式用,而且应用分层次用.为帮助同学理解这些特点,现对这道题进行解读,供同学们参考.     

关于[m,n,p_1,p_2,…,p_s]指数型群

本文给出了[m,n,p_1,p_2,…,p_s]指数型群的结构,改进了前人的结果.     

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.     

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.     

关于K2,s+1 VS,K1,n的Ramsey数

It is shown that the Ramsey number r(K2,s 1, K1,n) ≤ n √sn (s 3)/2 o(1) for large n, and r(K2,s 1, K1,n)∈{(q-1)2/s 1,-(q-1)2/s 2},wheren (q-1)2/s-q 2 and q is a prime power such that s|(q - 1).     

The [n 1,n 2, ...,n s]-th reduced KP hierarchy andW 1+∞ constraints

To every partition n=n₁+n₂+...+n_s one can associate a vertex operator realization of the Lie algebras a and gl_n. Using this construction, we obtain reductions of the s-component KP hierarchy, reductions which are related to these partitions. In this way we obtain matrix KdV-type equations. We show that the following two constraints on a KP -function are equivalent: (1) is a -function of the [n₁, n₂, ..., n_s]-th reduced KP hierarchy which satisfies the string equation L_–1=0; (2) satisfies the vacuum constraints of the W₁₊ algebra.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 32–42, July, 1995.     

Boundedness properties of the difference equation x n+1=(α+βx n +δx n−2)/(x n−1)

Consider the third-order difference equation x _n+1 = (α+βx _n +δx _n ? 2)/(x _n ? 1) with α ∈ [0,∞) and β,δ ∈ (0,∞). It is shown that this difference equation has unbounded solutions if and only if δ>β.     

An efficient algorithm for generating all k-subsets (1 ⩽ k ⩽ m ⩽ n) of the set [1, 2,…, n] in lexicographical order

We consider the problem of generating all k-subsets (1 ? k ? m ? n) of the set [1, 2, …, n] in lexicographical order. The running time per k-subset is shown to be constant. The experimental data suggest that our algorithm is about 25% faster than the algorithm proposed by A. Nijenhuis and H. S. Wilf (“Combinatorial Algorithms,” 2nd ed., Academic Press, New York, 1978).     

完全t部图 K(n1,n2,…,nt)的色唯一性

本文使用比较两个色等价图的色划分数的方法,得出了完全t部图的色等价图类仍为完全t部图的一般形式数值条件,进一步得出了K(n1,n2,n3)和K(n1,n2,n3,n4)为色唯一图的一般形式数值条件.     

A Note on the Recursive Sequence x n + 1 = p k x n + p k − 1 x n − 1 +...+ p 1 x n − k + 1

We present some comments on the behavior of solutions of the difference equation where p _i 0, i = 1,..., k, k N, and x _–k ,..., x _–1 R.     

More on the Difference Equation y n + 1 = ( p + y n −1 )/( qy n + y n −1 )

From previous studies of the equation in the title with positive parameters p and q and positive initial conditions we know that if q h 4 p + 1 then the equilibrium is a global attractor. We also know that if q > 4 p + 1 then every solution eventually enters and remains in the interval [ p / q , 1]. In this strip there exists a "unique" prime period two solution that is locally asymptotically stable. In this paper, we provide more insight as to the behavior of solutions of the equation in the title in the strip [ p / q , 1], where a one-dimensional stable manifold lives.     

关于完全t部图K(n1,n2,…,nt)的色唯一性

设P（G，λ）是图G的色多项式，如果对任意使P（G，λ）=P（H，λ）的图H都与G同构，则称G是色唯一图。这里通过比较图的特征子图的个数，讨论了由Koh和Teo在文献[1]中提出的问题（若｜ni-nj｜≤2，1≤i，j≤t且min{n1，n2，…，nt}充分大，K（n1，n2，…，nt）是否为色唯一图？）。证明了，若｜ni—nj｜≤2且t↑∑↑i=1 ni〉t^2/2＋t√t-1，则K（n1，n2，…，nt）是色唯一图；若αi=0或k，t↑∑↑i=1 n＋αi〉t^2k^2/8＋｜tk｜/2√t-1，则K（n＋α1，n＋α2，…，n＋αt）是色唯一图。其条件比文献[4]中的条件较好一些。     

Minimal Covers of Q +(2n + 1, q) by (n − 1)-Dimensional Subspaces

A t-cover of a quadric is a set C of t-dimensional subspaces contained in such that every point of is contained in at least one element of C.We consider (n – 1)-covers of the hyperbolic quadric Q ⁺(2n + 1, q). We show that such a cover must have at least q ^{n + 1} + 2q + 1 elements, give an example of this size for even q and describe what covers of this size should look like.