由本性随机阵导致的陈代数

A matrix of order n whose row sums are all equal to 1 is called an essentially stochastic matrix (see Johnsen [4]). We extend this notion as the following. Let F be a field of characteristic 0 or a prime greater than n. M_n(F) denotes the set of all n×n matrices over F. Let t be an elernent of F. A matrix A=(a_ij) in M_n(F) is called essentially t-stochastic' provided its row sums are each equal to t. We denote by R_n(t) the set of all essentially t-stochastic matrices over F. We shall mainly study R_n(0) and R_n(F)=(?)R_n(t). Our main references are Johnson [2,4] and Kim [5].     

On the dynamics of entire functions on the Julia set

Let f be an entire function and ?ⁿ its n-th iterate. Let P(?) denote the postcritical set of ? and J(?) the Julia set of ?. Suppose that the set E of all z ∈ J(?) with limsup_n→∞ dist (?ⁿ(z), P(?) U {∞}) > 0 has positive measure. It is proved that for a given set A ? ? of positive measure the set {n ∈ ?; ?ⁿ(z) ∈ A} is infinite for almost all z in the plane. From this follows that the forward orbit of almost all z ∈ ? is dense in the plane if E has positive measure.     

可换环上一些线性李代数的扩代数

设R是任意含单位元的可换环,gl(n,R)是R上n级一般线性李代数.t表示gl(n,R)中所有上三角矩阵组成的子代数,d表示gl(n,R)中所有对角矩阵组成的子代数.本文将分别确定t在gl(n,R)中的扩代数和d在t中的扩代数.     

Some inequalities of the V. A. Markov type

Let B be a domain in the complex plane, let p_n(z) and P_n(z) be polynomials of degree n where the zeros of P_n(z) lie in , let(z) be a finite function,(z) 0, z . We consider the problem of estimating from above the functions L[p_n(z)]=(z)p_n(z) – wp_n(z), z , if ¦p_n(z)¦ ¦P_n(z)¦ for zB. Under some very general conditions on B, z, (z), and w we prove the inequality ¦L[p_n(z)]¦ ¦L[P_n(z)]¦.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 431–440, April, 1968.     

Approximation of function by Euler means

Let 蒖_n $$(q, f, x) = \frac{1}{{(1 + q)^n }}\sum\limits_{k = 0}^n {(_k^n )q^{n - k} s_k (f, x)} $$ denote the Euler means of the Fourier series of the 2π-perodic function f(x). For an integer q>0 and a function f(x)∈H^ω?C([0, 2π]), the main term of deviationf(x)-蒖_n(q, f, x) is calculated in this note. Asymptoteaally exact order 3 of decrease of the upper bound of such deviations over the class H^ω is also obtained.     

关于p叶近于凸性的充分条件

Let Ap(n)(p, n∈N={1,2,...}) denote the class of functions of the form f(z)=zp+ap+nzp+n+... which are analytic in the unit disc E={z:|z|＜1}. By using the method of differential subordinations we give some sufficient conditions for a function f(z)∈Ap(n) to be a certain subclass Rp(n,k) of p-valently close-to-convexity functions.     

双特征的Beltrami方程和拟正则映射

设Ω为Ｒｎ上的一个区域，ｎ２，对于具有双特征矩阵Ｇ（ｘ），Ｈ（ｘ）∈Ｃｋ，α（Ω，Ｒｎ），ｋ１，０＜α＜１的Ｂｅｌｔｒａｍｉ方程（１．４），建立了在Ｓｏｂｏｌｅｖ空间Ｗ１，ｎｌｏｃ（Ω，Ｒｎ）上广义解的正则性：ｆ（ｘ）∈Ｃｋ＋１，δｌｏｃ（Ω），对某一δ：０＜δ＜１．     

Wachstumsordnung und Index der Lösungen linearer Differentialgleichungen mit rationalen Koeffizienten

Let z=∞ be an irregular singular point of the differential equation wⁿ+p_n?1(z)w^(n?1)+...+p₀(z)w=0 with rational coefficients. The functions of the canonical set of solutions relative to z=∞ are of the form $$w(z) = z^\rho \cdot \sum { d_m (z) (\log z)^m , } \rho \varepsilon \mathbb{C}$$ with univalent functions d_m(z) in a neighbourhood of z=∞. Let λ(w)=max {λ(d_m)} denote the maximal order of growth of an irregular solution relative to z=∞, then it is shown that there exists a branch of w in the plane cut along a half ray, which attains the maximal order λ(w). An important tool for the proof is the index of the branches of w.     

布尔矩阵行空间基数的存在点

设 Ｂｎ 表示所有的ｎ 阶布尔矩阵的集合, Ｒ（ Ａ）表示 Ａ∈ Ｂｎ 的行空间,｜ Ｒ（ Ａ）｜表示 Ｒ（ Ａ）的基数．设ｍ ,ｎ,ｋ 为正整数,本文证明了当ｎ≥９, ｎ＋ ５２ ≤ｋ≤ｎ－ ３ 时,对任意的 ｍ ,２ｋ≤ｍ ≤２ｋ＋ ２ｎ－ ｋ＋ ２＋ ２ｎ－ ｋ＋ １ ＋ …＋ ２３,存在 Ａ∈ Ｂｎ,使得｜ Ｒ（ Ａ）｜＝ ｍ ．     

Commutativity Conditions for Rings and Generalized 2-CN Rings

Firstly,the commutativity of rings is investigated in this paper.Let R be a ring with identity.Then we obtain the following commutativity conditions: (1) if for each x ∈ R\N(R) and each y ∈ R,(xy)k =xkyk for k =m,m + 1,n,n + 1,where m and n are relatively prime positive integers,then R is commutative;(2) if for each x ∈ R\J(R) and each y ∈ R,(xy)k =ykxk for k =m,m+ 1,m+2,where m is a positive integer,then R is commutative.Secondly,generalized 2-CN rings,a kind of ring being commutative to some extent,are investigated.Some relations between generalized 2-CN rings and other kinds of rings,such as reduced rings,regular rings,2-good rings,and weakly Abel rings,are presented.     

Asymptotic Elasticity in Atomic Monoids

Let M be a commutative atomic monoid (i.e. every nonzero nonunit of M can be factored as a product of irreducible elements). Let ρ(x) denote the elasticity of x ∈ M, R(M) = {ρ(x) | x ∈ M} the set of elasticities of elements in M, and ρ(M) = sup R(M) the elasticity of M. Define \overline{ρ}(x) = lim_n→∞ ρ(xⁿ) to be the asymptotic elasticity of x. We determine some basic properties of the function \overline{ρ} and determine its image for certain block monoids.     

Rees algebras with minimal multiplicity

Let (R,m) be a local ring. Let S_M denote the Rees algebra S=R[m^rt] localized at its unique maximal homogeneous ideal M=(m,m^rt). Let TN denote the extended Rees algebra T= R[m^rt, t^-1] localized at its unique maximal homogeneous idea N= (t^?1,m,m^r). Multiplicity formulas are developedfor S_M and T_N. These are used to find necessaIy and sufficient conditions on a Cohen-Macaulay local ring (R,m) and r so that S_M and T_N are Cohen-Macaulay with minimal multiplicity     

Explicit congruences for the partition function modulo every prime

Let p(n) denote the number of unrestricted partitions of the positive integer n, and let m be a prime $\geq 13$. We prove, for k = 1, explicit congruences of the form where $r_{m,k}, \delta_{m,k}$ are integers depending on m and k and $\phi_{m,k}(z)$ are explicitly computable level one holomorphic modular forms of small weight. We also give theoretical and numerical support that the congruences also hold for k > 1. Our main idea is a level reduction result for the modular forms which originated from Atkin-Lehner. From our result, we deduce periodicity properties for the partition function with short periods which improve upon recent results of K. Ono. Received: 24 January 2002     

Transcendental division algebras and simple Noetherian rings

In this paper we prove the following theorem: Let D be a division ring with center the field k, and let k (x ₁, …, x_n) denote the rational function field in n variables over k. If D contains a maximal subfield which has transcendence degree at least n over k, then D ⊗_k k (x₁, …, x_n) is a simple Noetherian domain of Krull and global dimensions n. Rather surprisingly, the preceding result can be used to determine the maximum transcendence degrees of the commutative subalgebras of several classically studied division rings. Using the theorem we prove, for example, that in the division ring of quotients of the Weyl algebra,A _n, every maximal subfield has transcendence degree at mostn over the center.     

GENERALISED MAXIMAL INDEPENDENCE AND CLIQUE NUMBERS OF GRAPHS

Abstract

A set B of vertices of a graph G = (V,E) is a k-maximal independent set (kMIS) if B is independent but for all ?-subsets X of B, where ? ? k—1, and all (? + 1)-subsets Y of V—B, the set (B—X) u Y is dependent. A set S of vertices of C is a k-maximal clique (kMc) of G iff S is a kMIS of [Gbar]. Let β_k, (G) (w_k(G) respectively) denote the smallest cardinality of a kMIS (kMC) of G—obviously β_k(G) = w_k([Gbar]). For the sequence m₁ ? m₂ ?…? m_n = r of positive integers, necessary and sufficient conditions are found for a graph G to exist such that w_k(G) = m_k for k = 1,2,…,n and w(G) = r (equivalently, β_k(G) = m_k for k = 1,2,…,n and β(G) = r). Define s_k(?,m) to be the largest integer such that for every graph G with at most s_k(?,m) vertices, β_k(G) ? ? or w_k(G) ? m. Exact values for s_k(?,m) if k ≥ 2 and upper and lower bounds for s₁(?,m) are de termined.     

关于无K—间隔的组合数

从排列在一条直线上的n个元素中选取m个元素,以f_k(n,m)表示任意两个被选元素的间隔均不为k之方式数。如果这n个元素排列在园周上,则相应的组合数以g_k(m,m)表示。关于这两类组合数,I.Kaplansky于1943年首先研究了k=0时的计数问题,J. Konvalina于1981年应用递归方法得到了k=1时的计数表达式。对于一般的自然数k,这一问题似乎更加复杂。     

Ranges of certain functionals in classes of regular functions

LetP _κ,n (λ,β) be the class of functions \(g(z) = 1 + \sum\nolimits_{v = n}^\infty {c_\gamma z^v }\) , regular in ¦z¦<1 and satisfying the condition $$\int_0^{2\pi } {\left| {\operatorname{Re} \left[ {e^{i\lambda } g(z) - \beta \cos \lambda } \right]} \right|} /\left( {1 - \beta } \right)\cos \lambda \left| {d\theta \leqslant \kappa \pi ,} \right.z = re^{i\theta } ,$$ , 0 < r < 1 (κ?2,n?1, 0?Β<1, -π<λ<π/2;M _κ,n (λ,β,α),n?2, is the class of functions \(f(z) = z + \sum\nolimits_{v = n}^\infty {a_v z^v }\) , regular in¦z¦<1 and such thatF _α(z)∈P _κ,n?1(λ,β), where \(F_\alpha (z) = (1 - \alpha )\frac{{zf'(z)}}{{f(z)}} + \alpha (1 + \frac{{zf'(z)}}{{f'(z)}})\) (0?α?1). Onr considers the problem regarding the range of the system {g ^(v?1)(z_?)/(v?1)!}, ?=1,2,...,m,v=1,2,...,N _?, on the classP _κ,1(λ,β). On the classesP _κ,n (λ,β),M _κ,n (λ,β,α) one finds the ranges of C_v, v?n, a_m, n?m?2n-2, and ofg(?),F _?(?), 0<¦ξ¦<1, ξ is fixed.     

A Note on Comaximal Graph of Non-commutative Rings

Let R be a ring with unity. The graph Γ(R) is a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra?+?Rb?=?R. Let Γ₂(R) be the subgraph of Γ(R) induced by the non-unit elements of R. Let R be a commutative ring with unity and let J(R) denote the Jacobson radical of R. If R is not a local ring, then it was proved that:

1. If $\Gamma_2(R)\backslash J(R)$ is a complete n-partite graph, then n?=?2.
2. If there exists a vertex of $\Gamma_2(R)\backslash J(R)$ which is adjacent to every vertex, then R????₂×F, where F is a field.

In this note we generalize the above results to non-commutative rings and characterize all non-local ring R (not necessarily commutative) whose $\Gamma_2(R)\backslash J(R)$ is a complete n-partite graph.     

Local derivations of full matrix rings

Let \({\mathcal{R}}\) be a unital commutative ring and \({\mathcal{M}}\) be a 2-torsion free central \({\mathcal{R}}\) -bimodule. In this paper, for \({n \geqq 3}\), we show that every local derivation from M _n (\({\mathcal{R}}\)) into M _n (\({\mathcal{M}}\)) is a derivation.     

Largest and smallest maximal sets of pairwise disjoint partitions

Let $\text{X}$ be a maximal set of pairwise disjoint partitions of n into t distinct parts. Let M_t(n) (resp. m_t(n)) denote the size of the largest (resp. smallest) such maximal set $\text{X}$. Upper and lower bounds for $\text{M}{}_{\mathrm{t}}\text{(n)}\text{n}$ and $\text{m}{}_{\mathrm{t}}\text{(n)}\text{n}$ are established.