On the structure of linear semigroups

Green's Lemma [1, Lemma 2.2] is one of the most important theorems in the theory of semigroups. The main purpose of this note is to establish a generalized Green's Lemma and a generalized Clifford and Miller's Theorem [1, p. 59] in linear semigroups. A generalized Green's Lemma describes the behavior of certain mappings between two distinct D-classes.     

QUALITATIVE ANALYSIS ON FITZHUGH'S NERVE CONDUCTION EQUATION

The existence of limit cycles of Fitzhugh's nerve conduction equation(1)was studiedin[1-4],but only a small range of parameter was considered.In this paper the authorsestablish a result(Theorem A)by using the qualitative method of O.D.E.,whicnimproves the results in[1-4]considerably.     

Random approximations and random fixed point theorems for random 1-setn-contractive non-self-maps in abstract cones

In this paper, we will prove that the random version of Fan's Theorem [6, Theorem 2] is true for a random hemicompact 1-set-contractive map defined on a closed ball, a sphere and an annulus in cones. This class of random 1-set-contractive map includes random condensing maps, random continuous semicontractive maps, random LANE maps, random nonexpansive maps and others. As applications of our theorems, some random fixed point theorems of non-self-maps are proved under various well-known boundary conditions. Our results are generalizations, improvements or stochastic versions of the recent results obtained by many authors     

A short proof of the preservation of the ωω‐bounding property

There are two versions of the Proper Iteration Lemma. The stronger (but less well‐known) version can be used to give simpler proofs of iteration theorems (e.g., [7, Lemma 24] versus [9, Theorem IX.4.7]). In this paper we give another demonstration of the fecundity of the stronger version by giving a short proof of Shelah's theorem on the preservation of the ω^ω‐bounding property. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Anstee的两个定理的订正

本文的目的有二个,其一是给出反例说明Anstee的两个定理是欠妥的,其二是订正这二个定理。为方便起见,我们沿用[2]中的有关记号和定义。 设R和S分别为m维和n维非负整数向量,P=(P_(ij))_(m×n)为每列至多有一个1的(0,1)-矩阵。令_p(R,S)是一切以R为行和向量、S为列和向量且覆盖(cover)P的(0,1)-矩阵组成的集合。一个列向量a若是_p(R,S)中某个矩阵的第k列,则称a为_p(R,S)的     

A note on random fixed point theorem in probabilistic analysis

In this paper we first prove that the assumptions between basic Theorem 1 and 2, hence main Theorem 3 and 4, in Chang [ 1 ] are equivalent. We then give several common fixed point theorems which are proved under considerably weaker conditions. We finally point out three mistakes in Chang [ 1 ] and correct them. The method used in this paper defers from the method used by Chang [ 1 ]     

An alternative proof of lower bounds for the first eigenvalue on manifolds

Recently, Andrews and Clutterbuck [1] gave a new proof of the optimal lower eigenvalue bound on manifolds via modulus of continuity for solutions of the heat equation. In this short note, we give an alternative proof of Theorem 2 in [1]. More precisely, following Ni's method (Section 6 of [5]), we give an elliptic proof of this theorem.     

Nonoscillation theorems of second order nonlinear differential equations

The main purpose of this paper is to study the existence of nonoscillatory solutions of the second order non-linear differential equation (1). The author first generalizes a Wintner's lemma [1,8] to nonlinear equations (i.e. the following Theorem 1 and 4), and then obtains the necessary and sufficient conditions for the existence of nonoscillatory solutions of (1). These theorems generalize the corresponding results of [1] to include nonlinear equations. Using the above results, the author further obtains a series of criterion theorems for the existence of nonoscillatory solutions and comparison theorems for the oscillation and nonoscillation of nonlinear equations.     

概率度量空间与映象的不动点定理

概率度量空间的概念首先由Menger[7]提出,以后许多人对这一空间的理论和应用曾进行过某些讨论(见[1-9])。本文的目的是进一步研究这一空间中映象的不动点定理。在本文的§2中,我们得出了一些新型的不动点定理,这些结果改进和加强了引文[2,3,8]中某些主要结果。     

Social choice theory without Pareto: The pivotal voter approach

This paper extends the pivotal voter approach pioneered by Barberá [Barberá, S., 1980. Pivotal voters: A new proof of Arrow’s Theorem. Economics Letters 6, 13–6; Barberá, S., 1983. Strategy-proofness and pivotal voters: A direct proof of the Gibbard-Satterthwaite Theorem. International Economic Review 24, 413–7] to all social welfare functions satisfying independence of irrelevant alternatives. Arrow’s Theorem, Wilson’s Theorem, and the Muller–Satterthwaite Theorem are all immediate corollaries of the main result. It is further shown that a vanishingly small fraction of pairs of alternatives can be affected in the group preference ordering by multiple individuals, which generalizes each of the above theorems.     

关于单叶函数系数之—基本引理及其应用

<正> 设函数 f(z)=z+a_2Z~2+…在单位圆|z|<1上是正则的单叶的.这种函数的全体形成一族 S.S 中满足条件|f(z)|1上是单叶的,除开极点ζ=∞是正则的.这种函数的全体形成一族∑.∑中满足条件|F(ζ)|>R的函     

除數問題(Ⅰ)

<正> 設dk(n)是n分解為k個因子的數目,設 R_k(x)=(a_(k,0)+a_(k,1)ln x +…+a_(k,k-1)ln~(k-1)x)x (x>0)是ζ~k(s)x~s/s在s=1的留数。定義 △_k(x)=D_k(x)-R_k(x). 當n=2時,下列的公式是大家熟悉的(參看[1]):     

Approximation of signals of class Lip(α, p) by linear operators

Mittal, Rhoades [5], [6], [7] and [8] and Mittal et al. [9] and [10] have initiated a study of error estimates E_n(f) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper we continue the work. Here we extend two theorems of Leindler [4], where he has weakened the conditions on {p_n} given by Chandra [2], to more general classes of triangular matrix methods. Our Theorem also partially generalizes Theorem 4 of Mittal et al. [11] by dropping the monotonicity on the elements of matrix rows, which in turn generalize the results of Quade [15].     

A topological approach to the theory of groups acting on trees

Bass and Serre recast the foundations of combinatorial group theory in [7]. Here we apply the allied notions of fundamental group and covering space to redevelop their theory in a less combinatorial fashion; for example the Bass—Serre Structure Theorem is proved with no a priori knowledge of the group theoretic structure of the fundamental group of a graph of groups. Van Kampen's Theorem is used only once, in its simplest form (in the proof of Theorem 7). Cancellation arguments and normal form theorems, such as Britton's Lemma, are completely avoided; indeed they are incidental corollaries from our viewpoint. The tree which plays a central role in [7] appears in Theorem 2 as the natural analogue of the “strecken komplexe” introduced by A. Speiser [8], and subsequently also employed by R. Nevanlinna [5], to describe certain simply connected Riemann surfaces occurring in value distribution theory.     

The transcendental rank of the formulas of an S1-categorical theory

In this paper we investigate the transcendental rank introduced by Morley. We introduce the concept of a fibering and prove a series of lemmas about fibering, on the basis of which we prove the main theorems: on an estimate for the rank of a formula (Theorem 3); on the summation of ranks (Theorem 2); and a reproof of Baldwin's theorem on the finiteness of the rank of S₁-categorical theories (Theorem 1).     

Fixed point theorems for nonexpansive mappings

In this paper, we shall extend two fixed point theorems of F. E. Browder [4, Corollary to Theorem 1] and J. T. Markin [6, Theorem 1].     

关于《M1-空间的和定理》一文的注记

《数学研究与评论》1988年第一期研究通讯栏目上刊登了论文《M_1-空间的和定理》。我们这篇简短的注记在于说明: (1)上述论文所引进的“强遗传闭包保持集族”的概念等价于“遗传闭包保持集族”的概念。 (2)上述论文中的定理3的结论是不正确的。 关于(1)若正则拓扑空间X的子集族B是X的遗传闭包保持集族,那么由B的元的闭包所组成的集族亦是X的遗传闭包保持集族。     

On the finiteness of higher knot sums

In this paper we show that any higher knot (n≥3) can be decomposed as a sum of irreducible knots and there is a finite upper bound on the number of summands. The case n=1 is due to Schubert [10]. A proof of this present case was published in [12] by Soninskii but subsequently Maeda showed that a crucial lemma was false [7]. The difficulty is to find a bound on decompositions of the knot group π₁. This is achieved here by applying Dunwoody's work in [3]. This results in two theorems, 1.6 and 1.7, which are of some interest in their own right. The rest of the paper follows as in Sosinskii's paper.We are grateful to G.A. Swarup for pointing out the possibility of this line of attack and to A. Bartholomew for an idea used in the proof of Theorem 1.6.     

黎曼ζ函數的一種推廣——Ⅲ.Z_(n,k)(s)的均值公式

<正> 本篇的目的就是要為Z_(n,k)(s)建立類似的公式. 當σ>kν時我們很容易為Z_(n,k)(s)建立類似(1.1)的公式,在這種情形下,我們可以把Z_(n,k)(s)表成絕對收斂級數的和: