关于Q（√4k^2n＋1）的理想类群的循环子群

设ｄ,ａ,ｋ,ｎ是适合４ｋ＾２ｎ＋１＝ｄａ＾２,ｋ〉１,ｎ〉２,ｄ无平方因子的正整数;又设Ｃ（Ｋ）和ｈ（Ｋ）分别是实二次域Ｋ＝Ｑ（√ｄ）的理想类群和类数。本文证明了：当ａ〈０．５ｋ＾０．５６ｎ时,则ｈ（Ｋ）≡０（ｍｏｄ ｎ）和Ｃ（Ｋ）必有ｎ阶循环子群。     

也谈实二次域类数的可除性

设ｄ无平方因子，ｈ（ｄ）是二次域Ｑ（ｄ）的类数，本文证明了：若１＋４ｋ２ｎ＝ｄａ２，ａ，ｋ＞１，ｎ＞２为正整数，且ａ＜０．９ｋ３５ｎ或ｎ的奇素因子ｐ和ｋ的素因子ｑ均适合（ｐ，ｑ－１）＝１，则除（ａ，ｄ，ｋ，ｎ）＝（５，４１，２，４）以外，ｈ（ｄ）≡０（ｍｏｄｎ）．同时，我们猜测：上述结果中的条件（ｐ，ｑ－１）＝１是不必要的．     

丢番图方程与实二次域类数的可除性

设ｄ无平方因子，ｈ（ｄ）是二次域的类数。本文证明了：在方程Ｕ￣２－ｄＶ￣２＝４，（Ｕ，Ｖ）＝１有整数解时，丢番图方程４ｘ￣（２ｎ）－ｄｙ￣２＝－１，ｎ＞２无｜ｙ｜＞１的整数解；如果正整数ａ，ｋ，ｎ满足，ｋ＞１，ｎ＞２且而是Ｐｅｌｌ方程ｘ￣２－ｄｙ￣２＝－１的基本解，则ｈ（ｄ）≡０（ｍｏｄｎ）。     

椭圆一个定理的又一初等证明

定理　椭圆Ｃ：ｘ２ａ２＋ｙ２ｂ２＝１（ａ＞ｂ＞０）有且仅有两条对称轴：直线ｘ＝０和ｙ＝０．文［１］指出,这个定理的证明一般要用到仿射几何知识,同时文［１］给出了一个初等证明．笔者再给出这个定理的又一种初等证明如下．定理的证明　易验证直线ｘ＝０和ｙ＝０均是椭圆Ｃ的对称轴．因点Ｂ（０,ｂ）关于直线ｘ＝ｋ（ｋ≠０）的对称点Ｂ′（２ｋ,ｂ）不在椭圆Ｃ图１上,故直线ｘ＝ｋ（ｋ≠０）不是椭圆Ｃ的对称轴．设Ｆ１,Ｆ２是椭圆Ｃ的两个焦点,椭圆Ｃ的长轴Ａ１Ａ２关于直线ｌ：ｙ＝ｋｘ＋ｎ（ｋ,ｎ至少有一个不等于零）的…     

等差数列前n 1项同次幂和的递归关系

等差数列前ｎ＋１项同次幂和的递归关系李朝星（湖北师范学院４３５００２）设ａ，ｄ是任意实数但ｄ≠０，ｋ为非负整数．用Ｓｋ（ｎ；ａ，ｄ）表示等差数列ａ，ａ＋ｄ，ａ＋２ｄ，…，ａ＋ｎｄ，…的前ｎ＋１项ｋ次幂的和，即Ｓｋ（ｎ；ａ，ｄ）＝ａｋ＋（ａ＋ｄ）ｋ＋（...     

平均单叶函数的相邻系数

设ｆ（ｚ）＝ｚ＋Σａｎｚ＾ｎ为单位园｜ｚ｜＜１内解析且平均单叶，记其族为Ｍ又设｛ｆ（ｚ）／ｚ｝＾λ＝１＋Σ＾∞ｎ＝１Ｄｎ（λ），λ＞０，本文说明了：定理一 若ｆ∈Ｍ，λ＞０，则：Σ＾∞ｋ＝１｛｜｜Ｄｋ（λ）｜－｜Ｄｋ－１（λ）｜｜／ｄｋ（λ）｝＾２≤Ａｎ，ｎ＝２，３，…其中Ａ为绝对常数。ｄｋ（ｈ）＝ｈ（ｈ＋１）…（ｈ＋ｋ－１）／ｋ！当λ＝１／２，ｆ∈ｓ时为Ｉ．Ｖ．Ｍｉｌｍ所证明。定理二 若ｆ∈Ｍ并     

截断和的弱大数律

投｛Ｘｎ，ｎ≥１｝ｉ．ｉ．ｄ．，Ｘｎ，１≤Ｘｎ，２≤…≤Ｘｎ，ｎ是Ｘ１，Ｘ２，…，Ｘｎ的次序统计量．对非负整数ｋ，ｒ，ｋ＋ｒ≤ｎ，令．本文研究当ｋ＝ｋｎ，ｒ＝ｒｎ满足ｍｉｎ（ｋ，ｒ）→∞，ｍａｘ（ｋ，ｒ）→０时截断和Ｓｎ（ｋ，ｒ）的弱大数律．设βｎ＞０，Ｃｎ∈Ｒ，文中给出了依概率收敛的充要条件．     

关于一类和式的估值

设｛ａｎ｝为递增的正项等差数列：ａｎ＝ａ１＋（ｎ－１）ｄ，ｎ∈Ｎ，其中ｄ，ａ１＞０，本文讨论和式ｎｋ＝１１ａｋ＝１ａ１＋１ａ２＋…＋１ａｎ的估值，并解决文［１］中遗留的问题．定理１设ｄ≤２ａ１，则对任意ｎ∈Ｎ有ｍｎ≤ｎｋ＝１１ａｋ＜Ｍｎ①其中Ｍｎ...     

两个不等式的简捷证法

下面给出的两类不等式问题,一般是通过代换的方法证明．本文给出直接简捷的证明．命题１　设ｘｉ∈Ｒ＋（ｉ＝１,２,…,ｎ）且ｘ２１１＋ｘ２１＋ｘ２２１＋ｘ２２＋…＋ｘ２ｎ１＋ｘ２ｎ＝ａ（０＜ａ＜ｎ）,求证：ｘ１１＋ｘ２＋ｘ２２１＋ｘ２２＋…＋ｘ２ｎ１＋ｘ２ｎ≤ａ（ｎ－ａ）①证　由题设易知：１１＋ｘ２１＋１１＋ｘ２２＋…＋１１＋ｘ２ｎ＝ｎ－ａ．由于　１１＋ｘ２ｋ＋ｎ－ａａ·ｘ２ｋ１＋ｘ２ｋ　　≥２１１＋ｘ２ｋ·ｎ－ａａ·ｘ２ｋ１＋ｋ２ｋ　　＝２ｎ－ａａ·ｘｋ１＋ｘ２ｋ）（ｋ＝１,２,…,ｎ）,此ｎ式相…     

数据插值和共轭函数

设Ｇ（ｚ）在｜ｚ｜＜ρ（ρ＞１）中解析，且数据Ｒｅ［Ｇ（ｅｊ２ｋπ／ｎ）］；ｋ＝０，１，…，ｎ－１已给出，其中ｎ＝２ν＋１，本文构造了一个ν次多项式Ｐν（ｚ）满足插值条件Ｒｅ［Ｐν（ｅｊ２ｋπ／ｎ）］＝Ｒｅ［Ｇ（ｅｊ２ｋπ／ｎ）］，ｋ＝０，１，…，ｎ－１．并估计了误差‖Ｇ（ｅｊω）－Ｐν（ｅｊω）‖．此外，还给出了一个Ｗａｌｓｈ类型的超收敛定理．     

Maximal inequalities for a continuous semimartingale

Let X =( X t ) t S 0 be a continuous semimartingale given by d X t = f ( t ) w ( X t )d d M ¢ t + f ( t ) σ ( X t )d M t , X 0 =0, where M =( M t , F t ) t S 0 is a continuous local martingale starting at zero with quadratic variation d M ¢ and f ( t ) is a positive, bounded continuous function on [0, X ), and w , σ both are continuous on R and σ ( x )>0 if x p 0. Denote X 𝜏 * =sup 0 h t h 𝜏 | X t | and J t = Z 0 t f ( s ) } ( X s )d d M ¢ s ( t S 0) for a nonnegative continuous function } . If w ( x ) h 0 ( x S 0) and K 1 | x | n σ 2 ( x ) h | w ( x )| h K 2 | x | n σ 2 ( x ) ( x ] R , n >0) with two fixed constants K 2 S K 1 >0, then under suitable conditions for } we show that the maximal inequalities c p , n log 1 n +1 (1+ J 𝜏 ) p h Á X 𝜏 * Á p h C p , n log 1 n +1 (1+ J 𝜏 ) p (0< p < n +1) hold for all stopping times 𝜏 .     

Improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set

We will establish the following improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set: For each k and d, 0 k d, define f(d,k) = d+1 if k = 0 and f(d,k) = max{d+1,2d–2k+2} if 1 k d.Theorem 1. Let S be a compact, connected, locally starshaped set in R^d, S not convex. Then for a k with 0 k d, dim ker S k if and only if every f(d, k) lnc points of S are clearly visible from a common k-dimensional subset of S.Theorem 2. Let S be a nonempty compact set in R^d. Then for a k with 0 k d, dim ker S k if and only if every f (d, k) boundary points of S are clearly visible from a common k-dimensional subset of S. In each case, the number f(d, k) is best possible for every d and k.     

带弦圈的最小2宽直径

设k为正整数,G是简单k连通图.图G的k宽直径,dk(G),是指最小的整数ι使得对任意两不同顶点x,y∈V(G),都存在k条长至多为ι的内部不交的连接x和y的路.用C(n,t)表示在圈Gn上增加t条边所得的图.定义h(n,t):min{d2(C(n,t))}.本文给出了h(n,2)=[n/2].而且,给出了当t较大时h(n,t)的界.     

关于虚二次域类数的可除性

设α＞１，ｂ＞１，（α，ｂ）＝１，ｈ（－αｂ）表虚二次域的类数。如果有正整数ｘ，ｙ，ｎ，ｋ满足（１）αｘ￣２＋ｂｙ￣２＝４ｋ￣ｎ，ｂ且；或（２）αｘ￣２＋ｂｙ￣２＝ｋ￣ｎ，ｘ｜α，ｙ｜ｂ且αｂ≡２（ｍｏｄ４），则本文证明了关于ｈ（－αｂ）的可除性的两个定理（见定理１，２），其中符号ｘ｜α表示ｘ的每一个素因子整除α。     

代數拓撲學中(μ,△,γ)-系統的正則同模論Ⅰ

<正> 設K與L為拓撲空間,又設f:K→L為連續映像.由f導出了準同模對應f~n:H~n(L,G)→H~n(K,G),n=1,2,…,其中H~n(L,G),H~n(K,G)表示上同調羣,而G表示係數環或域以γ_p~n(K)或者     

On the strongly spherical Sasakian metric of a spherical tangent bundle

Let Mⁿ denote an n-dimensional Riemannian manifold. Its metric is called -strongly spherical if at every point Q Mⁿ there exists a -dimensional subspace _Q T_QMⁿ such that the curvature operator of the metric of Mⁿ satisfies R(X, Y) Z = k(< Y, Z > X < X, Z > Y), where k = const > 0, Y _Q , X, Z #x2208; T_QMⁿ. The number is called the index of sphericity and k the exponent of sphericity. The following theorems are proved in the paper.THEOREM 1. Let the Sasakian metric of T₁Mⁿ be -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if M² has constant Gaussian curvature K 1 and k = K²/4; b) = 3 if and only if M² has constant curvature K = 1 and k = 1/4; c) = 0, otherwise.THEOREM 2. Let the Sasakian metric of T₁Mⁿ (n Mⁿ) be -strongly spherical with exponent of sphericity k. If k > 1/3 and k 1, then = 0. Let us denote by (Mⁿ, K) a space of constant curvatureK. THEOREM 3. Let the Sasakian metric of T₁(Mⁿ, K) (n 3) be -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if K = 1/4; b) = 0, otherwise. In dimension n = 3 Theorem 2 is true for k {1/4, 1}.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 150–159, 1992.     

Formulae and Recursions for the Joint Distribution of Success Runs of Several Lengths

Consider a sequence of n independent Bernoulli trials with the j-th trial having probability pj of success, 1 j n. Let M(n,K) and N(n, K) denote, respectively, the r-dimensional random variables (M(n, k1),..., M(n,kr) and (N(n,k1), ..., N(n, kr)), where K = (k1, k2, ..., kr) and M(n, s) [N(n, s)] represents the number of overlapping [non-overlapping] success runs of length s. We obtain exact formulae and recursions for the probability distributions of M(n, K) and N(n, K). The techniques of proof employed include the inclusion-exclusion principle and generating function methodology. Our results have potential applications to statistical tests for randomness.