Sums of the Numbers of Prime Factors of Positive Integers

Denote by ω(n) and Ω(n) the number of distinct prime factors of n and the total number of prime factors of n,respectively.For any positive integer ι,we prove that ∑↑2≤n≤x1/ω（n)=ι↑∑↑κ=0(ι↑∑↑i=κ（-1)^i-κCi^κF^(i-κ)(1)κ！x/（loglogx）^i 1 O(x/(loglogx)^ι 2) ∑↑2≤n≤xΩ（n)/ω（n)=x ι↑∑↑κ=0ι↑∑↑i=κ∑↑p1/p^κ 2-p^κ 1（-1)^i-κCi^κF^(i-κ）（1）κ!)x/(loglogx)^i 1 O(x/(loglogx)ι 2） where F(z)=1/г(z)pⅡ（1 z/p-1)(1-1/p)^z,and the constant O despends on ι.This improves previous result of R.L.Duncan and Chao Huizhong.     

自共轭四元数矩阵广义逆A{i,…，j;＊}的Loewner偏序

本文利用自共轭四元数矩阵广义逆的显公式，给出了在Loewner偏序下A{1；*；s；≤Aκ^(1),A{1;*;t;≥Aκ^(1)}（其中A∈H（n，*））；A{2；≥；t；≥Aκ^(2)},A{2;≥;s;≤Aκ^(2)},A{1;≥；t；≥As^(2)}（其中A∈H（n，≥））的表式。     

奇摄动非线性系统Robin边值问题

本文研究了非线性系统奇摄动问题：ε2y"－（x，y，y）＝0，0＜x＜1，0＜ε≤1，y（0）－py'（0）＝A，p＞0，y（1）＝B，其中y，f，A，B为n维向量．在相应的假设下，利用代数型边界层函数，证明了该问题存在一个解y（x，ε），并利用微分不等式方法得到了其解的渐近估计．     

两个四阶奇异微分算子积的自伴性

本文在区间[a,∞)上研究由具有任意亏指数的对称常微分算式ly:=y(4)-(py′)′+qy生成的两个四阶奇型微分算子Li(i=1,2)的积L2L1的自伴性.在0∈Π(L0(l))及l2在L2[0,∞)中是部分分离的假设条件下,借助实参数解对自共轭域的描述定理,获得两个四阶微分算子乘积自伴的充要条件,同时证明若L1和L2自伴,则L=L2L1自伴的充要条件是L1=L2,其中-∞a∞,2≤d≤4,Π(L0(l))是l在L2[a,∞)中产生的最小算子L0(l)的正则型域.     

一类具有内部奇异点的微分算式乘积的自伴域

本文研究一类带有内部奇异点的n阶复值系数对称微分算式ty=Σna_j（t）y^（j）（t）乘积的自共轭域描述问题.通过构造相应的直和空间,应用直和空间的相关理论,在直和空间上生成的相应于l的最小算子T₀（l）的正则型域Π（T₀（l））满足（-r,r）■Π（T₀（l））∩R及l²在直和空间中是部分分离的条件下,利用微分方程ly=±λy的解给出l²的自共轭域的完全解析描述,并且确定自共轭边界条件的矩阵仅由微分方程的解在正则点的初始值决定,其中0相似文献   

一个代数不等式的指数推广

文 [1 ]中 ,程龙海先生证明了下面不等式 :若 0≤ x,y≤ 1 ,则x2 y2 ( 1 - x) 2 y2 x2 ( 1 - y) 2 ( 1 - x) 2 ( 1 - y) 2≤ 2 2 . ( 1 )本文将 ( 1 )式作如下推广定理　若 0≤ x,y≤ 1 ,n≥ 2 ,n∈ N,则n xn yn n ( 1 - x) n yn n xn ( 1 - y) n n ( 1 - x) n ( 1 - y) n≤ 2 n 2 . ( 2 )引理　若 u≥υ≥ 0 ,n≥ 2 ,n∈ N,则n un υn ≤ u ( n 2 - 1 )υ. ( 3)证明　因为 u≥υ≥ 0 ,所以[u ( n 2 - 1 )υ]n=un ∑ni=1Cinun- i( n 2 - 1 ) ivi≥ un ∑ni=1Cin( n 2 - 1 ) iυn=un [∑ni=0Cin(…     

m个微分算式乘积的自伴边界条件

本文假设n阶正则对称微分算式l(y)的幂算式lm(y)在L2[a,∞)中是部分分离的,首先刻画了由幂算式lm(y)生成的微分算子T(lm)的自伴边界条件.然后,在L2[a,∞)中,借助T(lm)的自伴域的这种刻画形式,研究了m个由n阶微分算式l(y)生成的微分算子Tk(l)(k=1,2,…,m;m∈z,m≥2)乘积的自伴性问题,获得了乘积算子Tm(l)…T2(l)T1(l)是自伴算子的充要条件.     

m个微分算式乘积的自伴边界条件

本文假设n阶正则对称微分算式l(y)的幂算式l~m(y)在L~2[α,∞)中是部分分离的,首先刻画了由幂算式l~m(y)生成的微分算子T(l~m)的自伴边界条件．然后,在L~2[α,∞)中,借助T(l~m)的自伴域的这种刻画形式,研究了m个由n阶微分算式l(y)生成的微分算子T_k(l)(k=1,2,……,m；m∈z,m≥2)乘积的自伴性问题,获得了乘积算子T_m(l)…T_2(l)T_1(l)是自伴算子的充要条件．     

具有正则型点的奇异微分算子的自共轭扩张

在П(L0)n R≠θ的条件下,本文讨论了具有中间亏指数的对称微分算式l(y)的自共轭域,其中П(L0)是由l(y)生成的最小算子L0的正则型域.使用方程l(y)=λ0y,(λ0∈П(L0)∩R)的实参数L2-解,我们对最大算子域DM进行新的分解,由此得到l(y)的自共轭域新的完全解析刻画,其中自共轭边界条件中矩阵M,N的确定与l(y)=λ0y在无穷远点的性质无关,仅与其在t=0点初始值的选择有关.由于自共轭箅子谱是实的,使用实参数λ0不仅有利于我们找到方程的显解,更重要的是可以得到谱的有关信息.     

一类滞后差分方程解的渐近性

考虑时滞差分方程xn－xn-1＝F（－f（xn）＋g（xn－k）），这里k是正整数，F，f，g是R→R的连续函数，F和f在R上单调增加，且对所有的u≠0，uF（u）＞0.我们证明了如果对所有的y∈R，有f（y）≥g（y）（f（y）≤g（y）），则方程的每个解趋于一个常数或－∞（∞）．进一步，如果对所有的y∈R，有f（y）≡g（y）；则方程的每个解当n→∞时趋于常数．     

Instructions for Authors

<正>Submission Authors must use LaTeX for typewriting,and visit our website www.actamath.com to submit your paper.Our address is Editorial Office of Acta Mathematica Sinica,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,P.R.China.     

THE 8~(th) INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS

正August 10-14,2015Beijin,China The International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

FRACTIONAL INTEGRALS OF THE WEIERSTRASS FUNCTIONS： THE EXACT BOX DIMENSION

The present paper investigates the fractal structure of fractional integrals of Weierstrass functions. The ezact box dimension for such functions many important cases is established. We need to point out that, although the result itself achieved in the present paper is interesting, the new technique and method should be emphasized. These novel ideas might be useful to establish the box dimension or Hausdorff dimension (especially for the lower bounds) for more general groups of functions.     

English Series

正1 Aims and Scope Acta Mathematicae Applicatae Sinica(English Series)is a quarterly journal established by the Chinese Mathematical Society.The journal publishes high quality research papers from all branches of applied mathematics,particularly welcomes those from partial differential equations,computational mathematics,applied probability,mathematical finance,statistics,dynamical systems,optimization and management science.     

CHINESE ANNALS OF MATHEMATICS SERIES B Volume 35 · 2014 TOTAL CONTENTS

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Congruences,ideals and annihilators in standard QBCC-algebras

We characterize congruence lattices of standard QBCC-algebras and their connection with the congruence lattices of congruence kernels. Work on the paper was supported by Council of Czech Government No J14/98:153100011.     

A decomposition of continuity via ideals

A new class of sets in ideal topological spaces is introduced and using these sets, a decomposition of continuity is given.      

When a compact (countably compact) set is closed. II

We obtain (a) necessary and sufficient conditions and (b) sufficient conditions for a compact (countably compact) set to be closed in products (sequential products) and subspaces (sequential subspaces) of normal spaces. As a consequence of these, sufficient conditions are obtained for (i) the closedness of arbitrary (countable) union of closed sets and (ii) the equality of the union of the closures and the closure of the union of arbitrary (countable) families of sets in these spaces. It is also shown that these results do not hold for quotients of even T ₄,-spaces.     

ASYMPTOTIC BEHAVIOR OF ECKHOFF＇S METHOD FOR FOURIER SERIES CONVERGENCE ACCELERATION

The current paper considers the problem of recovering a function using a limited number of its Fourier coefficients. Specifically, a method based on Bernoulli-like polynomials suggested and developed by Krylov, Lanczos, Gottlieb and Eckhoff is examined. Asymptotic behavior of approximate calculation of the so-called ＂jumps＂ is studied and asymptotic L2 constants of the rate of convergence of the method are computed.