高阶奇异积分的Hadamard主值

应用Euler径向微分算子D=z1 z1+…+zn zn研究复n维超球面 B≡{ζ∈Cn|ζ=(ζ1,…,ζn),|ζ1|2+…+|ζn|2=1}上两类高阶奇异积分的Hadamard主值.本文得到置换和合成公式并讨论了它们的拓广以及在偏微分奇异积分方程上的应用.     

一类离散型奇异积分算子

设｛αｋ｝∞ｋ＝－∞为正数缺项序列，满足ｉｎｆｋαｋ＋１／ｄｋ＝α＞１，Ω（ｙ′）为Ｂｅｓｏｖ空间Ｂ０，１１（Ｓｎ－１）上的函数，其中Ｓｎ－１为Ｒｎ（ｎ２）上的单位球面．本文证明：若∫Ｓｎ－１Ω（ｙ′）ｄσ（ｙ′）＝０，则离散型奇异积分ＴΩ（ｆ）（ｘ）＝∑∞ｋ＝－∞∫Ｓｎ－１ｆ（ｘ－αｋｙ′）Ω（ｙ′）ｄσ（ｙ′）和相关的极大算子ＴΩ（ｆ）（ｘ）＝ｓｕｐＮ∑∞ｋ＝Ｎ∫Ｓｎ－１ｆ（ｘ－αｋｙ′）Ω（ｙ′）ｄσ（ｙ′）均在Ｌ２（Ｒｎ）上有界．上述结果推广了Ｄｕｏａｎｄｉｋｏｅｔｘｅａ和ＲｕｂｉｏｄｅＦｒａｎｃｉａ［１］在Ｌ２情形下的一个结果     

一个排列数公式在数列求和中的应用

在排列组合中,由组合数性质Ｃｍｎ＋１＝Ｃｍｎ＋Ｃｍ－１ｎ,容易推出：Ｃｎｎ＋Ｃｎｎ＋１＋Ｃｎｎ＋１＋Ｃｎｎ＋２＋…＋＋Ｃｎｎ＋ｍ　＝Ｃｎ＋１ｎ＋ｍ＋１．于是　ＣｎｎＰｎｎ＋Ｃｎｎ＋１Ｐｎｎ＋Ｃｎｎ＋２Ｐｎｎ＋…＋Ｃｎｎ＋ｍＰｎｎ＝Ｃｍ＋１ｎ＋ｍ＋１Ｐｎｎ＝１ｎ＋１Ｃｎ＋１ｎ＋ｍ＋１Ｐｎｎ＋１．由ＣｎｋＰｎｎ＝Ｐｎｋ,得：Ｐｎｎ＋Ｐｎｎ＋１＋Ｐｎｎ＋２＋…＋Ｐｎｎ＋ｍ＝１ｎ＋１Ｐｎ＋１ｎ＋ｍ＋１．本文应用这个公式来研究现行高中教材《代数》下册中的一些数列前ｎ项和,不仅可以简化运算,而且为这些数列前…     

1999年全国高中数学联合竞赛试题及参考答案

一、选择题１．给定公比为ｑ（ｑ≠１）的等比数列｛ａｎ｝,设ｂ１＝ａ１＋ａ２＋ａ３,ｂ２＝ａ４＋ａ５＋ａ６,…,ｂｎ＝ａ３ｎ－２＋ａ３ｎ－１＋ａ３ｎ,…,则数列｛ｂｎ｝（　　）．　（Ａ）是等差数列　　（Ｂ）是公比为ｑ的等比数列　（Ｃ）是公比为ｑ３的等比数列　（Ｄ）既非等差数列又非等比数列解　由题设,ａｎ＝ａ１ｑｎ－１,则　ｂｎ＋１ｂｎ＝ａ３ｎ＋１＋ａ３ｎ＋２＋ａ３ｎ＋３ａ３ｎ－２＋ａ３ｎ－１＋ａ３ｎ＝ａ１ｑ３ｎ＋ａ１ｑ３ｎ＋１＋ａ１ｑ３ｎ＋２ａ１ｑ３ｎ－３＋ａ１ｑ３ｎ－２＋ａ１ｑ３ｎ－１＝ａ１ｑ３…     

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…     

自然数分数幂数列近似求和初探

自然数方幂求和问题：即Ｓｐ＝１ｐ＋２ｐ＋…＋ｎｐ求和,两千多年来,为人们关注和熟知．三百多年前,贝努利用二项式定理及递归方法,对每个自然数ｐ,可逐个求出Ｓｐ．今天,Ｓｐ的求法仍在不断被改进、创新．这在许多著作及刊物中均可找到．我们知道：ｐ＜－１时,Ｓｐ收敛．例如熟知　ｌｉｍｎ→∞（１１２＋１２２＋…＋１ｎ２）＝π２６．当ｐ≥－１时,Ｓｐ发散．（ｐ＝－１时Ｓｐ＝１１＋１２＋…＋１ｎ,即调和级数,可用递归型公式求和）．当ｐ为非负整数时,熟知Ｓ０＝ｎ,Ｓ１＝ｎ（ｎ＋１）２,Ｓ２＝１６ｎ（ｎ＋１）（２ｎ…     

数学问题解答

１９９９年１０月号数学问题解答（解答由问题提供人给出）１２１６．设复数α１、α２、…、αｎ,ｎ≥３,有｜α１｜＝｜α２｜＝…＝｜αｎ｜＝１;试证：ｚ＝（α１＋α２）（α２＋α３）…（αｎ＋α１）α１α２…αｎ是实数;证明　因为ｚ＝〔（α１＋α２）（α２＋α３）…（αｎ＋α１）α１α２…αｎ〕＝（α１＋α２）（α２＋α３）…（αｎ＋α１）α１α２…αｎ＝〔（α１＋α２）α１α２〕〔（α２＋α３）α２α３〕…〔（αｎ＋α１）αｎα１〕α１α２α３…αｎ·α１α２·α２α３·…·αｎα１＝（α１＋α…     

解平面弹性力学问题的边界积分方程的机械求积方法及其外推

１．引 言 考虑平面弹性力学内或外位移边值问题和内或外应力边值问题这里Ω是平面有界开集,Ωｃ是闭包Ω的补集,Γ是Ω或Ωｃ的边界,ｕ＝（ｕ１,ｕ２）是位移,ｎ＝（ｎ１,ｎ２）是Γ的外法向单位向量,δｉｊ＝（ｕｉ,ｊ＋ｕｊ,ｉ）／２是应变张量,λ和μ是Ｌａｍｅ常数,并且按张量计算规则：重复下标蕴含对该下标从１到２的求和． 使用直接边界元方法（１．１）与（１．２）皆可被转换为边界积分方程组这里αｉｊ（ｙ）是取决于ｙ∈Γ的常数,当ｙ是Γ的光滑点时,;式中是ｋｅｌｖｉｎ基本解,有以下表达式［５,７］这里ｒ＝…     

有限域上一类方程的解数公式

本文给出有限域Ｆｑ上一类方程ａ１ｘ１ｄ１１…ｘｎｄ１ｎ＋ａ２ｘ１ｄ２１…ｘｎｄ２ｎ＋…＋ａｓｘ１ｄｓ１…ｘｎｄｓｎ＝ｂ的解数公式，这里ｄｉｊ＞０，ａｉ∈Ｆｑ，ｉ＝１，…，ｓ，ｊ＝１，…，ｎ．特别当ｓ＝ｎ，ｇｃｄ（｜ｄｉｊ｜，ｑ－１）＝１时，得到了简明的解数公式．     

因子分解的故事（续）

４　因子分解和解析数论我们再简单介绍一下解析数论,它的起源可以上溯到欧拉对无限求和以及无穷乘积的收敛性研究;每个学过微积分的人都知道,当ｓ是大于１的实数时,级数ζ（ｓ）＝１＋１２ｓ＋１３ｓ＋…＋１ｎｓ＋…＝∑∞ｎ＝１ｎ－ｓ是收敛的,而ｓ＝１时,此级数发散（即其和１＋１２＋１３＋…＋１ｎ＋…为正无穷大）;对于ｓ＞１,欧拉于１７３７年把这个级数写成无穷乘积的形式：∑∞１ｎ－ｓ＝Πｐ１１－ｐ－ｓ＝Πｐ１＋１ｐｓ＋１ｐ２ｓ＋…＋１ｐｍｓ＋…,其中ｐ过所有素数,这个等式的正确性是基于算术基本定理：将上式右…     

Regularity of Positive Solutions for an Integral System on Heisenberg Group

In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, \begin{equation} \left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.\end{equation} for $x\in \mathbb{H}^n$, where $0<\alpha 1$ satisfying $\frac{1}{p+1} $+ $\frac{1}{q+1} + \frac{1}{r+1} = \frac{Q+α+β}{Q}.$ We show that positive solution triples $(u,v,w)\in L^{p+1}(\mathbb{H}^n)\times L^{q+1}(\mathbb{H}^n)\times L^{r+1}(\mathbb{H}^n)$ are bounded and they converge to zero when $|x|→∞.$     

BMO ESTIMATES FOR MULTILINEAR FRACTIONAL INTEGRALS

In this paper,the authors prove that the multilinear fractional integral operator T A 1,A 2 ,α and the relevant maximal operator M A 1,A 2 ,α with rough kernel are both bounded from L p (1 p ∞) to L q and from L p to L n/(n α),∞ with power weight,respectively,where T A 1,A 2 ,α (f)(x)=R n R m 1 (A 1 ;x,y)R m 2 (A 2 ;x,y) | x y | n α +m 1 +m 2 2 (x y) f (y)dy and M A 1,A 2 ,α (f)(x)=sup r0 1 r n α +m 1 +m 2 2 | x y | r 2 ∏ i=1 R m i (A i ;x,y)(x y) f (y) | dy,and 0 α n, ∈ L s (S n 1) (s ≥ 1) is a homogeneous function of degree zero in R n,A i is a function defined on R n and R m i (A i ;x,y) denotes the m i t h remainder of Taylor series of A i at x about y.More precisely,R m i (A i ;x,y)=A i (x) ∑ | γ | m i 1 γ ! D γ A i (y)(x y) r,where D γ (A i) ∈ BMO(R n) for | γ |=m i 1(m i 1),i=1,2.     

Oscillation Theorems for a Nonlinear System of Differential Equations

We shall establish some theorems analogous to Kneser's oscillation theorems for the linear differential equation of second order, which give sufficient conditions for oscillation of all solutions of the system of differential equations of the type u′i = |u3-i|λi sgn u3-i + (-1)i-1bi(t)ui (i = 1,2), where the functions bi (i = 1, 2) are nonnegative and summable on each finite segment of the interval [0, + ∞) and λi > 0 (i = 1, 2).     

A note on parameterized Marcinkiewicz integrals with variable kernels

In this paper,the parameterized Marcinkiewicz integrals with variable kernels defined by μΩ^ρ（f）（x）=（∫0^∞│∫│1-y│≤t Ω（x,x-y）/│x-y│^n-p f（y）dy│^2dt/t1＋2p）^1/2 are investigated.It is proved that if Ω∈ L∞（R^n） × L^r（S^n-1）（r〉（n-n1p＇/n） is an odd function in the second variable y,then the operator μΩ^ρ is bounded from L^p（R^n） to L^p（R^n） for 1 〈 p ≤ max{（n＋1）/2,2}.It is also proved that,if Ω satisfies the L^1-Dini condition,then μΩ^ρ is of type（p,p） for 1 〈 p ≤ 2,of the weak type（1,1） and bounded from H1 to L1.     

Comparison theorems for oscillation of second-order half-linear differential equations

Summary We establish new comparison theorems on the oscillation of solutions of a class of perturbed half-linear differential equations. These improve the work of Elbert and Schneider [6] in which connections are found between half-linear differential equations and linear differential equations. Our comparison theorems are not of Sturm type or Hille--Wintner type which are very famous. We can apply the main results in combination with Sturm's or Hille--Wintner's comparison theorem to a half-linear differential equation of the general form (|x'|^α-1x')' + a(t) |x|^α-1x = 0.     

On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients

For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln(ψ) = n ∑ k=0 akψ(k), where the constant coefficients ak ∈ R may be adapted to f . We prove that for each f ∈ C(n)(I), there is a selection of coefficients {a1, ,an} and a corresponding linear combination Sn( f ,t) = n ∑ k=1 bkeλkt of functions ψk(t) = eλkt in the nullity of L which satisfies the following Jackson’s type inequality: f (m) Sn(m )( f ,t) ∞≤ |an|2n|Im|1/1q/ep|λ|λn|n|I||nm1 Ln( f ) p, where |λn| = mka x|λk|, 0 ≤ m ≤ n 1, p,q ≥ 1, and 1p + q1 = 1. For the particular operator Mn(f) = f + 1/(2n) f(2n) the rate of approximation by the eigenvalues of Mn for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.     

P-叶星像函数类的一些充分条件

令Hn(p)表示形如f(z)=zp ∑ ∞k=n pakzk,且在单位圆U={z;|z|<1}内解析的函数f(z) 的全体所成的函数类.本文应用微分从属技巧得到了p-叶β级星像函数的一些充分条件,所得结果推广了一些作者的相关结果.     

Asymptotic formulas for nonoscillatory solutions of conditionally oscillatory half-linear equations

We establish asymptotic formulas for nonoscillatory solutions of a special conditionally oscillatory half-linear second order differential equation, which is seen as a perturbation of a general nonoscillatory half-linear differential equation

Rough bilinear fractional integrals with variable kernels

We study the rough bilinear fractional integral

Sufficient conditions for absolute asymptotic stability of linear equations in a Banach space

For a linear differential equation of the type (1) $$\frac{{dx}}{{dt}} = A_0 x(t) + A_1 x(t - \Delta _1 ) + ... + A_n x(t - \Delta _n )$$ we establish the followingTHEOREM. If $$\overline {\left| {z_1 } \right| = ...\underline{\underline \cup } \left| z \right|_n = 1\sigma \left( {A_0 + \sum\nolimits_{k = 1}^n {z_k A_k } } \right)} \subset \left\{ {\lambda :\operatorname{Re} \lambda< 0} \right\}$$ then system (1) is absolutely asymptotically stable.