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…     

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     

一类离散动力系统的稳定性

研究一类作 为基因选择模 型的离散动力系 统ｙ ｎ ＋ １ ＝ ｙｎ ｅｂ（ １ － ２ ｙ ｎ － ｋ ）１－ ｙｎ ＋ ｙｎ ｅｂ（ １ － ２ ｙ ｎ － ｋ ） ,　ｎ ＝ ０,１,… ,的稳定性 ,其中 ｂ∈（０,∞）, Ｋ∈ ｛１,２,…｝     

圆锥曲线弦的中点与斜率的关系及其应用

关于圆锥曲线弦的中点问题,许多文章已有论述,本文综其为一体,给出圆锥曲线弦的一个重要性质．定理　圆锥曲线Ａｘ２＋Ｃｙ２＋Ｄｘ＋Ｅｙ＋Ｆ＝０的弦的斜率为ｋ,弦的中点为（ｘ０,ｙ０）,同有Ａｘ０＋Ｃｋｙ０＋１２Ｄ＋１２ｋＥ＝０．证　设弦的两端点为（ｘ１,ｙ１）,（ｘ２,ｙ２）斜率为ｋ,则有Ａｘ２１＋Ｃｙ２１＋Ｄｘ１＋Ｅｙ１＋Ｆ＝０,Ａｘ２２＋Ｃｙ２２＋Ｄｘ２＋Ｅｙ２＋Ｆ＝０．两式相减,得Ａ（ｘ２１－ｘ２２）＋Ｃ（ｙ２１－ｙ２２）＋Ｄ（ｘ１－ｘ２）　＋Ｅ（ｙ１－ｙ２）＝０．两边同除以ｘ１－ｘ２,注意到…     

一类线性errors－in－variables模型的参数强相合估计及其a．s．收敛速度

本文讨论了如下一类线性ｅｒｒｏｒｓ－ｉｎ－ｖａｒｉａｂｌｅｓ模型——多元线性结构关系模型β′ｘｋ＋α＝０，ξｋ＝ｘｋ＋εｋ．｛ｋ＝１，２，…，ｎ．其中，｛ｘｋ：ｋ＝１，２，…，ｎ｝为一组ｉ．ｉ．ｄ．的ｍ维随机向量，｛εｋ：ｋ＝１，２，…，ｎ｝是ｉ．ｉ．ｄ．的随机误差，Ｅ（ε１）＝０，Ｖａｒ（ε１）＝σ２Ｉｍ．且｛ｘｋ：ｋ＝１，２，…，ｎ｝与｛εｋ：ｋ＝１，２，…，ｎ｝相互独立．在一些条件下，我们证明了估计量β，α，σ２的强相合性、唯一性，并给出了估计量的收敛速度为ｏ（ｎ－１－１ｑ），这里ｑ∈［１，２）．对于Ｅ（ｘ１）ｕ１和Ｖａｒ（ｘ１）Ｖｘ的估计也得出了同样的结果     

广义本原指数的缺数系

运用数论和图论技巧，得到了当λ（Ｄ）３时本原有向图Ｄ的广义指数ｅｘｐ（Ｄ，ｋ）的界，这里λ（Ｄ）表示Ｄ中不同长的圈的类数，还证明了对任何整数ｎ，ｔ，不存在ｎ阶本原有向图Ｄ，使得ｎ２－ｔｎ＋１４（ｔ＋１）２＋ｋ－２＜ｅｘｐ（Ｄ，ｋ）＜ｎ２－（ｔ－１）ｎ＋ｔ＋ｋ－３．     

高维空间中半线性波动方程的Sobolev指数

ＧｕｓｔａｖｏＰｏｎｃｅ与ＴｈｏｍａｓＣ．Ｓｉｄｅｒｉｓ［４］猜测对一些具有特殊非线性项的半线性波动方程，如ｕｔ－△ｕ＝ｕｋ（Ｄｕ）α（ｘ∈Ｒｎ，ｋ∈Ｚ＋，ｌ＝｜α｜２），其中Ｓｏｂｏｌｅｖ指数会在ｎ２与（ｎ２＋１）之间．文［４］中，在ｘ∈Ｒ３时，回答了这一问题．本文在ｎ３维空间中，得到了半线性波动方程ｕｔ－△ｕ＝ｕｋ（Ｄｕ）α（ｘ∈Ｒｎ，ｋ∈Ｚ＋，ｌ＝｜α｜２）的Ｓｏｂｏｌｅｖ指数为ｍａｘ｛ｎ２＋１２，（ｎ２－１）·ｌ－３ｌ－１＋２｝，此数确实在区间［ｎ２＋１２，ｎ２＋１］中．     

数学问题解答

数学问题解答１９９８年２月号问题解答（解答由问题提供人给出）１１１６设ａｉ０（１ｉｎ），ｎｉ＝１ａｉ＝１（ｎ２），并记ａｎ＋１＝ａ１，则对ｋ∈Ｎ，有不等式：（３）ｋｎ１－ｋｎｉ＝１（ａｉ２＋ａｉａｉ＋１＋ａｉ＋１２）ｋ２，且对左边不等...     

数学问题解答

１９９９年４月号问题解答（解答由问题提供人给出）１１８６设ｎ∈Ｎ,ｎ≥２,ｋ∈Ｒ＋,求函数ｙ＝ｘｎｘ－ｋ（ｘ∈（ｋ,＋∞））的最小值．解由均值不等式ｘ１＋ｘ２＋…＋ｘｎｎ≥ｎｘ１ｘ２…ｘｎ得ｘ１ｘ２…ｘｎ≤（ｘ１＋ｘ２＋…＋ｘｎｎ）ｎ∵ｙ＝ｘｎｘ－ｋ...     

反差分计算在数列求和上的应用

反差分计算在数列求和上的应用常丰（蚌埠教育学院，蚌埠２３３０１０）一、差分设ｆ（ｎ）是非负整数ｎ＝０，１，２，３，…的函数，其差分Δｆ（ｎ）定义为Δｆ（ｎ）＝ｆ（ｎ＋１）－ｆ（ｎ）．例如，设阶乘幂ｆ（ｎ）＝ｎ（ｋ）＝ｎ（ｎ－１）…（ｎ－ｋ＋１），ｋ≥...     

关于一类具有中间亏指数的正系数的对称微分算子

本文研究了一类具有中间亏指数的正系数的对称微分算式     

Qualitative analysis of a fourth order difference equation

In this paper, we will investigate some qualitative behavior of solutions of the following fourth order difference equation $x_{n+1}=ax_{n-1}+\frac{bx_{n-1}}{cx_{n-1}-dx_{n-3}},$ \ $n=0,1,...,$ where the initial conditions $x_{-3,}x_{-2},\ x_{-1}$\ and\ $x_{0}\ $are arbitrary real numbers and the values $a,\ b,\ c\ $and$\;d$ are\ defined as positive real numbers.     

Transformations z(t) = L(t)y(ϕ(t)) of ordinary differential equations

The paper describes the general form of an ordinary differential equation of an order n + 1 (n ≥ 1) which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form $f\left( {s,w_{00} \upsilon _0 ,...,\sum\limits_{j = 0}^n {w_{nj\upsilon _j } } } \right) = \sum\limits_{j = 0}^n {w_{n + 1j\upsilon j} + w_{n + 1n + 1} f\left( {x,\upsilon ,\upsilon _1 ,...,\upsilon _n } \right),}$ where $w_{n + 10} = h\left( {s,x,x_1 ,u,u_1 ,...,u_n } \right),w_{n + 11} = g\left( {s,x,x_1 ,...,x_n ,u,u_1 ,...,u_n } \right){\text{ and }}w_{ij} = a_{ij} \left( {x_i ,...,x_{i - j + 1} ,u,u_1 ,...,u_{i - j} } \right)$ for the given functions a _ij is solved on $\mathbb{R},u \ne {\text{0}}$ .     

关于一类Hermite-Fejér插值算子的平均收敛

本文给出基于{xk}_(k=0)~(n+1)的Hermite-Fejér插值算子平均收敛的一些新结论,这里x0=1,xn+1=-1,xk（k=1,2,…,n）是n阶Jacobi多项式的零点.     

关于一类Hermite-Fejér插值算子的平均收敛

本文给出基于{xk}_(k=0)~(n+1)的Hermite-Fejér插值算子平均收敛的一些新结论,这里x0=1,xn+1=-1,xk（k=1,2,…,n）是n阶Jacobi多项式的零点.     

Dynamical behavior and solution of nonlinear difference equation via Fibonacci sequence

In this paper, we study the behavior of the difference equation $x_{n+1}=ax_{n}+\dfrac{bx_{n}x_{n-1}}{cx_{n-1}+dx_{n-2}},~n=0,1,\ldots,$ where the initial conditions $x_{-2},\ x_{-1},\ x_{0}$ are arbitrary positive real numbers and $a,b,c,d$ are positive constants. Also, we give the solution of some special cases of this equation.     

On the divisibility properties concerning sums of binomial coefficients

For any integer \(n> 1,\) we prove

$$\begin{aligned} 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(3k+1){2k\atopwithdelims ()k}^3(-8)^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(6k+1){2k\atopwithdelims ()k}^3(-512)^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(42k+5){2k\atopwithdelims ()k}^3 4096^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(20k^2+8k+1){2k\atopwithdelims ()k}^5(-4096)^{n-1-k}. \end{aligned}$$

The first three results confirm three divisibility properties on sums of binomial coefficients conjectured by Z.-W. Sun.

     

Oscillation Criteria for First Order Delay Difference Equations

Oscillation criteria for all solutions of the first order delay difference equation of the form where {p_n} is a sequence of nonnegative real numbers and k is a positive integer are established especially in the case that the well-known oscillation conditions are not satisfied. Dedicated to Professor Y.G. Sficas on the occasion of his 60h birthday     

Existence of Positive Solutions for N-term Non-autonomous Fractional Differential Equations

Existence of positive solutions for the nonlinear fractional differential equation D ^αu = f(x,u), 0 < α < 1 has been given (S. Zhang. J. Math. Anal. Appl. 252 (2000), 804–812) where D ^α denotes Riemann–Liouville fractional derivative. In the present work we extend this analysis for n-term non autonomous fractional differential equations. We investigate existence of positive solutions for the following initial value problem

Arithmetic properties of the values of a set of functions satisfying a system of differential equations

General results were presented in [2] and [3] concerning arithmetic properties of the values at algebraic points of a class of analytic functions satisfying linear differential equations. In the present note we consider the application of these results to the set of functions $$\begin{gathered} ^f (\alpha _k z) = \sum\nolimits_{n = 0}^\infty {\frac{{ \mu (\mu + 1)... (\mu + n - 1) }}{{\lambda (\lambda + 1)... (\lambda + n - 1)}}} (\alpha _k z)^n (k = 1,2,...,m,) \hfill \\ \lambda \ne 0, - 1, - 2,...), \hfill \\ \end{gathered}$$ where α₁, ..., α_m are algebraic numbers; λ and μ are rational numbers; and the functions satisfy a system of linear differential equations.