与三个数论函数有关的两个方程的解

讨论了与数论函数Euler函数φ(m),函数Ω(m)和函数ω(m)相关联的两个方程φ(m)=2~(Ω(m)+ω(m))3~(Ω(m)+ω(m))与φ(m)=2~(Ω(m)-ω(m))3~(Ω(m)-ω(m))的可解性,利用这三个数论函数的相关性质以及初等方法,给出这两个方程的解,其中函数Ω(m)为m的质因子个数函数,函数ω(m)为m的相异质因子个数函数.     

一个包含Euler函数的方程

对任意自然数n≥1,著名的Euler函数ψ(n)定义为不大于n且与n互素的正整数的个数.本文的主要目的是研究方程ψ(ψ(ψ(n)))=2ω(n)的可解性,其中ω(n)表示n的所有不同素因子的个数,并给出了该方程的所有正整数解.     

一道调研试题解法的思考

(2011年1月襄阳市高中调研考试理科第21题)已知函数f(x)=px-p/x-2lnx.(1)若p=2,求曲线f(x)在点(1,f(1))处的切线方程;(2)若函数f(x)在其定义域内为增函数,求     

齐型空间中分数次积分交换子的加权端点估计

该文得到齐型空间中分数次积分交换子[b,I_α]的加权端点估计ω({x∈X:|[b,I_α]f(x)|t})≤Cψ(∫_xA(||b||_*(|f(x)|/t)■(ω(x))dμ(x))其中b∈BMO(X,d,μ),A(t)=tlog(e+t),ψ(t)=[tlog(e+t~α)]~(1/(1-α)),■(t)=t~(1-α)log(e+t~(-α)).     

有限域上的绝对迹函数和原根

设p是一个素数,m>0整数,GF(p~m)是有p~m个元素的有限域,T(x)=x+x~p+…+x~(p~(m-1))为GF(p~m)上的绝对迹。R_(p~m)(h)表示T(x)=h(h∈GF(p))在GF(p~m)中的本原根解数。本文证明了如下结果: (ⅰ)R_(p~m)(o)≥φ(p~m—1)/p(p~m—1){p~m—(2~(ω(p-1/p-1))-1)(p-1)p~m(1/2)-p}, (ⅱ)R_(p~m)(h)≥φ(p~m-1)/p(p~m-1){p~m-(2~(ω(p-1/p-1))-1)p~m(1/2)-(2~(ω(p-1))-2~(ω(p-1/p-1))p~(m+1)},其中h≠0,ω(n)表示n的不同素因子个数,φ(n)表示通常的Euler函数。 (ⅲ)当m≥3时,对任给的p和h≠0,h∈GF(p)。除p~m=11~3外,总有R_(p~m)(h)>0。     

有关数论函数φ(m)的一方程整数解的讨论

令φ(m)是Euler函数,其中m是一正整数.讨论包含Euler函数φ(m)的方程φ(xyz)=7(φ(x)+φ(y)+φ(z))的可解性,利用初等的方法以及Euler函数φ(m)的有关性质,给出了该方程的全部的87组正整数解.     

用二次曲线定义解无理方程

解无理方程的常用方法是使方程有理化,但对于一些特殊的无理方程,如果盲目乘方,往往会招致繁琐的运算。这就需要根据题中的一些特殊条件,采用特殊的解法。而利用二次曲线的定义,将无理方程转化为二次曲线的标准方程是值得注意的解题方法,现举几例介绍如下: 例1 解方程 x~2-10(3~(1/2))x+80+(1/2)x~2+10(3~(1/2))x+80=20 解:原方程可化为:(x-5(3~(1/2))~2+5~(1/2)+(x+5(3~(1/2)))~2+5~(1/2)=20令y~2=5,则原方程为:(x-5(3~(1/2))~2+y~2)~(1/2)+(x+5(3~(1/2))~2+y~2~(1/2)=20。此方程表示动点P(x,y)到两定点(5(3~(1/2)),0)、(-5(3~(1/2)),0)的距离之和为20,故它表示椭圆。     

与Euler函数φ(n)有关的两个方程

设φ(n)是Euler函数.主要研究了方程φ(xy)=k(φ(x)+φ(y))的可解性问题,其中k=4,6,利用初等的方法给出了这2个方程的所有正整数解.     

有关Euler函数(n)的方程的正整数解

设(n)是Euler函数.主要研究了方程(xy)=3((x)+(y))的可解性问题,利用初等的方法给出了这一方程的所有的35组正整数解.对于任意素数k>3,(x,y)=(3k,4k),(4k,3k)是方程(xy)=k((x)+(y))的2个正整数解.证明了更为一般的结论:对于任意奇数k>3,当gcd(k,3)=1时,(x,y)=(3k,4k),(4k,3k)是方程(xy)=k((x)+(y))的2个正整数解.     

解方程(2+(2+(2+X)~(1/2))~(1/2))~(1/2)=2所想到的

探求方程(2+(2+(2+x)~(1/2))~(1/2))~(1/2)=2殊解法时,联想到方程(a±(a±…±(a±x)~(1/2))~(1/2))~(1/2)=x与力程x=(a±x)~(1/2)是否等价的问题。如果结论成立,则x=1/2((4a+1)~(1/2))±1)就是方程 (a±(a±…±(a±x)~(1/2))~(1/2))~(1/2)=x的根,这样不仅可使这种形式的方程有了较为简捷的求解公式,而且也为形如(a±(a±…±(a±x)~(1/2))~(1/2))~(1/2)=b的方程提供了一种极为简便的解法。事实上,若a>0,x>0,则     

Qualitative Analysis of Bifurcating Solutions in the Lengyel-Epstein Model

One of the most fundamental problems in theoretical biology is to explain the mechanisms by which patterns and forms are created in the＇living world. In his seminal paper ＂The Chemical Basis of Morphogenesis＂, Turing showed that a system of coupled reaction-diffusion equations can be used to describe patterns and forms in biological systems. However, the first experimental evidence to the Turing patterns was observed by De Kepper and her associates（1990） on the CIMA reaction in an open unstirred reactor, almost 40 years after Turing＇s prediction. Lengyel and Epstein characterized this famous experiment using a system of reaction-diffusion equations. The Lengyel-Epstein model is in the form as follows     

A REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.     

(0,1 ;0)-INTERPOLATION ON INFINITE INTERVAL (-∞, +∞)

In this paper, we study the explicit representation and convergence of (0, 1; 0)-interpolation on infinite interval, which means to determine a polynomial of degree ≤ 3n - 2 when the function values are prescribed at two set of points namely the zeros of Hn(x) and H′n(x) and the first derivatives at the zeros of H′n(x).     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

CONTENTS OF VOLUME 30

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Instructions for contributions

正Applied Mathematics-A Journal of Chinese Universities,Series B(Appl.Math.J.Chinese Univ.,Ser.B)is a comprehensive applied mathematics journal jointly sponsored by Zhejiang University,China Society for Industrial and Applied Mathematics,and Springer-Verlag.It is a quarterly journal with     

Journal of Mathematical Research with Applications Guide for Authors

正Journal overview:Journal of Mathematical Research with Applications(JMRA),formerly Journal of Mathematical Research and Exposition(JMRE)created in 1981,one of the transactions of China Society for Industrial and Applied Mathematics,is a home for original research papers of the highest quality in all areas of mathematics with applications.The target audience comprises:pure and applied mathematicians,graduate students in broad fields of sciences and technology,scientists and engineers interested in mathematics.     

Staircase transportation problems with superadditive rewards and cumulative capacities

A cumulative-capacitated transportation problem is studied. The supply nodes and demand nodes are each chains. Shipments from a supply node to a demand node are possible only if the pair lies in a sublattice, or equivalently, in a staircase disjoint union of rectangles, of the product of the two chains. There are (lattice) superadditive upper bounds on the cumulative flows in all leading subrectangles of each rectangle. It is shown that there is a greatest cumulative flow formed by the natural generalization of the South-West Corner Rule that respects cumulative-flow capacities; it has maximum reward when the rewards are (lattice) superadditive; it is integer if the supplies, demands and capacities are integer; and it can be calculated myopically in linear time. The result is specialized to earlier work of Hoeffding (1940), Fréchet (1951), Lorentz (1953), Hoffman (1963) and Barnes and Hoffman (1985). Applications are given to extreme constrained bivariate distributions, optimal distribution with limited one-way product substitution and, generalizing results of Derman and Klein (1958), optimal sales with age-dependent rewards and capacities.To our friend, Philip Wolfe, with admiration and affection, on the occasion of his 65th birthday.Research was supported respectively by the IBM T.J. Watson and IBM Almaden Research Centers and is a minor revision of the IBM Research Report [6].