On the Decay of Solution of Some Nonlinear Hyperbolic Equation

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＲｅｓｕｌｔＩｎｔｈｉｓａｒｔｉｃｌｅｗｅａｒｅｃｏｎｃｅｒｎｅｄｗｉｔｈｔｈｅｄｅｃａｙｏｆｇｌｏｂａｌｓｏｌｕｔｉｏｎｏｆｔｈｅｉｎｉｔｉａｌ－ｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｆｏｒｔｈｅｆｏｌｌｏｗｉｎｇｎｏｎｌｉｎｅａｒｈｙｐｅｒｂｏｌｉｃｅｑｕａｔｉｏｎｕｔｔ＋Ａｕ＋｜ｕｔ｜αｕｔ＝ｆ（ｘ,ｔ）　　　　　ｉｎΩ×Ｒ＋,（１）ｕ（ｘ,０）＝ｕ０（ｘ）,ｕｔ（ｘ,０）＝ｕ１（ｘ）　　ｘ∈Ω,（２）ｕ（ｘ,ｔ）＝０　　　　　　　　　　　　（ｘ,ｔ…     

Boundedness of the Maximal Operator on Lipα（R^n）（0〈α〈1）

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＡｌｏｃａｌｌｙｉｎｔｅｇｒａｂｌｅｆｕｎｃｔｉｏｎｆ（ｘ）ｂｅｌｏｎｇｓｔｏＬｉｐα（Ｒｎ）,ｉｆｔｈｅｒｅｉｓａｃｏｎｓｔａｎｔＣ,ｓｕｃｈｔｈａｔｆｏｒｅｖｅｒｙｘ,ｙ∈Ｒｎ｜ｆ（ｘ）－ｆ（ｙ）｜≤Ｃ｜ｘ－ｙ｜α　　ＴｈｅｓｍａｌｌｅｓｔｃｏｎｓｔａｎｔＣｓａｔｉｓｆｉｅｓａｂｏｖｅｉｓｃａｌｌｅｄＬｉｐｓｃｈｉｔｚｎｏｒｍｏｆｆａｎｄｉｓｄｅｎｏｔｅｄｂｙｙｆｙ∧α．Ｂｙ［１］,ｆ∈Ｌｉｐα（Ｒｎ）ｅｑｕｉｖａｌｅｎｔｔｏｆ∈εα,２,ｗｈｅｒｅεα,２＝…     

一类拟线性椭圆型偏微分方程的先验界的估计

近几年对边值问题－ｄｉｖ（｜Ｄｕ｜^p-2Ｄｕ）＝λｆ（ｕ）｝在Ω上ｕ｜(?)Ω＝０正解方面已经得到了许多结果．这里λ＞０，Ω是有界区域和对ｓ≥０，ｆ（ｓ）≥０．在本文中在条件Ｎ≥ｐ＞１，Ω＝Ｂ_１＝｛ｘ∈Ｒ^N，｜ｘ｜＜１｝和ｆ∈Ｃ¹（０，∞）∩Ｃ⁰（［０，∞）），ｆ（０）＝０，研究了这类问题的正对称解的先验界估计．     

具临界指数和临界非线性边界条件的拟线性椭圆型方程Neumann问题

本文给出ＲＮ（Ｎ３）中有界光滑区域Ω上的拟线性椭圆型方程：－∑Ｎｉ＝１ｘｉ·｜Ｄｕ｜ｐ－２ｕｘｉ＝λ｜ｕ｜ｐ－２ｕ＋ａ（ｘ）｜ｕ｜ｐ－２ｕ＋ｆ（ｘ，ｕ），ｘ∈Ω（λ＞０，ｐ＝Ｎｐ／（Ｎ－ｐ），２ｐ＜Ｎ）在边界条件：－｜Ｄｕ｜ｐ－２Ｄνｕ｜Ω＝ψ（ｘ）｜ｕ｜ｑ－２ｕ（ｑ＝（Ｎ－１）ｐ／（Ｎ－ｐ））下的多解性结果．     

Asymptotic Behavior of Nodal Solutions for Semilinear Elliptic Equationsin R~n

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＲｅｃｅｎｔｌｙ，ｔｈｅｓｅｍｉｌｉｎｅａｒｅｌｌｉｐｔｉｃｅｑｕａｔｉｏｎｓ△ｕ＋ｆ（ｕ）＝０（１．１）ｕ（ｘ）→０ａｓ｜ｘ｜→∞（１．２）ｉｎＲｎｗｅｒｅｃｏｎｓｉｄｅｒｅｄｗｉｄｅｌｙ（ｓｅｅ［１］－［７］）．Ｉｎｔｈｅｎｉｃｅｐａｐｅｒｓ［１］ａｎｄ［２］，ｉｔｗａｓｐｒｏｖｅｄｔｈａｔａｎｙｐｏｓｉｔｉｖｅｓｏｌｕｔｉｏｎｏｆ（１．１）ｍｕｓｔｂｅｒａｄｉａｌｉｎｃａｓｅｆ（ｕ）∈Ｃ１＋δ（δ＞０）．Ｔｈｅｒｅｆｏｒｅ，ａｎｙｐｏｓｉｔｉｖｅｓｏｌｕｔｉ…     

一道不等式题的简捷证明

题　已知二次函数ｆ（ｘ）＝ａｘ２＋ｂｘ＋ｃ,当－１≤ｘ≤１时,有－１≤ｆ（ｘ）≤１．求证：当－２≤ｘ≤２时,有　－７≤ｆ（ｘ）≤７．这是文［１］例３,原给出的证明较繁,现简证如下．证明　∵　ｆ（１）＝ａ＋ｂ＋ｃ,ｆ（０）＝ｃ,ｆ（－１）＝ａ－ｂ＋ｃ,∴　２ａ＝ｆ（１）＋ｆ（－１）－２ｆ（０）,∴　｜２ａ｜≤｜ｆ（１）｜＋｜ｆ（－１）｜＋２｜ｆ（０）｜≤１＋１＋２＝４,且　｜ｃ｜＝｜ｆ（０）｜≤１．若ｘ∈［－２,２］,则　ｘ′＝ｘ２∈［－１,１］,于是可得　｜ｆ（ｘ）｜＝｜ｆ（２ｘ′）｜＝｜２ｆ（…     

一类椭圆方程很弱解关于区域的连续性

本文讨论了二阶椭圆型方程－Δｕ＝ｆ（ｘ,ｕ）,ｘ∈Ω的Ｄｉｒｉｃｈｌｅｔ问题ｕ｜Ω＝０的很弱解ｕ∈Ｗ０１,ｒ（Ω）（１＜ｒ＜２）关于区域Ω的连续性及很弱边值问题的很弱解的唯一性．     

奇阶中立型时滞微分方程解的振动性

１．引言考虑奇阶非线性泛函微分方程［ｘ（ｔ）－ｃｘ（ｔ—(）］（ｎ）＋ｐ（ｔ）ｆ（ｘ（ｔ－σ））＝０（１）对方程（１）我们作如下假设（Ｈ）：（Ｈ１）ｎ＞１是奇整数,ｐ∈Ｃ（（ｔ０,∞）,（ｔ０,∞））;（Ｈ２）τ＞０,σ＞０且０≤ｃ≤１;（Ｈ３）ｆ∈Ｃ（Ｒ,Ｒ）是单调增加,ｘｆ（ｘ）＞０,Ｘ≠０且当｜ｘ｜→∞时有｜ｆ（ｔ）｜→∞．设δ＝ｍａｘ｛τ,σ｝,∈Ｃ（［Ｔ－δ,Ｔ］,Ｒ）．方程（１）在［Ｔ,∞）上的解是指函数ｘ∈Ｃ（［Ｔ,∞）,Ｒ）,使得ｘ（ｔ）＝(（ｔ）,Ｔ－δ≤ｔ≤Ｔ,［ｘ（ｔ）－ｃｘ…     

一类具有双重奇性的抛物型方程的整体解和有限熄灭

本文讨论具有双重奇性的抛物型方程ｕｔ＝ ｄｉｖ（｜△ｕ~ａ｜ｐ~－２△ｕ~ａ）,（ｘ,ｔ） ∈ Ｒ~ｎ ×（０,∞）,其中Ｐ＞ １,ａ＞ ０,ｎ≤ ２．证明当１＜ Ｐ＜ｎ（ａ＋１）／（ａｎ＋１）时,存在整体自相似解ｕｇｓ（·,ｔ） ∈ Ｌ~ｑ（Ｒ~ｎ）（ｑ＞ｓ＝~△ｎ［１－ａ（ｐ－１）］／ｐ）,但是ｕｇｓ∈~／Ｌ~ｓ（Ｒ~ｎ）（定理２．１）;同时存在有限熄灭的自相似解ｕｌｓ满足相同的积分条件（定理 ３．１）．     

关于迭代函数方程f~2(x)=af(x) bx的通解

设λ的二次三项式λ２－ａλ－ｂ的两个零点为λ１＝ｒ，λ２＝ｓ（ａ及ｂ为实数）．对０＜ｒ＜ｓ，ｒ＜０＜ｓ≠－ｒ及ｒ＝ｓ≠０这三种情形，Ｊ．Ｍａｔｋｏｗｓｋｉ与ＷｅｉｎｉａｎＺｈａｎｇ在“Ｍｅｔｈｏｄｏｆｃｈａｒａｃｔｅｒｉｓｔｉｃｓｆｏｒｆｕｎｃｔｉｏｎａｌｅｑｕａｔｉｏｎｓｉｎｐｏｌｙｎｏｍｉａｌｆｏｒｍ”一文中给出了迭代函数方程ｆ２（ｘ）＝ａｆ（ｘ）＋ｂｘ，对任ｘ∈Ｒ；ｆ∈Ｃ０（Ｒ，Ｒ）（１）的通解，并证明了当ｒ及ｓ非实数时方程（１）无解．对ｒ＝－ｓ≠０的情形，Ｍ．Ｋｕｃｚｍａ已给出了方程（１）的通解．本文则对ｒ＜ｓ＜０及ｒｓ＝０这两种情形给出了方程（１）的通解．此外，本文还给出了ｒ＜０＜ｓ≠－ｒ时关于方程（１）的通解的一个简洁的证明     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP ——In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

A unified approach to certain problems of best local approximation

In this paper we study best local quasi-rational approximation and best local approximation from finite dimensional subspaces of vectorial functions of several variables. Our approach extends and unifies several problems concerning best local multi-point approximation in different norms.     

Boundedness for commutators of multipliers on Hardy type spaces

In this paper, we study the commutators generalized by multipliers and a BMO function. Under some assumptions, we establish its boundedness properties from certain atomic Hardy space Hb^p（R^n） into the Lebesgue space L^p with p 〈 1.     

Towards Excellence in Science Chinese Academy of Sciences

<正>May 26,2014,Beijing Science is a human enterprise in the pursuit of knowledge.The scientific revolution that occurred in the 17th Century initiated the advances of modern science.The scientific knowledge system created by     

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

<正>August 10-14,2015Beijing,ChinaThe 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     

Bounds for the zeros of polynomials

Let P(z)=∑↓j=0↑n ajx^j be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrǒm-Kakeya theorem.     

Boundedness of commutators related to Marcinkiewicz integrals on hardy type spaces

In this paper, the authors study the boundedness of the operator [μΩ, b], the commutator generated by a function b ∈ Lipβ(Rn)(0 ＜β≤ 1) and the Marcinkiewicz integrals μΩ, on the classical Hardy spaces and the Herz-type Hardy spaces in the case Ω∈ Lipα(Sn-1)(0 ＜α≤ 1).     

Jacobi polynomials used to invert the laplace transform

Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.     

Generalization of the interaction between Haar approximation and polynomial operators to higher order methods

In applications it is useful to compute the local average empirical statistics on u. A very simple relation exists when of a function f（u） of an input u from the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation.