一类泛函微分方程解的振动定理

给出了二阶泛函数微分方程 x"(t)+f(t,x(g(t,x(t)))=0 t≥ t_0 其中 f(t,u)(?)C([t_0,∞)×R,R),f(t,0)=0 和 g(t,v)(?)C([t_0,∞)×R,R),(?)(t,v)=∞的一切解均为振动的必要条件。     

关于Sz''''asz的一个问题

1981年Sz′asz提出了如下的问题: “In which ring are the distinct subrings always non-isomorphic?” 为讨论此问题,先引入如下的 定义 一个(结合)环R,叫做内同构环(inner isomorphic ring),若R的所有真子环都是同构的。 一个(结合)环R,叫做内异环(inner non-isomorphic ring),若R的不同子环也不同构。 本文共分三节,在§§1—2中,分别给出了一个环R是内同构环和内异环的充要条件,并且也容易看它们的结构;在§3中还给出了一个有限多单环R是内异环的一个充重条件。 下面的环,都是给合环。     

关于Hermite-Fejér插值多项式算子的一点注记

本注记分二部分.在§1中我们给出了崔明根、邓中兴[1]的新近结果的多方面改进.在§2中我们讨论了扩充Hermite—Fejer插值过程,给出了有关收敛性的充要条件,改进了R.Bojanic[2]的相应结果,特别地给出了谢庭藩教授在郑州举行的第五届函数逼近论会议期间向作者提出的一个问题的一个问答.     

一个Gauss型函数方程

给出了任意两个正实数的几何-调和平均值的一个积分表示式,并由此去探讨了函数方程f(ab,2ab/a+b)=f(a,b),a,b＞0其中f:R+×R+→R是此方程的一个未知函数.     

三维Minkowski空间中的极小曲面

给出了三维Minkowski空间中类空曲面活动标价的四元素表示,并利用四元素既适合于旋转结构的侵入又适合于2×2矩阵处理极小曲面的分析特性,经研究得到了R12中的极小曲面Weierstrass表示和曲面的可积条件.     

四元整数环上的二维射影线性群的自同构

设K是有理四元数体,它含有子环 R={(a+bi+cj+dk)/2|a,b,c,d同为奇数或同为偶数}, R是个非交换欧几里得环。由[1]知道,R的乘法可逆元集合是 U={±1,±i,±j,±k,(±1±i±j±k)/2},     

一类具耗散项的拟线性方程初值问题整体光滑解的存在性

§1 序言本文考虑下述方程:这里 a>0是固定常数,σ:R→R,g:[0,+∞)×R→R,及 y_0,y_1:R→R 是给定的光滑函数,并假定:(σ):σ∈C~2(R),σ(o)=0,σ′(ξ)≥ε>0 (ξ∈R;ε>0)且有σ″(ξ)≠0.(g):g,g_x∈C([0,∞)×R),g(t)=(?)|g(t,x)|∈L~∞(0,∞)∩ L′(0,∞),     

利用调和积分核求解二维狄氏问题

本文给出调和积分核的定义,证明它与狄氏问题的几个关系,然後应用於解决具体问题. §1.有调和函数U(r,θ),它的值在半径为R的圆周上已给定为f(θ),在圆内部之值U(r,θ)由柏桑公式给出     

斜群环及其上的模

§1.定义与准备所讨论的环均为有单位元的结合环,子环均与原来环有相同的单位元,模均为右酉模。先给出群模与斜群模的定义。设R为环,G为乘群,θ:G→Aut(R)为群同态,M为R-模。     

漏斗问题

现有一张圆形的马口铁皮 ,半径为R ,若要焊制一只漏斗 ,应该怎样下料才能使做好的漏斗有最大的容量 ?图 1分析　假设做得的圆锥体形状的漏斗如图1所示 ,若圆口半径为x ,高为h ,因为它是半径为R的铁皮制成的 ,故h =R2 -x2 ,所以圆锥的体积v =13 πx2 h =13 πx2 R2 -x2 .所以v2 =π32 ·x4 ·(R2 -x2 ) =π218·x2 ·x2 ·(2R2 -2x2 ) .又由于x2 +x2 + (2R2 -2x2 ) =2R2 为定值 ,运用平均不等式知x2 =2R2 -2x2 即x =63 R时 ,漏斗有最大的容量 ,于是角α的弧度为α=2πR -2πxR =2πR(R -63 R) =2π3 (3 -6) ,该角为 66°4′.当截去中心…     

Navier-Stokes方程与Euler方程的稳定性比较

比较了Navier-Stokes 方程和Euler方程的稳定性;并以它们的典型初值问题为例,分析了Navier-Stokes方程和Euler方程稳定性不同的原因．     

Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations

We obtain new gauge-invariant forms of two-dimensional integrable systems of nonlinear equations: the Sawada-Kotera and Kaup-Kuperschmidt system, the generalized system of dispersive long waves, and the Nizhnik-Veselov-Novikov system. We show how these forms imply both new and well-known twodimensional integrable nonlinear equations: the Sawada-Kotera equation, Kaup-Kuperschmidt equation, dispersive long-wave system, Nizhnik-Veselov-Novikov equation, and modified Nizhnik-Veselov-Novikov equation. We consider Miura-type transformations between nonlinear equations in different gauges. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 160, No. 1, pp. 35–48, July, 2009.     

Validity of the KdV equation for the modulation of periodic traveling waves in the NLS equation

Bethuel et al.  and  and Chiron and Rousset [3] gave very nice proofs of the fact that slow modulations in time and space of periodic wave trains of the NLS equation can approximately be described via solutions of the KdV equation associated with the wave train. Here we give a much shorter proof of a slightly weaker result avoiding the very detailed and fine analysis of ,  and . Our error estimates are based on a suitable choice of polar coordinates, a Cauchy–Kowalevskaya-like method, and energy estimates.     

Solutions of the three-dimensional sine-Gordon equation

We obtain exact solutions U(x, y, z, t) of the three-dimensional sine-Gordon equation in a form that Lamb previously proposed for integrating the two-dimensional sine-Gordon equation. The three-dimensional solutions depend on arbitrary functions F(α) and ϕ(α,β), whose arguments are some functions α(x, y, z, t) and β(x, y, z, t). The ansatzes must satisfy certain equations. These are an algebraic system of equations in the case of one ansatz. In the case of two ansatzes, the system of algebraic equations is supplemented by first-order ordinary differential equations. The resulting solutions U(x, y, z, t) have an important property, namely, the superposition principle holds for the function tan(U/4). The suggested approach can be used to solve the generalized sine-Gordon equation, which, in contrast to the classical equation, additionally involves first-order partial derivatives with respect to the variables x, y, z, and t, and also to integrate the sinh-Gordon equation. This approach admits a natural generalization to the case of integration of the abovementioned types of equations in a space with any number of dimensions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 370–377, March, 2009.     

The stability of Cauchy's gamma-beta functional equation

We obtain the super stability of Cauchy's gamma-beta functional equation

B(x,y)f(x+y)=f(x)f(y),     

Modified method of simplest equation and its application to nonlinear PDEs

We search for traveling-wave solutions of the class of PDEswhere A_p(Q),B_r(Q),C_s(Q),D_u(Q) and F(Q) are polynomials of Q. The basis of the investigation is a modification of the method of simplest equation. The equations of Bernoulli, Riccati and the extended tanh-function equation are used as simplest equations. The obtained general results are illustrated by obtaining exact solutions of versions of the generalized Kuramoto-Sivashinsky equation, reaction-diffusion equation with density-dependent diffusion, and the reaction-telegraph equation.     

Symbolic computation of solutions for a forced Burgers equation

In this paper we give exact solutions for a forced Burgers equation. We make use of the generalized Cole-Hopf transformation and the traveling wave method.     

On reduction of some classes of partial differential equations to equations with fewer variables and exact solutions

We establish a connection between the fundamental solutions to some classes of linear nonstationary partial differential equations and the fundamental solutions to other nonstationary equations with fewer variables. In particular, reduction enables us to obtain exact formulas for the fundamental solutions of some spatial nonstationary equations of mathematical physics (for example, the Kadomtsev-Petviashvili equation, the Kelvin-Voigt equation, etc.) from the available fundamental solutions to one-dimensional stationary equations.