三组谱的Sturm-Liouville反问题

考虑与三组谱关联的逆Sturm-Liouville问题,证明了若对于给定的两组数列,在一定条件下,可划分为三组数列,使其分别为区间[0,a]上三个Sturm-Liouville问题的部分特征值,则通过三组谱的部分特征值能唯一确定区间[0,a]上的势函数q(x).     

参数边界条件下奇型Sturm-Liouville算子的半逆问题

Sturm-Liouville算子的半逆问题讨论由一组谱和半区间上势函数唯一确定整个区间上势函数q(x).本文利用Koyunbakan和Panakhov的方法和[13]的结论,讨论(0,π)上的奇型Sturm-Liouville问题满足-y″+[q(x)-1/4sin2x]y=λy,参数边界条件y(0,λ)=0或y′(0,λ)-hy(0,λ)=0和y′(π,λ)+(aλ+b)y(π,λ)=0,证明一组谱和(π/2,π)上的势函数q(x)唯一确定(0,π)上的势函数q(x).     

Sturm-Liouville问题的特征值对势函数的依赖性

研究了定义在[0,1]上的Sturm-Liouville问题的特征值对势函数的连续依赖性.应用比较定理和定义区间单调性证明了:当部分区间[x0,1]上的势函数趋于无穷大时,[0,1]区间上的特征值渐进趋近于[0,x0]区间上的某个特征值.推广了一些作者对Sturm-Liouville问题研究的相应结果,并为其相应问题的研究提供了一个新的视角.     

关于缺失有限个特征值的逆Sturm-Liouville问题

本文考虑由三组谱确定势函数和边值条件的问题,即证明Sturm-Liouville问题的势函数可以由[0,1]区间上的一组整谱和[0,α],[α,1](0<α<1)区间上的两组部分谱唯一确定.特别地,在这两个区间内,分别可以缺失任意-个特征值,势函数仍可以被唯一确定.     

扩散方程系数的半逆问题

一般地,扩散方程的系数q(x)与p(x)是由两组谱或者一组谱及其标准常数唯一确定的.运用Hochstadt与Lieberman的方法证明了:(a)如果给定区间[π/2,π]上的p(x)及区间[0,π]上的q(x),则扩散方程的一组谱可唯一确定另一半区间[0,π/2]上系数p(x);(b)如果给定区间[π/2,π]上的q(x)及区间[0,π]上的p(x),则扩散方程的一组谱可唯一确定另一半区间[0,π/2]上系数q(x).     

扩散方程系数的半逆问题水

一般地,扩散方程的系数q(x)与p(x)是由两组谱或者一组谱及其标准常数唯一确定的.运用Hochstadt与Lieberman的方法证明了:(a)如果给定区间[π/2,π]上的p(x)及区阿[0,π]上的q(x),则扩散方程的一组谱可唯一确定另一半区间[0,π/2]上系数p(x);(b)如果给定区间[π/2,π]上的g(x)及区间[0,π]上的p(x),则扩散方程的一组谱可唯一确定另一半区间[0,π/2]上系数q(x).     

奇型Sturm-Liouville算子的半逆问题

奇型Sturm-Liouville问题的势函数q(x)由两组谱唯一确定的,Sturm-Liouville算子的半逆问题讨论由一组谱和一半势函数唯一确定势函数q(x).本文利用Koyunbakan和Panakhov的方法,证明一组谱和(π/2,π)上的势函数q(x)唯一确定(0,π)上Sturm-Liouville方程含有奇型的1sin2x的势函数q(x).     

参数边界条件下Sturm-Liouville算子的逆谱问题

本文研究参数边界条件下Sturm-Liouville算子的逆谱问题.利用Weyl函数的结果,证明对固定的n,n∈N_0,及不同的b_k,谱集合{λ_n(q,b_k)}_(k=1)~(+∞)能够唯一确定[0,1]上的势函数q(x),这个定理是文献[4]结果的本质推广.     

二阶线性方程边值问题解的变号性质

一、引言 我们研究的是边值问题的解在[0,1]上的变号次数。其中q(x),f(x)。在[0,1]上连续,a,a′β,β′保证(1),(2)的解存在[1]。主要结果是当q(x)≤π~2-d,d>0时。(1),(2)的解u(x)在[0,1]上变号的次数最多不超过f(x)在[0,1]上变号的次数加两次变号,即线性系统因外力引起的响应函数的震荡次数不超过外力振荡次数与固有振荡次数之和。     

边界条件含有特征参数的Sturm-Liouville算子的唯一性定理

建立了一类Sturm-Liouville问题的唯一性定理.对于固定的n∈Z,证明了该Sturm-Liouville问题的第n个特征值λ_n（q,a）关于a是严格单调的.对不同系数的a_k,如果能够测得第n个特征值的谱集合{λ_n（q,a_k）}_k=1^+∞,则谱集合{λ_n（q,a_k）}_k=1^+∞能够唯一确定[0,π]上的势函数q（x）.     

THE CONSTRUCTION OF DENUMERABLE q-PROCESSES IN RANDOM ENVIRONMENTS SATISFYING (F) OR (B)

This article is a continuation of [9]. Based on the discussion of random Kol-mogorov forward (backward) equations, for any given q-matrix in random environment,Q(θ) = (q(θ; x, y), x, y ∈ X), an infinite class of q-processes in random environments sat-isfying the random Kolmogorov forward (backward) equation is constructed. Moreover,under some conditions, all the q-processes in random environments satisfying the random Kolmogorov forward (backward) equation are constructed.     

THE PERIODIC HARRY-DYM EQUATION

The development of the inverse scattering transform(I.S.T)has made it possible tosolve certain physically significant nonlinear evolution equations with periodic boundaryconditions.Date and Tanaka have considered kdv equation;Ma and Ablowitz havediscussed the cubic Schrodinger equation.In this paper,following closely the analysis in[2,3]the author considers Harry-Dym eqution(q~2)_t=-2r_(xxx)(Ⅰ)where q(x,t)is periodic in x with period π for all time q(x,t)=q(x π,t),q(x,t)=r~(-1)(x,t)>0     

Recursion Operators for a Class of Integrable Third-Order Evolution Equations

We consider   u_t = u ^α u_xxx + n ( u ) u_xu_xx + m ( u ) u ³ _x + r ( u ) u_xx + p ( u ) u ² _x + q ( u ) u_x + s ( u )  with  α= 0  and  α= 3  , for those functional forms of   m , n , p , q , r , s   for which the equation is integrable in the sense of an infinite number of Lie-Bäcklund symmetries. Recursion operators which are x - and t -independent that generate these infinite sets of (local) symmetries are obtained for the equations. A combination of potential forms, hodograph transformations, and x -generalized hodograph transformations are applied to the obtained equations.     

A note on sphere containment orders

Using the ideas of Scheinerman and Wierman [1] and of Hurlbert [2] we give a very short proof that the infinite order [2]×[3]× cannot be represented by containment of Euclidean balls in ad-dimensional space for anyd. Also we give representations of the orders [2]×[2]× and [3]×[3]×[3] by containment of circles in the plane.The work was financially supported by the Russian Foundation of Fundamental Research, Grant 93-011-1486     

一类三次Pisot数的一些性质

本文研究了当q〉1为三次Pisot数，利用递归的方法构造一个无穷序列，通过对此序列，得到mR[q]∩Z[q]与此序列间和mR^-[q]与mR[q]∩Z[q]之间的一些关系．     

积分中值定理中间点比较及有关平均不等式

中值定理中间点是区间端点的平均.设f (x)、g(x)在同一区间[a,b]内严格单调并可积,p(x)、q(x)恒正可积,按积分中值定理各有唯一的中间点ξf ,p(a,b)和ξg,q(a,b) .当f递增(减)且f (g- 1)凸(凹)时,有ξg,p(a,b) <ξf,p(a,b) ;当p(x)q(x) 递增(减)且q(x) ∫bap(x) dx >( <) 0时,有ξf,q(a,b) <ξf ,p(a,b) .由此可证明和发现一系列有关平均的不等式.     

Absolute and Convective Instability for Evolution PDEs on the Half-Line

We analyze evolution PDEs exhibiting absolute (temporal) as well as convective (spatial) instability. Let  ω( k )  be the associated symbol, i.e., let  exp[ ikx −ω( k ) t ]  be a solution of the PDE. We first study the problem on the infinite line with an arbitrary initial condition   q ₀( x )  , where   q ₀( x )  decays as  | x | → ∞  . By making use of a certain transformation in the complex k -plane, which leaves  ω( k )  invariant, we show that this problem can be analyzed in an elementary manner. We then study the problem on the half-line, a problem physically more realistic but mathematically more difficult. By making use of the above transformation, as well as by employing a general method recently introduced for the solution of initial-boundary value problems, we show that this problem can also be analyzed in a straightforward manner. The analysis is presented for a general PDE and is illustrated for two physically significant evolution PDEs with spatial derivatives up to second order and up to fourth order, respectively. The second-order equation is a linearized Ginzburg–Landau equation arising in Rayleigh–Bénard convection and in the stability of plane Poiseuille flow, while the fourth-order equation is a linearized Kuramoto–Sivashinsky equation, which includes dispersion and which models among other applications, interfacial phenomena in multifluid flows.     

Exact expansions of Hankel transforms and related integrals

The Ramanujan Journal - The Hankel transform $$\mathcal {H}_n [f(x)](q)=\int _0^{\infty } \!\! \, x f(x) J_n(q x) \mathrm{d}x$$ is studied for integer $$n\geqslant -1$$ and positive parameter q. It...     

Two theorems on boundary properties of minimal surfaces in nonparametric form

Let D be a region with rectifiable Jordan boundary , and let z=f(x, y) be a minimal surface defined over D. This paper establishes that: 1) function z=f(x, y) almost everywhere on has finite or infinite angular boundary values; 2) if region D is the exterior of a circle then, almost everywhere on boundary , function z=f(x, y) can be continued by continuity.Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 551–556, April, 1977.     

一阶中立型时滞微分方程的振动性

分别获得了中立型时滞微分方程「ｘ（ｔ）＋ｐｘ（ｔ－γ」’＋ｑ（ｔ）ｘ（ｔ－σ）＝０的振动性和非振动性的充分条件,其中ｐ〉０,ｑ（ｔ）是一个正的周期函数。