非线性扩散方程Cauchy问题解的唯一性和稳定性

研究方程au/at=△A(u)+Σ^N_i=1 abⁱ(u) / ax^#em/em#+c(u)Cauchy问题解的唯一性。证明了如果u∈BV_x且A(u)严格递增,则解是唯一的。     

附加广义函数的几个导数

作者证明了广义函数（ｘ±ｉ０）^λｌｎ^k（ｘ±ｉ０）的表示定理,给出了附加广义函数的导数：（ｌｎ^kｘ_±）＇,（ｘ^λ_±ｌｎ^kｘ_±）＇,（ｘ^-n_±ｌｎ^k＇ｘ_±）＇,（ｄ／ｄｘ）｛（ｘ±ｉ０）^λｌｎ^k（ｘ±ｉ０     

一类修正的KdV方程的特征问题

本文将一类修正的Kdv方程u_t+auu_x + au²u_x +μu_x³+β(1+u²)(1+u_{x^{2)(1+ut2)(1+ux32)(ux3+σux}}     

乘积的平方性

本文证明了乘积 (f(x)=ax²+bx+c∈Z[x]是二次不可约多项式)在n充分大时不是平方数.       

算子方程的解及算子张量积

本文讨论Hilbert空间上一类三阶二元算子方程组A^*AC = αA^*A² + βAA^*A;AA^*C = λA^*A² + γAA^*A,给出所有重交换的解(A,C).作为应用,得到算子张量积A(?)B+C(?)D和A₁(?)A₂(?)…(?)A_n为拟正规算子的充分必要条件.     

关于Diophantine 方程xn+2kyn=pz2

设p 为奇素数且对任意的整数m, d, p≠(2^m±1)=/d², 则对任意的素数n > p^8p2, 方程xⁿ+2^kyⁿ=pz², k≥2 没有整数解(x, y, z) 使得x, y, z 两两互素且均不为0.     

Wilson一般化定理的精确刻画 *

给定任意正整数集合K及正整数λ ,令c(K ,λ)表示最小的正整数 ,使得v∈B(K ,λ)对任意整数v≥c(K ,λ)成立 ,且满足同余关系式λv(v -1)≡ 0 (modβ(K) )和λ(v-1)≡ 0 (modα(K) ) .设K₀ 是K的等价集 ,k和k* 分别是K₀ 中最小和最大的整数 .证明了c(K ,λ)≤expexp{Q₀},这里 ,Q0 =max { 2 ( 2p(K₀) ² -k+k² log₄ k)p(K₀) ⁴,(k^k2 42^y-k-2)(^y₂) } ,p(K₀ ) =∏l∈K₀l,y =k *+k(k- 1 ) + 1 .     

具无限时滞的差分方程的一致渐近稳定性

本文考虑了如下形式的具无限时滞的差分方程x(n＋1)-x(n)＝F(n,x_n),n∈Z⁺ ,F:Z⁺×C_d(M)→R,这里C_d(M)＝(?),获得了零解－致稳定与一致渐近稳定的充分条件．     

几个非线性演化方程的解析解

本文我们求出了K—P方程u_xt+6(uu_x)_x+u_xxxx+3k²u_yy=0和Boussinesq方程u_tt-u_xt-6(u²)_xx+u_xxxx=0的孤立波解族.求出了广义Schr?dinger方程iu_t+u_xx-u相似文献   

关于P2(C)全纯曲线唯一性问题的注记*

本文讨论P²(C)中全纯曲线相交处于次一般位置超平面的唯一性.设f₁, f₂, · · · , f_λ为P²(C)中线性非退化的全纯曲线,H₁, H₁, · · · , H_q为P²(C)上处于m-次一般位置的超平面,满足A_j :f₁^-1(H_j) = · · · =f_λ^-1(H_j) (1 ≤ j ≤ q)且A_i ∩ A_j = ?(i = j).假设存在整数l (2 ≤ l ≤ λ),使得f_j1(z) ∧ f_j2(z) ∧ · · · ∧ f_jl(z) = 0 (z ∈ A_j)对任意l个指标1 ≤ j₁ < j₂ < · · · < j_l < λ成立.那么当 q > 2λ/λ-l+1 + 3/2 m时, f₁ ∧ · · · ∧ f_λ ≡ 0.关键技术是第二基本定理中不等式改进为: ∥(q - 3m/2)T_ft(r)≤ Pjq=1N²(f_t,H_j )(r, 0) + o(T_ft(r))(1 ≤ t ≤ λ).     

Resonance phenomena for asymmetric weakly nonlinear oscillator

We establish the coexistence of periodic solution and unbounded solution, the infinity of largeamplitude subharmonics for asymmetric weakly nonlinear oscillator x" + a2x+ - b2x- + h(x) = p(t) with h(±∞) - 0 and xh(x) → +∞(x →∞), assuming that M(τ ) has zeros which are all simple and M(τ ) 0respectively, where M(τ ) is a function related to the piecewise linear equation x" + a2x+ - b2x- = p(t).``     

Geometrical solutions of some quadratic equations with non-real roots

This note gives geometrical/graphical methods of finding solutions of the quadratic equation ax ² + bx + c = 0, a p 0, with non-real roots. Three different cases which give rise to non-real roots of the quadratic equation have been discussed. In case I a geometrical construction and its proof for finding the solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when a,b,c ] R, the set of real numbers, are presented. Case II deals with the geometrical solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when b ] R, the set of real numbers; and a,c ] C, the set of complex numbers. Finally, the solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when a,c ] R, the set of real numbers, and b ] C, the set of complex numbers, are presented in case III.     

A Jacobi spectral Galerkin method for the integrated forms of fourth‐order elliptic differential equations

This article analyzes the solution of the integrated forms of fourth‐order elliptic differential equations on a rectilinear domain using a spectral Galerkin method. The spatial approximation is based on Jacobi polynomials P (x), with α, β ∈ (?1, ∞) and n the polynomial degree. For α = β, one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for α = β = ?½, α = β = 0, the Chebyshev of the first and second kinds and Legendre polynomials respectively; and for the nonsymmetric Jacobi polynomials, the two important special cases α = ?β = ±½ (Chebyshev polynomials of the third and fourth kinds) are also recovered. The two‐dimensional version of the approximations is obtained by tensor products of the one‐dimensional bases. The various matrix systems resulting from these discretizations are carefully investigated, especially their condition number. An algebraic preconditioning yields a condition number of O(N), N being the polynomial degree of approximation, which is an improvement with respect to the well‐known condition number O(N⁸) of spectral methods for biharmonic elliptic operators. The numerical complexity of the solver is proportional to N^d+1 for a d‐dimensional problem. This operational count is the best one can achieve with a spectral method. The numerical results illustrate the theory and constitute a convincing argument for the feasibility of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Finite-gap solutions of the Davey-Stewartson equations

Summary We describe the finite-gap solutions of different modifications of the Davey-Stewartson (DS) equations. The restrictions on the spectral data which give us solutions of the real forms DS1 and DS2⁺ of DS are the same as those in the case of KP1 and KP2 of the Kadomtsev-Petviashvily equation. But for DS2⁻ the restrictions that we regard have no analogues in other integrable systems. We describe also the restrictions that provide regularity of those solutions for DS1 and DS2^±. The finite-gap solutions include rational and soliton solutions. We give some classes of those solutions. The well-known dromions for DS1 are examples of that kind.     

Multi-dimensional versions of a formula of Popoviciu

In this paper, an explicit formulation for multivariate truncated power functions of degree one is given firstly. Based on multivariate truncated power functions of degree one, a formulation is presented which counts the number of non-negative integer solutions of s×(s 1) linear Diophantine equations and it can be considered as a multi-dimensional versions of the formula counting the number of non-negative integer solutions of ax by = n which is given by Popoviciu in 1953.     

A polynomial <Emphasis Type="Italic">p</Emphasis>-adic dynamical system

We completely describe the Siegel discs and attractors for the p-adic dynamical system f(x) = x ²ⁿ⁺¹ + axⁿ⁺¹ on the space of complex p-adic numbers.