方程f(x)=P-x与f-1(x)=P-x的两根之和一定为P吗

常见题目:①设方程10x=p-x的根为x1,方程lgx=p-x的根为x2,则x1 x2=p;②设方程x3=p-x的根为x1,方程3x=p-x的根为x2,则x1 x2=p.可以用数形结合法或函数的单调性证明,此略.我们类比猜想:方程f(x)=p-x与f-1(x)=p-x的两根之和一定为p(p为实常数)吗?经过探究发现,此结论不一定成立.一     

一类两点边值问题的多重非负解

该文考虑两点边值问题[1/q(t)][q(t)y′(t)]′+p(t)f(y(t))= 0,λ_1 y(α)-λ_2y′(α)=0 and y(β)=B非负解的存在性, 其中p(t)可能在t=α或t=β附近具有奇异性, f(0)≥0, lim_(y→+∞)f(y)/y=+∞, 并且存在y>0, 使得f(y)<0.      

随机变量序列的大数定律及常返性定理

对一类有界独立或相依的随机变量序列|ξn|,获得了它的伯努利大数定律、波雷尔强大数定律及常返性定理.作为应用,得出了Loève专著[1]中的推广的伯努利大数定律、常返性定理,改进了[1]中的推广的波雷尔强大数定律.     

非线性椭圆方程自由边值问题球对称解的存在性

给出了如下的非线性椭圆方程自由边值问题-Δu=λu+(1+ε)u+p,x∈B Rn,u|Ω=μ,∫Ωnu=-M(1)在C[0,1]中的球对称解的存在性.并得到比上述问题更一般的非线性椭圆方程自由边值问题-Δu=h(u),x∈B Rn,u|Ω=μ,∫Ωun=-M,在C[0,1]中的球对称解的存在性,其中B为Rn中的单位球,p>1,λ>0,μ<0,M>0,ε>0;λ,μ,M,ε均为常数,n为正整数.     

脉冲微分方程的上、下解及其在常微分方程上的应用

In this paper, we establish the existence of upper and lower solutions for a periodic boundary value problems (PBVP for short) of impulsive differential equations. which guarantees the existence of at least one solution for the problem. As an application, these results are applied to PBVP of ODE and some examples are given to illustrate our results.     

拟线性椭圆变分不等式的障碍最优控制问题

对拟线性椭圆变分不等式的障碍最优控制问题(即以障碍为控制变量)进行了研究．指标泛函为Lagrange型,其中含有控制变量二阶导数的p次幂,这使得最优性条件的推导颇为不易．对所考虑的问题给出了最优控制的存在性定理以及必要条件．     

三阶泛函微分方程的周期解的存在性

本文利用重合度理论研究了一类三阶泛函微分方程方程的2π-周期解的问题,获得了该方程存在2π-周期解的若干新结论,改进推广了有关文献中的已有结果．     

QUATERNION GENERALIZED SINGULAR VALUE DECOMPOSITION AND ITS APPLICATIONS

This paper derives a theorem of generalized singular value decomposition of quaternion matrices （QGSVD）,studies the solution of general quaternion matrix equation AXB -CYD= E,and obtains quaternionic Roth＇s theorem. This paper also suggests sufficient and necessary conditions for the existence and uniqueness of solutions and explicit forms of the solutions of the equation.     

Mackey Theory for p-adic Lie Groups

This paper gives a p-adic analogue of the Mackey theory, which relates representations of a group of type G - H × t A to systems of imprimitivity.     

Gain Characteristics of Er^3＋-Doped Phosphate Glass Fibres

An erbium-doped phosphate glass fibre has been drawn by the rod-in-tube technique in our laboratory. The gain for the Er^3＋-doped phosphate glass fibre with different pump powers and with different input signal wavelengths is investigated. The 2.2-cm-long fibre, pumped by a single-mode 980-nm fibre-pigtailed laser diode, can provide a net gain per unit length greater than 1.SdB/cm. The pump threshold is about 50mW at the wavelength of 1534nm, and below 70roW at 1550nm. The gain linewidth of the Er^3＋-doped phosphate glass fibre is greater than 34 nm and can cover the C band in optical communication networks.