关于“非线性交调的频率设计”的评注——A题的解答和有关情况

<正>一、问题的背景 A题是一道关于非线性交调的频率设计问题,其工程背景广泛存在于通信系统中。例如,人造卫星通信中的频率配置问题就与本题有关。众所周知,人造卫星转发器的能源大多依赖于太阳能,因而功率是非常有限的,而行波管放大器的输入输出关系便是非线性的,倘若要求工作在线性区域内则会使本来功率就非常有限的放大器的输出信号更加微弱。因此,为了获得最大的输出功率就要克服工作在非线性区域内带来的许多问题,其中之一就是由非线性(幅度、相位)引出的交扰调制(Intermodulation),简称交调。设想对非线性器件输入υ(t)=cos2πf_1t+cos2πf_2t(f_1≠f_2),而输入输出关系为y(t)=υ(t)+υ~2(t),则y(t)的展式中不仅包含有原信号频率f_1和F_2,而且包含有2f_1,f_1±f_2等新的频率成分,称为交调。如果这些交调出现在f_1和f_2的接收带内就会形成干扰。工程设计中的一项任务就是在允许的范围内调整(f_1,f_2),使得各交调对信号不构成干扰,或者是弱干扰。     

一类解析函数族子集的三阶哈达玛乘积

设P[A,B]为著名的Janowski函数类.定义函数族的三阶哈达玛乘积为Q_1*Q_2*Q_3={f_1*f_2*f_3(z):f_1∈Q_1,f_2∈Q_2,f_3∈Q_3}.讨论并得到了P(A_1,B_1)*P(A_2,B_2)*P(A_3,B_3)=P(X,Y)的充要条件.     

非对称振子的拟周期运动

考虑跳跃非线性的微分方程(?)+ax+-bx-+φ(x)=p(t),其中a,b>0,p(t)∈c(R/2πZ)且φ:R→R是一无界函数.我们证明了方程有无穷多的拟周期解且方程的所有解均是有界的(参见文[1—19]).     

n阶非线性常微分方程组边值问题的正解

运用不动点指数理论,研究以下$n$阶非线性常微分方程组边值问题正解的存在性和多重正解的存在性\[\left\{\ay\begin{array}{l}-u^{(n)}=f_1(x,u,v),\q-v^{(n)}=f_2(x,u,v),\\[2mm]u^{(i)}(0)=u^{(p)}(1)= v^{(i)}(0)=v^{(p)}(1)=0.\end{array}\right. \] 这里$n\geq 2$, $i = 0,1,2,\cdots,n-2$, $p \in \{1,2,\cdots,n-1\}$, $f_i\in C([0,1]\times\mathbb R^+\times\mathbb R^+,\mathbb R^+)~(i=1,2)$. 用凹函数刻画非线性项$f_1,f_2$的耦合行为, 因而非线性项 $f_i(i=1,2)$ 既可以都是超线性的, 也可以都是次线性的,还可以是混合非线性的(即其中一个是超线性的, 另一个是次线性的).     

关于M序列反馈函数的构造方法

n级非奇异移位寄存器的反馈函数f(x_1,x_2,…,x_n), f(x_1,x_2,…,x_n)=x_1( )f_0(x_2,…,x_n)的重量ω(f),是指n-1个变元的布尔函数f_0(x_2,…,x_n)的重量ω(f_0),即f_0(x_2,…,x_n)取值为1的点的个数。设f(x_1,x_2,…,x_n)是n级M序列的反馈函数,我们知道,当n>2时,有     

两类二元函数芽的一个共同性质及其应用

本文主要研究二元C~∞函数芽环中函数芽的性质问题.利用Mather有限决定性定理和C~∞函数的右等价关系,获得了带有任意4次至k次齐次多项式p_i(x,y),q_i(x,y)(i=4,5,···,k)k k的两类函数芽f_1=x~2y+sum from i=4 to k(p_i(x,y)),f_2=xy~2+sum from i=4 to k(q_i(x,y))(k≥5)的一个共同性质:若M_2~k?M_2J(f_j)(j=1,2)且f_1,f_2的轨道切空间的余维分布均为c_i=1(i=4,5,···,k-1),则对这里的i,p_i(x,y)中xy~(i-1),yi的系数和q_i(x,y)中x~(i-1)y,x~i的系数均为零.最后,利用该性质,给出了f_1,f_2和一类余维数为7的二元函数芽的标准形式.     

高阶齐次线性微分方程解的零点

研究了一类高阶齐次线性微分方程解的零点收敛指数,并得到当方程的系数A_0为整函数,其泰勒展式为缺项级数,并且A_0起控制作用时,方程f~((k))+A_(k-2)f~((k-2))+…+A_1f′+A_0f=0的任意两个线性无关解f_1,f_2满足max{λ(f_1),λ(f_2)}=∞,其中λ(f)表示亚纯函数.f的零点收敛指数.     

求三角函数的周期

“问题:确定下列函数的周期: 1) f_1(x)=cos 3x/2-sin x/3 2) f_2(x)=cos 2x-tgx。解:用P表示函数f_1(x)的周期,那末根据周期函数的定义有: cos 3x/2-sin x/3==cos 3/2(x+P)-sin 1/3(x+P)……(1)等式(1)对任何x值都成立。当x=0,就得到: 1-0=cos 3/2P-sin P/3……(2)可知当P=12π时,适合等式(2)。所以函数f_1(x)的周期为12π。类似地可求出f_2(x)的周期。”对于这样的解答,不能使我们满意。第一。“猜测”方程(2)的最小正数解和求出函数f_1(x)最小正周期是同样困难的(或容易的)。因此求出方程(2)来,不能使解答容易。     

二次函数的极值和二次函数的分类

用初等数学的方法求函数的极值,陈振宜、范会国两同志分别著有小册子作了专門的叙述,其中都提到了二次函数的极值及其求法。本文旨在用同样初等的方法,对二次函数的极值問題作更深入一步的討論。有趣的是,通过极值存在与否的討論,还可以得到函数的分类;从几何的观点来看,就是曲綫的分类。陈振宜同志已經提出:如果x的函数y可以化成 f_1(y)x~2+f_2(y)x+f_3(y)=0 (1)形式,其中f_1,f_2,f_3是y的函数,那么,由于x必須是实数,即(1)必須有实根,其判別式 f_2(y)~2-4f_1(y)f_3(y)≥0。(2)只要这一不等式可解,y的极值就可以求得(見“极大与极小”第13頁)。从中学生的知識水平来看,如果(2)的左边部分是y的二次函数,那么不等式(2)总是容易解出的,而且有权現成的方法。下面三种情况都是属于这种情形     

复振荡理论中关于超级的角域分布

设f_1和f_2是微分方程f″+A(z)f=0的两个线性无关的解,其中A(z)是无穷级整函数且超级σ_2(A)=0．令E=f_1f_2．本文研究了微分方程f″+A(z)f=0的解在角域中的零点分布,得出E的超级为+∞的Borel方向与零点聚值线的关系．     

On Global Asymptotic Stability of Second Order Nonlinear Differential Systems

This paper investigates the global asymptotic stability of the autonomous planar systems $ \dot {x} = p_2(y)q_2(x)y $ , $ \dot {y} = p_3(y)q_3(x)x + p_3(y)q_4(x)y $ and $ \dot {x} = f_1(x) + h_2(x)y $ , $ \dot {y} = f_3(x) + h_4(x)y $ , under the assumption that all functions involved in the equations are continuous and that the origin is a unique equilibrium. We present necessary and sufficient conditions for the origin to be globally asymptotically stable.     

Hyers–Ulam–Rassias Stability of Derivations in Proper JCQ*–triples

In this paper, we investigate stability of derivations in proper JCQ*–triples associated to the following Pexiderized functional equation $$f(x + y + z) = f_{0}(x) + f_{1}(y) + f_{2}(z)$$ .     

Kinetic equation in the theory of nonequilibrium phase transitions induced by “colored” multiplicative noise

Conclusions The formalism presented in this paper for deriving a kinetic equation for a dynamical system with multiplicative colored noise is to quite a degree universal. It can be used to investigate nonequilibrium phase transitions induced by both additive and multiplicative colored noise. The formalism can be used to describe, the behavior of a system under the influence of several stationary noise sources. The formalism can be generalized to the case of a system of equations with noise sources. One can investigate in its framework systems of the form ,i.e., systems subject simultaneously to noise and a deterministic force.V. A. Steklov Mathematics Institute, USSR Academy of Sciences. Translated from Teoreticheskaya i matematicheskaya Fizika, Vol. 85, No. 2, pp. 288–301, November, 1990.     

Almansi decomposition for Dunkl operators

Let Ω be a G-invariant convex domain in ℝ^N including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (Δh)ⁿf = 0 for some integer n. Here_333-01is the Dunkl Laplacian, and D_j is the Dunkl operator attached to the Coxeter group G,

$$\mathcal{D}_j f(x) = \frac{\partial }{{\partial x_j }}f(x) + \sum\limits_{v \in R_ + } {\kappa _v \frac{{f(x) - f(\sigma _v x)}}{{\left\langle {x,v} \right\rangle }}} v_j ,$$

where K_v is a multiplicity function on R and σ_v is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form

$$f(x) = f_0 (x) + \left| x \right|^2 f_1 (x) + \cdots + \left| x \right|^{2(n - 1)} f_{n - 1} (x), \forall x \in \Omega ,$$

where f_j are Dunkl harmonic functions, i.e. Δ_hf_j = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.

     

Some further stability and boundedness results of some differential equations of the fourth order

In this paper sufficient conditions (Theorem 1 and Corollary 1) for the asymptotic stability (in the large) of the trivial solution x=0 of the differential equations $$D_1 (x) = x^{(4)} + f_1 (\ddot x)\dddot x + f_2 (\dot x,\ddot x) + g(\dot x) + h(x,\dot x) = 0$$ , and $$D_2 (x) = x^{(4)} + F_1 (\ddot x)\dddot x + F_2 (\dot x,\ddot x)\ddot x + G(\dot x)\dot x + H(x,\dot x)x = 0$$ are given. A result (Theorem 2) on the boundedness of the solutions of the differential equations D₁(x)=p₁(t) and D₂(x)=p₂(t) is also established. Further, the results which we obtain reduce to results which are more general than those obtained by Ezeilo [1] for the differential equation $$x^{(4)} + f_1 (\ddot x)\dddot x + a_2 \ddot x + g(\dot x) + a_4 x = p(t)$$ .     

Positive solutions to singular semipositone boundary value problems of second order coupled differential systems

We establish the existence of positive solutions for the second order singular semipositone coupled Dirichlet systems $$\left\{ \begin{aligned} &x{''} +f_1 \bigl(t,y(t)\bigr)+e_1(t)=0, \\ &y{''} +f_2\bigl(t,x(t) \bigr)+e_2(t)=0, \\ &x(0)=x(1)=0,\qquad y(0)=y(1)=0. \end{aligned} \right. $$ The proof relies on Schauder’s fixed point theorem.     

LIMIT CYCLES OF THE GENERALIZED POLYNOMIAL LINARD DIFFERENTIAL SYSTEMS

Using the averaging theory of first and second order we study the maximum number of limit cycles of generalized Linard differential systems{x = y + εh_l~1(x) + ε~2h_l~2(x),y=-x- ε(f_n~1(x)y~(2p+1) + g_m~1(x)) + ∈~2(f_n~2(x)y~(2p+1) + g_m~2(x)),which bifurcate from the periodic orbits of the linear center x = y,y=-x,where ε is a small parameter.The polynomials h_l~1 and h_l~2 have degree l;f_n~1and f_n~2 have degree n;and g_m~1,g_m~2 have degree m.p ∈ N and[·]denotes the integer part function.     

一类具时滞的Liénard型方程的周期解

利用重合度理论研究了一类具时滞的Liénard型 方程x'+f_1(x)|x'|^2+f_2(t,x(t),x(t-\delta(t)))x'+g(t,x(t-\tau(t)))=p(t).获得了该方程存在T-周期解的若干新结论, 改进推广了有关文献中的已有结果.     

Proximal and syndetical properties in nonautonomous discrete systems

This paper is mainly concerned with a class of nonautonomous discrete systems $(X, f_{1,\infty})$. New definitions of proximity relations and sensitivity in nonautonomous discrete systems are given. Some relations among $P(f_{1,\infty})$, $L(f_{1,\infty})$, $R(f_{1,\infty})$, $S(f_{1,\infty})$ and $P(f_{1,\infty})(x)$ are derived. And some chaotic properties of $f_{1,\infty}$ are proved.     

Existence of optimal controls for some nonlinear processes

We prove the existence of optimal controls for an ordinary differential system which is nonlinear in the state functionx but is linear in the control functionu, that is, $$dx/dt = f_1 (t,x) + f_2 (t,x)u$$ Rather weak regularity assumptions are made on the right-hand side of the above system, the constraints, and the cost functional.