一个描述血吸虫病的数学模型的周期解

研究一个描述血吸虫病的周期微分方程模型dx/dt=-rx+A/S(t)y,dy/dt=-δ(t)y+B(S(t)-y) x2/1+ x.数值计算发现该系统同时具有渐近稳定的零解和一个正周期解.通过证明该系统解的有界性,并在一个函数空间上构造单调有界序列,进而证明了在一定条件下正周期解的存在性.     

函数单调性的应用例谈

函数单调性是函数的重要性质,有着极为广泛的应用,本文举例加以说明. 一、函数单调性在解方程中的应用若函数f(x)在区间I上是单调函数,则方程f(x)=f(y)在区间I上有解的充要条件是:x=y.     

导数问题中指数类型函数的变形问题

<正>指数函数是高中阶段非常重要的一种函数类型,跟指数函数有关的不等式恒成立问题,方程有解问题都是常见题型.画图像时往往先求导找单调区间,当遇到形为y=e^x+f(x)的函数,其导函数是y=ex+f(x)的函数,其导函数是y=e^x+f′(x).求单调区间时候往往需要解超越不等式,那么可能需要二次求导甚     

一类广义Liénard型系统周期解的存在性和不存在性

讨论了一类广义Liénard型系统.x=p(y)k(x),.y=-f(x,y)p(y)q(y)-g(x)h(y)非零周期解的存在性和不存在性,给出了非零周期解的存在和不存在的一类充分条件.     

构造单调函数解题

函数的单调性是函数的一个重要性质,对有些数学问题,根据题目条件及结构特征,恰当地构造单调函数,利用函数的单调性,常能获得简捷、直观的解法.1.求值例1设x,y为实数,且满足(x-1)3 2003(x-1)=-1(y-1)3 2003(y-1)=1.则x y=.解原方程组化为(x-1)3 2003(x-1)=-1(1-y)3 2003(1-y)=-1.构造函数f(t)=t3 3t,易知函数f(t)=t3 3t在(-∞, ∞)上单调递增,而f(x-1)=-1=f(1-y),所以x-1=1-y,即x y=2.2.确定大小例2若(log23)x (log35)y≥(log35)-x (log23)-y,则()A.x-y≥0B.x y≥0C.x-y≤0D.x y≤0解由条件得(log23)x-(log53)x≥(log23)-y-(log53)-y,设函…     

一类广义Liénard型系统解振荡的充要条件

给出了一类广义Liénard型系统(x)=p(y)k(x),(y)=-f(x,y)p(y)q(y)-g(x)h(y).解振荡的充要条件,文中的引理也有助于研究这类系统周期解的存在性.     

用点到直线距离公式解决与函数相关的问题

点P(x,y)到直线Ax By C=0距离为d=|Ax By C|/A~2 B~2,当P(x,y)在函数y=f(x)上时,该公式变为d=|Ax Bf(x) C|/A~2 B~2,本文通过引进函数y=f(x),借助该公式解决一些与函数相关的问题.1.求函数单调性例1求f(x)=|x 2-1-x2|的单调区间及单调性.分析把函数f(x)作为点线间距离,借助图象,看x变大时,该距离如何变?图1例1图解函数的定义域是-1≤x≤1,令y=1-x2,即x2 y2=1,y≥0.如图1,所以f(x)=|x 2-y|=|x 2-y|2×2,几何意义:半圆上动点M(x,y)到定直线l:x-y 2=0的距离的2倍.由图1知使OB⊥l时,B到l的距离最小,显然OB:y=-x,由x2 y2=1,(y≥0),y=-x,…     

一类非自治非线性微分方程周期解的存在性

本文讨论非自治非线性微分方程组■=ф(y)-f(x),■=-g(x)+e(t) (1)周期解的存在性.N.Levinson 曾给出■(y)≡y、g(x)≡x 时系统(1)存在周期解的条件,井竹君推广了文[1]的工作.本文给出方程组(1)存在周期解的一组充分条件,进一步推广了文[2]的结果.     

一类随机最优控制问题的单调控制解

讨论了一类可允许控制策略满足单调非降条件的随机最优控制问题,给出了值函数v(t,x,y,)满足一类受梯度限制的Hamilton-Jacobi-Bellman(HJB)方程:max{Lv(t,x,y), v(t,x,y)/ y}=0,其中Lv(t,x,y)= v/ t b(t,x,y,) v/ x 1/2σ2(t,x,y) 2v/ x2 f(t,x,y).借助粘性解的思想,定义了该类HJB方程的粘性解并在此意义下证明了v(t,x,y)是唯一粘性解,这类方程在随机控制,金融数学等领域内有重要应用.     

挖掘隐含条件完善解题过程

高中学生在解题时,如何充分利用已知条件,特别是如何从题意中分离出隐含条件,找到有效的解题方法,完善解题过程是一个值得注意的问题.一、函数中的几个问题例1设函数f(x)=loga(1-ax)在[1,2]上单调递增,求实数a的取值范围.解:由题意可知:a>0,∴g(x)=1-ax在[1,2]上单调递减.要使f(x)在[1,2]上单调递增只需:0g(<2)a<>10即:01-<2aa<>10∴a∈0,21其实,问题的关键在挖掘对数要求真数大于0这一隐含条件.例2已知,x+2y=2,(x≥0,y≥0)求x2+y2的最值.解:以x=2-2y代入x2+y2为x2+y2=(2-2y)2+y2=5y2-8y+4=5y-452+54∵yx≥≥00∴2y-≥20y≥0∴0≤y≤1∴x2+y…     

Nearly Comonotone Approximation of Periodic Functions

Suppose that a continuous 27r-periodic function f on the real axis changes its monotonicity at points y_1:-π≤ y_(2s)y_(2s-1)… y_1 π,s ∈ IN.In this PaPer,for each n≥N,a trigonometric polynomial P_n of order cn is found such that:P_n has the same monotonicity as f,everywhere except,perhaps,the small intervals(y_i-π/n,y_i+π/n)and‖f-P_n‖c(s)ω_3(f,π/n),where N is a constant depending only on mini=1,...,2s {y_i-y_(i+1)},c,c(s) are constants depending only on s,ω_3(f_1,·) is the modulus of smoothness of the 3-rd order of the function f,and ||·|| is the max-norm.     

Monotonicity of the ratios of two Abelian integrals for Hamiltonian systems with parameters

We study the monotonicity of the ratios of two Abelian integrals $\oint_{\gamma_{i}(h)}ydx$ $\backslash$ $\oint_{\gamma_{0i}(h)}xydx$ over three period annuli $\{\gamma_i(h)\}$, for $i=1, 2, 3$, defined by a seventh-degree hyperelliptic Hamiltonian $H(x,y)=y^2+\Psi(x)$ with a parameter. The parameter makes the problem more challenging to analyze. To over the difficulty, we apply some criterion with the help of transformations, tools in computer algebra such as boundary polynomial theory to determine the monotonicity of the ratios. Our results establish the existence and uniqueness of limit cycle bifurcated from each period annulus.     

一类函数的对数完全单调性

通过对参数λ,μ的讨论,主要利用函数的单调性理论,已有对数完全单调函数的性质以及幂函数的积分表达式研究了函数Gλ,μ(x)及函数[Gλ,μ(x)]-1的对数完全单调性,并在此基础上得到了一定条件下函数Gλ,μ(x)及[Gλ,μ(x)]-1对数完全单调的充要条件.     

Monotonicity of the ratio of two Abelian integrals for a class of symmetric hyperelliptic Hamiltonian systems

In this paper we study the monotonicity of the ratio of two hyperelliptic Abelian integrals $I_0(h)=\oint_{\Gamma_h}ydx$ and $I_1(h)=\oint_{\Gamma_h}xydx$ for which $\Gamma_h$ is a continuous family of periodic orbits of a Newtonian system with Hamiltonian function of the form $H(x,y)=\frac{1}{2}{y^2}\pm \Psi(x)$, where $\Psi$ is an arbitrary even function of degree six.     

Attractive Periodic Orbits in Nonlinear Discrete-time Neural Networks with Delayed Feedback

We consider the discrete-time system x ( n )= g x ( n m 1)+ f ( y ( n m k )), y ( n )= g y ( n m 1)+ f ( x ( n m k )), n ] N describing the dynamic interaction of two identical neurons, where g ] (0,1) is the internal decay rate, f is the signal transmission function and k is the signal transmission delay. We construct explicitly an attractive 2 k -periodic orbit in the case where f is a step function (McCulloch-Pitts Model). For the general nonlinear signal transmission functions, we use a perturbation argument and sharp estimates and apply the contractive map principle to obtain the existence and attractivity of a 2 k -periodic orbit. This is contrast to the continuous case (a delay differential system) where no stable periodic orbit can occur due to the monotonicity of the associated semiflow.     

On The Generalized Cocycle Equation of Ebanks and Ng

I show that in order to solve the functional equation $$F_{1}(x+y,z)+F_{2}(y+z,x)F_{3}(z+x,\ y)+F_{4}(x,y)+F_{5}(y,z)+F_{6}(z,x)=0$$ for six unknown functions (x,y,z are elements of an abelian monoid, and the codomain of each F _j is the same divisible abelian group) it is necessary and sufficient to solve each of the following equations in a single unknown function $$\matrix{\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad G(x+y,\ z)- G(x,z)- G(y,z)=G(y+z,x)- G(y,x)- G(z,x)\cr \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad H(x+y,\ z)- H(x,z)- H(y,x)+H(y+z,\ x)- H(y,x)- H(z,x)\cr +H(z+x,\ y)- H(z,y)- H(x,y)=0.}$$      

g-拟凸函数的一个判别准则

证明了如下结果:设g∶H→H,C H是非空开的g-凸集,g(C)是凸集,f是C上的上半连续函数且存在α∈(0,1),使得f(αg(x)+(1-α)g(y))m ax{f。g(x),f。g(y)},x,y∈C,则f为C上的g-拟凸函数.     

Mal’tsevh 0-algebras

Let Φ be an associative commutative ring with unity, 1/6 ∈ Φ, write A for a Mal’tsev algebra over Φ, suppose that on A, the function h(y, z, t, x, x)=2[{yz, t, x}x+{yx, z, x}t], where {x, y, z}=(xy)z−(xz)y+2x(yz), is defined, and assume that H(A) is a fully invariant ideal of A generated by the function h. The algebra A satisfying an identity h(y, z, x, x, x)=0 [h(y, z, t, x, x)=0] is called a Mal’tsev h₀-algebra (h-algebra). We prove that in any Mal’tsev h₀-algebra, the inclusion H(A)·A₂⊆Ann A holds withAnnA the annihilator of A. This means that any semiprime h₀-algebra A is an h-algebra. Every prime h₀-algebra A is a central simple algebra over the quotient field Λ of the center of its algebra of right multiplications, R(A), and is either a 7-dimensional non-Lie algebra or a 3-dimensional Lie algebra over Λ. Supported by RFFR grant No. 94-01-00381-a. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 214–227, March–April, 1996.     

The Method of Moving Planes for Integral Equation in an Extremal Case

In this paper, we study the symmetry results and monotonicity of solutions for an integral equation $$u(x)=-c_N∫_{\mathbb{R}^N}e^{u(y)}log|x-y|dy$$ in an external case.     

On the reversible quadratic centers with monotonic period function

This paper is devoted to studying the period function of the quadratic reversible centers. In this context the interesting stratum is the family of the so-called Loud's dehomogenized systems, namely

We determine several regions in the parameter plane for which the corresponding center has a monotonic period function. To this end we first show that any of these systems can be brought by means of a coordinate transformation to a potential system. Then we apply a monotonicity criterium of R. Schaaf.