用等价无穷小替换求极限的几点注释

在这篇文章里讨论了几个函数和与差的等价无穷小问题,对高等数学教材中利用等价无穷小替换求极限的适用范围进行了拓广,并得到了可用等价无穷小替换求极限的新的充要条件.     

商的导数情形下的等价无穷小替换

对于复变函数中高阶极点留数的计算,常会碰到商的导数的极限,给出了商的导数情形下等价无穷小替换的一种方法,简化了计算.     

关于二重变上限积分的等价无穷小

将二重变上限积分看作是一类特殊的一元诱导函数,本文给出了两种二重变上限积分的定义方式,分别对积分限和被积函数做相应的等价无穷小量替换.在一定的条件下,替换后的二重变上限积分与替换前的二重变上限积分是等价无穷小,从而得到一类求极限的方法,并给出了应用实例.     

等价转化求解函数的零点问题

<正>将复杂问题逐步等价转化为简单问题是解决问题的基本策略.下面通过函数零点问题说明等价转化方法的应用.一、已知函数,求解函数的零点个数.例1求函数f(x)=sinx~2-1/3,x∈[-π,π]的零点个数.解因为f(x)是偶函数,所以我们可以先     

能否用极坐标方法求二重极限

对在二重极限存在性的讨论中能否使用极坐标的方法给予澄清.即,只要在替换过程中充分注意到变量趋于既定点的方式是任意的,也就是替换后在极径趋于0的同时极角是任意的,使用极坐标的方法是可以的,否则容易导致错误.     

关于一元含参量变上限积分的等价无穷小

本文应用等价无穷小量理论,对一类一元含参量变上限积分,在满足一定条件下,分别对积分限和被积函数做相应的等价无穷小量替换.,使替换后的一元含参量变上限积分与替换前的一元含参量变上限积分是等价无穷小.     

G-极限点集和G-链等价集的研究

根据度量空间中极限点和链等价点的定义,给出度量G-空间中G-极限点和G-链等价点的概念,并在度量G-空间中研究了它们的动力学性质,得到了G-极限点和G-链等价点的一些结果,这些结果丰富了度量G-空间中G-极限点和G-链等价点的理论.     

利用函数中间变量等价无穷小代换求极限

研究了一类极限,极限中函数的中间变量是无穷小,但函数本身并不是无穷小,利用中间变量等价无穷小的代换得到了极限的简化计算方法.     

用等价无穷小替换求一类和式极限

利用等价无穷小替换,得到了一类和式极限的简单求法,并给出若干实例.     

泰勒公式在含三层复合函数的不定式极限计算中的应用

本文探讨了一个含三层复合函数(由两层幂指函数复合而得的)的0/0型不定式极限,除了L′Hospital法则、等价无穷小替换、带Peano余项的泰勒公式等经典方法外,还需结合使用一阶带Lagrange余项的泰勒公式进行求解.     

Stability of Periodic Points of Diffeomorphisms of Multidimensional Space

We study the diffeomorphism of a multidimensional space into itself with a hyperbolic fixed point at the origin and a nontransversal homoclinic point. From the works of Sh. Newhouse, B.F. Ivanov, L.P. Shilnikov, and other authors, it follows that there is a method of tangency for the stable and unstable manifold such that the neighborhood of a nontransversal homoclinic point can contain an infinite set of stable periodic points, but at least one of the characteristic exponents of those points tends to zero as the period increases. In this paper, we study diffeomorphisms such that the method of tangency for the stable and unstable manifold differs from the case studied in the works of the abovementioned authors. This paper continues previous works of the author, where diffeomorphisms are studied such that their Jacobi matrices at the origin have only real eigenvalues. In those previous works, we find conditions such that the neighborhood of a nontransversal homoclinic point of the studied diffeomorphism contains an infinite set of stable periodic points with characteristic exponents separated from zero. In the present paper, it is assumed that the Jacobi matrix of the original diffeomorphism at the origin has real eigenvalues and several pairs of complex conjugate eigenvalues. Under this assumption, we find conditions guaranteeing that a neighborhood of a nontransversal homoclinic point contains an infinite set of stable periodic points with characteristic exponents separated from zero.     

A Barrier Function Method for the Nonconvex Quadratic Programming Problem with Box Constraints

In this paper a barrier function method is proposed for approximating a solution of the nonconvex quadratic programming problem with box constraints. The method attempts to produce a solution of good quality by following a path as the barrier parameter decreases from a sufficiently large positive number. For a given value of the barrier parameter, the method searches for a minimum point of the barrier function in a descent direction, which has a desired property that the box constraints are always satisfied automatically if the step length is a number between zero and one. When all the diagonal entries of the objective function are negative, the method converges to at least a local minimum point of the problem if it yields a local minimum point of the barrier function for a sequence of decreasing values of the barrier parameter with zero limit. Numerical results show that the method always generates a global or near global minimum point as the barrier parameter decreases at a sufficiently slow pace.     

Projective Approximation of Double Limit Points for Nonlinear Problems

In [2], general approximation results for the solution in a neighborhood of a simple limit point are given. In this paper we give projective approximation results for the solutions in a neighborhood of a double limit point. Application of these results to a nonlinear partial differential equation and numerical results are given.     

Generalization of a theorem on the quasi-analyticity of a function

It is a known fact that if a function, together with all of its derivatives, vanishes at a point, then the function will be zero in a neighborhood of the point if its successive derivatives satisfy certain estimates. We show that even if the function does not have a priori all of its derivatives but is such that its first derivative has a special sequence of majorizing functions, then in this case also the function will be equal to zero. We use our results to obtain theorems concerning the uniqueness of the solution of an abstract Cauchy problem.     

An analysis of neighborhood functions on generic solution spaces

The effectiveness of local search algorithms on discrete optimization problems is influenced by the choice of the neighborhood function. A neighborhood function that results in all local minima being global minima is said to have zero L-locals. A polynomially sized neighborhood function with zero L-locals would ensure that at each iteration, a local search algorithm would be able to find an improving solution or conclude that the current solution is a global minimum. This paper presents a recursive relationship for computing the number of neighborhood functions over a generic solution space that results in zero L-locals. Expressions are also given for the number of tree neighborhood functions with zero L-locals. These results provide a first step towards developing expressions that are applicable to discrete optimization problems, as well as providing results that add to the collection of solved graphical enumeration problems.     

AN INTERIOR TRUST REGION ALGORITHM FOR NONLINEAR MINIMIZATION WITH LINEAR CONSTRAINTS

AbstractAn interior trust-region-based algorithm for linearly constrained minimization problems is proposed and analyzed. This algorithm is similar to trust region algorithms for unconstrained minimization: a trust region subproblem on a subspace is solved in each iteration. We establish that the proposed algorithm has convergence properties analogous to those of the trust region algorithms for unconstrained minimization. Namely, every limit point of the generated sequence satisfies the Krush-Kuhn-Tucker (KKT) conditions and at least one limit point satisfies second order necessary optimality conditions. In addition, if one limit point is a strong local minimizer and the Hessian is Lipschitz continuous in a neighborhood of that point, then the generated sequence converges globally to that point in the rate of at least 2-step quadratic. We are mainly concerned with the theoretical properties of the algorithm in this paper. Implementation issues and adaptation to large-scale problems will be addressed in a     

TOWARDS A STANDARD FOR THE INDIVIDUAL‐BASED MODELING OF PLANT POPULATIONS: SELF‐THINNING AND THE FIELD‐OF‐NEIGHBORHOOD APPROACH

ABSTRACT. In classical theoretical ecology there are numerous standard models which are simple, generally applicable, and have well‐known properties. These standard models are widely used as building blocks for all kinds of theoretical and applied models. In contrast, there is a total lack of standard individual‐based models (IBM's), even though they are badly needed if the advantages of the individual‐based approach are to be exploited more efficiently. We discuss the recently developed ‘field‐of‐neighborhood’ approach as a possible standard for modeling plant populations. In this approach, a plant is characterized by a circular zone of influence that grows with the plant, and a field of neighborhood that for each point within the zone of influence describes the strength of competition, i.e., growth reduction, on neighboring plants. Local competition is thus described phenomenologically. We show that a model of mangrove forest dynamics, KiWi, which is based on the FON approach, is capable of reproducing self‐thinning trajectories in an almost textbook‐like manner. In addition, we show that the entire biomass‐density trajectory (bdt) can be divided into four sections which are related to the skewness of the stem diameter distributions of the cohort. The skewness shows two zero crossings during the complete development of the population. These zero crossings indicate the beginning and the end of the self‐thinning process. A characteristic decay of the positive skewness accompanies the occurrence of a linear bdt section, the well‐known self‐thinning line. Although the slope of this line is not fixed, it is confined in two directions, with morphological constraints determining the lower limit and the strength of neighborhood competition exerted by the individuals marking the upper limit.     

WEAK LIMITS OF EMPIRICAL MEASURES FOR A FAMILY OF DIFFUSION PROCESSES

This paper considers diffusion processes {X^∈(t)} on R^2, which are pertur-bations of dynamical system {X(t)} (dX(t) = b(X(t))dt) on R^2. By means of weakconvergence of probability measures, the authors characterize the limit behavior for em-pirical measures of {X^∈(t)} in a neighborhood domain of saddle point of the dynamicalsystem as the perturbations tend to zero.     

To the question of stability of periodic points of three-dimensional diffeomorphisms

Self-diffeomorphisms of three-dimensional space with a hyperbolic fixed point at the origin and a nontransversal point homoclinic to it are considered. It is assumed that the Jacobian matrix of the initial diffeomorphism has complex eigenvalues at the origin. It is shown that, under certain conditions imposed mainly on the character of tangency of the stable and unstable manifolds, a neighborhood of the nontransversal homoclinic point contains an infinite set of stable periodic points whose characteristic exponents are bounded away from zero.     

Delay nonlinear boundary conditions as limit of reactions concentrating in the boundary

In this work, we are interested in the dynamic behavior of a parabolic problem with nonlinear boundary conditions and delay in the boundary. We construct a reaction–diffusion problem with delay in the interior, where the reaction term is concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary, as a parameter ? goes to zero. We analyze the limit of the solutions of this concentrated problem and prove that these solutions converge in certain continuous function spaces to the unique solution of the parabolic problem with delay in the boundary. This convergence result allows us to approximate the solution of equations with delay acting on the boundary by solutions of equations with delay acting in the interior and it may contribute to analyze the dynamic behavior of delay equations when the delay is at the boundary.