Nonlinear properties of fractal growth of thin films at low temperature

Introduction 'The physics of thin film growth has been a topic of much current illterest. The application ofimproved and novel surfaCe science techniques such as field ion microscopy (FIM) and scanningtunneling microscopy (STM), has indeed revealed many atomic details of the film growthlll.A variety of fractal and dendritic structures have been observed in thin-film systemst2-7]. Forexample, STM images Of An on Ru(0001) grown at room temperatureI2] and Pt or Ag growthon Pt(111) at low…     

Mathematical Study of Very High Voltage Power Networks III: The Optimal AC Power Flow Problem

This paper shows how to apply the perturbation theory for nonlinear programming problems to the study of the optimal power flow problem. The latter is the problem of minimizing losses of active power over a very high voltage power networks. In this paper, the inverse of the square root of the reference voltage of the network is viewed as a small parameter. We call this scheme the very high voltage approximation.After some proper scaling, it is possible to formulate a limiting problem, that does not satisfy the Mangasarian-Fromovitz qualification hypothesis. Nevertheless, it is possible to obtain under natural hypotheses the second order expansion of losses and first order expansion of solutions. The latter is such that the computation of the active and reactive parts are decoupled. We also obtain the high order expansion of the value function, solution and Lagrange multiplier, assuming that interactions with the ground are small enough. Finally we show that the classical direct current approximation may be justified and improved using the framework of very high voltage approximation.     

Positive solutions of nonlinear problems involving the square root of the Laplacian

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.     

Numerical algorithms for solutions of Korteweg–de Vries equation

The nonlinear Korteweg–de Vries (KdVE) equation is solved numerically using both Lagrange polynomials based differential quadrature and cosine expansion‐based differential quadrature methods. The first test example is travelling single solitary wave solution of KdVE and the second test example is interaction of two solitary waves, whereas the other three examples are wave production from solitary waves. Maximum error norm and root mean square error norm are computed, and numerical comparison with some earlier works is done for the first two examples, the lowest four conserved quantities are computed for all test examples. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Slow growth of an isolated disk-shaped bubble of constant eccentricity in the presence of a distributed gas source

In this paper we consider the diffusion-controlled (small Péclet number) growth of an isolated, oblate-spheroidal (disk-shaped) bubble of constant eccentricity (aspect ratio) in a medium that actively produces the volatile substance via a distributed source, but does not itself offer significant resistance to growth. Oblate spheroidal bubbles are predicted to grow faster than spherical ones, due to the higher surface area to volume ratio; yet, bubbles of all eccentricities grow proportionally to the square root of time, as expected for a diffusive process. In the presence of a distributed source, however, the growth time becomes dependent on the square-root of the source strength, in the limit as the boundary forcing, i.e., the degree of super-saturation, becomes negligible. Furthermore, we demonstrate that the previously known spherical solution is contained within the more general spheroidal solution. In addition, we produced new expression to describe the growth of a disk in terms of the evolution of the radius of a volume-equivalent sphere and another simple expression relating the growth time of a disk to that of a sphere.     

并列双方柱绕流的大涡模拟及频谱分析

应用二阶全展开ETG有限元离散格式与大涡模拟相结合的方法,对间距比为1.5情况下的并列双方柱绕流进行了数值模拟．由王小华、何钟怡提出的二阶全展开ETG有限元方法通过对N-S方程中的时变项进行Taylor展开,从而把时间导数用空间导数来代替,其作用相当于引入了人工粘性．计算得到了不同时刻的流线图,给出了两方柱的阻力系数、升力系数以及两对称点上流向速度随时间的变化历程,并采用谱分析的方法研究了对称边界条件下并列双方柱绕流的频谱对称性问题．为了消除初始条件的影响,在所取样本中去除了计算中初始段的数据,分别分析了阻力系数、升力系数以及两对称点上流向速度的频谱．结果表明:对称边界条件下,双方柱绕流运动参量的时域过程虽然是不对称的,但频域过程却是对称的．     

Numerical studies of a nonlinear heat equation with square root reaction term

Interest in calculating numerical solutions of a highly nonlinear parabolic partial differential equation with fractional power diffusion and dissipative terms motivated our investigation of a heat equation having a square root nonlinear reaction term. The original equation occurs in the study of plasma behavior in fusion physics. We begin by examining the numerical behavior of the ordinary differential equation obtained by dropping the diffusion term. The results from this simpler case are then used to construct nonstandard finite difference schemes for the partial differential equation. A variety of numerical results are obtained and analyzed, along with a comparison to the numerics of both standard and several nonstandard schemes. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Approximate methods based on order reduction for the periodic solutions of nonlinear third-order ordinary differential equations

Four approximate methods based on order reduction, the introduction of a book-keeping parameter and power series expansions for the solution and the frequency of oscillation are used to analyze three autonomous, nonlinear, third-order ordinary differential equations which have analytical periodic solutions. The first method introduces the velocity in both sides of the equation if this (linear) term is not present. The second one is based on the first one but employs a new independent variable, whereas, in the third and fourth techniques, a term equal to the velocity times the square of the unknown frequency of oscillation is introduced in both sides of the equation. The third method uses the original independent variable, whereas the fourth one employs a new independent variable which depends linearly on the unknown frequency of oscillation. It is shown that the second method provides accurate solutions only for initial velocities close to unity, whereas the third one is found to yield very accurate results for the first and second equations considered here and only for large initial velocities for the third one. The fourth technique provides as accurate results as or more accurate results than parameter-perturbation techniques which deal with the third-order equations directly and are based on the expansion of certain constants that appear in the differential equations in terms of a book-keeping parameter.     

On the Application of the Perturbation Method in the Buckling Optimization of a Heavy Elastic Column of Constant Aspect Ratio

This paper addresses the problem of maximizing the first Euler load of a column of constant aspect ratio (moment of inertia varies with the square of the cross-sectional area), considering the column self-weight, subject to the constraint of fixed volume of column material and height. The coupled nonlinear integro-differential equations of optimality and stability, generated through variational principles, have been solved using the method of parameter perturbation. The optimal column has cross-sectional area that follows a perturbed two-thirds power law; the first Euler load is up to 87% larger than that of the corresponding column of uniform cross-section having the same volume and height.     

Nonlinear microbeam model based on strain gradient theory

The nonlinear governing equation of microbeam based on the strain gradient theory is derived by using a combination of the strain gradient theory and the Hamilton’s principle, and the nonlinear static bending deformation, the post-bucking problem and the nonlinear free vibration are analyzed. The nonlinear term in the nonlinear governing equation is associated with the mean axial extension of the microbeam. The static bending deformation of the clamped–clamped microbeam subjected to transverse force, the critical buckling loads and the nonlinear frequencies of the simple supported microbeam with initial lateral displacement are discussed. It is shown that the size effect is significant when the ratio of characteristic thickness to internal material length scale parameters is approximately equal to one or two, but is diminishing with the increase of the ratio. The results also indicate that the nonlinearity has a great effect on the static and dynamic behavior of microbeam. To attain accurate and reliable characterization of the static and dynamic properties of microbeam, therefore, both the micro structure dependent parameters and the nonlinear term have to be incorporated in the design of micro structures in MEMS or NEMS.     

功率放大器非线性特性及预失真建模研究

针对无记忆功率放大器的非线性特性及预失真建模的问题,首先建立了多项式模型、极坐标Saleh模型和基于正交三角函数的模型并利用MATLAB对其进行了求解,然后给出了无记忆多项式预失真处理器特性函数表达式及最小二乘解.针对记忆功率放大器的非线性特性及预失真建模的问题,首先建立了记忆多项式模型并对其进行了求解,然后建立了相应的有记忆多项式预失真模型并利用最小二乘法进行了求解,并提出了联合功率放大器特性和输入信号幅值范围的有记忆功放自适应预失真模型.最后求出所给输入信号、输出信号以及加入预失真后线性系统的输出信号的功率谱密度,并计算和比较了信道的带外失真参数ACPR;结果显示,加入预失真后大大提升了系统的性能,线性特性明显加强.     

A generalized sub-equations rational expansion method for nonlinear evolution equations

In this paper, based on symbolic computation and the idea of rational expansion method, a generalized sub-equations rational expansion method (GSRE) is devised to uniformly construct a series of exact complexiton solutions for nonlinear evolution equations. Compared with most existing tanh function methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general solutions which include many new types of complexiton solutions: the combination of hyperbolic (and square form) function and elliptic function, trigonometric (and square form) function and elliptic function. The efficiency of the method can be demonstrated on (2 + 1)-dimensional Burgers equations.     

A Haar wavelet collocation approach for solving one and two-dimensional second-order linear and nonlinear hyperbolic telegraph equations

We have developed a new numerical method based on Haar wavelet (HW) in this article for the numerical solution (NS) of one- and two-dimensional hyperbolic Telegraph equations (HTEs). The proposed technique is utilized for one- and two-dimensional linear and nonlinear problems, which shows its advantage over other existing numerical methods. In this technique, we approximated both space and temporal derivatives by the truncated Haar series. The algorithm of the method is simple and we can implement easily in any other programming language. The technique is tested on some linear and nonlinear examples from literature. The maximum absolute errors (MAEs), root mean square errors (RMSEs), and computational convergence rate are calculated for different number of collocation points (CPs) and also some 3D graphs are also drawn. The results show that the proposed technique is simply applicable and accurate.     

Symmetry and the golden ratio in the analysis of a regular pentagon

The regular pentagon had a symbolic meaning in the Pythagorean and Platonic philosophies and a subsequent important role in Western thought, appearing also in arts and architecture. A property of regular pentagons, which was probably discovered by the Pythagoreans, is that the ratio between the diagonal and the side of these pentagons is equal to the golden ratio. Here, we will study some relations existing between a regular pentagon and this ratio. First, we will focus on the group of fivefold rotational symmetry, to find the position in the complex plane of the vertices of this geometric figure. Then, we will propose an analytic method to solve the same problem based on the Cartesian coordinates, a method where we find the golden ratio without any specific geometric consideration. This study shows a comparison of the use of complex numbers, symmetries and analytic methods, applied to a subject which can be interesting for general education in mathematics. In fact, the proposed approach can convey and link several concepts, requiring only a general pre-college education, showing at the same time the richness that mathematics can offer in solving geometric problems.     

Complexity analysis of a full-Newton step interior-point method for linear optimization

This paper concerns a short-update primal-dual interior-point method for linear optimization based on a new search direction. We apply a vector-valued function generated by a univariate function on the nonlinear equation of the system which defines the central path. The common way to obtain the equivalent form of the central path is using the square root function. In this paper we consider a new function formed by the difference of the identity map and the square root function. We apply Newton’s method in order to get the new directions. In spite of the fact that the analysis is more difficult in this case, we prove that the complexity of the algorithm is identical with the one of the best known methods for linear optimization.     

On optimizing CSMA for wide area ad hoc networks

The recent deployment of data-rich smart phones has led to a fresh impetus for understanding the performance of wide area ad hoc networks. The most popular medium access mechanism for such ad hoc networks is CSMA/CA with RTS/CTS. In CSMA-like mechanisms, spatial reuse is achieved by implementing energy-based guard zones. We consider the problem of simultaneously scheduling the maximum number of links that can achieve a given signal to interference ratio (SIR). In this paper, using tools from stochastic geometry, we study and maximize the medium access probability of a typical link. Our contributions are two-fold: (i) We show that a simple modification to the RTS/CTS mechanism, viz., changing the receiver yield decision from an energy-level guard zone to an SIR guard zone, leads to performance gains; and (ii) We show that this combined with a simple modification to the transmit power level—setting it inversely proportional to the square root of the link gain—leads to significant improvements in network throughput. Further, this simple power-level choice is no worse than a factor of two away from optimal over the class of all “local” power level selection strategies for fading channels, and further is optimal in the non-fading case. The analysis relies on an extension of the Matérn hard core point process which allows us to quantify both these SIR guard zones and this power control mechanism.     

A Volterra integral formulation for determining the periodic solutions of some autonomous, nonlinear, third-order ordinary differential equations

A Volterra integral formulation based on the introduction of a term proportional to the velocity times the square of the (unknown) frequency of oscillation, a new independent variable equal to the original one times the (unknown) frequency of oscillation, the method of variation of parameters and series expansions of both the solution and the frequency of oscillation, is used to determine the periodic solutions to three nonlinear, autonomous, third-order, ordinary differential equations. It is shown that the first term of the series expansion of the frequency of oscillation coincides with that determined from a first-order harmonic balance procedure, whereas the two-term approximation to the frequency of oscillation is shown to be more accurate than that of a parameter perturbation procedure for the second equation considered in this paper. For the third equation, it is shown that the two-term approximation presented in this paper is more accurate than the corresponding one for one of the parameter perturbation methods, and for initial velocities less than one, for the other parameter perturbation approach.     

On the Numerical Identification of System Properties in a Square Frame

In this paper the influencing factors of the nonlinear behavior of a highly deformable square frame are aimed to highlight and identify. Therefore, in a first step a square frame consisting of four slender members of equal length has been investigated experimentally. Subsequently to these investigations, a numerical model is set up with a beam finite element model, which is based on the absolute nodal coordinate formulation (ANCF), including geometrical and possible material nonlinearities. The influencing factors as well as the system properties are identified by using a genetic optimization algorithm. The final results show that the highly nonlinear behavior of the square frame can be reproduced accurately in case the nonlinear characteristics of the joints are taken into account carefully. The numerical model combined with the system parameter identification can be utilized for further numerical investigations of non-standard structures undergoing large deformations. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hitting, returning and the short correlation function

We consider a stochastic process with the weakest mixing condition: the so called α. For any fixed n-string we prove the following results. (1) The hitting time has approximately exponential law. (2) The return time has approximately a convex combination between a Dirac measure at the origin and an exponential law. In both cases the parameter of the exponential law is λ(A)ℙ(A) where ℙ(A) is the measure of the string and λ(A) is a certain autocorrelation function of the string. We show also that the weight of the convex combination is approximately λ(A). We describe the behavior of this autocorrelation function. Our results hold when the rate of mixing decays polinomially fast with power larger than the golden number.     

一种新的迭代收敛阶数的证明与推广

迭代方法是求解非线性方程近似根的重要方法.本文基于隐函数存在定理,提出了一种新的迭代方法收敛性和收敛阶数的证明方法,并分别对牛顿(Newton)和柯西(Cauchy)迭代方法迭代收敛性和收敛阶数进行了证明.最后,利用本文提出的证明方法,证明了基于三次泰勒(Taylor)展式构成的迭代格式是收敛的,收敛阶数至少为4,并提出猜想,基于n次泰勒展式构成的迭代格式是收敛的,收敛阶数至少为(n+1).