Toward determination of the eigenvalues of problems on description of the rotation of deep-water drill columns

This paper presents a method for computing the complex eigenvalues for a boundary-value problem describing the rotation of deep-water drill columns for drilling at depths of thousands of meters. A solution is sought for a singularly perturbed boundary-value problem in the form of an asymptotic expansion in a small parameter ε for two parts of the of the drill column: a lower compressed length, and the upper length. The method can be used to investigate the stability of drill columns used for deep-water drilling. Translated fromDinamicheskie Sistemy. Vol. 12. pp. 29–36, 1993.     

Rational solutions of fifth-order evolutionary equations for describing waves on water

A method is proposed for obtaining the exact solutions of evolutionary equations in the form of a rational function. Invariant manifolds of the equations are used which have the same form of dependence on the required function and its derivatives as the generalized Riccati equations. Using fifth-order Kawahara and Korteweg–de Vries equations as an example, it is shown that their known particular solutions can be obtained using this method. New solutions of a non-linear fifth-order equation, which is encountered when describing long waves on water, are obtained.     

Numerical approximation of the spectra of Phan-Thien Tanner liquids

The linear stability of the linear Phan-Thien Tanner (PTT) fluid model is investigated for plane Poiseuille flow. The PTT model involves parameters that can be used to fit shear and extensional data, which makes it suitable for describing both polymer solutions and melts. The base flow is determined using a Chebyshev-tau method. The linear stability equations are also discretized using Chebyshev approximations to furnish a generalized eigenvalue problem. The spectrum is shown to comprise a continuous part and a discrete part. The theoretical and numerical results are validated for the UCM and Oldroyd-B models, which are special cases of the PTT model, by comparing with results in the literature. It is demonstrated that the linear extensional and elasticity parameters considered. The computational efficiency and accuracy of the numerical method are also investigated.     

Numerical Approximation of the Spectra of Phan-Thien Tanner Liquids

The linear stability of the linear Phan-Thien Tanner (PTT) fluid model is investigated for plane Poiseuille flow. The PTT model involves parameters that can be used to fit shear and extensional data, which makes it suitable for describing both polymer solutions and melts. The base flow is determined using a Chebyshev-tau method. The linear stability equations are also discretized using Chebyshev approximations to furnish a generalized eigenvalue problem. The spectrum is shown to comprise a continuous part and a discrete part. The theoretical and numerical results are validated for the UCM and Oldroyd-B models, which are special cases of the PTT model, by comparing with results in the literature. It is demonstrated that the linear PTT fluid is stable to infinitesimal disturbances with respect to the range of shear-thinning, extensional and elasticity parameters considered. The computational efficiency and accuracy of the numerical method are also investigated.     

Stability of viscoelastic systems under stochastic excitation

An effective method for the investigation of the stability for viscoelastic systems under a parametric stochastic excitation is proposed. The parametric force is assumed in the form of a Gaussian stationary colored noise. The method is based on the simulation of random processes, the numerical solution of differential equations, describing the perturbed motion of the considered system, and the calculation of top Liapunov exponents. The considered method makes it possible to estimate the almost sure stability and the stability with respect to statistical moments of the different order. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The stability of the equilibrium ellipsoids of a rotating liquid

The stability of the Riemann equilibrium ellipsoids, in the class of perturbations which satisfy the Dirichlet assumptions, is investigated using Rumyantsev's method [1]. It is shown that the equations of motion of the Dirichlet liquid ellipsoid are Hamiltonian on each combined level of the moment and circulation integrals, which corresponds to well-known results [2], although it is not a consequence of them. This fact provides, generally speaking, additional possibilities for solving the problem of determining the instability region. In the parameter space of each of the two families [3, 4] of Riemann ellipsoids, the region U for which almost all the equilibrium ellipsoids belonging to it are unstable is determined in explicit analytical form. It is shown that the stability region can be specified in explicit analytical form in both cases.     

A representation of the equations of multicomponent multiphase seepage

A mathematical model of non-isothermal multicomponent flows in a porous medium is investigated. A general case is considered when the model can be used to describe processes with an arbitrary number of components and phases. A general form of the system of mixed-type equations describing the flow, which is similar to the Godunov form for hyperbolic systems is proposed. The equations obtained are applicable to flows with gas, liquid and solid phases. The thermodynamic properties of the medium are determined solely by a single multivalued function, by changing which one can obtain models of different flows in a porous medium. A clear geometrical interpretation of the solutions of the equations is proposed. An equation for the entropy is obtained, and it is shown that in order that the model should not contradict the second law of thermodynamics, it is necessary to take into account, in the energy equation, the work of the gravity force, which is often neglected when investigating seepage.     

Estimation of nonstrict Archimedean copulas and its application to quantum networks

Bivariate nonstrict Archimedean copulas form a subclass of Archimedean copulas and are able to model the dependence structure of random variables that do not take on low quantiles simultaneously; i.e. their domain includes a set, the so‐called zero set, with positive Lebesgue measure but zero probability mass. Standard methods to fit a parametric Archimedean copula, e.g. classical maximum likelihood estimation, are either getting computationally more involved or even fail when dealing with this subclass. We propose an alternative method for estimating the parameter of a nonstrict Archimedean copula that is based on the zero set and the functional form of its boundary curve. This estimator is fast to compute and can be applied to absolutely continuous copulas but also allows singular components. In a simulation study, we compare its performance to that of the standard estimators. Finally, the estimator is applied when modeling the dependence structure of quantities describing the quality of transmission in a quantum network, and it is shown how this model can be used effectively to detect potential intruders in this network. Copyright © 2014 John Wiley & Sons, Ltd.     

Nonlinear electrohydrodynamic Kelvin-Helmholtz and Marangoni instabilities near the marginal state

The effects of surface tension and adsorption on the electrohydrodynamic Kelvin-Helmholtz instability are studied. The system is stressed by a normal electric field such that it allows for the presence of surface charges at the interface. The method used is that of multiple scales. The nonlinear Schrödinger equation describing the behavior of the disturbed system is derived. The stability of the perturbed system is discussed both analytically and numerically and the stability diagrams are obtained. At the critical point, a generalized formulation of the evolution equation is developed, which leads to the nonlinear Klein-Gordon equation. The various stability criteria are derived from this equation.     

Stable finite-element calculation of incompressible flows using the rotation form of convection

Conforming finite-element approximations are considered forthe incompressible Navier–Stokes equations with nonlinearterms written in the convection or rotation forms. Implicittime integration results in nice stability properties of auxiliaryproblems which can be solved by efficient numerical algorithms.The original nonlinear system admits relatively simple stabilizationstrategies. The paper presents in a unified form the convergenceanalysis, including the design of stabilization parameters,for linearized equations in both convection and rotation forms.Moreover, it is shown that a Galerkin discretization of thepressure-regularized Oseen problem with skew-symmetric termsin rotation form possesses better stability properties and,being much easier to solve, can be used as a predictor in implicitcalculations.     

COMPETITION AND COOPERATION IN A DYNAMICAL MODEL OF NATURAL RESOURCES

Abstract In this paper, we propose a model describing the commercial exploitation of a common renewable resource by a population of strategically interacting agents. Players can cooperate or compete; cooperators maximize the payoff of their group while defectors maximize their own profit. The partition of the players into two groups, defectors and cooperators, results from the players' choices, so it is not predetermined. This partition is decided as a Nash equilibrium of a static game. It is shown that different types of players can exist in an equilibrium; more precisely, depending on the parameter values such as resource stock, cost, and so on, there might be equilibria only with defectors, cooperators, or with a combination of cooperators and defectors. In any case the total harvest depends on the renewable resource stock, so it influences agents' positions. It is assumed that at each time period the agents harvest according to Nash equilibrium, which can be combined with a dynamic model describing the evolution of fish population. A complete analysis of the equilibria is presented and their stability is analysed. The effect of the different Nash equilibria on the stability of the fish stock, showing that full cooperation is the most stable case, is examined.     

Weight Functions Method in Stability Study of Systems

A new method for systems stability analysis is presented. This method is called weight functions method and it replaces the problem of Liapunov function finding with a problem of finding a number of functions (weight functions) equal to the number of first order differential equations describing the system. It is known that there are not general methods for finding Liapunov functions. The weight functions method is simpler than the classical method since one function at a time has to found. Conditions of solution stability for linear and nonlinear systems and some examples are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dispersion and stability properties of radial basis collocation method for elastodynamics

Strong form collocation with radial basis approximation, called the radial basis collocation method (RBCM), is introduced for the numerical solution of elastodynamics. In this work, the proper weights for the boundary collocation equations to achieve the optimal convergence in elastodynamics are first derived. The von Neumann method is then introduced to investigate the dispersion characteristics of the semidiscrete RBCM equation. Very small dispersion error (< 1%) in RBCM can be achieved compared to linear and quadratic finite elements. The stability conditions of the RBCM spatial discretization in conjunction with the central difference temporal discretization are also derived. We show that the shape parameter of the radial basis functions not only has strong influence on the dispersion errors, it also has profound influence on temporal stability conditions in the case of lumped mass. Further, our stability analysis shows that, in general, a larger critical time step can be used in RBCM with central difference temporal discretization than that for finite elements with the same temporal discretization. Our analysis also suggests that although RBCM with lumped mass allows a much larger critical time step than that of RBCM with consistent mass, the later offers considerably better accuracy and should be considered in the transient analysis. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

首次积分与临界情形运动稳定性

本文指出1946年苏联学者Четаев提出用首次积分的线性组合来构造李雅普诺夫函数后Четаев及其学生们解决保守系统的运动稳定性问题均用此方法.但是由于用试凑的方法解决起来比较麻烦,而且用他们的方法所得出的稳定性条件也不够完全,只能解决问题的纯虚根的情形,零根的情形却未考虑.本文提出利用降阶方法,也就是将微分方程通过消除循环坐标变换成标准形式,这样稳定性条件可直接由能量积分得出,用此方法计算起来不仅很简捷,而且零根情形亦可以考虑.因此对于具有两个循环坐标问题可以化成二阶系统并且很简捷地得到稳定性条件新结论.至于一个循环坐标问题,事实上Четаев及其学生们并未解决,例如外环为水平或任意角的陀螺仪的运动稳定性问题,但是用我们的方法却给出条件稳定与不稳定的条件.     

由纵向振动引起的质点与圆锥形杆碰撞问题的解析解

给出细长圆锥形的截面杆受到质点纵向碰撞时的精确解析解．提出了一种新方法用于分析质点-圆锥形杆碰撞,使用了叠加法给出杆的响应．其结果可验证数值解和其他解析解．所提出方法的优点之一是响应解的解析形式简洁．结论是质量比和一些描述杆几何形状的变量,如倾斜度、杆长和半径在撞击分析中具有重要作用．     

Nonlinear Evolution Equations and the Braiding of Weakly Transporting Flows Over Gravel Beds

The flow of a river that transports sediment in the form of gravel as bedload is investigated for the case when the transport is small. The linear stability of such flows is discussed and used to formulate some strongly nonlinear investigations describing the interaction of bar instabilities that are known to occur. The key spatial scales in the asymptotic limit of small transport are identified, and highly nonlinear evolution equations derived for each case. A generalized KDV equation is found to govern the nonlinear evolution at small wavenumbers, while at O(1) wavenumbers an infinite set of "triad-like" amplitude equations describes the flow. The interactions demonstrate the natural tendency of rivers of width significantly higher than the critical width at which instability first occurs to form complex patterns that may be associated with braided rivers. The weak transport limit used in our anaysis makes our work directly relevant to rivers experiencing flood conditions where the onset of a flood causes transport to begin. The results shown suggest that in the highly nonlinear stages, bars take the form of slabs tilted in the flow direction with steep edges. In addition, it is found that there is no equilibrium state. These findings are consistent with observations.     

基于遗传算法的工艺路径柔性调度算法

针对具有工艺路径柔性的车间调度问题,提出基于OR子图和子路径的工艺路径柔性描述方法,该描述方法形式简单且允许OR子图多层嵌套。以此为基础,设计了基于遗传算法的工艺路径柔性调度算法,并采用以工艺路径编码、机器编码和工件调度编码为基础的三维染色体编码策略,其中,工艺路径编码和机器编码分别通过最大子路径数量和最大机器数量随机产生,其优势在于任意染色体均表示可行解,并可以使用简单的交叉算子和变异算子实现遗传操作且其后代亦为可行解。最后通过实验证明了算法的优化能力。     

An affine scaling algorithm for linear programming problems with inequality constraints

A primal, interior point method is developed for linear programming problems for which the linear objective function is to be maximised over polyhedra that are not necessarily in standard form. This algorithm concurs with the affine scaling method of Dikin when the polyhedron is in standard form, and satisfies the usual conditions imposed for using that method. If the search direction is regarded as a function of the current iterate, then it is shown that this function has a unique, continuous extension to the boundary. In fact, on any given face, this extension is just the value the search direction would have for the problem of maximising the objective function over that face. This extension is exploited to prove convergence. The algorithm presented here can be used to exploit such special constraint structure as bounds, ranges, and free variables without increasing the size of the linear programming problem.This paper is in final form and no version of it will be submitted for publication elsewhere.     

Shadows and intersections: Stability and new proofs

We give a short new proof of a version of the Kruskal-Katona theorem due to Lovász. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a question of Mubayi. This in turn leads to another combinatorial proof of a stability theorem for intersecting families, which was originally obtained by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi by means of a purely combinatorial result of Frankl. We also give an algebraic perspective on these problems, giving yet another proof of intersection stability that relies on expansion of a certain Cayley graph of the symmetric group, and an algebraic generalisation of Lovász's theorem that answers a question of Frankl and Tokushige.     

Nonlinear Stability in Resonant Cases: A Geometrical Approach

Summary. In systems with two degrees of freedom, Arnold's theorem is used for studying nonlinear stability of the origin when the quadratic part of the Hamiltonian is a nondefinite form. In that case, a previous normalization of the higher orders is needed, which reduces the Hamiltonian to homogeneous polynomials in the actions. However, in the case of resonances, it could not be possible to bring the Hamiltonian to the normal form required by Arnold's theorem. In these cases, we determine the stability from analysis of the normalized phase flow. Normalization up to an arbitrary order by Lie-Deprit transformation is carried out using a generalization of the Lissajous variables. Received November 8, 2000; accepted January 6, 2001 Online publication March 23, 2001