On the class of proper linear differential systems with unbounded coefficients

Proper linear differential systems (whose coefficients are not necessarily bounded on the half-line) are defined as systems for which there exists a generalized Lyapunov transformation reducing them to a diagonal system with constant coefficients (Basov). We prove that Lyapunov’s original definition of a proper system and the Perron and Vinograd criteria hold for the class of proper systems as well as for the class of proper systems with uniformly bounded coefficients. We show that the Lyapunov properness criterion for a triangular system fails for systems with unbounded coefficients; namely, we construct an improper system with the following properties: the Lyapunov exponents of all nonzero solutions of that system are finite and exact, and for an arbitrary reduction of this system by a generalized Lyapunov transformation to triangular form, its diagonal coefficients have finite exact mean values, whose set with regard of multiplicities is independent of the choice of the transformation. In addition, we show that the main property of proper systems with uniformly bounded coefficients (preservation of conditional exponential stability as well as the dimension of the exponentially stable manifold and the exponent of the asymptotic behavior of solutions under perturbations of higher-order smallness) holds for proper systems with unbounded coefficients as well.     

Analysis of Caughley model with diffusion from mathematical ecology

We provide an analysis in function spaces of the nonlinear semigroup generated by the Caughley model with varied diffusion from mathematical ecology. The global long time asymptotic dynamics of the system of equations are well posed in the sense of an attractor. The behaviour of this attractor in small diffusion coefficients is studied. Two limit problems depending on the stability of the spatial domain in diffusion coefficients are obtained. An adequate scaling of the space variable yields a diffusion coefficients dependent spatial domain. The limit model equations are defined in the complete space of the domain and its diffusion coefficients are unitary. If the domain does not change with the diffusion coefficients, we obtain as a limit problem the system of equations with zero diffusion coefficients and no boundary conditions. The family of attractors in small diffusion coefficients is proved in the Hausdroff semidistance of sets to converge in the uniform topology of continuous functions.     

Integral formulae in the coupled problem of the thermoelasticity of an inhomogeneous body. Application in the mechanics of composite materials

A coupled unsteady problem of thermoelasticity for an inhomogeneous body, described by a system of four second-order partial differential equations with coefficients that vary depending on the coordinates, is considered, and the same problem for a homogeneous body of the same shape (the concomitant problem) is examined together with this original problem. Integral formulae are obtained that allow one to express the displacements and temperature in the original problem in terms of the displacements and temperature in the concomitant problem. Integral formulae are used to represent the solution of the original problem in the form of series over all possible derivatives of the solution of the concomitant problem. A system of recurrence problems is written for the coefficients of these series. Expressions are found for the coefficients of the concomitant problem (effective coefficients) and special boundary value problems are formulated, from the solution of which specific expressions are found for the effective thermoelasticity coefficients. A theorem concerning the fact that the effective coefficients satisfy the physicomechanical constraints imposed on the thermoelastic constants of real bodies is proved. The case of a layer that is inhomogeneous in its thickness is considered and explicit analytical expressions for all the thermoelasticity coefficients are obtained for it. The case when the thermoelasticity coefficients depend periodically on the coordinates is examined in detail.     

On the stability of a nonautonomous hamiltonian system under a parametric resonance of essential type

The problem of the stability of the equilibrium position of an nonautonomous Hamiltonian system with periodic coefficients, in which two multipliers of the linearized system are equal, is analyzed in a nonlinear setting. The stability in the finite approximation, and formal Liapunov stability or instability are proved, depending on the Hamiltonian's coefficients.     

"兰氏"平方律作战毁伤过程兵力损耗率系数战例求解理论

如何求得著名的“兰氏”平方律战斗动力学方程中双方兵力损耗率系数，这是作战模拟应用研究领域中一个久攻未克的难题。本文提出了以作战结果来逆向研究作战过程中双方兵力损耗率系数的思想。阐述了对于不变的作战双方在相同（相近）的作战环境与作战条件下相继进行的作战序列里，双方各自的兵力损耗率系数不变（波动不大）的公理，论证了揭示作战序列内部规律的两条定理。据此，建立了兵力损耗率系数的战例求解理论与方法。运用这一理论与方法，据以往发生的作战过程其数值特征可以求得未来相似或相同作战过程中双方兵力损耗率系数的具体取值，首次解决了作战模拟研究领域中兵力损耗率系数的具体取值这一难题。     

Arithmetic Continuation of Regular Roots of Formal Parametric Polynomial Systems

Given a regular system of polynomial equations with power series coefficients, an initial root is continued as a power series. With the ground domain as an arbitrary field, arithmetic alone is used for the root continuation over this field, and computation is quadratic in the number of computed coefficients. If the power series of the coefficients of the polynomial are geometrically bounded, then the coefficients of the power series of the root are also.     

Reversion of power series and the extended Raney coefficients

In direct as well as diagonal reversion of a system of power series, the reversion coefficients may be expressed as polynomials in the coefficients of the original power series. These polynomials have coefficients which are natural numbers (Raney coefficients). We provide a combinatorial interpretation for Raney coefficients. Specifically, each such coefficient counts a certain collection of ordered colored trees. We also provide a simple determinantal formula for Raney coefficients which involves multinomial coefficients.

     

Numerical solution of nonlinear inverse coefficient problems for ordinary differential equations

Parametric identification for a class of nonlinear objects with lumped parameters described by systems of ordinary differential equations is studied. The problem is to recover the coefficients of a dynamical system depending on the phase state. For that purpose, the phase space is subdivided into a finite set of subsets or zones in which the coefficients are assumed to be constant or linear functions of state. Once the coefficients in such a form are obtained, interpolation and approximation can be used to represent the coefficients as functions of the phase variables.     

一种辨识投入产出模型中主系数的方法

目前在投入产出模型中四种常用辨识主系数的方法都是针对国民经济主系统而言,而随着我国经济的发展,一些具有相同经济特征的部门组成的国民经济子系统正在国民经济中的地位日趋显著,辨识国民经济子系统的主系数也就显得尤为重要.本文依据直接消耗系数的剔除将会使子系统蕴含受影响对象的信息量遭受损失,没有被剔除的系数集应蕴含受影响对象最少的信息量,以此达到剔除主系数损失最大的思想提出了损失量法,用以辨识国民经济子系统的主系数,并对2002年中国投入产出表的农业系统作了实证分析.同时将损失量法扩展到辨识国民经济主系统的主系数方面,并和信息量法作了比较,证明了损失量法能够比信息量法以更少的主系数达到相同的阈值,即在相同的阈值下损失量法选出的主系数个数比信息量法少.     

Nonlocal limits in the study of linear elliptic systems arising in periodic homogenization

In the present paper, we obtain the two-scale limit system of a sequence of linear elliptic periodic problems with varying coefficients. We show that this system has not the same structure than the classical one, obtained when the coefficients are fixed. This is due to the apparition of nonlocal effects. Our results give an example showing that the homogenization of elliptic problems with varying coefficients, depending on one parameter, gives in general a nonlocal limit problem.     

Analysis of pattern formation in reaction diffusion models with spatially inhomogenous diffusion coefficients

We consider a reaction diffusion system in one spatial dimension in which the diffusion coefficients are spatially varying. We present a non-standard linear analysis for a certain class of spatially varying diffusion coefficients and show that it accurately predicts the behaviour of the full nonlinear system near bifurcation. We show that the steady state solutions exhibit qualitatively different behaviour to that observed in the usual case with constant diffusion coefficients. Specifically, the modified system can generate patterns with spatially varying amplitude and wavelength. Application to chondrogenesis in the limb is discussed.     

A new Euler matrix method for solving systems of linear Volterra integral equations with variable coefficients

This article develops an efficient solver based on collocation points for solving numerically a system of linear Volterra integral equations (VIEs) with variable coefficients. By using the Euler polynomials and the collocation points, this method transforms the system of linear VIEs into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Euler coefficients. A small number of Euler polynomials is needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for solving VIEs with variable coefficients.     

Lyapunov reducibility and stabilization of nonstationary systems with an observer

For a linear nonstationary control system with an observer, we assume that the coefficients are locally Lebesgue integrable and integrally bounded on ℝ and construct a linear feedback such that the closed-loop plant-controller system is Lyapunov reducible to the special triangular form corresponding to an independent shift of the diagonal coefficients in the original system and in the system of asymptotic estimation of the state by an arbitrary pregiven quantity. For a periodic system, we prove that the constructed controls and Lyapunov transformation are periodic. We obtain corollaries on the uniform stabilization and global controllability of the central and singular exponents of the system.     

ARMAX系统不外加输入激励的辨识

本文在不外加输入激励情况下，讨论了开环不稳定和非最小相位的ARMAX系统系数的一致估计．所用方法是用适应镇定的办法，使得闭环系统成为平稳可逆的ARMA过程，然后利用Yule-Walker方程给出闭环系统系数的一致估计，而把求开环系统系数的一致估计归结为解一组线性代数方程。     

A numerical approach to solve the model of an electromechanical system

In this paper, the model of an electromechanical system, which is a system of linear differential equations, is studied. Haar wavelet collocation method (HWCM) is applied for finding the approximate solution of the model. HWCM reduces the system of the model into a matrix‐vector form that contains the unknown Haar coefficients, and these coefficients are easily calculated. To demonstrate the validity and applicability of HWCM, numerical solutions of the system for different parameter values in the system are presented. The obtained results demonstrate the efficiency and accuracy of the method. All of the computations are performed via a program written in Mathematica.     

Approximate periodic solutions of a nonlinear parabolic system and an identification problem

This work continues our study in [L. Lei, Identification of parameters through the approximate periodic solutions of a linear parabolic system, preprint, 2005] on the identification problem for the coefficients for the lower order terms in a parabolic system, through its approximate periodic solutions. Different from the work in [L. Lei, Identification of parameters through the approximate periodic solutions of a linear parabolic system, preprint, 2005], our system now is nonlinear and the coefficients to be detected are from the first order term. From the application point of view, we now try to determine the diffusion coefficients for the system by the observation over a subregion of the physical domain. The existence and uniqueness problem of the approximate periodic solutions is studied in the first part of the paper.     

Continuous Observability for the Anisotropic Maxwell System

A boundary observability inequality for the homogeneous Maxwell system with variable, anisotropic coefficients is proved. The result implies uniqueness for an ill-posed Cauchy problem for Maxwell's system. Both results are so far known only in the special case of isotropic coefficients, i.e., when Maxwell's system reduces to a vector wave equation. Here the analysis has been carried out for the first-order system directly without references to the wave equation.     

Modeling and control of parafoils based on computational fluid dynamics

The determination of aerodynamic parameters of parafoil canopies has been a crucial issue because it affects the model precision. To calculate the aerodynamic coefficients of a canopy, the lifting-line theory has been used in the traditional method. However, because of the existence of leading-edge incisions, there are some restrictive assumptions in lifting-line theory when one is calculating the aerodynamic coefficients of a canopy. Therefore in this article we calculate the aerodynamic coefficients on the basis of computational fluid dynamics. As an improvement, the effects of a leading-edge incision and trailing-edge deflection are considered. Firstly, lift and drag coefficients are obtained by use of computational fluid dynamics. Then the least-squares method is used to identify incision and deflection factors. Furthermore, an eight-degrees-of-freedom mathematical model of a parafoil system is established on the basis of the parameters obtained. Finally, a novel control algorithm, generalized predictive control based on a characteristic model, is applied to the system. The precision of the model established and the effectiveness of the proposed control method are validated by simulation and airdrop testing.     

On Analytic Periodic Solutions to Nonlinear Differential Equations With Delay (Advance)

We study a system of the reaction–diffusion type, where diffusion coefficients depend in an arbitrary way on spatial variables and concentrations, while reactions are expressed as homogeneous functions whose coefficients depend in a special way on spatial variables. We prove that the system has a family of exact solutions that are expressed through solutions to a system of ordinary differential equations (ODE) with homogeneous functions in right-hand sides. For a special case of theODE systemwe construct a general solution represented by Jacobi higher transcendental functions. We also prove that these periodic solutions are analytic functions that can be expressed near each point on the period by convergent power series.     

Symbolic integration of meromorphic differential systems via Dirichlet functions

We present a symbolic technique for computing the exact or approximate solutions of linear differential systems with meromorphic coefficients. To any system of that form, we attach a non-commutative generating series F. The combinatorial decomposition of F allows to get the solution as a polynomial in Dirichlet functions, or hypergeometric functions, that are built from the coefficients of the system.