Linear Homogeneous System Research Articles

On periodic solutions of differential equations with piecewise constant argument

The periodic quasilinear system of differential equations with small parameter and piecewise constant argument of generalized type [M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. TMA, 66 (2007) 367–383, M.U. Akhmet, On the reduction principle for differential equations with piecewise argument of generalized type, J. Math. Anal. Appl. 336 (2007) 646–663] is addressed. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. One of the main auxiliary results of our paper is an analogue of Gronwall–Bellman Lemma for functions with piecewise constant and retarded–advanced type arguments. Dependence of solutions on the parameter is investigated. Appropriate examples are given to show our results.

On the Resonance and Radiation Characteristics of Multi-Layered Spherical Microstrip Antennas

The resonance and radiation characteristics of a spherical microstrip with its circular metallic patch located inside a coating, possessing an arbitrary number of layers, are investigated. The developed methodology combines the Legendre transform with a T-matrix method. The appropriate basis functions for the surface current on the patch are chosen according to the cavity model. The complex resonant frequencies are the singular points of a homogeneous linear system, resulting by following a Galerkin's technique. The numerical results of this article propose novel multi-layered spherical microstrip configurations. More precisely, a new type of excitation that concerns an amplifying layer located between core and patch is proposed. The control mechanism of the amplifying capability via the layer's thickness and dielectric constant is reported. Moreover, the behavior of a microstrip with two airgaps that are surrounded by a zero-index material is analyzed. Finally, a coating's continuous distribution, following a “shifted” Luneburg law, is treated by a step approximation of the radial function of its dielectric constant, and the achieved high quality factor of such a microstrip is discussed.

On the definition of the excess permittivity of a fluid mixture. II

In a previous work, an ideal fluid mixture is assumed to be a homogeneous linear and isotropic system that verifies the following conditions: (I) no interaction between molecules, (II) point-like molecules and (III) under fixed temperature and pressure conditions the number of molecules per unit volume is the same for both the individual components and the mixture. It was shown that for such ideal mixture the permittivity is the sum of the individual permittivities weighted with the mole fractions of the components. In this work it is shown that independently of the latter assumption the permittivity of the system can be expressed as the sum of the individual permittivities weighted with the volume fractions of the components.

Bounded and Periodic Solutions of Semilinear Impulsive Periodic System on Banach Spaces

AbstractA class of semilinear impulsive periodic system on Banach spaces is considered. First, we introduce the "Equation missing"-periodic PC-mild solution of semilinear impulsive periodic system. By virtue of Gronwall lemma with impulse, the estimate on the PC-mild solutions is derived. The continuity and compactness of the new constructed Poincaré operator determined by impulsive evolution operator corresponding to homogenous linear impulsive periodic system are shown. This allows us to apply Horn's fixed-point theorem to prove the existence of "Equation missing"-periodic PC-mild solutions when PC-mild solutions are ultimate bounded. This extends the study on periodic solutions of periodic system without impulse to periodic system with impulse on general Banach spaces. At last, an example is given for demonstration.

On the asymptotic degeneration of systems of linear inhomogeneous stochastic differential equations

Assuming the almost sure stability of a linear homogeneous system, we obtain sufficient conditions for the convergence to zero, in probability as well as pathwise, of solutions of the system of linear inhomogeneous stochastic differential equations. It is well known in the theory of deterministic systems that solutions of a linear inhomogeneous system approach zero if the homogeneous part of the system is exponentially stable and the inhomogeneous terms converge to zero as time goes to infinity. A similar result is considered in the current paper for the stochastic case. The author gives sufficient conditions for the convergence to zero, in probability and with probability one, of solutions of an inhomogeneous system of differential equations if the stochastic semigroup generated by the linear homogeneous part is stable with probability one. Consider a system of stochastic differential equations (1) dx(t) = ( A0x(t) + f0(t) ) dt+ m ∑ k=1 ( Akx(t) + fk(t) ) dwk(t), where Ak are (n× n) matrices; fk(t) = (fk1(t), . . . , fkn(t)), t ≥ 0, are vector functions; wr(t), t ≥ 0, are independent one-dimensional Wiener processes; x(t) = (x1(t), . . . , xn(t)), t ≥ 0, is a solution written as a row vector. Denote by {ei}i=1 an orthonormal basis in R and by 〈x, y〉 = n ∑ i=1 xiyi and ‖x‖ = 〈x, x〉 the scalar product and norm of vectors in R, respectively. Let H s be a stochastic semigroup of nondegenerate operators, H s = H t rH r s , 0 ≤ s ≤ r ≤ t, such that (see [7]) dH s = A0H t s dt+ m ∑ k=1 AkH t s dwk(t), H s s = I, s ≤ t. 2000 Mathematics Subject Classification. Primary 60H10; Secondary 34F05. c ©2008 American Mathematical Society 41 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 42 OLEKSANDER IL’CHENKO We assume that the semigroup H s is stable with probability one; that is, P { lim t→+∞ H 0x = 0 } = 1 for all x ∈ R. It is known (see [5, 7]) that the stability with probability one of the semigroup H s is equivalent to the exponential p-stability of H s for sufficiently small p > 0; that is, there are constants D = D(p) > 0 and λ = λ(p) > 0 such that (2) sup ‖x‖=1 E ∥∥Ht sx∥∥p ≤ De−λp(t−s), p ∈ (0, p0). A solution x(t), t ≥ 0, of system (1) can be represented in the following form: x(t) = H 0x+ ∫ t 0 ( H u )( f0(u)− m ∑ k=1 Akfk(u) ) du+ m ∑ k=1 H 0 ∫ t 0 ( H 0 )−1 fk(u) dwk(u) (3) (see [4, 6]). To study the behavior of x(t) as t → +∞, one needs to estimate the distributions of terms on the right hand side of (3). The following two auxiliary results contain necessary estimates. Lemma 1. Let condition (2) hold. If φ(t), t ≥ 0, is a continuous vector function, then, for arbitrary e > 0 and θ > 0, there exists a constant L e} ≤ L k ∑ i=0 e−(λ−θ)(k−i)p ( sup i≤u≤i+1 ‖φ(u)‖ )p for all k ∈ Z+. Proof of Lemma 1. We make use of the following obvious inequalities: M k ∑ i=0 e−θ(k+1−i) 0, k ∈ Z+, (4) ∥∥Ht sζ∥∥ ≤ n ∑ i=1 ∥∥Ht sei∥∥ ‖ζ‖, ζ = ζ(ω) ∈ R. (5) Taking into account (4) we obtain the bound (6) P { sup k≤t≤k+1 ∥∥∥∥∫ t 0 H uφ(u) du ∥∥∥∥ > e} ≤ P { sup k≤t≤k+1 ∥∥∥∥k−1 ∑

Projective re-normalization for improving the behavior of a homogeneous conic linear system

In this paper we study the homogeneous conic system \(F: Ax = 0, x \in C\setminus \{0\}\) . We choose a point \(\bar s \in {\rm int}C^*\) that serves as a normalizer and consider computational properties of the normalized system \(F_{\bar s} : Ax = 0, \bar s^T x = 1, x \in C\) . We show that the computational complexity of solving F via an interior-point method depends only on the complexity value \(\vartheta\) of the barrier for C and on the symmetry of the origin in the image set \(H_{\bar s} := \{Ax {\bar s}^Tx = 1, x \in C\}\) , where the symmetry of 0 in \(H_{\bar s}\) is$$ {\rm sym}(0, H_{\bar s}) := {\rm max}\{\alpha : y \in H_{\bar s} \Rightarrow -\alpha y \in H_{\bar s} \} .$$We show that a solution of F can be computed in \(O(\sqrt{\vartheta}{\rm ln}(\vartheta/{\rm sym}(0, H_{\bar s}))\) interior-point iterations. In order to improve the theoretical and practical computation of a solution of F, we next present a general theory for projective re-normalization of the feasible region \(F_{\bar s}\) and the image set \(H_{\bar s}\) and prove the existence of a normalizer \({\bar s}\) such that \({\rm sym}(0,H_{\bar s}) \ge 1/m\) provided that F has an interior solution. We develop a methodology for constructing a normalizer \({\bar s}\) such that \({\rm sym}(0, H_{\bar s}) \ge 1/m\) with high probability, based on sampling on a geometric random walk with associated probabilistic complexity analysis. While such a normalizer is not itself computable in strongly-polynomial-time, the normalizer will yield a conic system that is solvable in \(O(\sqrt{\vartheta}{\rm ln}(m\vartheta))\) iterations, which is strongly-polynomial-time. Finally, we implement this methodology on randomly generated homogeneous linear programming feasibility problems, constructed to be poorly behaved. Our computational results indicate that the projective re-normalization methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46% decrease in average IPM iterations for 100 randomly generated poorly-behaved problem instances of dimension 1,000 × 5,000.

Characterization of Forced Vibration for Difference Inclusions: A Lyapunov Approach

A Lyapunov approach to the characterization of forced vibration for continuous-time systems was recently proposed in. The objective of this paper is to derive counterpart results for discrete-time systems. The class of oscillatory input signals to be considered include sinusoidal signals, multitone signals, and periodic signals which can be described as the output of an autonomous system. The Lyapunov approach is developed for linear systems, homogeneous systems (difference inclusions), and nonlinear systems (difference inclusions), respectively. It is established that the steady-state gain can be arbitrarily closely characterized with Lyapunov functions if the output response converges exponentially to the steady state. We also evaluate other output measures such as the peak of the transient response and the convergence rate. Applying the results to linear difference inclusions, optimization problems with linear matrix inequality constraints can be formulated for the purpose of estimating the output measures. This paper's results can be readily applied to the evaluation of frequency responses of general nonlinear and uncertain systems by restricting the inputs to sinusoidal signals. Similar to continuous-time systems, it is observed that the peak of the frequency response can be strictly larger than the gain for a linear difference inclusion.

Generalization of Stanley's monster reciprocity theorem

By studying the reciprocity property of linear Diophantine systems in light of Malcev–Neumann series, we present in this paper a new approach to and a generalization of Stanley's monster reciprocity theorem. A formula for the “error term” is given in the case when the system does not have the reciprocity property. We also give an inductive proof of Stanley's reciprocity theorem for linear homogeneous Diophantine systems.

Free vibration of piles embedded in soil having different modulus of subgrade reaction

The soil that the pile is embedded in is idealized by a Winkler model and is assumed to be two layered. The part of the pile extending above the ground is called the first region, and the parts embedded in the soil are called the second and the third regions, respectively. The dynamic displacement function of the pile subjected to an axial force is obtained as a fourth order partial differential equation by taking account of the effects of bending moment and shear force. It is assumed that the behavior of the material is linearly elastic and axial force along the pile length to be constant. Shear effects are included in the differential equations by the second derivative of the elastic curve function with respect to shear deformation. Normalized natural circular frequencies of the pile are calculated using a carry-over matrix and the secant method for non-trivial solution of the linear homogeneous system of equations obtained for a specific value of the axial force, and for two combinations of boundary conditions: 1. One end fixed, and the other end free in displacement, but constrained against angular motion (henceforth referred to as a “sliding boundary condition”); 2. One end fixed, and the other end simply supported. Rotary inertia is considered. The results are presented in graphs.

Precise integration method without inverse matrix calculation for structural dynamic equations

The precise integration method proposed for linear time-invariant homogeneous dynamic systems can provide accurate numerical results that approach an exact solution at integration points. However, difficulties arise when the algorithm is used for non-homogeneous dynamic systems due to the inverse matrix calculation required. In this paper, the structural dynamic equalibrium equations are converted into a special form, the inverse matrix calculation is replaced by the Crout decomposition method to solve the dynamic equilibrium equations, and the precise integration method without the inverse matrix calculation is obtained. The new algorithm enhances the present precise integration method by improving both the computational accuracy and efficiency. Two numerical examples are given to demonstrate the validity and efficiency of the proposed algorithm.

Linear Impulsive Periodic System with Time-Varying Generating Operators on Banach Space

A class of the linear impulsive periodic system with time-varying generating operators on Banach space is considered. By constructing the impulsive evolution operator, the existence of -periodic -mild solution for homogeneous linear impulsive periodic system with time-varying generating operators is reduced to the existence of fixed point for a suitable operator. Further the alternative results on -periodic -mild solution for nonhomogeneous linear impulsive periodic system with time-varying generating operators are established and the relationship between the boundness of solution and the existence of -periodic -mild solution is shown. The impulsive periodic motion controllers that are robust to parameter drift are designed for a given periodic motion. An example given for demonstration.

A criterion for the existence of the unique invariant torus of a linear extension of dynamical systems

Under the assumption that a linear homogeneous system defined on the direct product of a torus and the Euclidean space is exponentially dichotomous on semiaxes, we obtain a necessary and sufficient condition for the existence of the unique invariant torus of the corresponding inhomogeneous linear system.

A Study on the Method of Vibration Analysis for Korean High Speed Train by the On-Line Test

The Korean High Speed Train (KHST) has been tested on the Kyongbu high speed line and the Honam conventional line since 2002. A data acquisition system was developed to test and prove the dynamic performance of the KHST, and the system has been found to be very efficient in acquiring multi-channel data from accelerometers located all over the train. Also presented in this paper is an analysis procedure which is simple and efficient in analyzing the acceleration data acquired during the on-line test of the KHST. The understanding of system vibration mode for a railway vehicle is essential to evaluate the characteristics of a dynamic system and to diagnose the dynamic problems of the vehicle system during tests and operations. Methods based on homogeneous linear systems are not realistic because real systems have nonlinear characteristics and are strongly dependent on environmental conditions. In this paper an efficient method of vibration analysis has been proposed and applied for the KHST to evaluate its vibration mode characteristics. The results show that this method is suitable to estimate the system vibration modes of the KHST.

Instability of an interface between air and a low conducting liquid subjected to charge injection

We study the linear stability of an interface between air and a low conducting liquid in the presence of unipolar injection of charge. As a consequence of charge injection, a volume charge density builds up in the air gap and a surface charge density on the interface. Above a certain voltage threshold the electrical stresses may destabilize the interface, giving rise to a characteristic cell pattern known as rose-window instability. Contrary to what occurs in the classical volume electrohydrodynamic instability in insulating liquids, the typical cell size is several times larger than the liquid depth. We analyze the linear stability through the usual procedure of decomposing an arbitrary perturbation into normal modes. The resulting homogeneous linear system of ordinary differential equations is solved using a commercial software package. Finally, an analytical method is developed that provides a solution valid in the limit of small wavenumbers.

Free vibration of semi-rigid connected and partially embedded piles with the effects of the bending moment, axial and shear force

Natural circular frequencies of piles partially embedded in the soil whose upper ends are semi-rigid connected are calculated using the method of initial values, the transfer matrix, the small-displacement theory and the Winkler model. It is assumed that the behavior of the material is linear-elastic, and that the axial force along the pile length and the modulus of subgrade reaction to be constant. The semi-rigid connections of the upper ends of the piles are modeled by elastic rotational springs. The transfer matrices are obtained due to the boundary conditions of the piles. Natural circular frequencies and relative stiffness of the piles are calculated for a non-trivial solution of a linear homogeneous system of equations obtained due to the terms of the transfer matrices of the piles, the stiffnesses of the linear-elastic rotational springs and ratio of embedded length of the pile to total length of the pile; the results are presented in graphs.

Integral equation analysis of microring and microdisk coupled-resonator optical waveguides

We present a rigorous integral equation (IE) method for analyzing propagation in microring and microdisk coupled-resonator optical waveguides (MR/MD-CROWs). The approach is based on Green's theorem to formulate the IE for the axial field component and the Galerkin technique to obtain a homogeneous linear system between the expansion coefficients of the resonator field in terms of cylindrical wave functions. The CROW propagation constants are determined by the nontrivial solutions of the system. TE and TM polarizations are treated in parallel. Examples for single-mode MR and whispering gallery mode (WGM) MD-CROWs are presented, including coupling with WGMs of higher radial order

Integral manifolds of differential equations with piecewise constant argument of generalized type

In this paper we introduce a general type of differential equations with piecewise constant argument (EPCAG). The existence of global integral manifolds of the quasilinear EPCAG is established when the associated linear homogeneous system has an exponential dichotomy. The smoothness of the manifolds is investigated. The existence of bounded and periodic solutions is considered. A new technique of investigation of equations with piecewise argument, based on an integral representation formula, is proposed. Appropriate illustrating examples are given.

An Efficient Re-Scaled Perceptron Algorithm for Conic Systems

The classical perceptron algorithm is an elementary row-action/relaxation algorithm for solving a homogeneous linear inequality system Ax > 0. A natural condition measure associated with this algorithm is the Euclidean width T of the cone of feasible solutions, and the iteration complexity of the perceptron algorithm is bounded by 1/T^2, see Rosenblatt 1962. Dunagan and Vempala have developed a re-scaled version of the perceptron algorithm with an improved complexity of O(n ln(1/T)) iterations (with high probability), which is theoretically efficient in T, and in particular is polynomial-time in the bit-length model. We explore extensions of the concepts of these perceptron methods to the general homogeneous conic system Ax is an element of a set int K where K is a regular convex cone. We provide a conic extension of the re-scaled perceptron algorithm based on the notion of a deep-separation oracle of a cone, which essentially computes a certificate of strong separation. We give a general condition under which the re-scaled perceptron algorithm is itself theoretically efficient; this includes the cases when K is the cross-product of half-spaces, second-order cones, and the positive semi-definite cone.

Projective Pre-Conditioners for Improving the Behavior of a Homogeneous Conic Linear System

We present a general theory for transforming a homogeneous conic system F: Ax = 0, x in C, x non-zero, to an equivalent system via projective transformation induced by the choice of a point in a related dual set. Such a projective transformation serves to pre-condition the conic system into a system that has both geometric and computational properties with certain guarantees. There must exist projective transformations that transform F to a system whose complexity is strongly-polynomial-time in m and the barrier parameter. We present a method for generating such a projective transformation based on sampling in the dual set, with associated probabilistic analysis. Finally, we present computational results that indicate that this methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46% decrease in average IPM iterations for 100 randomly generated poorly-behaved problem instances of dimension 1000 x 5000.

Modified precise time step integration method of structural dynamic analysis

The precise time step integration method proposed for linear time-invariant homogeneous dynamic systems can provide precise numerical results that approach an exact solution at the integration points. However, difficulty arises when the algorithm is used for non-homogeneous dynamic systems, due to the inverse matrix calculation and the simulation accuracy of the applied loading. By combining the Gaussian quadrature method and state space theory with the calculation technique of matrix exponential function in the precise time step integration method, a new modified precise time step integration method (e.g., an algorithm with an arbitrary order of accuracy) is proposed. In the new method, no inverse matrix calculation or simulation of the applied loading is needed, and the computing efficiency is improved. In particular, the proposed method is independent of the quality of the matrix H. If the matrix H is singular or nearly singular, the advantage of the method is remarkable. The numerical stability of the proposed algorithm is discussed and a numerical example is given to demonstrate the validity and efficiency of the algorithm.