Large-scale Stein and Lyapunov equations, Smith method, and applications

We consider the solution of large-scale Lyapunov and Stein equations. For Stein equations, the well-known Smith method will be adapted, with $A_k = A^{2^k}$ not explicitly computed but in the recursive form $A_k = A_{k-1}^{2}$ , and the fast growing but diminishing components in the approximate solutions truncated. Lyapunov equations will be first treated with the Cayley transform before the Smith method is applied. For algebraic equations with numerically low-ranked solutions of dimension n, the resulting algorithms are of an efficient O(n) computational complexity and memory requirement per iteration and converge essentially quadratically. An application in the estimation of a lower bound of the condition number for continuous-time algebraic Riccati equations is presented, as well as some numerical results.     

Asymptotic aspect of Drygas,quadratic and Jensen functional equations in metric abelian groups

The asymptotic stability behavior of Drygas, quadratic and Jensen functional equations is investigated. Indeed, we show that if these equations hold approximately for large arguments with an upper bound \({\varepsilon}\), then they are also valid approximately everywhere with a new upper bound which is a constant multiple of \({\varepsilon}\). These results will be applied to the study of asymptotic properties of Drygas, quadratic and Jensen functional equations. We also obtain some results of hyperstability character for these functional equations.     

Modeling,analysis and simulation of the human femur: prosthesis of femur head

We report the results obtained when carrying out the analysis of the strengths and strains that take place when prosthesis of femur head are implanted in the human femur. The modeling is made with tools provided by Computer Aided Geometry Design, using Non Uniform Rational B-Splines representations (NURBS). In the analysis of the strengths and strains, produced in the bone-cement-prosthesis interaction, partial differential equations of the coupled system are obtained for the displacements and deformations, that allow to analyze cases of physical and biological loads, under which will be work the implants during their useful life, which also is estimated; because of the frequent phenomenon of bony absorption and the consequent relaxation and loss of the replacement surgery. The obtained equations are solved by finite element method, with the corresponding estimation of the obtained errors. The simulations will be performed using high level software able to show realistic visualizations of the observed phenomena in real time. The fracture of the femur neck is a frequently observed trauma and has taken to the authors to make the present work. The obtained results will allow to make the modeling, analysis and simulation of other bony structures and the implants of the corresponding prosthesis. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms

This work is concerned with a system of nonlinear viscoelastic wave equations with nonlinear damping and source terms acting in both equations. We will prove that the energy associated to the system is unbounded. In fact, it will be proved that the energy will grow up as an exponential function as time goes to infinity, provided that the initial data are large enough. The key ingredient in the proof is a method used in Vitillaro (Arch Ration Mech Anal 149:155–182, 1999) and developed in Said-Houari (Diff Integr Equ 23(1–2):79–92, 2010) for a system of wave equations, with necessary modification imposed by the nature of our problem.     

Superconvergence of collocation methods for a class of weakly singular Volterra integral equations

We discuss the application of spline collocation methods to a certain class of weakly singular Volterra integral equations. It will be shown that, by a special choice of the collocation parameters, superconvergence properties can be obtained if the exact solution satisfies certain conditions. This is in contrast with the theory of collocation methods for Abel type equations. Several numerical examples are given which illustrate the theoretical results.     

Systematic Measures of Biological Networks II: Degeneracy,Complexity, and Robustness

This paper is part II of a two‐part series devoted to the study of systematic measures in a complex bionetwork modeled by a system of ordinary differential equations. In this part, we quantify several systematic measures of a biological network including degeneracy, complexity, and robustness. We will apply the theory of stochastic differential equations to define degeneracy and complexity for a bionetwork. Robustness of the network will be defined according to the strength of attractions to the global attractor. Based on the study of stationary probability measures and entropy made in part I of this series, we will investigate some fundamental properties of these systematic measures, in particular the connections between degeneracy, complexity, and robustness.© 2016 Wiley Periodicals, Inc.     

DISSIPATIVITY AND EXPONENTIAL STABILITY OF θ-METHODS FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH A BOUNDED LAG

This paper deals with analytic and numerical dissipativity and exponential stability of singularly perturbed delay differential equations with any bounded state-independent lag. Sufficient conditions will be presented to ensure that any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is dissipative and exponentially stable uniformly for sufficiently small ε > 0. We will study the numerical solution defined by the linear θ-method and one-leg method and show that they are dissipative and exponentially stable uniformly for sufficiently small ε > 0 if and only if θ = 1.     

Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite-dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.     

The Risk Model with Interest,Liquid Reserves and a Constant Dividend Barrier

In this paper, we consider the compound Poisson surplus model with interest, liquid reserves and a constant dividend barrier. When the surplus of an insurer is below a fixed level, the surplus is kept as liquid reserves, which does not earn interest. When the surplus attains the level, the surplus will receive interest at a constant rate. When the surplus hits another fixed higher lever, the excess of the surplus over this higher level will be distributed to the shareholders as dividends. We derive a system of integro-differential equations for the Gerber-Shiu discounted penalty function and obtain the solutions to these integro-differential equations. In the case where the claim sizes are exponential distributed, we get the exact solutions of zero discounted Gerber-Shiu function. We also get the integro-differential equation for the expectation of the discounted dividends until ruin which is the key to discuss the optimal dividend barrier. And we give the exact solution in the special case with exponential claim sizes.     

Parameter estimation in moving boundary problems

We describe an approximation scheme which can be used to estimate unknown parameters in moving boundary problems. The model equations we consider are fairly general nonlinear diffusion/reaction equations of one spatial variable. Here we give conditions on the parameter sets and model equations under which we can prove that the estimates obtained using the approximations will converge to best-fit parameters for the original model equations. We conclude with a numerical example.     

Degenerate difference-differential equations: Algebraic theory

In this paper we consider the algebraic aspects of the theory of degenerate difference-differential equations. It will be shown that the fundamental algebraic concepts to be used are module theoretic. We have to consider similarity of polynomial matrices in one or more indeterminates. In the case of systems with commensurable lags the underlying modules have a simple structure, because the corresponding ring of scalars is the principal ideal domain of polynomials in one indeterminate. This fact makes it possible to prove a structure theorem for degenerate difference-differential equations with commensurable lags. This theorem shows that degenerate systems of this type essentially are trivial in the sense of Henry [15], i.e., the characteristic quasipolynomial is a polynomial. Further it is shown that coordinate transforms with “time lag” play an essential role for the construction of degenerate equations. The power of the method is demonstrated by some examples, some of which are equations with incommensurable lags.     

Numerical treatment of oscillatory functional differential equations

In this paper we are concerned with oscillatory functional differential equations (that is, those equations where all the solutions oscillate) under a numerical approximation. Our interest is in the preservation of qualitative properties of solutions under a numerical discretisation. We give conditions under which an equation is oscillatory, and consider whether the discrete schemes derived using linear ?-methods will also be oscillatory. We conclude with some general theory.     

Two classes of piecewise-linear difference equations with eventual periodicity six

In this paper, we will study two classes of difference equations which are piecewise-linear and of similar forms. We will show that all non-trivial solutions of both equations are eventually periodic with prime period six.     

A theorem on energy integrals for linear second-order ordinary differential equations with variable coefficients

We aim at demonstrating a novel theorem on the derivation of energy integrals for linear second-order ordinary differential equations with variable coefficients. Namely, in this context, we will present a possible and consistent method to overcome the traditional difficulty of deriving energy integrals for Lagrangian functions that explicitly exhibit the independent variable. Our theorem is such that it appropriately governs the arbitrariness of the variable coefficients in order to have energy integrals ensured. In view of the theoretical framework in which the theorem will be embedded, we will also demonstrate that it can be applied as a mathematical method to solve linear second-order ordinary differential equations with variable coefficients. These results are expected to have a generalized fundamental character.     

A parameter estimation scheme for a class of sequential hybrid systems

It may happen that the equations governing the response of dynamical systems have some parameters whose values may not be known a priori and have to be obtained using parameter estimation schemes. In this article, we present a parameter estimation scheme for a class of sequential hybrid systems. By hybrid systems, we refer to those systems whose response is described by different governing equations corresponding to various regimes/modes of operation along with some criteria to switch between the same. In a sequential hybrid system, the different modes are arranged in a specific sequence and the system can switch from a given mode to either the previous mode or the following mode in this sequence. Here, we consider those systems whose governing equations consist of ordinary differential equations and algebraic equations. The conditions for switching between the various modes (referred to as transition conditions) are in the form of linear inequalities involving the system output. We shall first consider the case where the transition conditions are known completely. We present a parameter update scheme along with sufficient conditions that will guarantee bounded parameter estimation errors. Then, we shall consider the case where the transition conditions are not known in the sense that some parameters in these conditions are not known. We present a parameter estimation scheme for this case. We illustrate the performance of the parameter estimation scheme in both cases with some examples.     

Wave and Dirac operators,and representations of the conformal group

Let M be the flat Minkowski space. The solutions of the wave equation, the Dirac equations, the Maxwell equations, or more generally the mass 0, spin s equations are invariant under a multiplier representation U_s, of the conformal group. We provide the space of distributions solutions of the mass 0, spin s equations with a Hilbert space structure H_s, such that the representation U_s, will act unitarily on H_s. We prove that the mass 0 equations give intertwining operators between representations of principal series. We relate these representations to the Segal-Shale-Weil (or “ladder”) representation of U(2, 2).     

Domain Decomposition,Optimal Control of Systems Governed by Partial Differential Equations,and Synthesis of Feedback Laws

We present an iterative domain decomposition method for the optimal control of systems governed by linear partial differential equations. The equations can be of elliptic, parabolic, or hyperbolic type. The space region supporting the partial differential equations is decomposed and the original global optimal control problem is reduced to a sequence of similar local optimal control problems set on the subdomains. The local problems communicate through transmission conditions, which take the form of carefully chosen boundary conditions on the interfaces between the subdomains. This domain decomposition method can be combined with any suitable numerical procedure to solve the local optimal control problems. We remark that it offers a good potential for using feedback laws (synthesis) in the case of time-dependent partial differential equations. A test problem for the wave equation is solved using this combination of synthesis and domain decomposition methods. Numerical results are presented and discussed. Details on discretization and implementation can be found in Ref. 1.     

Polynomial solutions of the classical equations of Hermite,Legendre, and Chebyshev

The classical differential equations of Hermite, Legendre, and Chebyshev are well known for their polynomial solutions. These polynomials occur in the solutions to numerous problems in applied mathematics, physics, and engineering. However, since these equations are of second order, they also have second linearly independent solutions that are not polynomials. These solutions usually cannot be expressed in terms of elementary functions alone. In this paper, the classical differential equations of Hermite, Legendre, and Chebyshev are studied when they have a forcing term x ^M on the right-hand side. It will be shown that for each equation, choosing a certain initial condition is a necessary and sufficient condition for ensuring a polynomial solution. Once this initial condition is determined, the exact form of the polynomial solution is presented.     

Restrictions of Quadratic Forms to Lagrangian Planes,Quadratic Matrix Equations,and Gyroscopic Stabilization

We discuss the symplectic geometry of linear Hamiltonian systems with nondegenerate Hamiltonians. These systems can be reduced to linear second-order differential equations characteristic of linear oscillation theory. This reduction is related to the problem on the signatures of restrictions of quadratic forms to Lagrangian planes. We study vortex symplectic planes invariant with respect to linear Hamiltonian systems. These planes are determined by the solutions of quadratic matrix equations of a special form. New conditions for gyroscopic stabilization are found.