l-solutions and stability analysis of difference equations using the Kummer’s test

We address the p-summability and asymptotic stability properties in nonautonomous linear difference equations. We focus our discussion on two kind of difference equations. The first one is a first order system of linear nonautonomous difference equations, and our discussion involves the use of Kummer’s convergence test. The second one is a linear nonautonomous scalar higher order difference equation. In this case our discussion is based on a recently introduced transformation of a higher order system into a first-step recursion, where the companion matrices are well treatable from our point of view. We give insight on our ideas that are behind our methods, prove new results, and show applications.     

Control of the new 4th-order hyper-chaotic system with one input

We solve the problem of chaos suppression of Lü’s hyper-chaotic system via feedback control. We use only one control input and moreover the controller is a simple proportional feedback and uses the measurement of only one variable. We show that this simple control law suffices to stabilize the hyper-chaotic system to the zero equilibrium globally and asymptotically. We present stability proofs based on Lyapunov’s direct method and integration of solutions. As a corollary of our main result we draw the conclusion that the system is globally stabilizable by simply varying one parameter, when possible. Simulation experiments that show the effectiveness of our method are also presented.     

Existence of Solutions for a Fractional Derivative System of Equations

We study a fractional derivative system of equations. A Newton polygonal associated with this system is partially described. Under some additional assumptions, this Newton polygonal is fully described and L ² estimates are given, as well as an existence result. We finally discuss our assumptions.     

Gronwall's inequality for a nabla fractional difference system with a retarded argument and an application

ABSTRACT

We study the nabla fractional difference system with retarded argument. There are two major ingredients. A Gronwall's inequality for the nabla case is given. This allows us to evaluate the solution of nabla fractional difference system. We shall illustrate the validity of our results by means of examples.     

Energy drift in molecular dynamics simulations

In molecular dynamics, Hamiltonian systems of differential equations are numerically integrated using the Störmer–Verlet method. One feature of these simulations is that there is an unphysical drift in the energy of the system over long integration periods. We study this energy drift, by considering a representative system in which it can be easily observed and studied. We show that if the system is started in a random initial configuration, the error in energy of the numerically computed solution is well modeled as a continuous-time stochastic process: geometric Brownian motion. We discuss what in our model is likely to remain the same or to change if our approach is applied to more realistic molecular dynamics simulations.     

Stochastic Differential Games with Multiple Modes and Applications to Portfolio Optimization

Abstract

We study a zero-sum stochastic differential game with multiple modes. The state of the system is governed by “controlled switching” diffusion processes. Under certain conditions, we show that the value functions of this game are unique viscosity solutions of the appropriate Hamilton–Jacobi–Isaac' system of equations. We apply our results to the analysis of a portfolio optimization problem where the investor is playing against the market and wishes to maximize his terminal utility. We show that the maximum terminal utility functions are unique viscosity solutions of the corresponding Hamilton–Jacobi–Isaac' system of equations.     

Symmetry principles via interactions and symmetry-violations

In this work we would like to continue our ideas concerning the principle mentioned at the end of [Appl. Math. Comput. 113 (2000) 289]. From now on we will call this principle the principle of symmetry of states. We recall that:Principle of symmetry of states. Any system whose states have the same symmetry can take part in “interaction-chain”, until the symmetry of the states will be maximal and the system will be stable. In this case, the system has a minimal number of different states. Each “intermediate system” can be viewed like a homomorphic image of the anterior.We recall the fact that we have used the automata theory for modelling the systems. The “homomorphic image” words mentioned in the statement of our principle refer to automata-homomorphism. We mention also that we have applied this principle concerning radioactivity, making a conjecture, that we have considered each radioactive nucleus like an automaton and for each radioactive chain (viewed as an interaction-chain) applied our principle.Our purpose in this work is to try to find other examples of interactions which are in concordance with our principle (are all the interactions in concordance with this principle ?).     

Power processes and their application to reliability

We introduce the stochastic process we call a power process and show that under certain conditions it is stochastically monotone. We use it to model a deteriorative system where operating times and repair times are general discrete random variables, and illustrate our results with numerical examples.     

Stability of multi-server polling system with server limits

We consider a multi-server polling system with server limits, that is the number of servers that can attend a queue simultaneously is limited. Stability conditions are available when service policies are unlimited. The definition of stability conditions when both server limits and limited service policies apply remains an open problem. We postulate a conjecture for the stability condition in this case that is supported by our simulation results. The study of this particular variant of the multi-server polling system is motivated by the performance evaluation of next generation passive optical access networks.     

The FAN principle and weak König's lemma in herbrandized second-order arithmetic

We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically true. It is perhaps worthy of note that, in our interpretation, second-order variables are interpreted by finite sets of natural numbers.     

Strang-Type Preconditioners for Solving Linear Systems from Delay Differential Equations

We consider the solution of delay differential equations (DDEs) by using boundary value methods (BVMs). These methods require the solution of one or more nonsymmetric, large and sparse linear systems. The GMRES method with the Strang-type block-circulant preconditioner is proposed for solving these linear systems. We show that if a P _{k 1},k ₂-stable BVM is used for solving an m-by-m system of DDEs, then our preconditioner is invertible and all the eigenvalues of the preconditioned system are clustered around 1. It follows that when the GMRES method is applied to solving the preconditioned systems, the method may converge fast. Numerical results are given to illustrate the effectiveness of our methods.     

A Poisson–Charlier approximation for nonstationary queues

In this paper, we develop a new approximation for nonstationary multiserver queues with abandonment. Our method uses the Poisson–Charlier polynomials, which are a discrete orthogonal polynomial sequence that is orthogonal with respect to the Poisson distribution. We show that by appealing to the Poisson–Charlier polynomials that we can estimate the mean, variance, and probability of delay of our nonstationary queueing system with good accuracy. Lastly, we provide a numerical example that illustrates that our approximations are effective.     

Phase change problems with temperature dependent constraints for the volume fraction velocities

A class of nonlinear systems of parabolic PDEs is considered as a mathematical model for phase change phenomena arising in binary mixtures. We use the volume fraction of one of the components as an order parameter in our systems. Therefore, our system consists of two kinetic equations for the internal energy and order parameter which are derived in the non-smooth thermomechanics theory. In this paper we give a general treatment for our system and establish an existence result by applying the Schauder's fixed point theory with the subdifferential operator techniques.     

The low power architecture approach towards exascale computing

Energy efficiency is a first-order concern when deploying any computer system. From battery-operated mobile devices, to data centers and supercomputers, energy consumption limits the performance that can be offered.We are exploring an alternative to current supercomputers that builds on low power mobile processors. We present initial results from our prototype system based on ARM Cortex-A9, which achieves 120 MFLOPS/W, and discuss the possibilities to increase its energy efficiency.     

Convergence analysis of a hp-finite element approximation of the time-harmonic Maxwell equations with impedance boundary conditions in domains with an analytic boundary

We consider a nonconforming hp -finite element approximation of a variational formulation of the time-harmonic Maxwell equations with impedance boundary conditions proposed by Costabel et al. The advantages of this formulation is that the variational space is embedded in H¹ as soon as the boundary is smooth enough (in particular it holds for domains with an analytic boundary) and standard shift theorem can be applied since the associated boundary value problem is elliptic. Finally in order to perform a wavenumber explicit error analysis of our problem, a splitting lemma and an estimation of the adjoint approximation quantity are proved by adapting to our system the results from Melenk and Sauter obtained for the Helmholtz equation. Some numerical tests that illustrate our theoretical results are also presented. Analytic regularity results with bounds explicit in the wavenumber of the solution of a general elliptic system with lower order terms depending on the wavenumber are needed and hence proved.     

Fractional-step <Emphasis Type="Italic">θ</Emphasis>-method for solving singularly perturbed problem in ecology

We consider a predator-prey model arising in ecology that describes a slow-fast dynamical system. The dynamics of the model is expressed by a system of nonlinear differential equations having different time scales. Designing numerical methods for solving problems exhibiting multiple time scales within a system, such as those considered in this paper, has always been a challenging task. To solve such complicated systems, we therefore use an efficient time-stepping algorithm based on fractional-step methods. To develop our algorithm, we first decouple the original system into fast and slow sub-systems, and then apply suitable sub-algorithms based on a class of θ-methods, to discretize each sub-system independently using different time-steps. Then the algorithm for the full problem is obtained by utilizing a higher-order product method by merging the sub-algorithms at each time-step. The nonlinear system resulting from the use of implicit schemes is solved by two different nonlinear solvers, namely, the Jacobian-free Newton-Krylov method and the well-known Anderson’s acceleration technique. The fractional-step θ-methods give us flexibility to use a variety of methods for each sub-system and they are able to preserve qualitative properties of the solution. We analyze these methods for stability and convergence. Several numerical results indicating the efficiency of the proposed method are presented. We also provide numerical results that confirm our theoretical investigations.     

Bogoliubov Spectrum of Interacting Bose Gases

We study the large‐N limit of a system of N bosons interacting with a potential of intensity 1/N. When the ground state energy is to the first order given by Hartree's theory, we study the next order, predicted by Bogoliubov's theory. We show the convergence of the lower eigenvalues and eigenfunctions towards that of the Bogoliubov Hamiltonian (up to a convenient unitary transform). We also prove the convergence of the free energy when the system is sufficiently trapped. Our results are valid in an abstract setting, our main assumptions being that the Hartree ground state is unique and nondegenerate, and that there is complete Bose‐Einstein condensation on this state. Using our method we then treat two applications: atoms with “bosonic” electrons on one hand, and trapped two‐dimensional and three‐dimensional Coulomb gases on the other hand. © 2015 Wiley Periodicals, Inc.     

Asymmetric invariants for a class of strictly hyperbolic systems including the Timoshenko beam

We introduce a set of conserved quantities of energy‐type for a strictly hyperbolic system of two coupled wave equations in one space dimension. The system is subject to mechanical boundary conditions. Some of these invariants are asymmetric in the sense that their defining quadratic form contains second order derivatives in only one of the unknowns. We study their independence with respect to the usual energies and characterize their sign. In many cases, our results provide sharp well‐posedness and stability results. Finally, we apply some of our conservation laws to the study of a singular perturbation problem previously considered by J. Lagnese and J. L. Lions. Copyright © 2007 John Wiley & Sons, Ltd.     

Trajectories of lattice particles using the new approach to no integrability

We study two systems which lead to a lattice when an integration path is specified in “aesthetic field theory”. One of these cases involves nonsoliton type particles (magnitudes of maxima and minima oscillate in time). The other system is made up of soliton type particles. The two systems are intrinsically three-dimensional. We speak of the third dimension as “time”. In one of our solutions, the particles move on straight line trajectories, insofar as our numerical work indicates. In the other solution, the soliton type particles undergo what appears to be simple harmonic motion in both the x- and y-directions (loop motion). We then study these two systems using the new approach to integrability which involves a superposition principle and is characterized by a unique change function at each point. We still find multi maxima and minima. The systems are not as symmetric as the lattice. The soliton characteristic is preserved by the new method. We investigated the motion of lattice particles. We found evidence of maxima (minima) regions coalescing so that the location of the maxima (minima) became difficult to follow. The concept of location of particles may not even have a well-defined meaning here. We find examples of soliton particles appearing and disappearing. We conclude that the manner of integration in a no integrability theory can transform a system with well-defined trajectories into a system where particles can no longer be followed in time.     

Existence and uniqueness for a nonlinear parabolic/Hamilton–Jacobi coupled system describing the dynamics of dislocation densities

We study a mathematical model describing the dynamics of dislocation densities in crystals. This model is expressed as a 1D system of a parabolic equation and a first order Hamilton–Jacobi equation that are coupled together. We examine an associated Dirichlet boundary value problem. We prove the existence and uniqueness of a viscosity solution among those assuming a lower-bound on their gradient for all time including the initial time. Moreover, we show the existence of a viscosity solution when we have no such restriction on the initial data. We also state a result of existence and uniqueness of entropy solution for the initial value problem of the system obtained by spatial derivation. The uniqueness of this entropy solution holds in the class of bounded-from-below solutions. In order to prove our results on the bounded domain, we use an “extension and restriction” method, and we exploit a relation between scalar conservation laws and Hamilton–Jacobi equations, mainly to get our gradient estimates.