the dirichlet cauchy problem for nonlinear hyperbolic equations in a domain with edges abstract we study the dirichlet cauchy problem for nonlinear hyperbolic equation of second order in a domain with edges the aim of this paper is to prove the regularity of solution in weighted sobolev spaces

on approximate zeros and rootfinding algorithms for a complex polynomial on donne des criteres pour qu un nombre complexe soit une racine approchee d un polynome f par la methode de newton ou d euler d ordre k on construit des familles d algorithmes surement convergents

thermal effects in a two dimensional hybrid elastic structure in this paper we consider a two dimensional hybrid thermo elastic structure consisting of a thermo elastic plate which has a beam attached to its free end we show that the initial boundary value problem for the interactive system of partial differential equations which take account of the mechanical strains stresses and the thermal stresses in the plate and the beam can be associated with a uniformly bounded evolution operator it turns out that the interplay of parabolic dynamics due to the thermal effects in the hybrid structure and the hyperbolic dynamics associated with the elasticity of the structure yields analyticity for the entire system this result yields solvability for the problem under optimal initial freedom of the displacement velocity and temperature in the plate and the beam while uniform stability is readily available

stability analysis for genetic regulatory networks with delays the continuous time case and the discrete time case convergent dynamics are investigated in this paper for the genetic regulatory networks grns with delays in both the continuous time case and the discrete time form first by utilizing the lyapunov functional method and some inequality techniques sufficient criterion is derived to ensure the global exponential stability of the continuous time grns second by using the semi discretization technique and the email protected method as sources of reference two discrete time analogues of the continuous time grn model are formulated respectively moreover it is shown that under the same sufficient conditions derived earlier these two discrete time grn systems are also globally exponentially stable and the explicit exponential convergent rates are also given finally a numerical example is illustrated to show the validity of the obtained theoretical results and comparisons of the convergent rates for these two different discrete time analogues are also demonstrated

solving a polynomial equation some history and recent progress the classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present day computing we briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms we end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost

a fast parallel algorithm for determining all roots of a polynomial with real roots given a polynomial p z of degree n with m bit integer coefficients and an integer mu the problem of determining all its roots with error less than 2 mu is considered it is shown that this problem is in the class nc if p z has all real roots some very interesting properties of a sturm sequence of a polynomial with distinct real roots are proved and used in the design of a fast parallel algorithm for this problem using newton identities and a novel numerical integration scheme for evaluating a contour integral to high precision this algorithm determines good approximations to the linear factors of p z

blow up for quasi linear wave equations in three space dimensions we consider equations of the form u phi x t u u u 1a for a function u u x 1 x n t u x t here denotes the d alembertian square frac partial 2 partial t 2 vartriangle 1b and u represents the vector of first derivatives u that of second derivatives of u with respect to the x k and t on linearizing 1a shall go over into the classical wave equation u 0 this means that and its first derivatives with respect to u u u shall vanish for u u u 0

symmetric and arbitrarily high order birkhoff hermite time integrators and their long time behaviour for solving nonlinear klein gordon equations abstract the klein gordon equation with nonlinear potential occurs in a wide range of application areas in science and engineering its computation represents a major challenge the main theme of this paper is the construction of symmetric and arbitrarily high order time integrators for the nonlinear klein gordon equation by integrating birkhoff hermite interpolation polynomials to this end under the assumption of periodic boundary conditions we begin with the formulation of the nonlinear klein gordon equation as an abstract second order ordinary differential equation ode and its operator variation of constants formula we then derive a symmetric and arbitrarily high order birkhoff hermite time integration formula for the nonlinear abstract ode accordingly the stability convergence and long time behaviour are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi definite matrix subject to suitable temporal and spatial smoothness a remarkable characteristic of this new approach is that the requirement of temporal smoothness is reduced compared with the traditional numerical methods for pdes in the literature numerical results demonstrate the advantage and efficiency of our time integrators in comparison with the existing numerical approaches

canonical euler splitting method for nonlinear composite stiff evolution equations in this paper a new splitting method called canonical euler splitting method ces is constructed and studied which can be used for the efficient numerical solution of general nonlinear composite stiff problems in evolution equations of various type such as ordinary differential equations odes semi discrete unsteady partial differential equations pdes and ordinary or partial volterra functional differential equations vfdes and can significantly improve the computing speed on the basis of ensuring the computing quality stability consistency and convergence theories of this method are established a series of numerical experiments are given which check the efficiency of ces method and confirm our theoretical results

global classical solutions of nonlinear wave equations is a formally selfadjoint positive elliptic operator and f is a real c2 function with non negative primitive function we treat both the homogeneous initialboundary value problem over ir x f2 for a smooth open set 2clr and the cauchy problem over ir x 1r for the initial boundary value problem we show that any strong solution of 0 1 known to exist according to heinz and yon wahl 4 if 2rn f j u2 o lul 2j j 0 1 u oo

l c ℝn decay of classical solutions for nonlinear schrodinger equations on etudie le probleme aux valeurs initiales pour l equation de schrodinger non lineaire iu t u f u 2 u u o x x x r n t r 1 n 11 sous de bonnes hypotheses de regularite sur f et et des conditions de croissance et de signe sur f on montre que les normes maximum des solutions decroissent quand t au meme taux que celles des solutions de l equation de schrodinger libre

global existence and blow up of solutions for higher order viscoelastic wave equation with a nonlinear source term abstract the initial boundary value problem for some nonlinear higher order viscoelastic wave equation with a nonlinear source term in a bounded domain is studied the existence of global weak solutions for this problem is established by using the galerkin method meanwhile under suitable conditions on relaxation function g and the positive initial energy as well as non positive initial energy it is proved that the solution blows up in the finite time and the lifespan estimates of solutions are also given

complexity of bezout s theorem iv probability of success extensions we estimate the probability that a given number of projective newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n 1 complex variables of fixed degrees will find all the roots of the system we also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context further complexity theory is developed

generalized two mode harmonic oscillator dynamical group and squeezed states in this paper we generalize our previous results of the generalized one mode harmonic oscillator to the generalized two mode case systematic use is made of the so 3 2 dynamical group and we are able to write a general form for the exact time evolution operator in terms of squeezing operators of one and two modes a complete classification of the exact solutions is made and we derive them explicitly whenever possible the relevant results on algebraic decomposition coherent state generators and classification of the solutions are shown in tables a plethora of soluble hamiltonians already treated in the literature which appear to be particular cases of the general formalism presented herein are analysed as well as new cases which to the authors knowledge have not yet been considered

analysis and comparison of numerical methods for the klein gordon equation in the nonrelativistic limit regime we analyze rigourously error estimates and compare numerically temporal spatial resolution of various numerical methods for solving the klein gordon kg equation in the nonrelativistic limit regime involving a small parameter 0 varepsilon ll 1 which is inversely proportional to the speed of light in this regime the solution is highly oscillating in time i e there are propagating waves with wavelength of o varepsilon 2 and o 1 in time and space respectively we begin with four frequently used finite difference time domain fdtd methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size h and time step as well as the small parameter varepsilon based on the error bounds in order to compute correct solutions when 0 varepsilon ll1 the four fdtd methods share the same varepsilon scalability tau o varepsilon 3 then we propose new numerical methods by using either fourier pseudospectral or finite difference approximation for spatial derivatives combined with the gautschi type exponential integrator for temporal derivatives to discretize the kg equation the new methods are unconditionally stable and their varepsilon scalability is improved to o 1 and tau o varepsilon 2 for linear and nonlinear kg equations respectively when 0 varepsilon ll1 numerical results are reported to support our error estimates

on the complexity of polynomial zeros the parallel complexity of the simultaneous approximation to all the zeros of a polynomial is investigated by modifying and analyzing an algorithm given by householder it is possible to obtain a priori bounds to the number of iterations sufficient to yield a given accuracy and to the number of digits required in the finite arithmetic more classes of polynomials for which the simultaneous approximation to all the zeros can be carried out in polylogarithmic time are found some cases of polynomials customarily considered hard are easily solved the root finding problem for a polynomial of degree n having zeros z i i 1 cdots n is mathcal nc reduced to finding a polynomial a z such that a z i 1 a z i leq 1 1 n c where c is a constant

o 3 3 like symmetries of coupled harmonic oscillators in classical mechanics the system of two coupled harmonic oscillators is shown to possess the symmetry of the lorentz group o 3 3 or sl 4 r in the four dimensional phase space in quantum mechanics the symmetry is reduced to that of o 3 2 or sp 4 which is a subgroup of o 3 3 or sl 4 r respectively it is shown that among the six sp 4 like subgroups only one possesses the symmetry which can be translated into the group of unitary transformations in quantum mechanics

stability of imex runge kutta methods for delay differential equations stability of imex implicit explicit runge kutta methods applied to delay differential equations ddes is studied on the basis of the scalar test equation du dt lu t mu t t where t is a constant delay and l m are complex parameters more specifically p stability regions of the methods are defined and analyzed in the same way as in the case of the standard runge kutta methods a new imex method which possesses a superior stability property for ddes is proposed some numerical examples which confirm the results of our analysis are presented

on space time means and strong global solutions of nonlinear hyperbolic equations on demontre l existence de solutions globales appartenant a l loc r 2 l 2 2 r n pour des equations non lineaires de la forme t u u b x d u f u 0 t x r r n u 0 x 0 x t u 0 x 1 x x r n

the global cauchy problem for the non linear klein gordon equation on demontre l unicite de solutions faibles et l existence et l unicite de solutions globales fortement continues dans l espace energie

delay dependent stability of runge kutta methods for linear delay differential algebraic equations abstract this paper is concerned with the stability analysis of numerical solutions for linear delay differential algebraic equations by means of the argument principle the delay dependent stability of runge kutta methods is investigated several numerical examples are given to illustrate the effectiveness of the proposed results

stability analysis of numerical methods for delay differential equations this paper deals with the stability analysis of step by step methods for the numerical solution of delay differential equations we focus on the behaviour of such methods when they are applied to the linear testproblemu t u t u t with 0 and complex a general theorem is presented which can be used to obtain complete characterizations of the stability regions of these methods

deterministic improvement of complex polynomial factorization based on the properties of the associated resultant abstract we show an improvement of the known best deterministic approach to factorization of a univariate polynomial p x over the complex field previously such an improvement was achieved by means of a random perturbation of the coefficients of the input polynomial we now apply some new techniques which lead to a similar result without any randomization the approach is unusual in exploiting some properties of the resultant of p x and p x which is completely distinct from the techniques usually applied in order to compute complex polynomial factorization

coupled harmonic oscillator systems an elementary algebraic decoupling approach we present simple explicit coordinate transformations which serve to decouple the schrodinger equation for a pair of not necessarily identical harmonic oscillators in the presence of bilinear perturbing potentials we derive general conditions for the decoupling and give some examples of physical interest these include the much studied example with just a static perturbation the parallel problem with a dynamic coupling term and the classic example of an isotropic two dimensional oscillator in a transverse magnetic field first solved by fock 1928 by separation of variables

a remark on the equation of a vibrating plate on considere les equations de von karman decrivant une plaque vibrante a bord encastree ou completement libre on obtient dans les deux cas une solution classique unique on utilise un ensemble d equations integrales obtenues a partir de la formule des variations des constantes

additive semi implicit runge kutta methods for computing high speed nonequilibrium reactive flows this paper is concerned with time stepping numerical methods for computing stiff semi discrete systems of ordinary differential equations for transient hypersonic flows with thermo chemical nonequilibrium the stiffness of the equations is mainly caused by the viscous flux terms across the boundary layers and by the source terms modeling finite rate thermo chemical processes implicit methods are needed to treat the stiff terms while more efficient explicit methods can still be used for the nonstiff terms in the equations this paper studies three different semi implicit runge kutta methods for additively split differential equations in the form ofu f u g u wherefis treated by explicit runge kutta methods andgis simultaneously treated by three implicit runge kutta methods a diagonally implicit runge kutta method and two linearized implicit runge kutta methods the coefficients of up to third order accurate additive semi implicit runge kutta methods have been derived such that the methods are both high order accurate and strongly a stable for the implicit terms the results of two numerical tests on the stability and accuracy properties of these methods are also presented in the paper

symmetries of two mode squeezed states it is known that the symmetry of two mode squeezed states is governed by the group sp 4 which is locally isomorphic to the o 3 2 de sitter group it is shown that this complicated ten parameter group can be regarded as a product of two three parameter sp 2 groups it is shown also that two coupled harmonic oscillators serve as a physical basis for the symmetry decomposition it is shown further that the concept of entropy is needed when one of the two modes is not observed the entropy is zero when the system is uncoupled the system reaches thermal equilibrium when the entropy becomes maximal

the global cauchy problem for the critical non linear wave equation abstract we study the global cauchy problem for the non linear wave equation p 1 0 for the critical value p n 2 n 2 in space dimension n 3 we identify a weak space time integrability property stip of the solutions and prove that it is sufficient to ensure the uniqueness of weak solutions the global existence of finite energy solutions with the naturally associated stip and the global existence of regular solutions with some n dependent restrictions on the regularity for spherically symmetric solutions we prove that the previous crucial stip follows from the morawetz inequality actually in a much stronger form than necessary thereby proving that all the previous results hold in the spherically symmetric case

exponential stability of discrete time neural networks with delay and impulses abstract in this paper we investigate the exponential stability of discrete time neural networks with impulses and time varying delay the discrete time neural networks are derived by discretizing the corresponding continuous time counterparts with different discretization methods the impulses are classified into three classes input disturbances stabilizing and neutral type the impulses are neither helpful for stabilizing nor destabilizing the neural networks and then by using the excellent ideology introduced recently by chen and zheng w h chen w x zheng global exponential stability of impulsive neural networks with variable delay an lmi approach ieee trans circuits syst i 56 6 2009 1248 1259 the connections between the impulses and the utilized lyapunov function are fully explored with respect to each type of impulse novel techniques that used to realize the ideology in discrete time situation are proposed and it is shown that they are essentially different from the continuous time case several criteria for global exponential stability of the discrete time neural networks are established in terms of matrix inequalities and based on these theoretical results numerical simulations are given to compare the capability of different discretization methods

computing matrix eigenvalues and polynomial zeros where the output is real surprisingly simple corollaries from the courant fischer minimax characterization theorem enable us to devise a very effective algorithm for the evaluation of a set s interleaving the set e of the eigenvalues of an n x n real symmetric tridiagonal rst matrix tn as well as a point that splits e into two subsets of comparable cardinalities furthermore we extend this algorithm so as to approximate all the n eigenvalues of tn at nearly optimal sequential and parallel cost that is at the cost of staying within polylogarithmic factors from the straightforward lower bounds the resulting improvement of the known processor bound in nc algorithms for the rst eigenproblem is roughly by factor n our approach extends the previous works m ben or and p tiwari j complexity 6 1990 pp 417 442 and m ben or et al siam j comput 17 1988 pp 1081 1092 for the approximation of the zeros of a polynomial having only real zeros and our algorithm leads to an alternative and simplified derivation of the known record parallel and sequential complexity estimates for the latter problem

the dirac gamma matrices as relics of a hidden symmetry as fundamental representations of the algebra sp 4 r the relationship between the dirac gamma matrices and the fundamental representations of the algebra sp 4 r are clarified which is locally isomorphic to the de sitter algebra o 3 2 there are six sets of ten matrices among the fifteen traceless matrices which satisfy the algebra sp 4 r the representations can be identified with and transformed into one another through the permutation group of three objects s3 while each element of the invariant subgroup of s3 transforms an o 3 2 like sp 4 r representation into another o 3 2 like sp 4 r any other element of s3 transforms an o 3 2 like sp 4 r into an o 2 3 like sp 4 r and vice versa we discuss physical implications of this group theoretical analysis

p stability properties of runge kutta methods for delay differential equations recently the author and others have studied a class of runge kutta methods for ddes these methods make use of certain constrained meshes which allow to get optimal order results in this paper we study their asymptotic stability properties using the test equation begin gathered y t ay t by t tau t 0 hfill y t g t for tau leqq t leqq 0 hfill end gathered wherea b 0 andg t is continuous and complex valued it is known thaty t 0 ast if b re a in particular we show that any astable one step collocation method for odes inherits the same property when it is applied to ddes with a constrained mesh i e it is p stable

stability analysis for continuous time and discrete time genetic regulatory networks with delays this paper investigates the global exponential stability problem for the genetic regulatory networks grns with delays in both the continuous time case and the discrete time case first dini derivative method is applied for a new lyapunov functional to obtain a sufficient condition of the global exponential stability for a class of the continuous time grns which is given in the form of several elementary inequalities this sufficient condition is simple to be easily implemented on the computer second by using the semi discretization technique and the imex method two discrete time grn systems have been derived which can preserve the dynamical characteristics of the continuous time systems furthermore it is shown that under the same sufficient conditions obtained earlier these two discrete time grn systems are globally exponentially stable finally a pair of examples are given to show the validity of the obtained stability analysis results

imex runge kutta schemes for reaction diffusion equations a fundamental research is carried out into convergence and stability properties of imex implicit explicit runge kutta schemes applied to reaction diffusion equations it is shown that a fully discrete scheme converges if it satisfies certain conditions using a technique of the b convergence analysis developed by burrage hundsdorfer and verwer in 1986 stability of the schemes is also examined on the basis of a scalar test equation proposed by frank hundsdorfer and verwer in 1997

de sitter group as a symmetry for optical decoherence stokes parameters form a minkowskian 4 vector under various optical transformations as a consequence the resulting two by two density matrix constitutes a representation of the lorentz group the associated poincar e sphere is a geometric representation of the lorentz group since the lorentz group preserves the determinant of the density matrix it cannot accommodate the decoherence process through the decaying off diagonal elements of the density matrix which yields to an increase in the value of the determinant it is noted that the o 3 2 de sitter group contains two lorentz subgroups the change in the determinant in one lorentz group can be compensated by the other it is thus possible to describe the decoherence process as a symmetry transformation in the o 3 2 space it is shown also that these two coupled lorentz groups can serve as a concrete example of feynman s rest of the universe

arbitrarily high order time stepping schemes based on the operator spectrum theory for high dimensional nonlinear klein gordon equations abstract in this paper we explore arbitrarily high order lagrange collocation type time stepping schemes for effectively solving high dimensional nonlinear klein gordon equations with different boundary conditions we begin with one dimensional periodic boundary problems and first formulate an abstract ordinary differential equation ode on a suitable infinity dimensional function space based on the operator spectrum theory we then introduce an operator variation of constants formula which is essential for the derivation of our arbitrarily high order lagrange collocation type time stepping schemes for the nonlinear abstract ode the nonlinear stability and convergence are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi definite matrix under some suitable smoothness assumptions with regard to the two dimensional dirichlet or neumann boundary problems our new time stepping schemes coupled with discrete fast sine cosine transformation can be applied to simulate the two dimensional nonlinear klein gordon equations effectively all essential features of the methodology are present in one dimensional and two dimensional cases although the schemes to be analysed lend themselves with equal to higher dimensional case the numerical simulation is implemented and the numerical results clearly demonstrate the advantage and effectiveness of our new schemes in comparison with the existing numerical methods for solving nonlinear klein gordon equations in the literature

asymptotic preserving schemes for the klein gordon equation in the non relativistic limit regime we consider the klein gordon equation in the non relativistic limit regime i e the speed of light c c tending to infinity we construct an asymptotic expansion for the solution with respect to the small parameter depending on the inverse of the square of the speed of light as the first terms of this asymptotic can easily be simulated our approach allows us to construct numerical algorithms that are robust with respect to the large parameter c c producing high oscillations in the exact solution