global bifurcation of periodic solutions in nonlinear evolution problems with periodic forcing abstract this paper discusses the global bifurcation of 2 periodic solutions of u t p x u f t u with a homogeneous dirichlet boundary condition where p x is linear elliptic and the nonlinearity f is 2 periodic in t the main differences from existing theories devoted to this type of problem can roughly be summarized as follows i the bifurcation analysis makes no use of evolution operators or related concepts poincare maps floquet multipliers etc ii the bifurcation nonbifurcation points are characterized through an associated stationary problem iii the functional setting allows for nonlinearities f exhibiting time discontinuities among other things the results include various partial generalizations of the bifurcation from the principal eigenvalue theorem which unlike the classical version do not require linear parameter dependence

a unified floquet theory for discrete continuous and hybrid periodic linear systems abstract in this paper we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts e g hybrid dynamical systems we develop a comprehensive floquet theory including lyapunov transformations and their various stability preserving properties a unified floquet theorem which establishes a canonical floquet decomposition on time scales in terms of the generalized exponential function and use these results to study homogeneous as well as nonhomogeneous periodic problems furthermore we explore the connection between floquet multipliers and floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales finally we show this unified floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated but time varying system s poles in the complex plane we include several examples to show the utility of this theory

on the inversion of infinite moment matrices abstract motivated by 8 we study the existence of the inverse of an infinite hermitian positive definite matrix in short hpd matrix from the point of view of the asymptotic behaviour of the smallest eigenvalues of the finite sections we prove a sufficient condition to assure the inversion of an hpd matrix with square summable rows for infinite toeplitz matrices we introduce the notion of asymptotic toeplitz matrix and we show that under certain assumptions the inverse of an infinite toeplitz positive definite matrix is asymptotic toeplitz such inverses are computed in terms of the limits of the coefficients of the associated orthogonal polynomials we apply these results in the context of the theory of orthogonal polynomials in particular we show that for measures on the unit circle t verifying that the smallest eigenvalue of the finite sections of the corresponding moment matrix are away from zero in the limit we may assure the existence of all the limits of the coefficients of the orthonormal polynomials with respect to such measures

inertial manifolds and completeness of eigenmodes for unsteady magnetic dynamos we prove the emergence of asymptotic spatial patterns in magnetic dynamos generated by unsteady fluid flows the patterns emerge because solutions of the dynamo equation converge exponentially to a time dependent inertial manifold this inertial manifold exists for general time aperiodic velocity fields under a spectral gap condition on the associated stokes operator for time periodic velocity fields we show that the inertial manifold is spanned by floquet eigenmodes that are analogous to the strange eigenmodes observed in the mixing of diffusive tracers this result gives an affirmative answer to the long standing question of completeness of floquet solutions in time periodic dynamo problems

rotation invariant moment problems an important theorem of marcel riesz cf 14 states that the polynomials are dense in l2 x when x is a determinate measure on the real line in the indeterminate case riesz also characterized the measures x for which the polynomials are dense in l2 they are the so called nevanlinna extremal measures introduced in nevanlinna 11 it does not seem to be known whether the polynomials are dense in l2 g when x is a determinate measure on r d d 1 cf the expository paper by fuglede 7 as well as the research problems book 8 p 529 where devinatz poses the problem as question 1 and ascribes it to the physicist john challifour 1978 in this paper we shall settle the question in the negative there exist rotation invariant measures on r d d 1 which are determinate but for which the polynomials are not dense in l2 p such measures x are necessarily of the following very special form

the index of determinacy for measures and the norm of orthonormal polynomials for determinate measures having moments of every order we define and study an index of determinacy which checks the stability of determinacy under multiplication by even powers of it z for z a complex number using this index of determinacy we solve the problem of determining for which z e c the sequence p m z n m e n belongs to t2 where pn n is the sequence of orthonormal polynomials associated with the measure u

on minimal sets of scalar parabolic equations with skew product structures skew product semi flow i t x x y x x y which is generated by ut uxx f y t ix u ux t 0 0 x 1 y e y d or n boundary conditions is considered where x is an appropriate subspace of h2 0 1 y r is a minimal flow with compact phase space it is shown that a minimal set e c x x y of ilt is an almost 1 1 extension of y that is set yo y e y i card e c p 1 y i is a residual subset of y where p x x yy is the natural projection consequently if y r is almost periodic minimal then any minimal set e c x x y of ilt is an almost automorphic minimal set it is also proved that dynamics of it is closed in the category of almost automorphy that is a minimal set e c x x y of fit is almost automorphic minimal if and only if y r is almost automorphic minimal asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed examples of nonalmost periodic almost automorphic minimal sets are provided

immersed interface methods for stokes flow with elastic boundaries or surface tension a second order accurate interface tracking method for the solution of incompressible stokes flow problems with moving interfaces on a uniform cartesian grid is presented the interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids the interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed the stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper siam j numer anal 31 1994 pp 1019 1044 the resulting velocities are interpolated to the interface to determine the motion of the interface an implicit quasi newton method is developed that allows reasonable time steps to be used

orthogonal polynomials and analytic functions associated to positive definite matrices abstract for a positive definite infinite matrix a we study the relationship between its associated sequence of orthonormal polynomials and the asymptotic behaviour of the smallest eigenvalue of its truncation a n of size n n for the particular case of a being a hankel or a hankel block matrix our results lead to a characterization of positive measures with finite index of determinacy and of completely indeterminate matrix moment problems respectively

globally positive solutions of linear parabolic pdes of second order with robin boundary conditions it is shown that globally positive solutions of a linear second order parabolic partial differential equation on a bounded domain with robin boundary conditions are unique up to multiplication by a positive constant

truncated newton methods and the modeling of complex immersed elastic structures truncated newton minimization methods are combined with peskin s immersed boundary method to facilitate investigation of the dynamic interaction between a viscous incompressible fluid and immersed elastic objects of complex structure applications to aquatic animal locomotion and platelet aggregation during blood clotting are presented 1993 john wiley sons inc

convergence to periodic solutions in autonomous reaction diffusion systems in one space dimension we consider the autonomous reaction diffusion system ddt uv uv f uv uv vu t 0 0 x 1 with dirichlet boundary conditions where f c1 r2 r and is positive we show a poincare bendixson theorem which means that all solutions converge either to zero or to a periodic orbit furthermore we determine stability properties of the periodic orbits

strange eigenmodes and decay of variance in the mixing of diffusive tracers we prove the existence of asymptotic spatial patterns for diffusive tracers advected by unsteady velocity fields the asymptotic patterns arise from convergence to a time dependent inertial manifold in the underlying advection diffusion equation for time periodic velocity fields we find that the inertial manifold is spanned by a finite number of floquet solutions the strange eigenmodes observed first numerically by pierrehumbert these strange eigenmodes only admit a regular asymptotic expansion in the diffusivity if the velocity field is completely integrable

averaged dynamics of time periodic advection diffusion equations in the limit of small diffusivity we study the effect of advection and small diffusion on passive tracers the advecting velocity field is assumed to have mean zero and to possess time periodic stream lines using a canonical transform to action angle variables followed by a lie transform we derive an averaged equation describing the effective motion of the tracers an estimate for the time validity of the first order approximation is established for particular cases of a regularized vortical flow we present explicit formulas for the coefficients of the averaged equation both at first and at second order numerical simulations indicate that the validity of the above first order estimate extends to the second order

an overview of periodic elliptic operators the article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas close attention is paid to studying analytic properties of bloch and fermi varieties which influence significantly most properties of such operators the approaches described are applicable not only to the standard model example of schrodinger operator with periodic electric potential v x but to a wide variety of elliptic periodic equations and systems equations on graphs operator and other operators on abelian coverings of compact bases many important applications are mentioned however due to the size restrictions they are not dealt with in details

exponential separation and principal floquet bundles for linear parabolic equations on general bounded domains nondivergence case we consider the dirichlet problem for linear nonautonomous second order parabolic equations of nondivergence type on general bounded domains with bounded measurable coefficients under such minimal regularity assumptions we establish the existence of a principal floquet bundle exponentially separated from a complementary invariant bundle as a special case of our main theorem assuming the coefficients are time periodic we obtain a new result on the existence of a principal eigenvalue of an associated time periodic parabolic eigenvalue problem we also show the existence of a uniform spectral gap between the principal eigenvalue and the rest of the spectrum for a class of time periodic uniformly parabolic operators finally we prove the uniqueness of positive entire solutions in the class of solutions whose supremum norms do not grow superexponentially as time goes to negative infinity

unconditionally stable discretizations of the immersed boundary equations the immersed boundary ib method is known to require small timesteps to maintain stability when solved with an explicit or approximately implicit method many implicit methods have been proposed to try to mitigate this timestep restriction but none are known to be unconditionally stable and the observed instability of even some of the fully implicit methods is not well understood in this paper we prove that particular backward euler and crank nicolson like discretizations of the nonlinear immersed boundary terms of the ib equations in conjunction with unsteady stokes flow can yield unconditionally stable methods we also show that the position at which the spreading and interpolation operators are evaluated is not relevant to stability so as long as both operators are evaluated at the same location in time and space we further demonstrate through computational tests that approximate projection methods which do not provide a discretely divergence free velocity field appear to have a stabilizing influence for these problems and that the implicit methods of this paper when used with the full navier stokes equations are no longer subject to such a strict timestep restriction and can be run up to the cfl constraint of the advection terms

the blob projection method for immersed boundary problems a new finite difference numerical method for modeling the interaction between flexible elastic membranes and an incompressible fluid in a two dimensional domain is presented the method differs from existing methods in the way the forces exerted by the membranes on the fluid are modeled these are described by a collection of regularized point forces and the velocity field they induce is computed directly on a regular cartesian grid via a smoothed dipole potential comparisons between this method and the immersed boundary method of peskin and mcqueen are presented the results show that the method proposed here preserves volumes better and has a higher order of convergence

small eigenvalues of large hermitian moment matrices abstract we consider an infinite hermitian positive definite matrix m which is the moment matrix associated with a measure with infinite and compact support on the complex plane we prove that if the polynomials are dense in l 2 then the smallest eigenvalue n of the truncated matrix m n of m of size n 1 n 1 tends to zero when n tends to infinity in the case of measures in the closed unit disk we obtain some related results

floquet bundles for scalar parabolic equations for linear scalar parabolic equations such as u t u xx a t x u x b t x u on a finite interval 0 x with various boundary conditions we obtain canonical floquet solutions u n t x these solutions are characterized by the property that z u n t x n for all te where z denotes the zero crossing lap number of matano the coefficients a t x and b t x are not assumed to be periodic in t but if they are the solutions u n t x reduce to the standard floquet solutions our results may naturally be expressed in the language of linear skew product flows in this context we obtain for each n 1 an exponential dichotomy between the bundles span u n 1 and overline span u n cdot cdot n n 1 infty

invariant foliations near normally hyperbolic invariant manifolds for semiflows let m be a compact c1 manifold which is invariant and normally hyperbolic with respect to a c1 semiflow in a banach space then in an neighborhood of m there exist local c1 center stable and center unstable manifolds w cs and w cu respectively here we show that w cs and w cu may each be decomposed into the disjoint union of c1 submanifolds leaves in such a way that the semiflow takes leaves into leaves of the same collection furthermore each leaf intersects m in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time

analysis of stiffness in the immersed boundary method and implications for time stepping schemes the immersed boundary method is known to exhibit a high degree of numerical stiffness associated with the interaction of immersed elastic fibres with the surrounding fluid we perform a linear analysis of the underlying equations of motion for immersed fibres and identify a discrete set of fibre modes which are associated solely with the presence of the fibre this work generalises our results in a previous paper 1995 siam j appl math 55 1577 by incorporating the effect of spreading the singular fibre force over a finite smoothing radius corresponding to the approximate delta function used in the immersed boundary method we investigate the stability of the fibre modes their stiffness and their dependence on the problem parameters focusing on the influence of smoothing we then extend the analytical results to include the effect of time discretisation and draw conclusions about the time step restrictions on various explicit schemes as well as the convergence rates for an iterative semi implicit method we draw comparisons with computations and show how the results can be applied to help in choosing alternate time stepping schemes that are specially tailored to handle the stiffness in immersed fibres in particular we present numerical results that show how fully explicit runge kutta schemes perform in comparison with the best of the semi implicit schemes currently in use

globally positive solutions of linear parabolic partial differential equations of second order with dirichlet boundary conditions it is shown that globally positive solutions of a linear second order parabolic partial differential equation on a bounded domain with dirichlet boundary conditions are unique up to multiplication by a positive constant

improved volume conservation in the computation of flows with immersed elastic boundaries this paper introduces a recipe for the construction of a finite difference divergence operator the coefficients of which are completely determined once an interpolation scheme has been chosen substitution of this divergence operator for the previously used divergence based on central differences makes a dramatic improvement in the overall volume conservation that is observed in an immersed boundary computation this improvement is particularly important for computations in which an elastic boundary separates chambers containing fluid at substantially different pressures a situation which is prominent in cardiac fluid dynamics during the contraction of the ventricles

orthogonal matrix polynomials and higher order recurrence relations abstract it is well known that orthogonal polynomials on the real line satisfy a three term recurrence relation and conversely every system of polynomials satisfying a three term recurrence relation is orthogonal with respect to some positive borel measure on the real line we extend this result and show that every system of polynomials satisfying some 2 n 1 term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are n n matrices we apply this result to polynomials orthogonal with respect to a discrete sobolev inner product and other inner products in the linear space of polynomials as an application we give a short proof of krein s characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points

systems of differential delay equations floquet multipliers and discrete lyapunov functions abstract we define a discrete integer valued lyapunov function v for cyclic nearest neighbor systems of differential delay equations possessing a feedback condition this extends analogous definitions for cyclic systems of ode s and for scalar differential delay equations we relate the values of v to the real parts of the floquet multipliers for such linear periodic systems and thereby prove all floquet subspaces are at most two dimensional