krylov methods for the incompressible navier stokes equations methods are presented for time evolution steady state solving and linear stability analysis for the incompressible navier stokes equations at low to moderate reynolds numbers the methods use krylov subspaces constructed by the arnoldi process from actions of the explicit navier stokes right hand side and of its jacobian without inversion of the viscous operator time evolution is performed by a nonlinear extension of the method of exponential propagation steady states are calculated by inexact krylov newton iteration using orthores and gmres linear stability analysis is carried out using an implicitly restarted arnoldi process with implicit polynomial filters a detailed implementation is described for a pseudospectral calculation of the stability of taylor vortices with respect to wavy vortices in the couette taylor problem 9 1994 academic press inc

on helson matrices moment problems non negativity boundedness and finite rank we study helson matrices also known as multiplicative hankel matrices that is infinite matrices of the form m nm n m 1m nm n m 1 where is a sequence of complex numbers helson matrices are considered as linear operators on l2 n l2 n by interpreting helson matrices as hankel matrices in countably many variables we use the theory of multivariate moment problems to show that m is non negative if and only if is the moment sequence of a measure on r assuming that does not grow too fast we then characterize the non negative bounded helson matrices m as those where the corresponding moment measures are carleson measures for the hardy space of countably many variables finally we give a complete description of the helson matrices of finite rank in parallel with the classical kronecker theorem on hankel matrices

banach spaces of general dirichlet series abstract we study when the spaces of general dirichlet series bounded on a half plane are banach spaces and show that some of those classes are isometrically isomorphic between themselves in a precise way let n be a strictly increasing sequence of positive real numbers such that lim n n we denote by h n the complex normed space of all dirichlet series d s n b n n s which are convergent and bounded on the half plane re s 0 endowed with the norm d sup re s 0 d s if there exists q 0 such that inf n n 1 q n q 0 then h n is a banach space further if there exists a strictly increasing sequence r n of positive numbers such that the sequence log r n is q linearly independent n r for n p and n is the increasing rearrangement of the sequence n then h n is isometrically isomorphic to h b c 0 with this condition we explain more explicitly the optimal cases of the difference among the abscissas c b u and a

zeros of functions in hilbert spaces of dirichlet series the dirichlet hardy space fancyscript h 2 consists of those dirichlet series sum n a n n s for which sum n a n 2 1 2 is a necessary and sufficient condition for the existence of a nontrivial function f in fancyscript h 2 vanishing on a given bounded sequence the proof implies in fact a stronger result every function in the hardy space h 2 of the half plane text re s 1 2 can be interpolated by a function in fancyscript h 2 on such a blaschke sequence analogous results are proved for the hilbert space fancyscript d alpha of dirichlet series sum n a n n s for which sum n a n 2 d n alpha 1 2 partial results are then obtained for the zeros of functions in fancyscript h p l p analogues of fancyscript h 2 for 2 p infty based on certain contractive embeddings of fancyscript d alpha in fancyscript h p

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

norms of composition operators on the h2 space of dirichlet series abstract we consider composition operators c on the hardy space of dirichlet series h 2 generated by dirichlet series symbols we prove two different subordination principles for such operators one concerns affine symbols only and is based on an arithmetical condition on the coefficients of the other concerns general symbols and is based on a geometrical condition on the boundary values of both principles are strict in the sense that they characterize the composition operators of maximal norm generated by symbols having given mapping properties in particular we generalize a result of j h shapiro on the norm of composition operators on the classical hardy space of the unit disc based on our techniques we also improve the recently established upper and lower norm bounds in the special case that s c r 2 s a number of other examples are given

lower bounds for the maximum of the riemann zeta function along vertical lines let alpha in 1 2 1 be fixed we prove that begin aligned max 0 le t le t zeta alpha it ge exp left frac c alpha log t 1 alpha log log t alpha right end aligned for all sufficiently large t where we can choose c alpha 0 18 2 alpha 1 1 alpha the same result has already been obtained by montgomery with a smaller value for c alpha however our proof which uses a modified version of soundararajan s resonance method together with ideas of hilberdink is completely different from montgomery s this new proof also allows us to obtain lower bounds for the measure of those t in 0 t for which zeta alpha it is of the order mentioned above

the cesaro space of dirichlet series and its multiplier algebra abstract we consider the space h c e s p of all dirichlet series whose coefficients belong to the cesaro sequence space c e s p consisting of all complex sequences whose absolute cesaro means are in l p for 1 p it is a banach space of analytic functions for which we study the maximal domain of analyticity and the boundedness of point evaluations we identify the algebra of analytic multipliers on h c e s p as the wiener algebra of dirichlet series shifted to the vertical half plane c 1 q s c s 1 q where 1 p 1 q 1

addendum to a hilbert space of dirichlet series and systems of dilated functions in l 2 0 1 addendum to a hilbert space of dirichlet series and systems of dilated functions in l 2 0 1

contractive inequalities for bergman spaces and multiplicative hankel forms we consider sharp inequalities for bergman spaces of the unit disc establishing analogues of the inequality in carleman s proof of the isoperimetric inequality and of weissler s inequality for dilations by contractivity and a standard tensorization procedure the unit disc inequalities yield corresponding inequalities for the bergman spaces of dirichlet series we use these results to study weighted multiplicative hankel forms associated with the bergman spaces of dirichlet series reproducing most of the known results on multiplicative hankel forms associated with the hardy spaces of dirichlet series in addition we find a direct relationship between the two type of forms which does not exist in lower dimensions finally we produce some counter examples concerning carleson measures on the infinite polydisc

local properties of hilbert spaces of dirichlet series we show that the asymptotic behavior of the partial sums of a sequence of positive numbers determine the local behavior of the hilbert space of dirichlet series defined using these as weights this extends results recently obtained describing the local behavior of dirichlet series with square summable coefficients in terms of local integrability boundary behavior carleson measures and interpolating sequences as these spaces can be identified with functions spaces on the infinite dimensional polydisk this gives new results on the dirichlet and bergman spaces on the infinite dimensional polydisk as well as the scale of besov sobolev spaces containing the drury arveson space on the infinite dimensional unit ball we use both techniques from the theory of sampling in paley wiener spaces and classical results from analytic number theory c 2011 elsevier inc all rights reserved less

axisymmetric surface diffusion dynamics and stability of self similar pinchoff the dynamics of surface diffusion describes the motion of a surface with its normal velocity given by the surface laplacian of its mean curvature this flow conserves the volume enclosed inside the surface while minimizing its surface area we review the axisymmetric equilibria the cylinder sphere and the delaunay unduloid the sphere is stable while the cylinder is long wave unstable a subcritical bifurcation from the cylinder produces a continuous family of unduloid solutions we present computations that suggest that the stable manifold of the unduloid forms a separatrix between states that relax to the cylinder in infinite time and those that tend toward finite time pinchoff we examine the structure of the pinchoff showing it has self similar structure using asymptotic numerical and analytical methods in addition to a previously known similarity solution we find a countable set of similarity solutions each with a different asymptotic cone angle we develop a stability theory in similarity variables that selects the original similarity solution as the only linearly stable one and consequently the only observable solution we also consider similarity solutions describing the dynamics after the topological transition

riesz means in hardy spaces on dirichlet groups given a frequency lambda lambda n we study when almost all vertical limits of a mathcal h 1 dirichlet series sum a n e lambda ns are riesz summable almost everywhere on the imaginary axis equivalently this means to investigate almost everywhere convergence of fourier series of h 1 functions on so called lambda dirichlet groups and as our main technical tool we need to invent a weak type 1 infty hardy littlewood maximal operator for such groups applications are given to h 1 functions on the infinite dimensional torus mathbb t infty ordinary dirichlet series sum a n n s as well as bounded and holomorphic functions on the open right half plane which are uniformly almost periodic on every vertical line

on rotationally symmetric mean curvature flow the evolutionary motion of surfaces by their mean curvature has been studied by several authors from different points of view a measure theoretic approach is studied by k a brakke b the classical parametric problem is studied by g huisken hl non parametric mean curvature evolution with boundary conditions is treated in h2 for a detailed discussion see h3 we restrict our attention to the case in which the initial surface has rotational symmetry we shall show that under suitable initial conditions the solution degenerates in the sense that its curvature develops a singularity at exactly one point this confirms numerical results of the first author d for general surfaces in r3 see also the work of m a grayson gl p 286 g2 the proofs rely on the parabolic maximum principle the second author is gratefully indebted to s angencnt for bringing the problem to his attention and for pointing out the resemblance to so called quenching problems this research was financially via sfb 256 and sfb 123

intrinsic difficulties in stochastic control of unstable convolution operators on z stochastic control of causal linear unstable convolution operators on the full time axis is shown to be impossible the basic plant y gu and controller descriptions u cy via possibly unstable convolution operators or via their unstable transforms have been shown to possess severe limitations in stochastic control studies on the full time axis z and on z t in fact they have also similar limitations on n when g or c is noncausal and unstable such situations can appear in applications in which the independent variable is a spatial variable rather than time

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

basis properties of the p q sine functions we improve the currently known thresholds for basisness of the family of periodically dilated p q sine functions our findings rely on a beurling decomposition of the corresponding change of coordinates in terms of shift operators of infinite multiplicity we also determine refined bounds on the riesz constant associated with this family these results seal mathematical gaps in the existing literature on the subject

the implicit closest point method for the numerical solution of partial differential equations on surfaces many applications in the natural and applied sciences require the solutions of partial differential equations pdes on surfaces or more general manifolds the closest point method is a simple and accurate embedding method for numerically approximating pdes on rather general smooth surfaces however the original formulation is designed to use explicit time stepping this may lead to a strict time step restriction for some important pdes such as those involving the laplace beltrami operator or higher order derivative operators to achieve improved stability and efficiency we introduce a new implicit closest point method for surface pdes the method allows for large stable time steps while retaining the principal benefits of the original method in particular it maintains the order of accuracy of the discretization of the underlying embedding pde it works on sharply defined bands without degrading the accuracy of the method and it applies to general smooth surfaces it also is very simple and may be applied to a rather general class of surface pdes convergence studies for the in surface heat equation and a fourth order biharmonic problem are given to illustrate the accuracy of the method we demonstrate the flexibility and generality of the method by treating flows involving diffusion reaction diffusion and fourth order spatial derivatives on a variety of interesting surfaces including surfaces of mixed codimension

level set jet schemes for stiff advection equations the semijet method abstract many interfacial phenomena in physical and biological systems are dominated by high order geometric quantities such as curvature here a semi implicit method is combined with a level set jet scheme to handle stiff nonlinear advection problems the new method offers an improvement over the semi implicit gradient augmented level set method previously introduced by requiring only one smoothing step when updating the level set jet function while still preserving the underlying methods higher accuracy sample results demonstrate that accuracy is not sacrificed while strict time step restrictions can be avoided

projective methods for stiff differential equations problems with gaps in their eigenvalue spectrum we show that there exist classes of explicit numerical integration methods that can handle very stiff problems if the eigenvalues are separated into two clusters one containing the stiff or fast components and one containing the slow components these methods have large average step sizes relative to the fast components conventional implicit methods involve the solution of nonlinear equations at each step which for large problems requires significant communication between processors on a multiprocessor machine for such problems the methods proposed here have significant potential for speed improvement

complex symmetric composition operators on a hilbert space of dirichlet series abstract on the hilbert space of dirichlet series with square summable coefficients there are no non normal complex symmetric composition operators induced by non constant symbols this is in sharp contrast with the phenomenon on the classical hardy space over the unit disk

the explicit implicit null method a general method to remove the numerical instability of partial differential equations is presented two equal terms are added to and subtracted from the right hand side of the pde the first is a damping term and is treated implicitly the second is treated explicitly a criterion for absolute stability is found and the scheme is shown to be convergent the method is applied with success to the mean curvature flow equation the kuramoto sivashinsky equation and to the rayleigh taylor instability in a hele shaw cell including the effect of surface tension

removing the stiffness from interfacial flows with surface tension a new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two dimensional irrotational and incompressible fluids through the laplace young condition at the interface surface tension introduces high order terms both nonlinear and nonlocal into the dynamics this leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult this new formulation has all the nice properties for time integration methods that are associated with having a linear constant coefficient highest order term that is using this formulation we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in fourier space the approach is based on a boundary integral formulation and applies more generally even to problems beyond the fluid mechanical context here they are applied to computing with high resolution the motion of interfaces in hele shaw flows and the motion of free surfaces in inviscid flows governed by the euler equations one hele shaw computation shows the behavior of an expanding gas bubble over long time as the interface undergoes successive tip splittings and finger competition a second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification in euler flows the computation of a vortex sheet shows its roll up through the kelvin helmholtz instability this motion culminates in the late time self intersection of the interface creating trapped bubbles of fluid this is we believe a type of singularity formation previously unobserved for such flows in 2d finally computations of falling plumes in an unstably stratified boussinesq fluid show a very similar behavior

a general view of multipliers and composition operators this paper defines a multiplier pair which is a very general setting in which the notions of multipliers and composition operators can be studied we prove some results in this general setting and we present many new interesting examples in which these notions can be interpreted

a hardy space analysis of the baez duarte criterion for the rh abstract in this article methods from sub hardy hilbert spaces such as the de branges rovnyak spaces and local dirichlet spaces are used to investigate baez duarte s hilbert space reformulation of the riemann hypothesis rh

modified zeta functions as kernels of integral operators abstract the modified zeta functions n k n s where k n converge absolutely for re s 1 these generalise the riemann zeta function which is known to have a meromorphic continuation to all of c with a single pole at s 1 our main result is a characterisation of the modified zeta functions that have pole like behaviour at this point this behaviour is defined by considering the modified zeta functions as kernels of certain integral operators on the spaces l 2 i for symmetric and bounded intervals i r we also consider the special case when the set k n is assumed to have arithmetic structure in particular we look at local l p integrability properties of the modified zeta functions on the abscissa re s 1 for p 1