analysis of multiple scattering iterations for high frequency scattering problems i the two dimensional case we present an analysis of a recently proposed integral equation method for the solution of high frequency electromagnetic and acoustic scattering problems that delivers error controllable solutions in frequency independent computational times within single scattering configurations the method is based on the use of an appropriate ansatz for the unknown surface densities and on suitable extensions of the method of stationary phase the extension to multiple scattering configurations in turn is attained through consideration of an iterative neumann series that successively accounts for further geometrical wave reflections as we show for a collection of two dimensional cylindrical convex obstacles this series can be rearranged into a sum of periodic orbits of increasing period each corresponding to contributions arising from waves that reflect off a fixed subset of scatterers when these are transversed sequentially in a periodic manner here we analyze the properties of these periodic orbits in the high frequency regime by deriving precise asymptotic expansions for the currents i e the normal derivative of the fields that they induce on the surface of the obstacles as we demonstrate these expansions can be used to provide accurate estimates of the rate at which their magnitude decreases with increasing number of reflections which defines the overall rate of convergence of the multiple scattering series moreover we show that the detailed asymptotic knowledge of these currents can be used to accelerate this convergence and thus to reduce the number of iterations necessary to attain a prescribed accuracy

on a class of functionals invariant under a zn action abstract this document considers a system of ordinary differential equations of the form q v sub q t q f t where f and v are periodic in t v is periodic in the components of q q sub 1 q sub m and the mean value of f vanishes by showing that a corresponding functional is invariant under a natural z sub n action a simple variational argument yields at least n 1 distinct periodic solutions of more general versions of are also treated as is a class of neumann problems for semilinear elliptic partial differential equations

a stable runge kutta methods for semilinear evolution equations we consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of hilbert spaces in this setting we prove the existence of solutions which are temporally smooth in the norm of the lowest rung of the scale for an open set of initial data on the highest rung of the scale under the same assumptions we prove that a class of implicit a stable runge kutta semidiscretizations in time of such equations are smooth as maps from open subsets of the highest rung into the lowest rung of the scale under the additional assumption that the linear part of the evolution equation is normal or sectorial we prove full order convergence of the semidiscretization in time for initial data on open sets our results apply in particular to the semilinear wave equation and to the nonlinear schrodinger equation 2012 elsevier inc

energy concentration in band limited extrapolation the authors present a novel bandlimited extrapolation method that incorporates qualitative prior knowledge of the distribution of signal energy in both the time domain spatial domain and the frequency domain the extrapolation is easily computed by simple noniterative algorithm the prior information is incorporated by means of a weighted energy measure and the resulting solution is derived by straightforward hilbert space optimization the algorithm accommodates complex valued sequences nonuniform sampling and arbitrary spectral support and is applicable in one or several dimensions many existing extrapolation algorithms are noted as special cases examples of signal extrapolation and spectrum estimation using the new algorithm are given in one and two dimensions algorithm modification for noisy data is investigated and error bounds are presented

proof of a mckean conjecture on the rate of convergence of boltzmann equation solutions the present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous boltzmann equation for maxwellian molecules under really mild conditions on the initial datum and a weak physically consistent angular cutoff hypothesis our main result theorem 1 contains the first precise statement that the total variation distance between the solution and the limiting maxwellian distribution admits an upper bound of the form c e varlambda b t varlambda b being the least negative eigenvalue of the linearized collision operator and c a constant depending only on the initial datum the validity of this quantification was conjectured about fifty years ago by henry p mckean as to the proof of our results we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory highlighted by mckean

semilinear pseudodifferential equations in spaces of tempered ultradistributions abstract we study a class of semilinear elliptic equations on spaces of tempered ultradistributions of beurling and roumieu type assuming that the linear part of the equation is a pseudodifferential operator of infinite order satisfying a suitable ellipticity condition we prove a regularity result in the functional setting above for weak sobolev type solutions

a constructive approach to regularity of lagrangian trajectories for incompressible euler flow in a bounded domain the 3d incompressible euler equations are an important research topic in the mathematical study of fluid dynamics not only is the global regularity for smooth initial data an open issue but the behaviour may also depend on the presence or absence of boundaries for a good understanding it is crucial to carry out besides mathematical studies high accuracy and well resolved numerical exploration such studies can be very demanding in computational resources but recently it has been shown that very substantial gains can be achieved first by using cauchy s lagrangian formulation of the euler equations and second by taking advantage of analyticity results of the lagrangian trajectories for flows whose initial vorticity is holder continuous the latter has been known for about 20 years serfati in j math pures appl 74 95 104 1995 but the combination of the two which makes use of recursion relations among time taylor coefficients to obtain constructively the time taylor series of the lagrangian map has been achieved only recently frisch and zheligovsky in commun math phys 326 499 505 2014 podvigina et al in j comput phys 306 320 342 2016 and references therein here we extend this methodology to incompressible euler flow in an impermeable bounded domain whose boundary may be either analytic or have a regularity between indefinite differentiability and analyticity non constructive regularity results for these cases have already been obtained by glass et al ann sci ec norm sup 45 1 51 2012 using the invariance of the boundary under the lagrangian flow we establish novel recursion relations that include contributions from the boundary this leads to a constructive proof of time analyticity of the lagrangian trajectories with analytic boundaries which can then be used subsequently for the design of a very high order cauchy lagrangian method

initial value problems for creeping flow of maxwell fluids abstract we consider the flow of nonlinear maxwell fluids in the unsteady quasistatic case where the effect of inertia is neglected we study the well posedness of the resulting pde initial boundary value problem locally in time this well posedness depends on the unique solvability of an elliptic boundary value problem we first present results for the 3d case with sufficiently small initial data and for a simple shear flow problem with arbitrary initial data after that we extend our results to some 3d flow problems with large initial data we solve our problem using an iteration between linear subproblems the limit of the iteration provides the solution of our original problem

off spectral analysis of bergman kernels the asymptotic analysis of bergman kernels with respect to exponentially varying measures near emergent interfaces has attracted recent attention such interfaces typically occur when the associated limiting bergman density function vanishes on a portion of the plane the off spectral region this type of behavior is observed when the metric is negatively curved somewhere or when we study partial bergman kernels in the context of positively curved metrics in this work we cover these two situations in a unified way for exponentially varying weights on the complex plane we obtain a uniform asymptotic expansion of the coherent state of depthn rooted at an off spectral point which we also refer to as the root function at the point in question the expansion is valid in the entire off spectral component containing the root point and protrudes into the spectrum as well this allows us to obtain error function transition behavior of the density of states along the smooth interface previous work on asymptotic expansions of bergman kernels is typically local and valid only in the bulk region of the spectrum which contrasts with our non local expansions

an optimal basis of band limited functions for signal analysis and design this paper studies signal concentration in the time and frequency domains using the general constrained variational method of franks the minimum k th k 0 2 4 moment time duration measure for band limited signals is formulated a complete orthonormal set of band limited functions in l2 w w with the minimum fourth moment time duration measure is obtained numerical investigations of our optimal 4th moment functions show 1 less energy concentration in the main lobe than the prolate spheroidal wave functions pswf 2 higher energy concentration in the main lobe than gabor s function and 3 a larger main lobe than both pswf and gabor applications for our basis functions include 1 radar systems and high resolution communication systems and 2 representation and approximation to any band limited signal in a given time interval

partial linearization for planar nonautonomous differential equations abstract we prove a partial linearization theorem for planar nonautonomous differential equations with one center like direction and one hyperbolic direction gap conditions are formulated in terms of the dichotomy spectral intervals the result is applied to the duffing van der pol oscillator under a nonautonomous parametric perturbation

optimum waveform signal sets with amplitude and energy constraints it is desirable to choose the waveforms making up a signaling alphabet so that they are maximally separated one from another this problem is considered in the space of square integrable functions for signals which have finite duration and are constrained in the ranges of their values as well as in energy corresponding to each of the following cases we establish sharp bounds for the minimum distance and for the average distance between elements of a fixed size signal set and construct sets of signals that attain both bounds simultaneously begin list item em case a energy constraint only the average energy of the waveforms in the signal set is at most sigma where 0 leq sigma item em case b energy and peak amplitude constraints the average energy of the waveforms in the signal set is leq sigma 0 leq sigma and the absolute value of each waveform is at most 1 item em case c energy and value constraints the average energy of the waveforms in the signal set is at most b 2 sigma a 2 1 sigma and each waveform takes values in the set a b where 0 leq a and 0 leq sigma leq 1 end list cases a and b are applicable to signal design for communication in channels with additive noise say gaussian and case c is applicable to signal design for optical channels where the signal represents the intensity of a photon stream the general character of the results is that the minimum distance behaves like gamma sigma in cases a and b and like gamma sigma 1 sigma in case c with gamma a suitable constant

solutions blowing up on any given compact set for the energy subcritical wave equation abstract we consider the focusing energy subcritical nonlinear wave equation t t u u u p 1 u in r n n 1 given any compact set k r n we construct finite energy solutions which blow up at t 0 exactly on k the construction is based on an appropriate ansatz the initial ansatz is simply u 0 t x t a x 2 p 1 where a 0 vanishes exactly on k which is a solution of the ode h h p we refine this first ansatz inductively using only ode techniques and taking advantage of the fact that for suitably chosen a space derivatives are negligible with respect to time derivatives we complete the proof by an energy argument and a compactness method

asymptotic expansion of the dulac map and time for unfoldings of hyperbolic saddles local setting abstract in this paper we study unfoldings of planar vector fields in a neighbourhood of a hyperbolic resonant saddle we give a structure theorem for the asymptotic expansion of the local dulac time as well as the local dulac map with the remainder uniformly flat with respect to the unfolding parameters here local means close enough to the saddle in order that the normalizing coordinates provided by a suitable normal form can be used the principal part of the asymptotic expansion is given in a monomial scale containing a deformation of the logarithm the so called roussarie ecalle compensator especial attention is paid to the remainder s properties concerning the derivation with respect to the unfolding parameters

natural statistics for spectral samples spectral sampling is associated with the group of unitary transformations acting on matrices in much the same way that simple random sampling is associated with the symmetric group acting on vectors this parallel extends to symmetric functions k statistics and polykays we construct spectral k statistics as unbiased estimators of cumulants of trace powers of a suitable random matrix moreover we define normalized spectral polykays in such a way that when the sampling is from an infinite population they return products of free cumulants

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

exterior powers of the adjoint representation and the weyl ring of e8 abstract i derive explicitly all polynomial relations in the character ring of e 8 of the form k e 8 p k 1 8 0 where k e 8 is an arbitrary exterior power of the adjoint representation and i is the ith fundamental character this has simultaneous implications for the theory of relativistic integrable systems seiberg witten theory quantum topology orbifold gromov witten theory and the arithmetic of elliptic curves the solution is obtained by reducing the problem to a large but finite dimensional linear problem which is amenable to an efficient solution via distributed computation

on the periodic nonlinearity and the multiplicity of solutions abstract this document studies the multiplicity of solutions for semilinear elliptic systems as well as hamiltonian systems in which the nonlinear terms are periodic in certain variables the cuplength for cohomology rings of the torus is used our results generalize and unify several recent works by conley zehnder rabinowitz mawhin willem pucci serrin etc in particular the resonance problems and indefinite problems are studied keywords critical point neumann problem periodic solution

sobolev regularity of the equation on the hartogs triangle the regularity of the overline partial problem on the domain left z 1 right left z 2 right 1 in mathbb c 2 is studied using l 2 methods estimates are obtained for the canonical solution in weighted l 2 sobolev spaces with a weight that is singular at the point 0 0 in particular the singularity of the bergman projection for the hartogs triangle is contained at the singular point and it does not propagate

smooth invariant manifolds in banach spaces with nonuniform exponential dichotomy we establish the existence of smooth stable manifolds for semiflows defined by ordinary differential equations v a t v f t v in banach spaces assuming that the linear equation v a t v admits a nonuniform exponential dichotomy our proof of the ck smoothness of the manifolds uses a single fixed point problem in the unit ball of the space of ck functions with holder continuous kth derivative this is a closed subset of the space of continuous functions with the supremum norm by an apparently not so well known lemma of henry see proposition 3 the estimates showing that the functions maintain the original bounds when transformed under the fixed point operator are obtained through a careful application of the faa di bruno formula for the higher derivatives of the compositions see 31 and 35 as a consequence we obtain in a direct manner not only the exponential decay of solutions along the stable manifolds but also of their derivatives up to order k when the vector field is of class ck

ultradistributional boundary values of harmonic functions on the sphere abstract we present a theory of ultradistributional boundary values for harmonic functions defined on the euclidean unit ball we also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions to this end we obtain explicit estimates for partial derivatives of spherical harmonics which are of independent interest and refine earlier estimates by calderon and zygmund we apply our results to characterize the support of ultradistributions on the sphere via abel summability of their spherical harmonic expansions

smooth center manifolds for nonuniformly partially hyperbolic trajectories abstract we establish the existence of unique smooth center manifolds for ordinary differential equations v a t v f t v in banach spaces assuming that v a t v admits a nonuniform exponential trichotomy this allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories in addition we prove that the center manifolds are as regular as the vector field our proof of the c k smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space to the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting

concavity of the mutual information rate for input restricted memoryless channels at high snr we consider a memoryless channel with an input markov process supported on a mixing finite type constraint we continue the development of asymptotics for the entropy rate of the output hidden markov chain and deduce that at high signal to noise ratio the mutual information rate of such a channel is concave with respect to almost all input markov chains of a given order

an optimum complete orthonormal basis for signal analysis and design derives a new set of basis functions which have high energy concentration in both time and frequency the approach is based on transforming a signal energy maximization problem into an equivalent eigenvalue problem for a certain integral operator and then showing that the eigenvectors of this integral operator form a complete orthonormal basis this basis set is in some sense optimal since each basis function has the highest energy concentration within a certain subspace of spl lscr sup 2 furthermore this basis can be employed to analyze and design communication signals for which the energy concentration within the essential duration and essential bandwidth is important and some other optimum characteristics would also be required

a new algorithm for computing the multivariate faa di bruno s formula abstract a new algorithm for computing the multivariate faa di bruno s formula is provided we use a symbolic approach based on the classical umbral calculus that turns the computation of the multivariate faa di bruno s formula into a suitable multinomial expansion we propose a maple procedure whose computational times are faster compared with the ones existing in the literature some illustrative applications are also provided

comparison of fast boundary element methods on parametric surfaces we compare fast black box boundary element methods on parametric surfaces in r3 these are the adaptive cross approximation the multipole method based on interpolation and the wavelet galerkin scheme the surface representation by a piecewise smooth parameterization is in contrast to the common approximation of surfaces by panels nonetheless parametric surface representations are easily accessible from computer aided design cad and are recently topic of the studies in isogeometric analysis especially we can apply two dimensional interpolation in the multipole method a main feature of this approach is that the cluster bases and the respective moment matrices are independent of the geometry this results in a superior compression of the far field compared to other cluster methods

approximate solutions and micro regularity in the denjoy carleman classes abstract we begin with a sequence m of positive real numbers and we consider the denjoy carleman class c m we show how to construct m approximate solutions for complex vector fields with c m coefficients we then use our construction to study micro local properties of boundary values of approximate solutions in general m involutive structures of codimension one where the approximate solution is defined in a wedge whose edge where the boundary value exists is a maximally real submanifold we also obtain a c m version of the edge of the wedge theorem

divided differences of implicit functions under general conditions the equation g x y 0 implic itly denes y locally as a function of x in this article we express divided dierences of y in terms of bivariate divided dierences of g generalizing a recent result on divided dierences of inverse functions

a generalized saddle point theorem abstract the well known saddle point theorem is extended to the case of functions defined on a product space x v where x is a banach space and v is a compact manifold under some linking conditions the existence of at least cuplength v 1 critical points is proved the abstract theorems are applied to the existence problems of periodic solutions of hamiltonian systems with periodic nonlinearity and or resonance

derivatives of entropy rate in special families of hidden markov chains consider a hidden markov chain obtained as the observation process of an ordinary markov chain corrupted by noise recently zuk et al showed how in principle one can explicitly compute the derivatives of the entropy rate of at extreme values of the noise namely they showed that the derivatives of standard upper approximations to the entropy rate actually stabilize at an explicit finite time we generalize this result to a natural class of hidden markov chains called black holes we also discuss in depth special cases of binary markov chains observed in binary symmetric noise and give an abstract formula for the first derivative in terms of a measure on the simplex due to blackwell

on the computation of classical boolean and free cumulants abstract this paper introduces a simple and computationally efficient algorithm for conversion formulae between moments and cumulants the algorithm provides just one formula for classical boolean and free cumulants this is realized by using a suitable polynomial representation of abel polynomials the algorithm relies on the classical umbral calculus a symbolic language introduced by rota and taylor g c rota b d taylor the classical umbral calculus siam j math anal 25 2 1994 694 711 that is particularly suited to be implemented by using software for symbolic computations here we give a maple procedure comparisons with existing procedures especially for conversions between moments and free cumulants as well as examples of applications to some well known distributions classical and free end the paper

asymptotic expansion of multivariate conservative linear operators in this paper we study the asymptotic expansion of the partial derivatives of a sequence of linear conservative operators we find sufficient conditions on the sequence that guarantee that such an expansion can be derived from the asymptotic formula for the nondifferentiated sequence our results are valid even if the operators of the sequence present difficult conservative properties different form the classical preservation of the usual convexities we use our theorems to obtain the complete asymptotic expansions and voronovskaja formulae for the partial derivatives of multivariate versions of the meyer konig and zeller operators and the bleimann butzer and hahn operators

analysis of the domain mapping method for elliptic diffusion problems on random domains in this article we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains in particular based on the decay of the karhunen loeve expansion of the domain perturbation field we establish decay rates for the derivatives of the random solution that are independent of the stochastic dimension for the implementation of a related approximation scheme like quasi monte carlo quadrature stochastic collocation etc we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field this simplifies the implementation and yields a non intrusive approach having this machinery at hand we can easily transfer it to stochastic interface problems the theoretical findings are complemented by numerical examples for both stochastic interface problems and boundary value problems on random domains

a multivariate faa di bruno formula with applications a multivariate faa di bruno formula for computing arbitrary partial derivatives of a function composition is presented it is shown by way of a general identity how such derivatives can also be expressed in the form of an infinite series applications to stochastic processes and multivariate cumulants are then delineated

periodic solutions of spatially periodic hamiltonian systems abstract this work is concerned with the study of existence and multiplicity of periodic solutions of hamiltonian systems of ordinary differential equations z j hz z t f t when the hamiltonian h z t h p q t is periodic in the variable q and superlinear in the variable p by imposing a growth condition on the derivative of h we obtain the existence of at least n 1 periodic solutions where n is the dimension of the system the existence of periodic solutions is obtained by using a saddle point theorem recently proved by lui we consider a functional over e x m where e is a hilbert space and m is a compact manifold satisfying a saddle point condition on e uniformly on m we present a proof of this saddle point theorem using standard minimax techniques based on the cup length of m

a kam theorem for finitely differentiable hamiltonian systems abstract given l 2 2 d 4 we prove the persistence of a cantor family of kam tori of measure o e 1 2 l for any non degenerate nearly integrable hamiltonian system of class c l d t d where 1 is the diophantine power of the frequencies of the persistent kam tori and d r d is a bounded domain provided that the size e of the perturbation is sufficiently small this extends a result by d salamon in 25 according to which we do have the persistence of a single kam torus in the same framework moreover as for the persistence of a single torus the regularity assumption is essentially optimal

the fbi transforms and their use in microlocal analysis abstract using a more general class of fbi transforms introduced by s berhanu and j hounie in 12 we completely characterize regularity and microregularity in denjoy carleman non quasi analytic classes which includes the gevrey classes and m christ version of the fbi transform defined in 22 as examples we also exhibit a result on microlocal regularity for solutions of first order partial differential equations in these classes that do not seem possible to prove using the classical fbi transform

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

center manifolds for infinite delay we obtain a c1 center manifold theorem for perturbations of delay difference equations in banach spaces with infinite delay our results extend in several directions the existing center manifold theorems besides considering infinite delay equations we consider perturbations of nonuniform exponential trichotomies and generalized trichotomies that may exhibit stable unstable and central behaviors with respect to arbitrary asymptotic rates ec n for some diverging sequence n this includes as a very special case the usual exponential behavior with n n