birkhoff normal form for splitting methods applied to semilinear hamiltonian pdes part ii abstract splitting we consider hamiltonian pdes that can be split into a linear unbounded operator and a regular non linear part we consider abstract splitting methods associated with this decomposition where no discretization in space is made we prove a normal form result for the corresponding discrete flow under generic non resonance conditions on the frequencies of the linear operator and on the step size and under a condition of zero momentum on the nonlinearity this result implies the conservation of the regularity of the numerical solution associated with the splitting method over arbitrary long time provided the initial data is small enough this result holds for rounded numerical schemes avoiding at each step possible high frequency energy drift we apply these results to nonlinear schrodinger equations as well as the nonlinear wave equation

birkhoff normal form for some nonlinear pdes we consider the problem of extending to pdes birkhoff normal form theorem on hamiltonian systems close to nonresonant elliptic equilibria as a model problem we take the nonlinear wave equation with dirichlet boundary conditions on 0 g is an analytic skewsymmetric function which vanishes for u 0 and is periodic with period 2 in the x variable we prove under a nonresonance condition which is fulfilled for most g s that for any integer m there exists a canonical transformation that puts the hamiltonian in birkhoff normal form up to a reminder of order m the canonical transformation is well defined in a neighbourhood of the origin of a sobolev type phase space of sufficiently high order some dynamical consequences are obtained the technique of proof is applicable to quite general semilinear equations in one space dimension

time quasi periodic unbounded perturbations of schrodinger operators and kam methods we eliminate by kam methods the time dependence in a class of linear differential equations in l2 subject to an unbounded quasi periodic forcing this entails the pure point nature of the floquet spectrum of the operator h0 ep t for e small here h0 is the one dimensional schrodinger operator p2 v v x x 2 for x the time quasi periodic perturbation p may grow as x 2 2 and the frequency vector is non resonant the proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on kuksin s estimate of solutions of homological equations with non constant coefficients

containment regions for zeros of polynomials from numerical ranges of companion matrices containment regions for the zeros of a monic polynomial are given with the aid of results for containment regions for the numerical range of certain bordered diagonal matrices which are applied to different types of companion matrices of the polynomial

spatial low discrepancy sequences spherical cone discrepancy and applications in financial modeling in this paper we introduce a reproducing kernel hilbert space defined on r d 1 as the tensor product of a reproducing kernel defined on the unit sphere s d in r d 1 and a reproducing kernel defined on 0 we extend stolarsky s invariance principle to this case and prove upper and lower bounds for numerical integration in the corresponding reproducing kernel hilbert space the idea of separating the direction from the distance from the origin can also be applied to the construction of quadrature methods an extension of the area preserving lambert transform is used to generate points on s d 1 via lifting sobol points in 0 1 d to the sphere the d th component of each sobol point suitably transformed provides the distance information so that the resulting point set is normally distributed in r d numerical tests provide evidence of the usefulness of constructing quasi monte carlo type methods for integration in such spaces we also test this method on examples from financial applications option pricing problems and compare the results with traditional methods for numerical integration in r d

a generalized discrepancy and quadrature error bound an error bound for multidimensional quadrature is derived that includes the koksma hlawka inequality as a special case this error bound takes the form of a product of two terms one term which depends only on the integrand is defined as a generalized variation the other term which depends only on the quadrature rule is defined as a generalized discrepancy the generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the l p star discrepancy and p that arises in the study of lattice rules

invariant cantor manifolds of quasi periodic oscillations for a nonlinear schrodinger equation this paper was written while both authors were guests of the forschungsinstitut fur mathematik at the eth zurich and we thank the institute for its hospitality pleasant working atmosphere and helpful staff in particular we thank jiirgen moser for numerous stimulating discussions on the subject we also benefitted from a remark by sigurd angenent concerning the analyticity of the solutions the second author also thanks the deutsche forschungsgemeinschaft for their financial support through a heisenberg grant

newton s method and periodic solutions of nonlinear wave equations we prove the existence of periodic solutions of the nonlinear wave equation satisfying either dirichlet or periodic boundary conditions on the interval o the coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation using a version of newton s method we show that this equation has solutions provided the nonlinearity g x u satisfies certain generic conditions of nonresonance and genuine nonlinearity 1993 john wiley sons inc

on geometric properties of the numerical range abstract we investigate the shape of the numerical range a criterion for the numerical range of a matrix to be an elliptical disk is given the result is applied to show that there exist neither 3 by 3 nor 4 by 4 nilpotent matrices whose numerical range is an elliptical noncircular disk sufficient conditions for n by n tridiagonal matrices to have elliptical numerical range are obtained the boundary of the numerical range near a sharp point is examined finally the numerical range of a reducible matrix is compressed and its geometric properties are discussed

good path generation methods in quasi monte carlo for pricing financial derivatives the quasi monte carlo qmc method is a powerful numerical tool for pricing complex derivative securities whose accuracy is affected by the smoothness of the integrands the payoff functions of many financial derivatives involve two types of nonsmooth factors an indicator function called jump structure and a positive part of a smooth function called kink structure this paper develops a good path generation method pgm for recovering the superiority of the qmc method on problems involving multiple such structures this is achieved by realigning these structures such that the associated nonsmooth surfaces are parallel to as many coordinate axes as possible the proposed method has the advantage of addressing different structures according to their importance we also offer a systematic measurement of different structures for quantifying and then ranking their importance numerical experiments demonstrate that the proposed method is more efficient than traditional pgms for pricing exotic options suc

boundary generating curves of the c numerical range abstract let a be an n n matrix and c c 1 c 2 c n be a real n tuple the c numerical range of a is the set w c a j 1 n c j x j ax j x 1 x 2 x n is an orthonormal basis of c n we obtain parametric representations of the boundary generating curve of the c numerical range of a matrix applying this result we generalize the result of anderson to the c numerical range furthermore we give a description of the boundary generating curve of the c numerical range of certain types of nilpotent toeplitz matrices a sufficient condition for the boundary generating curve to be rational is obtained finally we explicitly compute the boundary generating curves of the numerical ranges for several concrete matrices and classify the rationality of the curves

the c numerical range of tridiagonal matrices abstract we give sufficient conditions for the c numerical range of a tridiagonal matrix to be an elliptic disc which generalize known results on the classical numerical range

qmc designs optimal order quasi monte carlo integration schemes on the sphere we study equal weight numerical integration or quasi monte carlo qmc rules for functions in a sobolev space hs sd with smoothness parameter s d 2 defined over the unit sphere sd in rd 1 focusing on n point configurations that achieve optimal order qmc error bounds as is the case for efficient spherical designs we are led to introduce the concept of qmc designs these are sequences of n point configurations xn on s d such that the worst case error satisfies sup f h s f hs 1 1 n x xn f x sd f x d d x o n s d n with an implied constant that depends on the hs sd norm but is independent of n here d is the normalized surface measure on s d we provide methods for generation and numerical testing of qmc designs an essential tool is an expression for the worst case error in terms of a reproducing kernel for the space hs sd with s d 2 as a consequence of this and a recent result of bondarenko et al on the existence of spherical designs with appropriate number of points we show that minimizers of the n point energy for this kernel form a sequence of qmc designs for hs sd furthermore without appealing to the bondarenko et al result we prove that point sets that maximize the sum of suitable powers of the euclidean distance between pairs of points form a sequence of qmc designs for hs sd with s in the interval d 2 d 2 1 for such spaces there exist reproducing kernels with simple closed forms that are useful for numerical testing of optimal order quasi monte carlo integration numerical experiments suggest that many familiar sequences of point sets on the sphere equal area points spiral points minimal coulomb or logarithmic energy points and fekete points are qmc designs for appropriate values of s for comparison purposes we show that configurations of random points that are independently and uniformly distributed on the sphere do not constitute qmc designs for any s d 2 if xn is a sequence of qmc designs for h s sd we prove that it is also a sequence of qmc designs for hs sd for all s d 2 s this leads to the question of determining the supremum of such s here called the qmc strength of the sequence for which we provide estimates based on computations for the aforementioned sequences received by the editor august 15 2012 and in revised form february 26 2013 2010 mathematics subject classification primary 65d30 65d32 secondary 11k38 41a55

the numerical range of banded biperiodic toeplitz operators abstract the numerical range of an operator is a well studied concept with many applications in several areas of mathematics in this paper the numerical range of banded biperiodic toeplitz operators is investigated performing a reduction to the 2 2 case namely the parametric equations of the boundary generating curves are deduced and two algorithms for the numerical generation of the numerical range are presented

on the numerical range of tridiagonal operators it is shown that the numerical range of a tridiagonal operator with 0 main diagonal is symmetric with respect to the origin some related specialized results under stronger hypotheses are included in particular if the order of a matrix is 3 then its numerical range is an elliptical disk and is explicitly described

circularity of the numerical range abstract an equivalent condition on a 3 square complex or a 4 square real upper triangular matrix is found for its numerical range to be a circular disk centered at the origin sufficient conditions for the circularity of the numerical range of certain sparse matrices are also given in terms of graphs

monte carlo variance of scrambled net quadrature hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter this paper studies the variance of one such hybrid scrambled nets by applying a multidimensional multiresolution wavelet analysis to the integrand the integrand is assumed to be measurable and square integrable but not necessarily of bounded variation in simple monte carlo every nonconstant term of the multiresolution contributes to the variance of the estimated integral for scrambled nets certain low dimensional and coarse terms do not contribute to the variance for any integrand in l2 the sampling variance tends to zero faster under scrambled net quadrature than under monte carlo sampling as the number of function evaluations n tends to infinity some finite n results bound the variance under scrambled net quadrature by a small constant multiple of the monte carlo variance uniformly over all integrands f latin hypercube sampling is a special case of scrambled net quadrature

birkhoff normal form for splitting methods applied to semilinear hamiltonian pdes part i finite dimensional discretization we consider discretized hamiltonian pdes associated with a hamiltonian function that can be split into a linear unbounded operator and a regular nonlinear part we consider splitting methods associated with this decomposition using a finite dimensional birkhoff normal form result we show the almost preservation of the actions of the numerical solution associated with the splitting method over arbitrary long time and for asymptotically large level of space approximation provided the sobolev norm of the initial data is small enough this result holds under generic non resonance conditions on the frequencies of the linear operator and on the step size we apply these results to nonlinear schrodinger equations as well as the nonlinear wave equation

lower dimensional invariant tori with prescribed frequency for the nonlinear schrodinger equation abstract in this paper one dimensional 1d nonlinear schrodinger equation i u t u x x m u u 2 u f u 2 u 0 subject to dirichlet boundary conditions is considered where the nonlinearity f is a real analytic function near u 0 with f 0 f 0 0 it is proved that for each given constant potential m and each prescribed integer b 1 the above equation admits a whitney smooth family of small amplitude time quasi periodic solutions whose b dimensional frequencies are just small dilation of a prescribed diophantine vector accordingly we obtain the existence of lower dimensional invariant kam tori with tangential frequencies constrained to a given diophantine direction in an infinite dimensional phase space setting

a lower bound for the worst case cubature error on spheres of arbitrary dimension this paper is concerned with numerical integration on the unit sphere sr of dimension r 2 in the euclidean space r 1 we consider the worst case cubature error denoted by e qm hs sr of an arbitrary m point cubature rule qm for functions in the unit ball of the sobolev space hs sr where s and show that the positive constant cs r in the estimate depends only on the sphere dimension r 2 and the index s of the sobolev space hs sr this result was previously only known for r 2 in which case the estimate is order optimal the method of proof is constructive we construct for each qm a bad function fm that is a function which vanishes in all nodes of the cubature rule and for which our proof uses a packing of the sphere sr with spherical caps as well as an interpolation result between sobolev spaces of different indices

perturbations of lower dimensional tori in the resonant zone for reversible systems consider a nearly integrable system which is reversible with respect to an involution of n n type for a given l resonant surface 1 l n we show that there is a subset of it with positive measure such that in each resonant torus corresponding to a point of this set at least one dimensional invariant submanifold can survive any small perturbations

spectrum for quantum duffing oscillator and small divisor equation with large variable coefficient by establishing a new estimate of the solutions for the small divisor equation with large variable coefficients and using the kam technique we prove a reduction theorem that entails the pure point nature of the floquet spectrum of the quantum duffing oscillator with a small perturbation temporal quasi periodic with nonresonant frequencies 2010 wiley periodicals inc

numerical ranges of unbounded operators arising in quantum physics creation and annihilation operators are used in quantum physics as the building blocks of linear operators acting on hilbert spaces of many body systems in quantum physics pairing operators are defined in terms of those operators in this paper spectral properties of pairing operators are studied the numerical ranges of pairing operators are investigated in the context of matrix theory the results give the numerical ranges of certain infinite tridiagonal matrices

quasi periodic solutions of derivative nonlinear schrodinger equations with a given potential abstract it is proved that for a prescribed potential v x there are many quasi periodic solutions of derivative nonlinear schrodinger equation i u t u x x v x u u x 2 u subject to dirichlet boundary condition by means of a kam theorem to a reversible system

fields of values and iterative methods abstract the performance of an iterative scheme to solve x t x c t c n n c c n is often judged by spectral properties of t if t is not normal it is however well known that only conclusions about the assymptotic behavior of an iterative method can be drawn from spectral information to anticipate the progress of the iteration after a finite number of steps the knowledge of the eigenvalues alone is often useless in addition the spectrum of t may be highly sensitive to perturbations if t is not normal an iterative method which on the basis of some spectral information is predicted to converge rapidly for t may well diverge if t is slightly perturbed in practice the convergence of the iteration x m t x m 1 c is therefore frequently measured by some norm verbar t verbar rather than by the spectral radius t but apart from the fact that norms lead to error estimates which are often too pessimistic they cannot be used to analyze more general schemes such as e g the chebyshev iterative methods here we discuss another tool to analyze the behavior of an iterative method namely the field of values w t the collection of all rayleigh quotients of t w t contains the eigenvalues of t and the numerical radius t max z w t z defines a norm on c n n the field of values represents therefore an intermediate concept to judge an iterative scheme by it is related to the spectral approach but has also certain norm properties

birkhoff normal form for partial differential equations with tame modulus we prove an abstract birkhoff normal form theorem for hamiltonian partial differential equations pdes the theorem applies to semilinear equations with nonlinearity satisfying a property that we call tame modulus such a property is related to the classical tame inequality by moser in the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times we also develop a general scheme to apply the abstract theory to pdes in one space dimensions and we use it to study some concrete equations nonlinear wave nlw equation nonlinear schrodinger nls equation with different boundary conditions an application to an nls equation on the d dimensional torus is also given in all cases we deduce bounds on the growth of high sobolev norms in particular we get lower bounds on the existence time of solutions