shape preserving rational cubic fractal interpolation function abstract a new type of c 1 fractal interpolation function fif is developed using the iterated function system ifs which contains the rational spline the numerator of this rational spline contains cubic polynomial and the denominator of the rational spline contains quadratic polynomial we find uniform error bound between the original function which belongs to the class c 2 and the fif we described suitable conditions on scaling factors and shape parameters such that they preserve the shape properties which are inherited in the data

semiclassical analysis of low lying eigenvalues ii tunneling we discuss the leading asymptotics of eigenvalue splittings of 2 x2v in the limit as x o and where v is a non negative potential with several zeros for example if eo x e1 x are the two lowest eigenvalues in a situation where v has precisely two zeros a and b related by a symmetry then limx x lin e1 x eo x is given as the distance from a to b in a certain riemann metric

on monotone and convex spline interpolation this paper is concerned with the problem of existence of monotone and or convex splines having degree n and order of continuity k which interpolate to a set of data at the knots the interpolating splines are obtained by using bernstein polynomials of suitable continuous piecewise linear functions they satisfy the inequality k n k the theorems presented here are useful in developing algorithms for the construction of shape preserving splines interpolating arbitrary sets of data points earlier results of mcallister passow and roulier can be deduced from those given in this paper

a hybrid 2 d 3 d multilevel green s function interpolation method for electrically large multilayered problems the multilevel green s function interpolation method has been implemented to solve the general electrically large multilayered problems a new hybrid 2 d and 3 d h2 d 3 d multilevel interpolation method and a cross level translation scheme are devised for the specific problems from which a quad tree is constructed the interpolation of the multilayered medium green s function using radial basis function is then studied to further accelerate the computation a subdomain fast fourier transform scheme and a sparse approximated inverse preconditioner with a new discarding strategy are also incorporated into the algorithm as numerical examples typical microstrip antenna arrays a metasurface antenna and electromagnetic bandgap integrated antenna arrays are analyzed with the proposed method and excellent performance is observed in both accuracy and efficiency

sobolev spaces and capacities theory on path spaces over a compact riemannian manifold we introduce sobolev spaces and capacities on the path space p m 0 m over a compact riemannian manifold m we prove the smoothness of the ito map and the stochastic anti development map in the sense of stochastic calculus of variation we establish a sobolev norm comparison theorem and a capacity comparison theorem between the wiener space and the path space p m 0 m moreover we prove the tightness of r p capacities on p m 0 m which generalises a result due to airault malliavin and sugita on the wiener space finally we extend our results to the fractional holder continuous path space

splitting in large dimension and infrared estimates ii moment inequalities this is the continuation of notes written for the nato asi conference in il ciocco september 96 consisting of the analysis of the links between estimating the splitting between the two first eigenvalues for the schrodinger operator h and the proof of infrared estimates for quantities attached to gaussian type measures these notes were mainly reporting on the old contributions of dyson frohlich glimm jaffe lieb simon and spencer in the 1970s in connection with more recent contributions of pastur khoruzhenko barbulyak and kondrat ev which treat in general more sophisticated models here we concentrate on the simplest model related to field theory and extend the results of barbulyak and kondrat ev by mixing ideas coming from pastur and khozurenko related to the use of bogolyubov s inequality with classical inequalities due to ginibre lebowitz sokal and others or in the case when the temperature t is zero by applying rather elementary estimates on schrodinger operators in order to find

semiclassical asymptotics of eigenvalues for schrodinger operators with magnetic fields abstract we consider schrodinger operators with magnetic fields on a two dimensional compact manifold or on r 2 the purpose is to study the semiclassical asymptotics of the eigenvalues by two different methods we obtain some facts on the harmonic oscillators under uniform magnetic fields and we also discuss them

corrections to comparison of interpolating functions and interpolating points in full wave multilevel green s function interpolation method aug 10 2691 2699 in the above titled paper ibid vol 58 no 8 pp 2691 2699 aug 10 table ii contained errors errors also appeared in equations 2 and 3 corrections are presented here

an improved multilevel green s function interpolation method with adaptive phase compensation an improved multilevel green s function interpolation method mlgfim with adaptive phase compensation apc is proposed the difficulty in applying interpolation approaches to the fast varying phase term in the integral equation kernel for full wave electromagnetic em simulations is eradicated by using the phase compensation and adaptive direction separation ads the multilevel tree structure in mlgfim keeps the number of direction separation invariant at all levels attributing to the recursive interpolation with multilevel phase compensation the proposed mlgfim apc in conjunction with the lagrange chebyshev interpolation yield an o n log n cpu time and o n computer memory requirement for surface scattering problems by introducing a transition level the mlgfim apc can adaptively incorporate interpolation techniques of conventional interpolation ci transition interpolation ti and phase compensation interpolation pi corresponding to electromagnetic simulation of problems of small moderate and large electrical sizes respectively large scale microstrip antenna arrays are simulated to illustrate the accuracy and efficiency of the proposed method it is found that the cpu time scales better than o n log n for these co planar problems

fractals in physics applications and theoretical developments abstract we report about the developments in two subjects in which we have been recently active the first one refers to a reanalysis of the distribution of galaxies and clusters with the methods of modern statistical mechanics contrary to the usual picture of a relatively smooth distribution at large scale we find that fractal and multifractal properties extend up to the observational limits this result should have deep implications also in relation to the recently observed anisotropy of the microwave background radiation the second subject concerns the attempts to develop a theory of fractal growth based on the new concept of the fixed scale transformation we discuss the instructive example of the percolating cluster in which the fst program can be fully implemented we then compare this example with the more complex cases of dla and dbm for which the identification of the scale invariant growth rules is much more complex in this respect the fst points to the key concept that is the scale invariance of screening and noise reduction the state of the art and the main open problems will be critically discussed

new approach to the semiclassical limit of quantum mechanics i multiple tunnelings in one dimension we propose a new approach for the estimate of the rate of degeneracy of the lowest eigenvalues of the schrodinger operator in the presence of tunneling based on the theory of diffusion processes our method provides lower and upper bounds for the energy splittings and the rates of localization of the wave functions and enables us to discuss cases which as far as we know have never been treated rigorously in the literature in particular we give an analysis of the effect on eigenvalues and eigenfunctions of localized deformations of 1 symmetric double well potentials 2 potentials periodic and symmetric over a finite interval theses situations are characterized by a remarkable dependence on such deformations our probabilistic techniques are inspired by the theory of small random perturbations of dynamical systems

trace formulas a golden thompson inequality and classical limit in boson fock space trace formulas for the heat semi groups of second quantization operators and their perturbations in the abstract boson fock space are given in terms of path functional integral representations as applications an inequality of golden thompson type and a classical limit are derived for the trace of the heat semi group of a perturbed second quantization operator the abstract results are applied to a model ofp type in quantum field theory

spectral and scattering theory of spatially cut off p ϕ 2 hamiltonians we study spatially cut off p 2 hamiltonians we show the local finiteness of the pure point spectrum outside of thresholds the limiting absorption principle and asymptotic completeness of scattering for such hamiltonians our results imply the absence of singular continuous spectrum

logarithmic sobolev inequalities on loop groups let g be a compact lie group denote by m the brownian bridge measure on the loop group y g c 0 1 g g 0 g 1 e the finite energy subgroup of y determines in a natural way a gradient operation for functions on y the following logarithmic sobolev inequality is proven f2 log f dm gradf y 2 v y f y 2 dm f 2log f wherein f denotes the l2 m norm and v is a potential which is quadratic in the associated lie algebra valued brownian motion the inequality is derived by a method of inheritance from the known inequality for the g valued brownian motion

finite dimensional approximations to wiener measure and path integral formulas on manifolds abstract certain natural geometric approximation schemes are developed for wiener measure on a compact riemannian manifold these approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi group on riemannian manifolds the path space is approximated by finite dimensional manifolds h p m consisting of piecewise geodesic paths adapted to partitions p of 0 1 the finite dimensional manifolds h p m carry both an h 1 and a l 2 type riemannian structures g 1 p and g 0 p respectively it is proved that 1 z i p e 1 2 e d vol g i p i d as mesh p 0 where e is the energy of the piecewise geodesic path h p m and for i 0 and 1 z i p is a normalization constant vol g i p is the riemannian volume form relative to g i p and is wiener measure on paths on m here 1 1 and 0 exp 1 6 1 0 scal s ds where scal is the scalar curvature of m these results are also shown to imply the well known integration by parts formula for the wiener measure

positivity preserving interpolation of positive data by rational cubics this work is a contribution towards the graphical display of data when it is positive the data are required to be represented in such a way that its visual display looks smooth and pleasant its positive shape is preserved everywhere and the computation cost is economical a c 1 piecewise rational cubic function in its most general form has been utilized for this objective the method is implemented for the 1d data initially and then it is extended to an interpolating rational bicubic form for the data arranged over a rectangular grid simple sufficient conditions are developed on the free parameters in the description of the rational function to visualize the positive data in the form of positive curves and surfaces

krein s formula and one dimensional multiple well abstract it is shown that all the discrete eigenvalues of a one dimensional schrodinger operator with a multiple well potential possess an asymptotic expansions in power of h 1 2 when h 0 a formula for all the coefficients of these expansions is given the method uses two main tools perturbation by boundary conditions and exponential decay of eigenfunctions which are developed in this article as a by product of this work the exponential localization of eigenvectors when h goes to zero can be proved

the fast multipole method we study integral methods applied to the resolution of the maxwell equations where the linear system is solved using an iterative method which requires only matrix vector products the fast multipole method fmm is one of the most efficient methods used to perform matrix vector products and accelerate the resolution of the linear system a problem involving n degrees of freedom may be solved in cniternlogn floating operations where c is a constant depending on the implementation of the method in this article several techniques allowing one to reduce the constant c are analyzed this reduction implies a lower total cpu time and a larger range of application of the fmm in particular new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms several numerical tests are also described these confirm the efficiency and the theoretical complexity of the fmm

logarithmic sobolev inequalities and exponential integrability we prove the exponential integrability of lipschitz functions in abstract wiener spaces by using logarithmic sobolev inequalities we introduce two notions of lipschitz continuity and apply the method of davis and simon j funct anal 59 1984 335 395 to prove our main theorem moreover we give several conditions under which the function is exponentially integrable

semi classical limit of the lowest eigenvalue of a schrodinger operator on a wiener space ii p ϕ 2 model on a finite volume abstract we determine the semi classical limit of the lowest eigenvalue of a p 2 hamiltonian on a finite volume interval

convexity preserving piecewise rational cubic interpolation a twice continuously differentiable piecewise rational cubic interpolant f to data s x j y j j 1 cdots n delta j 2 ne 0 x j 1 x j is defined in terms of turn parameters tau j j 1 cdots n 1 and end conditions f x 1 f x n 0 the necessary and sufficient condition f x j delta j 2 geqq 0 j 2 cdots n 1 is derived which ensures that the interpolant f preserves the local convexity concavity properties of s a fast simple and robust procedure for generating f is described and the results obtained for a specific data set s are compared with some standard cubic spline interpolants

tunneling for spatially cut off p phi _2 hamiltonians we study the asymptotic behavior of low lying eigenvalues of spatially cut off p 2 hamiltonian in the semi classical limit we determine the semi classical limit of the lowest eigenvalue of the hamiltonian in terms of the hessian of the potential function of the corresponding classical equation moreover we prove that the gap of the lowest two eigenvalues goes to 0 exponentially fast in the semi classical limit when the potential function is double well type in fact we prove that the exponential decay rate is greater than or equal to the agmon distance between two zero points of the symmetric double well potential function also we study basic properties of the agmon distance and instanton

semiclassical limit of the lowest eigenvalue of a schrodinger operator on a wiener space abstract we study a semiclassical limit of the lowest eigenvalue of a schrodinger operator on a wiener space key results are semiboundedness theorem of the schrodinger operator laplace type asymptotic formula and ims localization formula we also make a remark on the semiclassical problem of a schrodinger operator on a path space over a riemannian manifold

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

witten s complex and infinite dimensional morse theory we investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space the results are applied to an infinite dimensional problem involving the symplect c action function

preserving convexity through rational cubic spline fractal interpolation function we propose a new type of c 1 rational cubic spline fractal interpolation function fif for convexity preserving univariate interpolation the associated iterated function system ifs involves rational functions of the form p n x q n x where p n x are cubic polynomials determined through the hermite interpolation conditions of the fif and q n x are preassigned quadratic polynomials with two shape parameters the rational cubic spline fif converges to the original function f as rapidly as the rth power of the mesh norm approaches to zero provided f r is continuous for r 1 or 2 and certain mild conditions on the scaling factors are imposed furthermore suitable values for the rational ifs parameters are identified so that the property of convexity carries from the data set to the rational cubic fifs in contrast to the classical non recursive convexity preserving interpolation schemes the present fractal scheme is well suited for the approximation of a convex function f whose derivative is continuous but has varying irregularity

the log sobolev inequality on loop space over a compact riemannian manifold abstract we obtain a log sobolev inequality with a neat and explicit potential for the gradient on a based loop space over a compact riemannian manifold the potential term relies only on the curvature of the manifold and the hessian of the heat kernel and is l p integrable for all p 1 the log sobolev inequality is derived by a martingale representation theorem for the differentiable functions on loop space which is a variation of the clark ocone haussmann formula

on the spectrum of semi classical witten laplacians and schrodinger operators in large dimension we investigate the low lying spectrum of witten laplacians on forms of arbitrary degree in the semi classical limit and uniformly in the space dimension we show that under suitable assumptions implying that the phase function has a unique local minimum one obtains a number of clusters of discrete eigenvalues at the bottom of the spectrum moreover we are able to count the number of eigenvalues in each cluster we apply our results to certain sequences of schrodinger operators having strictly convex potentials and show that some well known results of semi classical analysis hold also uniformly in the dimension

the implementation of multilevel green s function interpolation method for full wave electromagnetic problems we extend the multilevel green s function interpolation method mlgfim developed for quasi electrostatic problems to full wave simulations the difficulty in applying the interpolation approach lies in the additional rapidly changing phase term associated with the full wave green s functions to enhance the efficiency and accuracy of the full wave green s function interpolation a scattered point set consisting of two staggered tartan grids in conjunction with radial basis function interpolation is employed to further reduce the computational complexity the qr factorization technique is applied to compress the low rank green s function matrices for electromagnetic scattering from pec spheres up to a diameter of eight wavelengths the proposed method compares well with mie s scattering in accuracy and shows the o nlogn efficiency as the method is kernel independent its extension to structures in layered medium is straightforward in the numerical simulations of finite microstrip patch arrays up to 11 by 11 elements the proposed method demonstrates very favorable dependencies of cpu time and memory storage requirement versus the number of unknowns

a cameron martin type quasi invariance theorem for brownian motion on a compact riemannian manifold abstract let m be a compact manifold without boundary o be a fixed base point in m g be a riemannian metric on m and be a g compatible covariant derivative on tm the tangent space to m assume the torsion t of satisfies the skew symmetry condition g t x y y 0 for all vector fields x and y on m for example take to be the levi civita covariant derivative on m g also let denote the wiener measure on w0 m c 0 1 m 0 o and let h s denote stochastic parallel translation relative to along the path w0 m up to time s 0 1 given a c1 function h 0 1 t0m it is shown that the differential equation t h t h with initial condition 0 id w m w m has a solution r maps w m w m the measurable maps from w m to w m this function is a flow on w m i e for all t r t t furthermore t has the quasi invariance property the law t of t with respect to the wiener measure is equivalent to for all t r this result is used to prove an integration by parts formula for the h derivative hf defined by h f d dt 0 f t where f is a c2 cylinder function on w m

generalized cubic spline fractal interpolation functions we construct a generalized cr fractal interpolation function cr fif f by prescribing any combination of r values of the derivatives f k k 1 2 dots r at boundary points of the interval i x0 xn our approach to construction settles several questions of barnsley and harrington j approx theory 57 1989 pp 14 34 when construction is not restricted to prescribing the values of f k at only the initial endpoint of the interval i in general even in the case when r equations involving f k x 0 and f k x n k 1 2 dots r are prescribed our method of construction of the c r fif works equally well in view of wide ranging applications of the classical cubic splines in several mathematical and engineering problems the explicit construction of cubic spline fif f delta x through moments is developed it is shown that the sequence f delta k x converges to the defining data function phi x on two classes of sequences of meshes at least as rapidly as the square of the mesh norm delta k approaches to zero provided that phi r x is continuous on i for r 2 3 or 4

leading large order asymptotics for φ4 2 perturbation theory we develop a rigorous semiclassical expansion to compute the radius of convergence of the borel transform for the pressure in 4 2 field theory this result gives a partial justification for the lipatov method of finding large order perturbation theory asymptotics in quantum field theory

semi classical limit of the bottom of spectrum of a schrodinger operator on a path space over a compact riemannian manifold abstract we determine the limit of the bottom of spectrum of schrodinger operators with variable coefficients on wiener spaces and path spaces over finite dimensional compact riemannian manifolds in the semi classical limit these are extensions of the results in s aida semiclassical limit of the lowest eigenvalue of a schrodinger operator on a wiener space j funct anal 203 2 2003 401 424 the problem on path spaces over riemannian manifolds is considered as a problem on wiener spaces by using ito s map however the coefficient operator is not a bounded linear operator and the dependence on the path is not continuous in the uniform convergence topology if the riemannian curvature tensor on the underling manifold is not equal to 0 the difficulties are solved by using unitary transformations of the schrodinger operators by approximate ground state functions and estimates in the rough path analysis

logarithmic sobolev inequalities on path spaces over riemannian manifolds let w o m be the space of paths of unit time length on a connected complete riemannian manifold m such that 0 o a fixed point on m and the wiener measure on w o m the law of brownian motion on m starting at o if the ricci curvature is bounded by c then the following logarithmic sobolev inequality holds

on the ornstein uhlenbeck operators on wiener riemannian manifolds abstract in this paper we study the properties of infinite dimensional ornstein uhlenbeck operators on submanifolds in wiener spaces we establish the hypoellipticity of the laplacians acting on the space of tensors by using this we show the essential selfadjointness of the laplacians acting on the space of 0 forms i e function spaces

a fast volume surface integral equation solver for scattering from composite conducting dielectric objects this paper presents a fast hybrid volume surface integral equation approach for the computation of electromagnetic scattering from objects comprising both conductors and dielectric materials the volume electric field integral equation is applied to the material region and the surface electric field integral equation is applied on the conducting surface the method of moments mom is used to convert the integral equation into a matrix equation and the precorrected fft p fft method is employed to reduce the memory requirement and cpu time for the matrix solution the present approach is sufficiently versatile in handling problems with either open or closed conductors and dielectric materials of arbitrary inhomogeneity due to the combination of the surface and volume electric field integral equations the application of the precorrected fft method facilitates the solving of much larger problems than can be handled by the conventional mom

c1 sign monotonicity and convexity preserving hermite polynomial splines of variable degree abstract this paper proposes an algorithm for constructing interpolatory hermite polynomial splines of variable degree which preserve the sign the monotonicity and the convexity of the data the polynomial segments are represented as bezier curves the degree of each segment plays the role of the tension parameter of the spline we discuss extensively the monotonicity and convexity criteria detailing a strict and a weak form of monotonicity preservation as well as their implications on the hermite interpolation we also propose a global method for estimating the nodal derivatives of the spline we evaluate the results of this method in an extensive set of examples comparing them with a number of local derivative estimation methods from the pertinent literature the algorithm is numerically stable simple to implement and it is of linear complexity since the related mathematical conditions can be expressed as linear inequalities with respect to the control points of the spline

comparison of interpolating functions and interpolating points in full wave multilevel green s function interpolation method the difficulty for the solution of full wave electromagnetic problems using multilevel green s function interpolation method mlgfim lies in applying interpolating approaches to efficiently and accurately approximate green s function with rapidly changing phase term we compare various interpolating schemes when radial basis function rbf is employed for the interpolation of scattered data of green s function we show that the infinitely smooth gaussian ga rbf has the best interpolation accuracy in order to improve the interpolation efficiency a new kind of staggered tartan grid is proposed a good calculation method for the shape parameter in ga rbf is given to solve its sensitivity to the group size and the number of interpolation points based on the analysis of variation of the number of interpolation points with electric length of the group adaptive choice of the types of interpolation functions and interpolation points are employed numerical examples show that the computational efficiency of this new interpolation scheme is much improved over the previously reported ones

renormalized differential geometry on path space structural equation curvature abstract the theory of integration in infinite dimensions is in some sense the backbone of probability theory on this backbone the stochastic calculus of variations has given rise to the flesh of differential calculus its first step is the construction at each point of the probability space of a cameron martin like tangent space in which the desired differential calculus can be developed this construction proceeds along the lines of first order differential geometry in this paper we address the following questions what could be the meaning of curvature of the probability space how and why how can curvatures be defined and computed why could a second order differential geometry be relevant to stochastic analysis we try to answer these questions for the probability space associated to the brownian motion of a compact riemannian manifold why a basicenergy identity for anticipative stochastic integralswill be obtained as a byproduct of our computation of curvature how there are essentially four bottlenecks in the development of differential geometry on wiener riemann manifolds i the difficulty of finding an atlas of local charts such that the changes of charts preserve the class of the wiener like measures together with their associated cameron martin like tangent spaces ii the difficulty of finding cylindrical approximations preserving the natural geometrical objects iii the difficulty of renormalizing the divergent series to which the summation operations of finite dimensional differential geometry give rise in the non intrinsic context of local charts iv the nonavailability of the computational procedures analogous to the local coordinates systems of the classical differential geometry in the context of path space the ito filtration provides a much richer structure than that available in the framework of an abstract wiener riemann manifold our work is a systematic attempt to replace the machinery of local charts with amethodology of moving frames in our context stochastic parallel transport provides a canonical moving frame on the path space the concept of a cylindrical approximation has to be reshaped in our new situation into some geometric limit theorems establishing that the riemannian geometric objects of the cylindrical approximations induce by a limiting procedure geometric objects on the path space those limit theorems are reminiscent of the classical theorems which say that a stratonovich sde is the limit of an appropriate sequence of ode the canonical coordinate system provided by the moving frame will make it possible to proceed to the needed renormalizations byintrinsicstochastic integrals in this context the anticipative stochastic integral theory of nualart and pardoux will play a decisive role finally the moving frame will provide aneffective algorithm of computation for this differential geometry in infinite dimensions in our study we encounter a new type of renormalization thehypoelliptic renormalization which corresponds to the fact that the bracket of smooth vector fields taking their values in the cameron martin space can get out of this cameron martin space this hypoelliptic problem induces the nonrenormalizability of some geometrical objects it leads also to a concept oftangent processesto probability spaces extending that based on the cameron martin theorem for tangent processes a formula of integration by parts still holds furthermore the tangent processes form a lie algebra under the bracket on the other hand tangent processes cannot be stochastically integrated this operation is well defined only for cameron martin type vector fields

the double well splitting of the low energy levels for the schrodinger operator of discrete φ4 models on tori the formula is obtained for the asymptotics of the splitting of the first two eigenvalues of the schrodinger operator that stands for the discrete multidimensional 4 model on tori the behavior of the main term is investigated in the limit of the large number of particles

fractal interpolation functions with partial self similarity abstract let a data set t i y i r y i 0 1 n be given where t 0 t 1 t 2 t n and y is a complete metric space in this article fractal interpolation functions fifs on i t 0 t n corresponding to the set are constructed by mappings w 1 w n each w k is of the form w k l k m k where l k j k i k is a homeomorphism and m k j k y y is continuous here i k t k 1 t k and j k t j k t l k j k l k 0 1 n are subintervals of i which depend on k in this construction the length of j k is not assumed to be larger than the length of i k and each l k is not supposed to be a contraction a fif established by this method has a property of self similarity between its graph on j k and on i k in this paper we give a construction of fifs with locally self similar graphs the stability and sensitivity of fifs established in this way are also discussed

prolate spheroidal wavefunctions as an alternative to chebyshev and legendre polynomials for spectral element and pseudospectral algorithms prolate spheroidal functions of order zero are generalizations of legendre polynomials which when the bandwidth parameter c 0 oscillate more uniformly on x 1 1 than either chebyshev or legendre polynomials this suggests that compared to these polynomials prolate functions give more uniform spatial resolution further when used as the spatial discretization for time dependent partial differential equations in combination with explicit time marching prolate functions allow a longer stable timestep than legendre polynomials we show that these advantages are real and further that it is almost trivial to modify existing pseudospectral and spectral element codes to use the prolate basis however improvements in spatial resolution are at most a factor of 2 approached slowly as n the timestep can be lengthened by several times but not by a factor that grows rapidly with n the prolate basis is not likely to radically expand the range of problems that can be done on a workstation however for production runs on the bleeding edge edge of arithmurgy such as numerical weather prediction the rewards for switching to a prolate basis may be considerable

an estimate of the gap of spectrum of schrodinger operators which generate hyperbounded semigroups abstract logarithmic sobolev inequalities with potential functions on loop spaces were proved by l gross 1991 j funct anal 102 265 313 s aida 1996 in proceedings of the fifth gregynog symposium stochastic analysis and applications pp 1 19 world scientific singapore and f z gong and z m ma 1998 j funct anal 599 623 the generators of the dirichlet forms are schrodinger operators on loop spaces and they generate hyperbounded semigroups recently f z gong m rockner and l m wu 2000 poincare inequality for weighted first order sobolev spaces on loop spaces submitted proved the existence of the gap of spectrum at the lowest eigenvalue of the schrodinger operators by using m hino s exponential decay estimates of the semigroup 2000 osaka j math 37 603 624 and the author s strong ergodicity property of the semi group s aida 1998 j funct anal 158 152 185 in this paper we will give a lower bound on the gap by using a weak poincare inequality which was introduced by m rockner and f y wang 2000 weak poincare inequalities and l 2 convergence rates of markov semigroups preprint also we will give estimates on the distribution function of ground states using the weak poincare inequality

an openmp parallelized multilevel green s function interpolation method accelerated by fast fourier transform technique a parallelized multilevel green s function interpolation method mlgfim accelerated by fast fourier transform fft technique is proposed the difficulties in applying various improved interpolation schemes to effectively reduce the number of interpolation points are overcome by using the fft technique in order to accelerate the convergence property of the iterative solution using the proposed algorithm a recently proposed preconditioning scheme i e multiplicative calderon preconditioner is adopted to transform the first kind integral operator to the second kind albeit an increase of computer memory storage requirement an openmp parallel implementation of the mlgfim fft algorithm on a share memory computer system is developed to analyze various electrically large electromagnetic scattering problems including a nasa almond a 20 wavelength cylinder capped with two half spheres and a 37 wavelength cylinder array numerical results illustrate good computational performance of the proposed algorithm