isotropic gauss markov currents a natural definition of the markov property for multi parameter random processes random fields is the following let xt t n be a multiparameter process for any set d in n let d denote the field generated by x t t d the field xt t d is said to be markov or markov of degree 1 6 or sharp markov if for any bounded open set d with smooth boundary d and d c are conditionally independent given d it has been known for some time that to find interesting examples of markov processes under this definition it is necessary to consider generalized random functions in this paper we show that a natural framework for the markov property of multiparameter processes is a class of generalized random differential forms i e random currents our principal objective is to relate the markovian nature of an isotropic gaussian current to its spectral properties

on the curvature of piecewise flat spaces we consider analogs of the lipschitz killing curvatures of smooth riemannian manifolds for piecewise flat spaces in the special case of scalar curvature the definition is due to t regge considerations in this spirit date back to j steiner we show that if a piecewise flat space approximates a smooth space in a suitable sense then the corresponding curvatures are close in the sense of measures

classical dirichlet forms on topological vector spaces the construction of the associated diffusion process given a minimal classical dirichlet form onl 2 e we construct the associated diffusion process heree is a locally convex topological vector space and is a not necessarily quasi invariant probability measure one the construction is carried out under certain assumptions one and which can be easily verified in many examples in particular we explicitly apply our results to time zero and space time quantum fields with or with out cut off

construction of convergent simplicial approximations of quantum fields on riemannian manifolds we construct simplicial approximations of random fields on riemannian manifolds of dimensiond we prove convergence of the fields to the continuum limit for arbitraryd in the gaussian case and ford 2 in the non gaussian case in particular we obtain convergence of the simplicial approximation to the continuum limit for quantum fields on riemannian manifolds with exponential interaction

on the increments of partial sum processes with multidimensional indices consider a set xn nd of iid random variables indexed by nd the positive integer d dimensional lattice points the erdos renyi 1970 law of large numbers is generalized to this case and some extensions are given the results include a theorem of chan 1976 for the d parameter brownian motion

the ultra weak variational formulation for elastic wave problems the ultra weak variational formulation has been used effectively to solve time harmonic acoustic and electromagnetic wave propagation in inhomogeneous media we develop the ultra weak variational formulation for elastic wave propagation in two space dimensions in order to improve the accuracy and stability of the method we find it necessary to approximate the s and p wave components of the solution in a balanced way some preliminary analysis is provided and numerical evidence is presented for the efficiency of the scheme in comparison to piecewise linear finite elements

on efficient preconditioners for iterative solution of a galerkin boundary element equation for the three dimensional exterior helmholtz problem the paper presents a galerkin numerical method for solving the hyper singular boundary integral equations for the exterior helmholtz problem in three dimensions with a neumann s boundary condition previous work in the topic has often dealt with the collocation method with a piecewise constant approximation because high order collocation and galerkin methods are not available due to the presence of a hypersingular integral operator this paper proposes a high order galerkin method by using singularity subtraction technique to reduce the hyper singular operator to a weakly singular one moreover we show here how to extend the previous work j appl numer math 36 4 2001 475 489 on sparse preconditioners to the galerkin case leading to fast convergence of two iterative solvers the conjugate gradient normal method and the generalised minimal residual method a comparison with the collocation method is also presented for the helmholtz problem with several wavenumbers

a fast high order algorithm for the solution of surface scattering problems basic implementation tests and applications we present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three dimensional space this algorithm evaluates scattered fields through fast high order solution of the corresponding boundary integral equation the high order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities the acceleration in turn results from use of a novel approach which based on high order two face equivalent source approximations reduces the evaluation of far interactions to evaluation of 3 d fast fourier transforms ffts this approach is faster and substantially more accurate and it runs on dramatically smaller memories than other fft and k space methods the present algorithm computes one matrix vector multiplication in o n6 5logn to o n4 3logn operations where n is the number of surface discretization points the latter estimate applies to smooth surfaces for which our high order algorithm provides accurate solutions with small values of n the former more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling further our approach exhibits super algebraic convergence it can be applied to smooth and nonsmooth scatterers and it does not suffer from accuracy breakdowns of any kind in this paper we introduce the main algorithmic components in our approach and we demonstrate its performance with a variety of numerical results in particular we show that the present algorithm can evaluate accurately in a personal computer scattering from bodies of acoustical sizes of several hundreds

representation and construction of multiplicative noise 251 the basic assumption in this paper in which we continue expand and in certain respects bring to conclusion the investigations from 2 is as follows we consider exclusively noise with values in some topological monoid semigroup which are multiplicative in distribution at dif ferent points since multiplication of independent random variables with values in a monoid corresponds to convolution it is clear that this is an invariance assumption we will now give a brief discussion of our results generally speaking one should look upon them as generalizations to random fields and therefore to a more general time parameter of the classical levy khinchin formulae and their consequences section 3 corresponds to processes with stationary and independent increments otherwise stationarity is not assumed in section 1 we prove a kind of daniell kolmogorov theorem to settle the relevant questions on consistency and existence furthermore support properties are discussed we introduce an additional basic assumption our noise is continuous on its support in the sense that it is non atomic in section 2 we restrict ourselves to noise with values in an abelian group a complete characterisation via the fourier transform is afforded by theorem 2 5 in section 3 we assume a general kind of invariance not necessarily given by translations also in the domain of definition then at least formally our random field is of the form x p where x t

a collocation method for high frequency scattering by convex polygons we consider the problem of scattering of a time harmonic acoustic incident plane wave by a sound soft convex polygon for standard boundary or finite element methods with a piecewise polynomial approximation space the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave recently chandler wilde and langdon proposed a novel galerkin boundary element method for this problem for which by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency here we propose a related collocation method using the same approximation space for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the galerkin scheme but with a substantially reduced computational cost

on the erdos renyi theorem for random fields and sequences and its relationships with the theory of runs and spacings we prove in this paper a law of erdos renyi type for arrays of independent and identically distributed random variables the relationships of our theorem with similar results obtained in the theory of runs and spacings are investigated applications include the evaluation of the rate of convergence of erdos renyi maxima in limiting cases and a generalization of a theorem of erdos and revesz on runs

stochastic multiplicative measures generalized markov semigroups and group valued stochastic processes and fields abstract we extend the well known relation between markov semigroups and processes with independent increments to the case of generalized group valued semigroups indexed by some measurable space and stochastic multiplicative measures we give a kolmogorov type theorem and a classification theorem connecting the stochastic multiplicative measures with vector valued levy khinchine fields the results have applications to the study of stochastic connections and differential equations for forms on lie groups

convergence of u 1 3 lattice gauge theory to its continuum limit it is shown that in three space time dimensions the pure u 1 lattice gauge theory with villain action and fixed coupling constant converges to the free electromagnetic field as the lattice spacing approaches zero the same holds for the wilson action on the electric sector

lattice systems with a continuous symmetry iii low temperature asymptotic expansion for the plane rotator model we prove that the expansion in powers of the temperature t of the correlation functions and the free energy of the plane rotator model on a d dimensional lattice is asymptotic to all orders in t the leading term in the expansion is the spin wave approximation and the higher powers are obtained by the usual perturbation series we also prove the inverse power decay of the pair correlation at low temperatures for d 3

abelian chern simons theory and linking numbers via oscillatory integrals a rigorous mathematical model of abelian chern simons theory based on the theory of infinite dimensional oscillatory integrals developed by albeverio and ho egh krohn is introduced a gauge fixed chern simons path integral is constructed as a fresnel integral in a certain hilbert space wilson loop variables are defined as fresnel integrable functions and it is shown in this context that the expectation value of products of wilson loops with respect to the chern simons path integral is a topological invariant which can be computed in terms of pairwise linking numbers of the loops as conjectured by witten furthermore a lattice chern simons action is proposed which converges to the continuum limit

analysis of a collocation method for integrating rapidly oscillatory functions a collocation method for approximating integrals of rapidly oscillatory functions is analyzed the method is efficient for integrals involving bessel functions j rx with a large oscillation frequency parameter r as well as for many other one and multi dimensional integrals of functions with rapid irregular oscillations the analysis provides a convergence rate and it shows that the relative error of the method is even decreasing as the frequency of the oscillations increases ko words oscillatory integrals collocation analysis

convergence of theu 1 4 lattice gauge theory to its continuum limit it is shown that in four space time dimensions the compactu 1 lattice gauge theory with general energy function converges to a renormalized free electromagnetic field on the current sector as the lattice spacing approaches zero provided the coupling constant is sufficiently large for the wilson energy function it is possible by judicious choice of the gibbs state to get convergence for arbitrary coupling strengths furthermore for all but a countable number of values of the coupling constant the limit exists and is independent of the particular state chosen to define the lattice model

a convergent lattice approximation for nonlinear electromagnetic fields in four dimensions a discrete version of the quaternionic cauchy riemann equation a f where f is gauss poisson white noise is discussed on the lattice z4 random variables f n and a n are constructed which approximate the corresponding random fields f and a respectively in the limit 0 in the gaussian case the random variables a n can be interpreted as the lattice approximation of the free electromagnetic euclidean potential field whereas in the non gaussian case one obtains an approximation of nonlinear interacting electromagnetic quantum fields convergence to the continuum limit is proven

improved erdos renyi and strong approximation laws for increments of partial sums let x 1 x 2 cdots be an i i d sequence with ex 1 0 ex 2 1 1 ee tx 1 infty t t 0 and partial sums s n x 1 cdots x n starting from some analogous results for the wiener process this paper studies the almost sure limiting behaviour of max 0 leq n leq n a n a 1 2 n s n a n s n as n rightarrow infty under various conditions on the integer sequence a n improvements of the erdos renyi law of large numbers for partial sums are obtained as well as strong invariance principle type versions via the komlos major tusnady approximation an appearing gap between these two results is also going to be closed

lattice systems with a continuous symmetry we prove that the expansion in powers of the temperature t of the correlation functions and the free energy of the plane rotator model on a d dimensional lattice is asymptotic to all orders in t the leading term in the expansion is the spin wave approximation and the higher powers are obtained by the usual perturbation series we also prove the inverse power decay of the pair correlation at low temperatures for d 3

ferromagneticity of simplicial fields on two dimensional compact manifolds smooth triangulations of a compact smooth connected two dimensional riemannian manifold m are considered the q simplicial fields are defined with values in the space of q cochains and a natural gaussian measure is defined giving their distribution with covariance defined essentially in terms of the combinatorial laplacian c in the continuum limit this measure for q 0 is the free quantum field measure over m in this case it is shown that for a certain collection of triangulations there exists a sequence of subdivisions of each triangulation such that the corresponding measure is ferromagnetic it is also shown that for sufficiently fine subdivisions m2i m 0 has nonpositive off diagonal elements the proofs are obtained by a result on triangulations by simplexes with acute angles it is also proven that the probability measures describing quantum fields on m with polynomial trigonometric or exponential interactions satisfy fkg inequalities

random fields as solutions of the inhomogeneous quaternionic cauchy riemann equation i invariance and analytic continuation we consider random fieldsa satisfying the quaternionic cauchy riemann equation a f wheref is white noise under appropriate conditions onf a is invariant under the proper euclidean group in four dimensions but in general not under time reflection the schwinger functions can be analytically continued to wightman functions satisfying the relativistic postulates on invariance specrral property and locality

on the evaluation of highly oscillatory integrals by analytic continuation we consider the integration of one dimensional highly oscillatory functions based on analytic continuation rapidly converging quadrature rules are derived for a general class of oscillatory integrals with an analytic integrand the accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency and for the case of an increasing number of points and fixed frequency these results are then used to obtain quadrature rules for more general oscillatory integrals i e for functions that exhibit some smoothness but that are not analytic the approach described in this paper is related to the steepest descent method but it does not employ asymptotic expansions it can be used for small or moderate frequencies as well as for very high frequencies the approach is compared with the oscillatory integration techniques recently developed by iserles and norsett

uniqueness and the global markov property for euclidean fields the case of general exponential interaction the uniqueness and the global markov property for the regular gibbs measure corresponding to the interaction u lambda varphi lambda int limits lambda d 2 x int d varrho alpha e alpha varphi 0 x for 0 d a probability measure with support in 2 sqrt pi 2 sqrt pi is proved

high frequency scattering by convex curvilinear polygons we consider the scattering of a time harmonic acoustic incident plane wave by a sound soft convex curvilinear polygon with lipschitz boundary for standard boundary or finite element methods with a piecewise polynomial approximation space the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to the frequency of the incident wave here we propose a novel galerkin boundary element method with a hybrid approximation space consisting of the products of plane wave basis functions with piecewise polynomials supported on several overlapping meshes a uniform mesh on illuminated sides and graded meshes refined towards the corners of the polygon on illuminated and shadow sides numerical experiments suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy need only grow logarithmically as the frequency of the incident wave increases

a wavenumber independent boundary element method for an acoustic scattering problem in this paper we consider the impedance boundary value problem for the helmholtz equation in a half plane with piecewise constant boundary data a problem which models for example outdoor sound propagation over inhomogeneous flat terrain to achieve good approximation at high frequencies with a relatively low number of degrees of freedom we propose a novel galerkin boundary element method using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that on each element the approximation space contains polynomials of degree nu multiplied by traces of plane waves on the boundary we prove stability and convergence and show that the error in computing the total acoustic field is cal o n nu 1 log 1 2 n where the number of degrees of freedom is proportional to n log n this error estimate is independent of the wavenumber and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity

boundary value problems and hardy spaces associated to the helmholtz equation in lipschitz domains abstract in this paper we introduce and discuss in the clifford algebra framework certain hardy like spaces which are well suited for the study of the helmholtz equation u k 2 u 0 in lipschitz domains of r n 1 in particular in the second part of the paper these results are used in connection with the classical boundary value problems for the helmholtz equation in lipschitz domains in arbitrary space dimensions in this setting existence uniqueness and optimal estimates are obtained by inverting the corresponding layer potential operators on l p for sharp ranges of p s also a detailed discussion of the helmholtz eigenvalues of lipschitz domains is presented

a hybrid numerical asymptotic method for scattering problems we develop a hybrid numerical asymptotic method for the helmholtz equation the method is a galerkin finite element method in which the space of trial solutions is spanned by asymptotically derived basis functions the basis functions are very efficient in representing the solution because each is the product of a smooth amplitude and an oscillatory phase factor like the asymptotic solution the phase is determined a priori by solving the eiconal equation using the ray method while the smooth amplitude is represented by piecewise polynomials the number of unknowns necessary to achieve a given accuracy with this new basis is dramatically smaller than the number necessary with a standard method and it is virtually independent of the wavenumber k we apply the method to the problems of scattering from a parabola and from a circle and compare the results with those of a standard finite element method

coupling of fast multipole method and microlocal discretization for the 3 d helmholtz equation we are concerned with an integral method applied to the solution of the helmholtz equation where the linear system is solved using an iterative method we need to perform matrix vector products whose time and memory requirements increase as a function of the wavenumber k many methods have been developed to speed up the matrix vector product calculation or to reduce the size of the system microlocal discretization methods enable one to consider new systems with reduced size another method the fast multipole method is one of the most efficient and robust methods used to speed up the calculation of matrix vector products in this paper a coupling of these two recently developed methods is presented this coupling enables one to reduce cpu time very efficiently for large wavenumbers satisfactory numerical tests are also presented to confirm the theoretical study within a new integral formulation results are obtained for a sphere with a size of 26 using a resolution based on a mesh with an average edge length of about 2 where is the wavelength results are also given for an industrial test case from dassault aviation the cetaf

a galerkin boundary element method for high frequency scattering by convex polygons in this paper we consider the problem of time harmonic acoustic scattering in two dimensions by convex polygons standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave here we present a novel galerkin boundary element method which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon we prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency numerical results demonstrate the same logarithmic dependence on the frequency for the galerkin method solution our boundary element method is a discretization of a well known second kind combined layer potential integral equation we provide a proof that this equation and its adjoint are well posed and equivalent to the boundary value problem in a sobolev space setting for general lipschitz domains