the transverse index theorem for proper cocompact actions of lie groupoids given a proper cocompact action of a lie groupoid we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes we prove a cohomological index formula for this pairing by applying the van est map and algebraic index theory finally we discuss in examples the meaning of the index pairing and our index formula

eta invariants with spectral boundary conditions we study the asymptotics of the heat trace where p is an operator of dirac type where f is an auxiliary smooth smearing function which is used to localize the problem and where we impose spectral boundary conditions using functorial techniques and special case calculations the boundary part of the leading coefficients in the asymptotic expansion is found

l2 analytic torsion abstract in this paper we introduce a new differential invariant called l2 analytic torsion for closed manifolds with positive decay and whose universal covers have trivial l2 cohomology l2 analytic torsion can be thought of as a suitable generalization of the ray singer analytic torsion we establish various functorial properties of l2 analytic torsion and also compute it for odd dimensional closed hyperbolic manifolds with the help of results from fried

proof of the symmetry of the off diagonal hadamard seeley dewitt s coefficients in c lorentzian manifolds by a local wick rotation completing the results achieved in a previous paper we prove the symmetry of hadamard seeley dewitt off diagonal coefficients in smooth d dimensional lorentzian manifolds this result is relevant because it plays a central role in physics in particular in the theory of the stress energy tensor renormalization procedure in quantum field theory in curved spacetime to this end it is shown that in any lorentzian manifold a sort of local wick rotation of the metric can be performed provided the metric is a locally analytic function of the coordinates and the coordinate are appropriate no time like killing field is necessary such a local wick rotation analytically continues the lorentzian metric in a neighborhood of any point more generally in a neighborhood of a space like cauchy hypersurface into a riemannian metric the continuation locally preserves geodesically convex neighborhoods in order to make rigorous the procedure the concept of a complex pseudo riemannian not hermitian or kahlerian manifold is introduced and some features are analyzed using these tools the symmetry of hadamard seeley dewitt off diagonal coefficients is proven in lorentzian analytical manifolds by analytical continuation of the symmetric riemannian heat kernel coefficients this continuation is performed in geodesically convex neighborhoods in common with both the metrics then the symmetry is generalized to c non analytic lorentzian manifolds by approximating lorentzian c metrics by analytic metrics in common geodesically convex neighborhoods

a fixed point formula and harish chandra s character formula the main result in this paper is a fixed point formula for equivariant indices of elliptic differential operators for proper actions by connected semisimple lie groups on possibly noncompact manifolds with compact quotients for compact groups and manifolds this reduces to the atiyah segal singer fixed point formula other special cases include an index theorem by connes and moscovici for homogeneous spaces and an earlier index theorem by the second author both in cases where the group acting is connected and semisimple as an application of this fixed point formula we give a new proof of harish chandra s character formula for discrete series representations

zeta function regularization the multiplicative anomaly and the wodzicki residue the multiplicative anomaly associated with the zeta function regularized determinant is computed for the laplace type operators v1 v1 and v2 v2 with v1 v2 constant in a d dimensional compact smooth manifold md making use of several results due to wodzicki and by direct calculations in some explicit examples it is found that the multiplicative anomaly is vanishing for d odd and for d 2 an application to the one loop effective potential of the o 2 self interacting scalar model is outlined

compatibility conditions for time dependent partial differential equations and the rate of convergence of chebyshev and fourier spectral methods abstract compatibility conditions for partial differential equations pdes are an infinite set of relations between the initial conditions the pde and the boundary conditions which are necessary and sufficient for the solution to be c that is infinitely differentiable everywhere on the computational domain including the boundaries since the performance of chebyshev spectral and spectral element methods is dramatically reduced when the solution is not c one would expect that the compatibility conditions would be a major theme in the spectral literature instead it has been completely ignored therefore we pursue three goals here first we present a proof of the compatibility conditions in a simplified form that does not require functional analysis second we analyze the connection between the compatibility conditions and the rate of convergence of chebyshev methods lastly we describe strategies for slightly adjusting initial conditions so that the compatibility conditions are satisfied

heat kernel coefficients of the laplace operator on the d dimensional ball we present a very quick and powerful method for the calculation of heat kernel coefficients it makes use of rather common ideas as integral representations of the spectral sum mellin transforms non trivial commutation of series and integrals and skillful analytic continuation of zeta functions on the complex plane we apply our method to the case of the heat kernel expansion of the laplace operator on a d dimensional ball with either dirichlet neumann or in general robin boundary conditions the final formulas are quite simple using this case as an example we illustrate in detail our scheme which serves for the calculation of an in principle arbitrary number of heat kernel coefficients in any situation when the basis functions are known we provide a complete list of new results for the coefficients b3 b10 corresponding to the d dimensional ball with all the mentioned boundary conditions and d 3 4 5

classification ofc algebras of real rank zero and unsuspendede equivalence types abstract in this article examples are given to prove that the graded scaled ordered k group is not the complete invariant for a c algebra in the class of unital separable nuclear c algebras of real rank zero and stable rank one even for a c algebra in the subclass which consists of those real rank zero stable rank one c algebras being expressed as inductive limits of k n i 1 m n i c x n i where x n i are two dimensional finite cw complexes and n i are positive integers in the case of simple such c algebras it has been proved that the above invariant is the complete invariant by george elliott and the author these examples prove that the classification conjecture of elliott for the case of non simple real rank zero c algebras should be revised one needs extra invariants the obstruction preventing two such c algebras with the same graded scaled ordered k group from being isomorphic is that they have different unsuspended e equivalence types a refinement of kk equivalence type of c algebras due to connes and higson in this article it is proved that for the above class of inductive limit c algebras the obstruction of unsuspended e equivalence type is the only obstruction i e if two c algebras in the class are unsuspended e equivalence then they are isomorphic it is a surprise that in the case of simple such c algebras or even the case of c algebras with finitely many ideals the obstruction will disappear see section 4

sum accelerated pseudospectral methods finite differences and sech weighted differences this work continues our previous studies of algorithms for accelerating the convergence of pseudospectral derivative series in order to obtain new differentiation schemes which have the sparsity and low cost of finite differences but the accuracy of spectral methods we develop a general theoretical framework for difference schemes finite differences are close to optimum but can be bettered by a new scheme we have dubbed sech weighted differences through numerical examples we show that sech weighted differences are effective in contrast non linear accelerations like pade approximants and the levin u transform so popular in other applications are inferior for approximating derivatives to linear accelerations like finite differences sech weighted differences and the euler method

linear convergence for vector sequences and some applications abstract in this paper we first give the relation between some definitions of linearly convergent vector sequences proposed by nievergelt osada and sadok and we choose the first one in order to generalize a theorem which gives the relation between the asymptotic behaviour of the ratios of the errors and that of the differences for linear convergent scalar sequences some properties of the matrix used in the definition of linear convergence are studied and a convergence acceleration result which generalizes some results given by nievergelt and sadok is given

optimal filter and mollifier for piecewise smooth spectral data we discuss the reconstruction of piecewise smooth data from its pseudo spectral information spectral projections enjoy superior resolution provided the function is globally smooth while the presence of jump discontinuities is responsible for spurious o 1 gibbs oscillations in the neighborhood of edges and an overall deterioration of the convergence rate to the unacceptable first order classical filters and mollifiers are constructed to have compact support in the fourier frequency and physical time spaces respectively and are dilated by the projection order or the width of the smooth region to maintain this compact support in the appropriate region here we construct a noncompactly supported filter and mollifier with optimal joint time frequency localization for a given number of vanishing moments resulting in a new fundamental dilation relationship that adaptively links the time and frequency domains not giving preference to either space allows for a more balanced error decomposition which when minimized yields an optimal filter and mollifier that retain the robustness of classical filters yet obtain true exponential accuracy

the summation of the ordinary and renormalized perturbation series for the ground state energy of the quartic sextic and octic anharmonic oscillators using nonlinear sequence transformations the strongly divergent rayleigh schrodinger perturbation expansion for the ground state energy of the quartic sextic and octic anharmonic oscillator and a corresponding renormalized perturbation expansion f vinette and j ci ek j math phys 32 3392 1991 are summed by pade approximants by levin s sequence transformation d levin int j comput math b 3 371 1973 and by a closely related sequence transformation which was suggested recently e j weniger comput phys rep 10 189 1989 it is shown that the renormalized perturbation expansion can be summed much more easily than the original perturbation expansion from which it was derived and that levin s sequence transformation diverges and is not able to sum the perturbation expansions the pade summation of the renormalized perturbation expansions gives relatively good results in the quartic and sextic case in the case of the octic anharmonic oscillator even the renormalized perturbation expansion is not pade summable the best results are clearly obtained by the new sequence transformation which for instance is able to sum the renormalized perturbation expansions for the infinite coupling limits of the quartic sextic and octic anharmonic oscillator and which produces at least in the quartic and sextic case extremely accurate results

improvement of accuracy in numerical methods for inverting laplace transforms based on the post widder formula abstract the paper proposes a new computational version for numerical inversion of laplace transforms at a point and on an interval based on the post widder formula in this version the original sequence of post widder approximants is calculated using operations on series and the approximate value of the sought function is constructed as a limit of this sequence using polynomial and rational extrapolation to the limit some procedures are proposed for improvement of accuracy in the problem of inversion at equidistant points of an interval without computing additional approximants an extension of the method to the two dimensional case is discussed the one dimensional version is illustrated with some probabilistic examples involving both explicit and implicit functions the accuracy achieved in these examples with using the standard double precision arithmetic evaluated in terms of relative errors is about 10 10 10 14 even in case of out of scale parameters computational aspects are discussed in comparison with previously known realizations of the post widder method

uniqueness theorem for bms invariant states of scalar qft on the null boundary of asymptotically flat spacetimes and bulk boundary observable algebra correspondence this work concerns some features of scalar qft defined on the causal boundary mathfrak j of an asymptotically flat at null infinity spacetime and based on the bms invariant weyl algebra mathcal w mathfrak j

extended procedures for extrapolation to the limit in this paper new procedures for the extrapolation to the limit of slowly convergent sequences and functions are proposed they are based on the notions of error estimates and annihilation operators we obtain generalizations of the discrete and confluent e transformation which are the most general sequence and function transformations known so far it is shown that many transformations studied in the literature are included in our formalism particular cases of these procedures are discussed then several extensions of the e algorithm are given finally the procedure q is applied to our procedures to produce new procedures for extrapolation to the limit

residues of the eta function for an operator of dirac type abstract we compute the asymptotics of tr l 2 pe tp 2 where p is a first order operator of dirac type this is equivalent to evaluating the residues of the eta function

stieltjes transforms and the stokes phenomenon recently berry olver and jones have found uniform asymptotic expansions for the exponentially small remainder terms that result when asymptotic expansions are optimally truncated these uniform expansions describe the rapid change in the behaviour of the remainders as a stokes line is crossed we show how such uniform expansions may be found when a function can be expressed as a stieltjes transform such an approach has the following advantages the uniform expansion is calculated directly non uniform expansions away from the stokes line are readily found and explicit error bounds may be established we illustrate the method by application to the modified bessel function k v z

on the stress energy tensor of quantum fields in curved spacetimes comparison of different regularization schemes and symmetry of the hadamard seeley dewitt coefficients we review a few rigorous and partly unpublished results on the regularisation of the stress energy in quantum field theory on curved spacetimes 1 the symmetry of the hadamard seeley dewitt coefficients in smooth riemannian a lorentzian spacetimes 2 the equivalence of the local function and the hadamard point splitting procedure in smooth static spacetimes 3 the equivalence of the dewitt schwinger and the hadamard point splitting pro cedure in smooth riemannian and lorentzian spacetimes

a lag averaged generalization of euler s method for accelerating series abstract euler s method is an efficient accelerator for an alternating series that is one whose terms a j are such that sign a j sign a j 1 however its effectiveness deteriorates rapidly as the period p of the oscillations in degree j increases to reduce this nonuniformity we introduce an algorithm which begins with the recursive computation of a triangular array of numbers the diagonal elements define a generalization of euler s method each horizontal row defines a delayed euler method which is even more effective the key element is that the array is generated by repeatedly averaging partial sums of the original series with a lag that is a distance in degree between the partial sums that are combined which optimally is half the period p of the oscillations in the series we illustrate the algorithm with the fourier series for a function with a shock wave jump discontinuity the terms a j 1 j sin jx vary from strictly alternating p 2 at x 0 to monotonic p as x the lag averaged euler scheme with and without delay is much more efficient than its classical counterpart especially near the discontinuities we present an analytical convergence theory for both the standard and generalized euler methods

local causal structures hadamard states and the principle of local covariance in quantum field theory in the framework of the algebraic formulation we discuss and analyse some new features of the local structure of a real scalar quantum field theory in a strongly causal spacetime in particular we use the properties of the exponential map to set up a local version of a bulk to boundary correspondence the bulk is a suitable subset of a geodesic neighbourhood of an arbitrary but fixed point p of the underlying background while the boundary is a part of the future light cone having p as its own tip in this regime we provide a novel notion for the extended algebra of wick polynomials on the aforesaid cone and on the one hand we prove that it contains the information of the bulk counterpart via an injective homomorphism while on the other hand we associate to it a distinguished state whose pull back in the bulk is of hadamard form the main advantage of this point of view arises if one uses the universal properties of the exponential map and of the light cone in order to show that for any two given backgrounds m and m and for any two subsets of geodesic neighbourhoods of two arbitrary points it is possible to engineer the above procedure such that the boundary extended algebras are related via a restriction homomorphism this allows for the pull back of boundary states in both spacetimes and thus to set up a machinery which permits the comparison of expectation values of local field observables in m and m

the generally covariant locality principle a new paradigm for local quantum field theory a new approach to the model independent description of quantum field theories will be introduced in the present work the main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity thus giving rise to the concept of a locally covariant quantum field theory such locally covariant quantum field theories will be described mathematically in terms of covariant functors between the categories on one side of globally hyperbolic spacetimes with isometric embeddings as morphisms and on the other side of algebras with unital injective monomorphisms as morphisms moreover locally covariant quantum fields can be described in this framework as natural transformations between certain functors the usual haag kastler framework of nets of operator algebras over a fixed spacetime background manifold together with covariant automorphic actions of the isometry group of the background spacetime can be re gained from this new approach as a special case examples of this new approach are also outlined in case that a locally covariant quantum field theory obeys the time slice axiom one can naturally associate to it certain automorphic actions called relative cauchy evolutions which describe the dynamical reaction of the quantum field theory to a local change of spacetime background metrics the functional derivative of a relative cauchy evolution with respect to the spacetime metric is found to be a divergence free quantity which has as will be demonstrated in an example the significance of an energy momentum tensor up to addition of scalar functions for the locally covariant quantum field theory furthermore we discuss the functorial properties of state spaces of locally covariant quantum field theories that entail the validity of the principle of local definiteness

a continuous euler transformation and its application to the fourier transform of a slowly decaying fuction the euler transformation is a linear sequence transformation to accelerate the convergence of an alternating series the sequence of weights of the transformation is extended to a continuous weight function which can accelerate fourier type integrals including hankel transforms with a slowly convergent integrand we show that the continuous weight function can also be used to compute the fourier transform of a slowly decaying function using fft

note on a theorem of h moscovici let be a discrete subgroup of a semisimple lie group g such that g has a finite volume using a theorem of moscovici we express the multiplicity of discrete series representations of g in the discrete spectrum of l2 g as the l2 index of a twisted dirac operator this result which extends a result of moscovici and of the author holds for all integrable discrete series and for infinitely many nonintegrable discrete series in particular up to computing l2 indices in the special rank one case it implies the osborne warner formula

harmonically induced representations of solvable lie groups soit un groupe de lie g d algebre de lie g et h une polarisation pour l orbite on etudie la representation obtenue quand h est totalement complexe non necessairement positive et g resoluble

local wick polynomials and time ordered products of quantum fields in curved spacetime in order to have well defined rules for the perturbative calculation of quantities of interest in an interacting quantum field theory in curved spacetime it is necessary to construct wick polynomials and their time ordered products for the noninteracting theory a construction of these quantities has recently been given by brunetti fredenhagen and kohler and by brunetti and fredenhagen but they did not impose any locality or covariance condition in their constructions as a consequence their construction of time ordered products contained ambiguities involving arbitrary functions of spacetime point rather than arbitrary parameters in this paper we construct an extended wick polynomial algebra large enough to contain the wick polynomials and their time ordered products by generalizing a construction of dutsch and fredenhagen to curved spacetime we then define the notion of a local covariant quantum field and seek a definition of local wick polynomials and their time ordered products as local covariant quantum fields we introduce a new notion of the scaling behavior of a local covariant quantum field and impose scaling requirements on our local wick polynomials and their time ordered products as well as certain additional requirements such as commutation relations with the free field and appropriate continuity properties under variations of the spacetime metric for a given polynomial order in powers of the field we prove that these conditions uniquely determine the local wick polynomials and their time ordered products up to a finite number of parameters these parameters correspond to the usual renormalization ambiguities occurring in minkowski spacetime together with additional parameters corresponding to the coupling of the field to curvature we also prove existence of local wick polynomials however the issue of existence of local time ordered products is deferred to a future investigation

existence of local covariant time ordered products of quantum fields in curved spacetime we establish the existence of local covariant time ordered products of local wick polynomials for a free scalar field in curved spacetime our time ordered products satisfy all of the hypotheses of our previous uniqueness theorem so our construction essentially completes the analysis of the existence uniqueness and renormalizability of the perturbative expansion for nonlinear quantum field theories in curved spacetime as a byproduct of our analysis we derive a scaling expansion of the time ordered products about the total diagonal that expresses them as a sum of products of polynomials in the curvature times lorentz invariant distributions plus a remainder term of arbitrarily low scaling degree

one loop stress tensor renormalization in curved background the relation between ζ function and point splitting approaches and an improved point splitting procedure we conclude the rigorous analysis of a previous paper v moretti commun math phys 201 327 1999 concerning the relation between the euclidean point splitting approach and the local function procedure to renormalize physical quantities at one loop in euclidean quantum field theory in curved space time the case of the stress tensor is now considered in general d dimensional closed manifolds for positive scalar operators v x results obtained formally in previous works in the case d 4 and v x r x m2 are rigorously proven and generalized it is also proven that in static euclidean manifolds the method is compatible with lorentzian time analytic continuations it is proven that the result of the function procedure is the same obtained from an improved version of the point splitting method which uses a particular choice of the term w0 x y in the hadamard expansion of the green s function given in terms of heat kernel coefficients this version of the point splitting procedure works

the dirac operator on homogeneous spaces and representations of reductive lie groups ii soit g un groupe de lie reductif reel connexe et k un sousgroupe compact maximal contenant un sousgroupe connexe compact donne l de g soit d v l operateur de dirac tordu soit kerd v l espace des spineurs harmoniques l une representation unitaire de g on etudie comment kerd v l se decompose en representations irreductibles unitaires de g quand l est un sous groupe compact

algebraic quantum field theory perturbation theory and the loop expansion the perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory we show that the algebra of interacting fields is additive i e fully determined by its subalgebras associated to arbitrary small subregions of minkowski space we also give an algebraic formulation of the loop expansion by introducing a projective system n of observables up to n loops where 0 is the poisson algebra of the classical field theory finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of green s functions

comments on the stress energy tensor operator in curved spacetime the technique based on a algebra of wick products of field operators in curved spacetime in the local covariant version proposed by hollands and wald is strightforwardly generalized in order to define the stress energy tensor operator in curved globally hyperbolic spacetimes in particular the locality and covariance requirement is generalized to wick products of differentiated quantum fields within the proposed formalism there is room to accomplish all of the physical requirements provided that known problems concerning the conservation of the stress energy tensor are assumed to be related to the interface between the quantum and classical formalism the proposed stress energy tensor operator turns out to be conserved and reduces to the classical form if field operators are replaced by classical fields satisfying the equation of motion the definition is based on the existence of convenient counterterms given by certain local wick products of differentiated fields these terms are independent from the arbitrary length scale and any quantum state and they classically vanish on solutions of the klein gordon equation considering the averaged stress energy tensor with respect to hadamard quantum states the presented definition turns out to be equivalent to an improved point splitting renormalization procedure which makes use of the nonambiguous part of the hadamard parametrix only that is determined by the local geometry and the parameters which appear in the klein gordon operator in particular no extra added by hand term g q and no arbitrary smooth part of the hadamard parametrix generated by some arbitrary smooth term 0 are involved the averaged stress energy tensor obtained by the point splitting procedure also coincides with that found by employing the local function approach whenever that technique can be implemented

conformal generally covariant quantum field theory the scalar field and its wick products in this paper we generalize the construction of generally covariant quantum theories given in bfv03 to encompass the conformal covariant case after introducing the abstract framework we discuss the massless conformally coupled klein gordon field theory showing that its quantization corresponds to a functor between two certain categories at the abstract level the ordinary fields could be thought of as natural transformations in the sense of category theory we show that the wick monomials without derivatives wick powers can be interpreted as fields in this generalized sense provided a non trivial choice of the renormalization constants is given a careful analysis shows that the transformation law of wick powers is characterized by a weight and it turns out that the sum of fields with different weights breaks the conformal covariance at this point there is a difference between the previously given picture due to the presence of a bigger group of covariance it is furthermore shown that the construction does not depend upon the scale appearing in the hadamard parametrix used to regularize the fields finally we briefly discuss some further examples of more involved fields

improved kinetic renormalisation group approach to diffusion limited aggregation an improved kinetic renormalisation group approach to diffusion limited aggregation is presented this approach is based on the growth process itself and accounts for the dispersity in the growth probabilities it yields the multifractal spectrum d q with better values at smaller q on the 2d square lattice the authors find df 1 694 for the fractal dimensions of the cluster and that of its interface and d 1 1 01 for the information dimension the former agrees with the simulation results df approximately 1 70 and the latter compares very well with the exact value d 1 1

l2 index and the selberg trace formula abstract a method is developed for computing the l 2 index of a locally symmetric elliptic differential operator d acting on a locally symmetric manifold m g k with g semisimple of real rank one and of finite co volume based on applying the selberg trace formula to the difference of the two heat kernels associated to d the applications include an extension of the osborne warner multiplicity formula to certain non integrable discrete series derived from the l 2 spinor formula and showing the existence in some cases of non invariant l 2 cohomology classes in the middle dimension via the l 2 signature formula

acceleration of algebraically converging fourier series when the coefficients have series in powers of 1 n if the coefficients in a fourier cosine series f x f email protected n 0 a ncos nx decrease as a small negative power of n then one may need millions of terms to sum the series to high accuracy we show that if the a n are known analytically and have a power series in 1 n then it is straightforward to approximate f x as a series of what we shall the lanczos krylov lk functions we describe the similar methodology for sine series general fourier series are merely the sum of a cosine series with a sine series and thus are implicitly handled too for cosine coefficients that involve only even powers of n and sine coefficients that are functions of odd powers of n the lk functions may be expressed in terms of bernoulli polynomials the lk functions for cosine coefficients involving odd powers of n and for sine coefficients in even powers of n are not known explicitly these are also known as clausen functions we provide rapidly convergent series to compute these clausen functions to high accuracy our method includes the endpoint subtraction ideas of lanczos and krylov but is more general the sum n 1 1 n 1 1 email protected cos nx where l 0 is a constant arises in phase transitions in absorbed monolayers on metal surfaces it is easily summed by our method which correctly incorporates the logarithmic singularities at x email protected

proof of the symmetry of the off diagonal heat kernel and hadamard s expansion coefficients in general c riemannian manifolds we consider the problem of the symmetry of the off diagonal heat kernel coefficients as well as the coefficients corresponding to the short distance divergent part of the hadamard expansion in general smooth analytic or not manifolds the requirement of such a symmetry played a central role in the theory of the point splitting one loop renormalization of the stress tensor in either riemannian or lorentzian manifolds actually the symmetry of these coefficients has been assumed as a hypothesis in several papers concerning these issues without an explicit proof the difficulty of a direct proof is related to the fact that the considered off diagonal heat kernel expansion also in the riemannian case in principle may be not a proper asymptotic expansion on the other hand direct computations of the off diagonal heat kernel coefficients are impossibly difficult in nontrivial cases and thus no case is known in the literature where the symmetry does not hold by approximating c metrics with analytic metrics in common totally normal geodesically convex neighborhoods it is rigorously proven that in general c riemannian manifolds any point admits a geodesically convex neighborhood where the off diagonal heat kernel coefficients as well as the relevant hadamard expansion coefficients are symmetric functions of the two arguments

an extended multistep shanks transformation and convergence acceleration algorithm with their convergence and stability analysis the molecule solution of an extended discrete lotka volterra equation is constructed from which a new sequence transformation is proposed a convergence acceleration algorithm for implementing this sequence transformation is found it is shown that our new sequence transformation accelerates some kinds of linearly convergent sequences and factorially convergent sequences with good numerical stability some numerical examples are also presented

compositions of hadamard type fractional integration operators and the semigroup property abstract this paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of hadamard in the space xcp of functions f on r 0 such that 0 u c f u p du u ess sup u 0 u c f u for c r in particular in the space lp 0 1 p the semigroup property and its generalizations are established for the generalized hadamard type fractional integration operators under consideration conditions are also given for the boundedness in xcp of these operators they involve kummer confluent hypergeometric functions as kernels

zeta functions formulas and applications abstract the existence conditions of the zeta function of a pseudodifferential operator and the definition of determinant thereby obtained are reviewed as well as the concept of multiplicative anomaly associated with the determinant and its calculation by means of the wodzicki residue exponentially fast convergent formulas valid in the whole of the complex plane and yielding the pole positions and residua that extend the ones by chowla and selberg for the epstein zeta function quadratic form and by barnes affine form are then given after briefly recalling the zeta function regularization procedure in quantum field theory some applications of these expressions in physics are described

numerical comparisons of nonlinear convergence accelerators as part of a continuing program of numerical tests of convergence accelerators we have compared the iterated aitken s a2 method wynn s algorithm brezinski s 0 algorithm and levin s u transform on a broad range of test problems linearly convergence alternating monotone and irregular sign series logarithmically convergent series power method and bernoulli method sequences alternating and monotone asymptotic series and some perturba tion series arising in applications in each category either the algorithm or the u transform gives the best results of the four methods tested in some cases differences among methods are slight and in others they are quite striking 1 introduction we reported in 19 on a numerical test program that compared eleven different scalar convergence accelerators on a broad range of infinite series problems in an attempt to find an across the board accelerator that would handle both linear and logarithmic convergence and both alternating and monotone series or sequences the best candidates to emerge from that study were levin s u transform 14 and brezinski s 0 algorithm 6 we also provided some theoretical support for that conclusion in terms of the wide range of series summed exactly by these methods and by proving that 02 and the levin transforms all accelerate linear convergence the other levin transforms t and v produced essentially equivalent numerical results to u except that t fails on logarithmic convergence we have continued the test program started with 19 to include monotone linearly convergent series series with irregular sign patterns specifically fourier series and sequences whose terms do not depend explicitly on the previous term and or the index e g bernoulli s method since convergence accelerators are often known to transform divergent series to convergent ones we have also included in our test program divergent power series asymptotic series and a few perturbation series the methods considered in this paper are the u and 0 algorithms noted above and

three algorithms for hadamard finite part integrals and fractional derivatives abstract three algorithms for the evaluation of the hadamard finite part integral of the form 1 1 f t 1 t 1 where is a positive non integer are described one algorithm is based on a knowledge of the chebyshev series expansion of f on 1 1 the other two on polynomial interpolations to f at the zeros of tn the chebyshev polynomial of the first kind convergence theorems are given for each algorithm and each algorithm is demonstrated numerically

explicit zeta functions for bosonic and fermionic fields on a non commutative toroidal spacetime explicit formulae for the zeta functions s corresponding to bosonic 2 and to fermionic 3 quantum fields living on a non commutative partially toroidal spacetime are derived formulae for the most general case of the zeta function associated with a quadratic linear constant form in z are obtained they provide the analytical continuation of the zeta functions in relation to the whole complex s plane in terms of series of bessel functions of fast exponential convergence thus being extended chowla selberg formulae as is well known this is the most convenient expression that can be found for the analytical continuation of a zeta function in particular the residua of the poles and their finite parts are explicitly given an important novelty is the fact that simple poles show up at s 0 as well as in other places simple or double depending on the number of compactified non compactified and non commutative dimensions of the spacetime where they had never appeared before this poses a challenge to the zeta function regularization procedure

an equivariant index for proper actions i abstract equivariant indices taking values in group theoretic objects have previously been defined in cases where either the group acting or the orbit space of the action is compact in this paper we define an equivariant index without assuming the group or the orbit space to be compact this allows us to generalise an index of deformed dirac operators defined for compact groups by braverman in parts ii and iii of this series we explore some properties and applications of this index

some properties of the sobolev laguerre polynomials abstract in this paper we study the case 0 of the sobolev laguerre polynomials we determine a generating function for the polynomials and an expansion formula

micro local approach to the hadamard condition in quantum field theory on curved space time for the two point distribution of a quasi free klein gordon neutral scalar quantum field on an arbitrary four dimensional globally hyperbolic curved space time we prove the equivalence of 1 the global hadamard condition 2 the property that the feynman propagator is a distinguished parametrix in the sense of duistermaat and hormander and 3 a new property referred to as the wave front set spectral condition wfssc because it is reminiscent of the spectral condition in axiomatic quantum field theory on minkowski space results in micro local analysis such as the propagation of singularities theorem and the uniqueness up toc of distinguished parametrices are employed in the proof we include a review of kay and wald s rigorous definition of the global hadamard condition and the theory of distinguished parametrices specializing to the case of the klein gordon operator on a globally hyperbolic space time as an alternative to a recent computation of the wave front set of a globally hadamard two point distribution on a globally hyperbolic curved space time given elsewhere by kohler to correct an incomplete computation in 32 we present a version of this computation that does not use a deformation argument such as that used in fulling narcowich and wald and is independent of the cauchy evolution argument of fulling sweeny and wald both of which are relied upon in kohler s proof this leads to a simple micro local proof of the preservation of hadamard form under cauchy evolution first shown by fulling sweeny and wald relying only on the propagation of singularities theorem in another paper 33 the equivalence theorem is used to prove a conjecture by kay that a locally hadamard quasi free klein gordon state on any globally hyperbolic curved space time must be globally hadamard

heat kernels and functional determinants on the generalized cone we consider zeta functions and heat kernel expansions on the bounded generalized cone in arbitrary dimensions using an improved calculational technique the specific case of a global monopole is analysed in detail and some restrictions thereby placed on thea 5 2 coefficient the computation of functional determinants is also addressed general formulas are given and known results are incidentally and rapidly reproduced

traces of semigroups associated with interacting particle systems abstract witten laplacians associated with interacting particle systems on infinite coverings of compact manifolds are considered the probabilistic representations of the corresponding heat kernels are given von neumann traces of the corresponding semigroups are computed

finite temperature casimir energy in closed rectangular cavities a rigorous derivation based on a zeta function technique we derive rigorously explicit formulae of the casimir free energy at finite temperature for massless scalar field and electromagnetic field confined in a closed rectangular cavity with different boundary conditions by a zeta regularization method we study both the low and high temperature expansions of the free energy in each case we write the free energy as a sum of a polynomial in temperature plus exponentially decay terms we show that the free energy is always a decreasing function of temperature in the cases of massless scalar field with the dirichlet boundary condition and electromagnetic field the zero temperature casimir free energy might be positive in each of these cases there is a unique transition temperature as a function of the side lengths of the cavity where the casimir energy changes from positive to negative when the space dimension is equal to two and three we show graphically the dependence of this transition temperature on the side lengths of the cavity finally we also show that we can obtain the results for a non closed rectangular cavity by letting the size of some directions of a closed cavity go to infinity and we find that these results agree with the usual integration prescription adopted by other authors

an extension of the chowla selberg formula useful in quantizing with the wheeler de witt equation the two dimensional inhomogeneous zeta function series with homogeneous part of the most general epstein type sigma m n in z am2 bmn cn2 q s is analytically continued in the variable s using zeta function techniques a simple formula is obtained which extends the chowla selberg formula to inhomogeneous epstein zeta functions the new expression is then applied to solve the problem of computing the determinant of the basic differential operator that appears in an attempt at quantizing gravity by using the wheeler de witt equation in 2 1 dimensional spacetime with the torus topology

a study of semiiterative methods for nonsymmetric systems of linear equations given a nonsingular linear systema x b a splittinga m n leads to the one step iteration 1 x m t x m 1 c witht m 1n andc m 1 b we investigate semiiterative methods sim s with respect to 1 under the assumption that the eigenvalues oft are contained in some compact set of with 1 there exist sim s which are optimal with respect to but except for some special sets such optimal methods are not explicitly known in general using results about maximal convergence of polynomials and uniformly distributed nodes from approximation and function theory we describe here sim s which are asymptotically optimal with respect to it is shown that euler methods extensively studied by niethammer varga nv are special sim s various algorithms for sim s are also derived here a 1 1 correspondence between euler methods and sim s generated by generalized faber polynomials is further established here this correspondence gives that asymptotically optimal euler methods are quite near the optimal sim s

zeta function determinant of the laplace operator on the d dimensional ball we present a direct approach for the calculation of functional determinants of the laplace operator on balls dirichlet and robin boundary conditions are considered using this approach formulas for any value of the dimension d of the ball can be obtained quite easily explicit results are presented here for dimensionsd 2 3 4 5 and 6

euler characteristics and gysin sequences for group actions on boundaries let g be a locally compact group let x be a universal proper g space and let open image in new window be a g equivariant compactification of x that is h equivariantly contractible for each compact subgroup open image in new window let open image in new window assuming the baum connes conjecture for g with coefficients open image in new window and c x we construct an exact sequence that computes the map on k theory induced by the embedding open image in new window this exact sequence involves the equivariant euler characteristic of x which we study using an abstract notion of poincare duality in bivariant k theory as a consequence if g is torsion free and the euler characteristic open image in new window is non zero then the unit element of open image in new window is a torsion element of order open image in new window furthermore we get a new proof of a theorem of luck and rosenberg concerning the class of the de rham operator in equivariant k homology

zeta function regularization is uniquely defined and well hawking s zeta function regularization procedure 1977 is shown to be rigorously and uniquely defined thus putting an end to the spreading lore about different difficulties associated with it basic misconceptions misunderstandings and errors which keep appearing in important scientific journals when dealing with this beautiful regularization method and other analytical procedures are clarified and corrected

weakly parametric pseudodifferential operators and atiyah patodi singer boundary problems this paper introduces a class of pseudodifferential operators depending on a parameter in a particular way the main application is a complete expansion of the trace of the resolvent of a dirac type operator with nonlocal boundary conditions of the kind introduced by atiyah patodi and singer aps this extends the partial expansion in g2 to a complete one and extends the complete expansion in gs 1 to the case where the dirac operator does not have a product structure near the boundary a secondary application is to obtain a complete expansion of the resolvent of a bdo on a compact manifold essentially reproving a result of agranovich agr the resolvent expansion yields immediately an expansion of the trace of the heat kernel and determines the singularities of the zeta function moreover a pseudodifferential factor can be allowed a major motive for these expansions is to obtain index formulas for elliptic operators there are many such applications in the physics and geometry literature the index formula comes from one particular term in the expansion but each term is a spectral invariant and they have been used for other purposes as well as for the index in particular branson and gilkey have a number of papers e g bg and gi analyzing these invariants and drawing geometric consequences interest in the asymptotic behavior of the resolvent goes back to carleman c more recently agmon agm developed it extensively for analytic applications he introduced the fundamental idea of treating the resolvent parameter essentially as another cotangent variable this idea was developed in s1 to analyze the singularities of the zeta function of an elliptic odo on a compact manifold and in 3 to analyze the resolvent of a differential operator with differential boundary conditions the technique works smoothly for differential operators producing so called local invariants integrals over the underlying

trigonometric approximation of periodic functions belonging to lip ω t p class analysis of signals or time functions are of great importance because it convey information or attributes of some phenomenon in this paper we consider the functions signals defined in terms of their integral modulus of continuity and approximate them by matrix means of their trigonometric fourier series the error of approximation obtained in this paper depends on the integral modulus of continuity of the signal we also discuss an example to show the application of the result in the situation when the fourier series of the signal has gibbs phenomenon we also discuss some corollaries of our theorems

a local to global singularity theorem for quantum field theory on curved space time we prove that if a reference two point distribution of positive type on a time orientable curved space time cst satisfies a certain condition on its wave front set the classp m g condition and if any other two point distribution i is of positive type ii has the same antisymmetric part as the reference modulo smooth function and iii has the same local singularity structure then it has the same global singularity structure in the proof we use a smoothing positivity preserving pseudo differential operator the support of whose symbol is restricted to a certain conic region which depends on the wave front set of the reference state this local to global theorem together with results published elsewhere leads to a verification of a conjecture by kay that for quasi free states of the klein gordon quantum field on a globally hyperbolic cst the local hadamard condition implies the global hadamard condition a counterexample to the local to global theorem on a strip in minkowski space is given when the classp m g condition is not assumed

spectral asymmetry on the ball and asymptotics of the asymmetry kernel let be the dirac operator on a d 2d dimensional ball with radius r we calculate the spectral asymmetry for d 2 and d 4 when local chiral bag boundary conditions are imposed with these boundary conditions we also analyse the small t asymptotics of the heat trace where p is an operator of dirac type and f is an auxiliary smooth smearing function

the epsilon algorithm and related topics the epsilon algorithm is recommended as the best all purpose acceleration method for slowly converging sequences it exploits the numerical precision of the data to extrapolate the sequence to its limit we explain its connections with pade approximation and continued fractions which underpin its theoretical base then we review the most recent extensions of these principles to treat application of the epsilon algorithm to vector valued sequences and some related topics in this paper we consider the class of methods based on using generalised inverses of vectors and the formulation specifically includes the complex case wherever possible

the l 2 index theorem for homogeneous spaces of lie groups the l2 index theorem for covering spaces of atiyah 3 and singer 20 asserts that given a discrete group f acting smoothly and freely on a manifold m with compact quotient m f m and an elliptic differential operator d on m which is f invariant and thus descends to an elliptic differential operator d on the compact manifold m then the p index indrd dimrkerd dimrkerd of d coincides with the ordinary index ind d dim ker d dim ker d of d here ker d is the space of l2 solutions of du 0 and dim r denotes the dimension function corresponding to the trace on the commutant of f acting on the hilbert space of l2 sections over m naturally associated to p the importance of this theorem lies in the fact that indd 0 implies the existence of nontrivial l2 solutions for the equation du 0 and as such it was used in a crucial way by atiyah and schmid 5 to construct explicit realizations of the discrete series representations for semisimple lie groups indeed if g is a lie group which possesses a discrete torsion free cocompact subgroup f and h is a compact subgroup of g then the l2 index theorem applied to the covering space m g h of m f g h combined with the index formula of atiyah singer 6 yields existence results for l2 solutions of g invariant elliptic equations on the homogeneous space g h it is relevant to note that with d denoting this time a g invariant elliptic differential operator on g h the ratio between the f index of d and the covolume of f gives a number independent of f this real number indg d can be in fact intrinsically defined as the difference of the two formal degrees corresponding to the representations of g on ker d and ker d respectively and hence makes sense for any unimodular lie group g even when it has no discrete cocompact subgroups which outside the semisimple case is the generic

complete determination of the singularity structure of zeta functions series of extended epstein type provide examples of non trivial zeta functions with important physical applications the regular part of their analytic continuation is seen to be a convergent or an asymptotic series their singularity structure is completely determined in terms of the wodzicki residue in its generalized form which is proven to yield the residua of all the poles of the zeta function and not just that of the rightmost pole obtainable from the dixmier trace the calculation is a very down to earth application of these powerful functional analytical methods in physics

on the issue of imposing boundary conditions on quantum fields an interesting example of the deep interrelation between physics and mathematics is obtained when trying to impose mathematical boundary conditions on physical quantum fields this procedure has recently been re examined with care comments on that and previous analysis are provided here together with considerations on the results of the purely mathematical zeta function method in an attempt at clarifying the issue hadamard regularization is invoked in order to fill the gap between the infinities appearing in the qft renormalized results and the finite values obtained in the literature with other procedures

a proof that the discrete singular convolution dsc lagrange distributed approximating function ldaf method is inferior to high order finite differences finite differences approximate the mth derivative of a function u x by a series j n nd j m u x j where the x j are the grid points the closely related discrete singular convolution dsc and lagrange distributed approximating function ldaf methods treated here as a single algorithm approximate derivatives in the same way as finite differences but with different numerical weights that depend upon a free parameter a by means of fourier analysis and error theorems we show that the dsc is worse than the standard finite differences in differentiating exp ikx for all k when a a f d where a f d 1 n 1 with n as the stencil width is the value of the dsc parameter that makes its weights most closely resemble those of finite differences for a

microlocal analysis and interacting quantum field theories renormalization on physical backgrounds we present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic curved space times we develop a purely local version of the stuckelberg bogoliubov epstein glaser method of renormalization by using techniques from microlocal analysis relying on recent results of radzikowski kohler and the authors about a formulation of a local spectrum condition in terms of wave front sets of correlation functions of quantum fields on curved space times we construct time ordered operator valued products of wick polynomials of free fields they serve as building blocks for a local perturbative definition of interacting fields renormalization in this framework amounts to extensions of expectation values of time ordered products to all points of space time the extensions are classified according to a microlocal generalization of steinmann scaling degree corresponding to the degree of divergence in other renormalization schemes as a result we prove that the usual perturbative classification of interacting quantum field theories holds also on curved space times finite renormalizations are deferred to a subsequent paper as byproducts we describe a perturbative construction of local algebras of observables present a new definition of wick polynomials as operator valued distributions on a natural domain and we find a general method for the extension of distributions which were defined on the complement of some surface

calculating normalization constants of closed queueing networks by numerically inverting their generating functions a new algorithm is developed for calculating normalization constants partition functions and moments of product form steady state distributions of closed queuing networks and related models the essential idea is to numerically invert the generating function of the normalization constant and related generating functions appearing in expressions for the moments it is known that the generating function of the normalization constant often has a remarkably simple form but numerical inversion evidently has not been considered before for p dimensional transforms as occur with queuing networks having p closed chains the algorithm recursively performs p one dimensional inversions the required computation grows exponentially in the dimension but the dimension can often be reduced by exploiting conditional decomposition based on special structure for large populations the inversion algorithm is made more efficient by computing large sums using euler summation the inversion algorithm also has a very low storage requirement a key ingredient in the inversion algorithm is scaling an effective static scaling is developed for multichain closed queuing networks with only single server and optionally infinite server queues an important feature of the inversion algorithm is a self contained accuracy check which allows the results to be verified in the absence of alternative algorithms

multidimensional extension of the generalized chowla selberg formula after recalling the precise existence conditions of the zeta function of a pseudodifferential operator and the concept of reflection formula an exponentially convergent expression for the analytic continuation of a multidimensional inhomogeneous epstein type zeta function of the general form with a the p p matrix of a quadratic form a p vector and q a constant is obtained it is valid on the whole complex s plane is exponentially convergent and provides the residua at the poles explicitly it reduces to the famous formula of chowla and selberg in the particular case p 2 q 0 some variations of the formula and physical applications are considered

strong ellipticity and spectral properties of chiral bag boundary conditions we prove strong ellipticity of chiral bag boundary conditions on even dimensional manifolds from a knowledge of the heat kernel in an infinite cylinder some basic properties of the zeta function are analysed on cylindrical product manifolds of arbitrary even dimension

a new algorithm for cauchy principal value and hadamard finite part integrals an algorithm for the approximate evaluation of integrals defined by cauchy principal value or by hadamard finite part has been proposed the convergence of the procedure is proved the stability of the algorithm is considered and some numerical examples are given

some numerical algorithms to evaluate hadamard finite part integrals abstract some algorithms are described for the numerical evaluation of hadamard finite part integrals of type 1 1 f x x t 2 v dx t where v is a jacobi weight convergence results and some numerical examples are given

the microlocal spectrum condition and wick polynomials of free fields on curved spacetimes quantum fields propagating on a curved spacetime are investigated in terms of microlocal analysis we discuss a condition on the wave front set for the correspondingn point distributions called microlocal spectrum condition sc on minkowski space this condition is satisfied as a consequence of the usual spectrum condition based on radzikowski s determination of the wave front set of the two point function of a free scalar field satisfying the hadamard condition in the kay and wald sense we construct in the second part of this paper all wick polynomials including the energy momentum tensor for this field as operator valued distributions on the manifold and prove that they satisfy our microlocal spectrum condition

numerical analysis 2000 vol ii interpolation and extrapolation multivariate polynomial interpolation is a basic and fundamental subject in approximation theory and numerical analysis which has received and continues receiving not deep but constant attention in this short survey we review its development in the rst 75 years of this century including a pioneering paper by kronecker in the 19th century c 2000 elsevier science b v all rights reserved