on bounded approximations of periodicity for computational homogenization of stokes flow in porous media by separation of scales and the homogenization of a flow through porous media a two scale problem arises where a darcy type flow is present on the macroscale and a stokes flow on the subscale in this paper the problem is given as the minimization of a potential additional constraints imposing periodicity in a weak sense are added using lagrange multipliers in particular the upper and lower energy bounds for the corresponding strongly periodic problem are produced quantifying the accuracy of the weakly periodic boundary conditions a numerical example demonstrates the evaluation of energy bounds and the performance of weakly periodic boundary conditions on a representative volume element

analytic torsion of a bounded generalized cone we compute the analytic torsion of a bounded generalized cone by generalizing the computational methods of m spreafico and using the symmetry in the de rham complex as established by m lesch we evaluate our result in lower dimensions and further provide a separate computation of analytic torsion of a bounded generalized cone over s 1 since the standard cone over the sphere is simply a flat disc

a summation formula over the zeros of a combination of the associated legendre functions with a physical application by using the generalized abel plana formula we derive a summation formula for the series over the zeros of a combination of the associated legendre functions with respect to the degree the summation formula for the series over the zeros of the combination of the bessel functions previously discussed in the literature is obtained as a limiting case as an application we evaluate the wightman function for a scalar field with a general curvature coupling parameter in the region between concentric spherical shells on a background of constant negative curvature space for the dirichlet boundary conditions the corresponding mode sum contains the series over the zeros of the combination of the associated legendre functions the application of the summation formula allows us to present the wightman function in the form of the sum of two integrals the first one corresponds to the wightman function for the geometry of a single spherical shell and the second one is induced by the presence of the second shell the boundary induced part in the vacuum expectation value of the field squared is investigated for points away from the boundaries the corresponding renormalization procedure is reduced to that for the boundary free part

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

small mass expansion of functional determinants on the generalized cone in this paper we compute the small mass expansion for the functional determinant of a scalar laplacian defined on the bounded generalized cone in the framework of zeta function regularization we obtain an expression for the functional determinant valid in any dimension for both dirichlet and robin boundary conditions in terms of the spectral zeta function of the base manifold moreover as a particular case we specify the base to be a d dimensional sphere and present explicit results for d 2 3 4 5

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

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

zeta regularized determinants for conic manifolds we study relative zeta regularized determinants of laplace type operators on compact conic manifolds we establish gluing formulae for relative zeta regularized determinants for arbitrary self adjoint extensions of the laplace beltrami operator we express the relative determinants for these as a ratio of the determinants of certain finite matrices for the self adjoint extensions corresponding to dirichlet and neumann conditions the formula is particularly simple and elegant

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

dirac operator on a disk with global boundary conditions we compute the functional determinant for a dirac operator in the presence of an abelian gauge field on a bidimensional disk under global boundary conditions of the type introduced by atiyah patodi singer we also discuss the connection between our result and the index theorem

antisymmetric tensor fields on spheres functional determinants and non local counterterms the hodge de rham laplacian on spheres acting on antisymmetric tensor fields is considered explicit expressions for the spectrum are derived in a quite direct way confirming previous results associated functional determinants and the heat kernel expansion are evaluated using this method new non local counterterms in the quantum effective action are obtained which can be expressed in terms of betti numbers

on the determinant of elliptic differential and finite difference operators in vector bundles over s 1 for an elliptic differential operatora overs 1 a sum limits k 0 n a k x d k witha k x in end r and as a principal angle the regularized determinant det a is computed in terms of the monodromy mapp a associated toa and some invariant expressed in terms ofa n anda n 1 a similar formula holds for finite difference operators a number of applications and implications are given in particular we present a formula for the signature ofa whena is self adjoint and show that the determinant ofa is the limit of a sequence of computable expressions involving determinants of difference approximation ofa

the spectral zeta function for laplace operators on warped product manifolds of the type i f n in this work we study the spectral zeta function associated with the laplace operator acting on scalar functions defined on a warped product of manifolds of the type i f n where i is an interval of the real line and n is a compact d dimensional riemannian manifold either with or without boundary starting from an integral representation of the spectral zeta function we find its analytic continuation by exploiting the wkb asymptotic expansion of the eigenfunctions of the laplace operator on m for which a detailed analysis is presented we apply the obtained results to the explicit computation of the zeta regularized functional determinant and the coefficients of the heat kernel asymptotic expansion

the very unusual properties of the resolvent heat kernel and zeta function for the operator d2 dr2 1 4r2 in this paper we analyze the resolvent the heat kernel and the spectral zeta function of the operator d2 dr2 1 4r2 over the finite interval the structural properties of these spectral functions depend strongly on the chosen self adjoint realization of the operator a choice being made necessary because of the singular potential present only for the friedrichs realization standard properties are reproduced for all other realizations highly nonstandard properties are observed in particular for k n we find terms like logt k in the small t asymptotic expansion of the heat kernel furthermore the zeta function has s 0 as a logarithmic branch point

determinants finite difference operators and boundary value problems we relate the determinants of differential and difference operators to the boundary values of solutions of the operators previous proofs of related results have involved considering one parameter families of such operators showing the desired quantities are equal up to a constant and then calculating the constant we take a more direct approach for a fixed operator we prove immediately that the two sides of our formulas are equal by using the following simple observation proposition 1 3 let u su n c write u in block form u left begin array 20 c u 11 u 12 u 21 u 22 end array right where u11and u22are square matrices then det u 11 overline det u 22

estimates and extremals for zeta function determinants on four manifolds leta be a positive integral power of a natural conformally covariant differential operator on tensor spinors in a riemannian manifold suppose thata is formally self adjoint and has positive definite leading symbol for example a could be the conformal laplacian yamabe operator l or the square of the dirac operator open image in new window within the conformal class left g e 2w g 0 w in c infty m right of an einstein locally symmetric background metricgo on a compact four manifoldm we use an exponential sobolev inequality of adams to show that bounds on the functional determinant ofa and the volume ofg imply bounds on thew2 2 norm of the conformal factorw provided that a certain conformally invariant geometric constantk k m g o a is strictly less than 32 2 we show for the operatorsl and open image in new window that indeedk 32 2 except when m g o is the standard sphere or a hyperbolic space form on the sphere a centering argument allows us to obtain a bound of the same type despite the fact thatk is exactly equal to 32 2 in this case finally we use an inequality of beckner to show that in the conformal class of the standard four sphere the determinant ofl or of open image in new window is extremized exactly at the standard metric and its images under the conformal transformation groupo 5 1

vector fields on a disk with mixed boundary conditions vector fields on a disk satisfying two types of boundary conditions are studied these boundary conditions are selected by brst invariance in electrodynamics they also appear in the de rham complex the construction of the harmonic expansion is presented the eigenfunctions of the vector laplace operator are expressed in terms of fields satisfying pure dirichlet or robin boundary conditions for the case of four dimensional disk several first coefficients of the heat kernel expansion are computed an error in the analytical expression by branson and gilkey is corrected

heat kernel expansion on noncompact domains and a generalized zeta function regularization procedure heat kernel expansion and zeta function regularization are discussed for laplace type operators with discrete spectrum in noncompact domains since a general theory is lacking the heat kernel expansion is investigated by means of several examples it is pointed out that for a class of exponential analytic interactions generically the noncompactness of the domain gives rise to logarithmic terms in the heat kernel expansion then a meromorphic continuation of the associated zeta function is investigated a simple model is considered for which the analytic continuation of the zeta function is not regular at the origin displaying a pole of higher order for a physically meaningful evaluation of the related functional determinant a generalized zeta function regularization procedure is proposed

the spectral determinant of the isotropic quantum harmonic oscillator in arbitrary dimensions we show that the spectral determinant of the isotropic quantum harmonic oscillator converges exponentially to one as the space dimension grows to infinity we determine the precise asymptotic behaviour for large dimension and obtain estimates valid for all cases with the same asymptotic behaviour in the large as a consequence we provide an alternative proof of a conjecture posed by bar and schopka concerning the convergence of the determinant of the dirac operator on s n determining the exact asymptotic behaviour for this case and thus improving the estimate on the rate of convergence given in the work of moller math ann 343 35 51 2009

computing variational bounds for flow through random aggregates of spheres known formulas for variational bounds on darcy s constant for slow flow through porous media depend on two point and three point spatial correlation functions certain bounds due to prager and doi depending only a two point correlation functions have been calculated for the first time for random aggregates of spheres with packing fractions up to 0 64 three radial distribution functions for hard spheres were tested for up to 0 49 1 the uniform distribution or well stirred approximation 2 the percus yevick approximation and 3 the semi empirical distribution of verlet and weis the empirical radial distribution functions of bennett and finney were used for packing fractions near the random close packing limit rcp 0 64 an accurate multidimensional monte carlo integration method vegas developed by lepage was used to compute the required two point correlation functions the results show that doi s bounds are preferred for 0 10 the upper bounds computed using the well stirred approximation actually become negative which is physically impossible as increases indicating the very limited value of this approximation the other two choices of radial distribution function give reasonable results for up to 0 49 however these bounds do not decrease with as fast as expected for large it is concluded that variational bounds dependent on three point correlation functions are required to obtain more accurate bounds on darcy s constant for large

conical casimir pistons with hybrid boundary conditions in this paper we compute the casimir energy and force for massless scalar fields endowed with hybrid boundary conditions in the setting of the bounded generalized cone by using spectral zeta function regularization methods we obtain explicit expressions for the casimir energy and force in arbitrary dimensions in terms of the zeta function defined on the piston our general formulas are subsequently specialized to the case in which the piston is modeled by a d dimensional sphere in this particular situation explicit results are given for d 2 3 4 5

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

functional determinants on spheres and sectors the scalar functional determinants on sectors of the two dimensional disc and spherical cap are determined for arbitrary angles rational factors of the whole sphere and hemisphere expressions are also given in low dimensions for both the ordinary and the conformal laplacian

a classification of finite 2 groups with exactly three involutions we finish here a classification of finite 2 groups with exactly three involutions we clear up this remaining very difficult situation and determine completely the structure of g in terms of two generators and relations

conformal invariants for determinants of laplacians on riemann surfaces for a riemann surface with smooth boundaries conformal weyl invariant quantities proportional to the determinant of the scalar laplacian operator are constructed both for dirichlet and neumann boundary conditions the determinants are defined by zeta function regularization the other quantities in the invariants are determined from metric properties of the surface as applications explicit representations for the determinants on the flat disk and the flat annulus are derived

spectral analysis and zeta determinant on the deformed spheres we consider a class of singular riemannian manifolds the deformed spheres s n k defined as the classical spheres with a one parameter family g k of singular riemannian structures that reduces for k 1 to the classical metric after giving explicit formulas for the eigenvalues and eigenfunctions of the metric laplacian delta s n k we study the associated zeta functions zeta s delta s n k we introduce a general method to deal with some classes of simple and double abstract zeta functions generalizing the ones appearing in zeta s delta s n k an application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions and in particular zeta 0 delta s n k and zeta 0 delta s n k we give explicit formulas for the zeta regularized determinant in the low dimensional cases n 2 3 thus generalizing a result of dowker 25 and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter k

zeta function for the laplace operator acting on forms in a ball with gauge boundary conditions the laplace operator acting on antisymmetric tensor fields in a d dimensional euclidean ball is studied gauge invariant local boundary conditions absolute and relative ones in the language of gilkey are considered the eigenfuctions of the operator are found explicitly for all values of d using in a row a number of basic techniques as mellin transforms deformation and shifting of the complex integration contour and pole compensation the zeta function of the operator is obtained from its expression in particular 0 and 0 are evaluated exactly a table is given in the paper for d 3 4 8 the functional determinants and casimir energies are obtained for d 3 4 6

zeta function regularization for a scalar field in a compact domain we express the zeta function associated with the laplacian operator on s1r m in terms of the zeta function associated with the laplacian on m where m is a compact connected riemannian manifold this gives formulae for the partition function of the associated physical model at low and high temperature for any compact domain m furthermore we provide an exact formula for the zeta function at any value of r when m is a d dimensional box or a d dimensional torus this allows a rigorous calculation of the zeta invariants and the analysis of the main thermodynamic functions associated with the physical models at finite temperature

analysis of an inhomogeneous generalized epstein hurwitz zeta function with physical applications the inhomogeneous generalized epstein hurwitz like multidimensional series em s a1 am c1 cm c2 n1 nm 0 a1 n1 c1 2 am nm cm 2 c2 s is investigated by means of a nontrivial asymptotic recurrence it is reduced to the one dimensional case f s a b2 n 0 n a 2 b2 s which is then studied in full detail in particular asymptotic expansions for f and its derivatives f s and f a together with analytical continuations of the same in the variable s are explicitly obtained using zeta function techniques several plots and tables of the numerical results are given some explicit applications to the regularization by means of hurwitz zeta functions of different problems that have appeared recently in the physical literature are considered

isolated involutions whose centraliser is soluble abstract we analyse a minimal counterexample to glauberman s z theorem from a local group theoretic point of view one of the main results is a group theoretic proof for the z theorem in the special case where the centraliser of an isolated involution is soluble

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

variationally consistent computational homogenization of transient heat flow a framework for variationally consistent homogenization combined with a generalized macrohomogeneity condition is exploited for the analysis of non linear transient heat conduction within this framework the classical approach of model based first order homogenization for stationary problems is extended to transient problems homogenization is then carried out in the spatial domain on representative volume elements rve which are in practice introduced in quadrature points in standard fashion along with the classical averages a higher order conservation quantity is obtained an iterative fe2 algorithm is devised for the case of non linear diffusion and storage coefficients and it is applied to transient heat conduction in a strongly heterogeneous particle composite parametric studies are carried out in particular with respect to the influence of the internal length associated with the second order conservation quantity copyright c 2009 john wiley sons ltd

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

functional determinants and geometry in this paper we study the zeta function determinant in the context of elliptic boundary value problems our main technique is to relate the determinant of an operator or a ratio of determinants to the boundary values of the solutions of the operator this has the advantage of restricting attention to the solutions of the operator which do not depend on the boundary conditions and can often be written down explicitly rather than the eigenvalues which are usually difficult to work with in addition the problem is reduced to a calculation over the boundary of the manifold which is a closed manifold of dimension one less than the original manifold this has special significance in the case that the manifold is a finite interval in this case the boundary is a pair of points and the determinant of an ordinary differential operator is expressed in terms of the determinant of a finite matrix

conformal transformations and the effective action in the presence of boundaries the conformal properties of the heat kernel expansion are used to determine the local form of the coefficients in a manifold with boundary the conformal transformation of the effective action is obtained a novel derivation of the boundary term in the gauss bonnet chern theorem is detailed

effective action in spherical domains the effective action on an orbifolded sphere is computed for minimally coupled scalar fields the results are presented in terms of derivatives of barnes functions and it is shown how these may be evaluated numerical values are shown an analytical heat kernel derivation of the cesaro fedorov formula for the number of symmetry planes of a regular solid is also presented

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

zeta determinant for laplace operators on riemann caps the goal of this paper is to compute the zeta function determinant for the massive laplacian on riemann caps or spherical suspensions these manifolds are defined as compact and boundaryless d dimensional manifolds deformed by a singular riemannian structure the deformed spheres considered previously in the literature belong to this class after presenting the geometry and discussing the spectrum of the laplacian we illustrate a method to compute its zeta regularized determinant the special case of the deformed sphere is recovered as a limit of our general formulas

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

symmetric tensor eigenspectrum of the laplacian on n spheres the eigenvalues and degeneracies of the covariant laplacian acting on symmetric tensors of rank m 2 defined on n spheres with n 3 are given