connections between kelvin functions and zeta functions with applications a very simple method for resuming infinite series of the type sup approximately l 1 or 1 l 1 l n 1 2 cosh la k n 1 2 lb is presented as a result a series expansion in powers of a and b is obtained relevant to many physical applications

generalized multidimensional epstein zeta functions explicit formulas are given for the expression of multiple sum zeta functions of the type l1 ln a1 l1 c1 2 an ln cn 2 m2 s where ai 0 m2 0 i 1 n in terms of riemann and hurwitz zeta functions and in terms of kelvin functions using different analytical continuation methods several formulas especially useful for different values of the involved parameters are obtained

fractional levy stable motion can model subdiffusive dynamics we show in this paper that the sample time average mean squared displacement msd of the fractional levy stable motion behaves very differently from the corresponding ensemble average second moment while the ensemble average msd diverges for 2 the sample msd may exhibit either subdiffusion normal diffusion or superdiffusion thus h self similar levy stable processes can model either a subdiffusive diffusive or superdiffusive dynamics in the sense of sample msd we show that the character of the process is controlled by a sign of the memory parameter d h 1 we also introduce a sample p variation dynamics test which allows to distinguish between two models of subdiffusive dynamics finally we illustrate a subdiffusive behavior of the fractional levy stable motion on biological data describing the motion of individual fluorescently labeled mrna molecules inside live e coli cells but it may concern many other fields of contemporary experimental physics doi 10 1103 physreve 82 021130

the heat kernel for deformed spheres we derive the asymptotic expansion of the heat kernel for a laplace operator acting on deformed spheres we calculate the coefficients of the heat kernel expansion on two and three dimensional deformed spheres as functions of deformation parameters we find that under some deformation the conformal anomaly for free scalar fields on r4 s2 and r6 s2 is cancelled

dual subspaces of operators in this note we introduce some new constructions of dual spaces of operators which are of course at the same time operator spaces in the sense of pisier 2003 12 we exemplify the utility of these constructs by establishing in this more general setting a curious and little known result from the theory of dual algebras namely from chevreau and pearcy 1991 11

inhomogeneous multidimensional epstein zeta functions the pole structure of the inhomogeneous multidimensional epstein zeta function em2n s a1 an n1 nn 1 a1n21 ann2n m2 s is determined using heat kernel techniques the poles of em2n s a1 an are found to be s n 2 n 1 2 1 2 2l 1 2 l n0 furthermore their residues and em2n p a1 an p n0 are given explicitly these results are used to find the high temperature expansion of the helmholtz free energy of a massive spin 0 and spin 1 2 gas subject to dirichlet boundary conditions on hypercuboids in a flat n dimensional space time

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

can one hear the shape of a drum revisited in a landmark paper mark kac in 1966 amer math monthly 73 pp 1 23 showed that geometric properties of regions in r 2 can be obtained by studying the asymptotic properties of the spectrum of the laplacian he also conjectured that the shape of a region might be completely determined by the spectrum we describe recent developments in this field and discuss the status of the kac conjecture as well as similar conjectures for operators other than the laplacian

a simple recurrence for the higher derivatives of the hurwitz zeta function a recurrent formula which allows the calculation of the asymptotic series expansion of any derivative m z a m z a zm of the hurwitz zeta function z a is obtained in particular the first terms of the series corresponding to n a in inverse powers of a are explicitly given for n 0 1 2 3 knowledge of these expressions is fundamental e g in the zeta function regularization procedure

quantum mass correction for the twisted kink we present an analytic result for the 1 loop quantum mass correction in semiclassical quantization for the twisted 4 kink on s1 without explicit knowledge of the fluctuation spectrum for this purpose we use the contour integral representation of the spectral zeta function by solving the bethe ansatz equations for the n 2 lam equation we obtain an analytic expression for the corresponding spectral discriminant we discuss the renormalization issues of this model an energetically preferred size for the compact space is finally obtained

fractional brownian motion versus the continuous time random walk a simple test for subdiffusive dynamics fractional brownian motion with hurst index less then 1 2 and continuous time random walk with heavy tailed waiting times and the corresponding fractional fokker planck equation are two different processes that lead to a subdiffusive behavior widespread in complex systems we propose a simple test based on the analysis of the so called p variations which allows distinguishing between the two models on the basis of one realization of the unknown process we apply the test to the data of golding and cox phys rev lett 96 098102 2006 describing the motion of individual fluorescently labeled mrna molecules inside live e coli cells it is found that the data does not follow heavy tailed continuous time random walk the test shows that it is likely that fractional brownian motion is the underlying process

on the zeta function regularization of a two dimensional series of epstein hurwitz type as a further step in the general program of zeta function regularization of multiseries expressions some original formulas are provided for the analytic continuation to any value of s of two dimensional series of epstein hurwitz type namely n1 n2 0 a1 n1 c1 2 a2 n2 c2 2 s where the aj are positive reals and the cj are not simultaneously nonpositive integers they come out from a generalization to hurwitz functions of the zeta function regularization theorem of the author and romeo phys rev d 40 436 1989 for ordinary zeta functions for s k 0 2 with k 1 2 3 the final results are in fact expressed in terms of hurwitz zeta functions only for general s they also involve bessel functions a partial numerical investigation of the different terms of the exact algebraic equations is also carried out as a by product the series n 0exp a n c 2 a c 0 is conveniently calculated in terms of them

simplified vacuum energy expressions for radial backgrounds and domain walls we extend our previous results of simplified expressions for functional determinants for radial schrodinger operators to the computation of vacuum energy or mass corrections for static but spatially radial backgrounds and for domain wall configurations our method is based on the zeta function approach to the gel fand yaglom theorem suitably extended to higher dimensional systems on separable manifolds we find new expressions that are easy to implement numerically for both zero and non zero temperatures

on the structure of contraction operators with applications to invariant subspaces on ameliore certaines conditions suffisantes pour un membre de la classe a d 0 on obtient de nouvelles conditions suffisantes pour qu une contraction sur un espace de hilbert dont le spectre contient le cercle unite ait des sous espaces invariants non triviaux

anomalous diffusion schemes underlying the cole cole relaxation the role of the inverse time α stable subordinator abstract the paper presents the random variable formalism of the anomalous diffusion processes the emphasis is on a rigorous presentation of asymptotic behaviour of random walk processes with infinite mean random time intervals between jumps we elucidate the role of the so called inverse time stochastic process the main mathematical tool that allows us to modify the dynamics of standard relaxation processes and give rise to the nonexponential decay of modes in particular we show that the brownian motion in combination with an appropriate inverse time process may lead not only to exponential but also to the nonexponential relaxation responses

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

xi and eta function resummation of finite series general case given a spectrum of positive numbers lambda m from which a xi function z s sigma m lambda sm can be constructed the reorganization of series of the type f s t identical to m lambda smf lambda mt into power series in t is examined in detail using the method of xi function resummation for summand functions f lambda mt having power series expansions in lambda mt with infinite radius of convergence and which satisfy other conditions of a rather general nature the author finds that f s t can be reorganized to f s t n antbn k cktdk in t r s t where r s t vanishes exponentially as t to 0 the numbers an bn cn dn can all be computed in terms of the xi function z s r s t is difficult to evaluate but important general features of this function can be determined the power series expansion of f s t can be regarded as a generalization of the heat kernel expansion for which f lambda mt exp lambda mt and s 0 to nonzero complex variable s which is useful and to many other summand functions f lambda mt remarkably the xi function resummation method can be applied as easily to divergent series f s t as it can to convergent ones the method is therefore both a rearrangement procedure for convergent series and a summation prescription for divergent series

on the quantum zeta function it is remarkable that the quantum zeta function defined as a sum over energy eigenvalues e admits of exact evaluation in some situations for which not a single e be known herein we show how to evaluate instances of z s and of an associated parity zeta function y s for various quantum systems for some systems both z n y n can be evaluated for infinitely many integers n such z y values can be used for example to effect sharp numerical estimates of a system s ground energy the difficult problem of evaluating the analytic continuation z s for arbitrary complex s is discussed within the contexts of perturbation expansions path integration and quantum chaos

on the distribution of eigenvalues of grand canonical density matrices using physical arguments and partition theoretic methods we demonstrate under general conditions that the eigenvalues w m of the grand canonical density matrix decay rapidly with their index m like w m exp b 1 ln m 1 1 where b and are positive constants o 1 which may be computed from the spectrum of the hamiltonian we compute values of b and for several physical models and confirm our theoretical predictions with numerical experiments our results have implications in a variety of questions including the behaviour of fluctuations in ensembles and the convergence of numerical density matrix renormalization group techniques

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

stochastic origins of the long range correlations of ionic current fluctuations in membrane channels an explicit stochastic representation of a stationary ionic current signal recorded from a single channel of a biological membrane is presented in the framework of the proposed approach we show how the dichotomous time structure of the signal leads to the non markovian character of the channel current the rescaled range hurst and detrended fluctuation analyses confirm the theoretical result to investigate the ionic current fluctuations we introduce the orey index as a statistical method providing additional information on the properties of stochastic processes in order to reveal any differences between the experimental and reconstructed signals we apply also the statistical tests to the model based simulations of the channel action

on the structure of contraction operators ii soit 7b h un espace de hilbert complexe separable de dimension infinie et 7b l 7b h l algebre de tous les operateurs lineaires bornes sur 7b h soit d le disque unite ouvert dans c d et n est l ensemble des entiers positifs il existe r 6 tel que chaque contraction t dans 7b l 7b h avec c t soit a un sous espace hyperinvariant non trivial soit appartient a la classe a 1 r

some analytic continuations of the barnes zeta function in two and higher dimensions abstract formulas for the analytic continuation of the barnes zeta function and some affine extensions thereof in two and more dimensions are constructed the expressions are used to deal with determinants of multidimensional harmonic oscillators an example is therewith obtained of the multiplicative anomaly or defect associated with the most common definition due to ray and singer d b ray reidemeister torsion and the laplacian on lens spaces adv math 4 1970 109 126 d b ray i m singer r torsion and the laplacian on riemannian manifolds adv math 7 1971 145 201 d b ray i m singer analytic torsion for complex manifolds ann math 98 1973 154 177 of determinant of a pseudodifferential operator admitting a zeta function m kontsevich s vishik geometry of determinants of elliptic operators in functional analysis on the eve of the 21st century vol 1 1995 pp 173 197

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

fractional brownian motion approach to polymer translocation the governing equation of motion we suggest a governing equation that describes the process of polymer chain translocation through a narrow pore and reconciles the seemingly contradictory features of such dynamics i a gaussian probability distribution of the translocated number of polymer segments at time t after the process has begun and ii a subdiffusive increase of the distribution variance t with elapsed time t t the latter quantity measures the mean squared number s of polymer segments that have passed through the pore t s t s t 0 2 and is known to grow with an anomalous diffusion exponent 1 our main assumption i e a gaussian distribution of the translocation velocity v t and some important theoretical results derived recently are shown to be supported by extensive brownian dynamics simulation which we performed in 3d we also numerically confirm the predictions made recently that the exponent changes from 0 91 to 0 55 to 0 91 for short intermediate and long time regimes respectively

detecting origins of subdiffusion p variation test for confined systems in this paper we propose a method to distinguish between mechanisms leading to single molecule subdiffusion in confinement we show that the method of p variation introduced in the recent paper m magdziarz et al phys rev lett 103 180602 2009 can be successfully applied also for confined systems we propose a test which allows distinguishing between heavy tailed continuous time random walk and fractional brownian motion in the presence of binding potentials and reflecting boundaries we apply our test to the experimental data describing motion of mrna molecules inside e coli cells the results of the test show that it is more likely that fractional brownian motion is the underlying process

estimation and testing of the hurst parameter using p variation progress in measurement tools and numerical simulations has led to an increase in attention towards complex systems recently developed methods based on the so called p variation appeared to be very effective in researching the stochastic origin of such phenomena in particular in fitting the widely used fractional brownian motion model in this paper we propose an estimator based on the p variation which allows us to determine the hurst parameter of a single fractional brownian motion other fractal dimension gaussian processes or a fractional levy stable motion trajectory additionally we present a statistical test designed to validate the results we explain thoroughly the procedure of analysing data and study its effectiveness during monte carlo simulations and real data applications

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

subnormal operators whose adjoints have rich point spectrum a generalized version of the glauber klauder basic formula of quantum optics is shown to be valid for any cyclic subnormal operator s whose adjoint has a rich point spectrum p s in the sense that a semispectral measure of s vanishes on c p s it is exhibited that such operators always have analytic models the point spectrum of the adjoint of a subnormal operator which satisfies a generalized version of the glauber klauder formula is proved to be rich in the above sense

infinite products partition functions and the meinardus theorem the meinardus theorem provides an asymptotic value for the infinite product generating functions of many different partition functions and thereby asymptotic values for these partition functions themselves under assumptions somewhat stronger than those of meinardus a more specific version of the theorem is derived then this is generalized from the spectrum of integers considered by meinardus to infinite products constructed from an arbitrary spectrum included are infinite products that generate partition functions able to count states in the quantum theory of extended objects there emerges a quite general and explicit method of which only limited details are given here for deriving the asymptotic growth of the state density function of multidimensional quantum objects one finds a universal geometry independent leading behavior with nonleading geometry dependent corrections that the method is capable of providing explicitly

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

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

stochastic models that separate fractal dimension and the hurst effect fractal behavior and long range dependence have been observed in an astonishing number of physical biological geological and socioeconomic systems time series profiles and surfaces have been characterized by their fractal dimension a measure of roughness and by the hurst coefficient a measure of long memory dependence both phenomena have been modeled and explained by self affine random functions such as fractional gaussian noise and fractional brownian motion the assumption of statistical self affinity implies a linear relationship between fractal dimension and hurst coefficient and thereby links the two phenomena this article introduces stochastic models that allow for any combination of fractal dimension and hurst coefficient associated software for the synthesis of images with arbitrary prespecified fractal properties and power law correlations is available the new models suggest a test for self affinity that assesses coupling and decoupling of local and global behavior

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

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

multiple harmonic oscillator zeta functions the partition function for a quantum mechanical system can at least formally be expressed as a high temperature series in terms of the zeta function of the system s hamiltonian this is done explicitly for a system of non interacting harmonic oscillators the hamilton zeta function belongs to a class of zeta functions sigma f ni n1 omega 1 n2 omega 2 nn omega n a s summed over non negative integers ni very little is known about these zeta functions and an investigation of some of their properties is presented

the zeta function constructed from the zeros of the bessel function this paper studies the function built from the zeros of the bessel function the known first eight terms of the mcmahon expansion with are used to construct an accurate approximation to the quality of this approximation is investigated numerically by comparison with a known but at least numerically little studied integral formula for excellent numerical agreement is found for fixed and variable real s and for fixed s and variable both formulae for therefore seem to work well our approximation also accurately reproduces known special values of important properties of are investigated for the first time including several of its zeros in addition some general theory is presented in two areas i perturbed spectra and ii the interrelationship between functions like representable as infinite products and the functions constructed from their infinite spectrum of zeros

a contribution to the structure theory of operators in the class a soit 7b h un espace de hilbert separable et soit t une contraction agissant sur 7b h on suppose que t est absolument continu soit 7b a t l etoile fermee algebre faible engendree par t si t a on dit que t appartient a la classe a 1 si pour chaque fonctionnelle etoile continue faible sur 7b a t il existe des vecteurs n y 7b h satisfaisant a a 7b a t on montre que chaque operateur t tel que t a et soit t n 0 soit t n o avec 1 n appartient a a 1

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

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

pathwise convergent higher order numerical schemes for random ordinary differential equations random ordinary differential equations rodes are ordinary differential equations odes with a stochastic process in their vector field they can be analysed pathwise using deterministic calculus but since the driving stochastic process is usually only holder continuous in time the vector field is not differentiable in the time variable so traditional numerical schemes for odes do not achieve their usual order of convergence when applied to rodes nevertheless deterministic calculus can still be used to derive higher order numerical schemes for rodes via integral versions of implicit taylor like expansions the theory is developed systematically here and applied to illustrative examples involving brownian motion and fractional brownian motion as the driving processes

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

general moment theorems for nondistinct unrestricted partitions a well known result from hardy and ramanujan aysmptotic formulae in combinatory analysis proc lond math soc 17 75 1918 gives an asymptotic expression for the number of possible ways to express an integer as the sum of smaller integers in this vein we consider the general partitioning problem of writing an integer n as a sum of summands from a given sequence of nondecreasing integers under suitable assumptions on the sequence we obtain results using associated zeta functions and saddle point techniques we also calculate higher moments of the sequence as well as the expected number of summands applications are made to various sequences including those of barnes and epstein types these results are of potential interest in statistical mechanics in the context of bose einstein condensation

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

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