certain families of series associated with the hurwitz lerch zeta function abstract the history of problems of evaluation of series associated with the riemann zeta function can be traced back to christian goldbach 1690 1764 and leonhard euler 1707 1783 many different techniques to evaluate various series involving the zeta and related functions have since then been developed the authors show how elegantly certain families of series involving the hurwitz lerch zeta function can be evaluated by starting with a single known identity for the hurwitz lerch zeta function some of the special cases of these series identities are also shown to lead to certain known families of summation formulas involving the hurwitz zeta function

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

unified treatment of several asymptotic expansions concerning some mathematical constants recently various approximation formulas for some mathematical constants have been investigated and presented by many authors in this paper we first find that the relationship between the coefficients pj and qj is such that xj 0qjxj ln xj 0pjxj x where is the logarithmic derivative of the gamma function often referred to as psi function and p0 q0 1 next by using this result we give a unified treatment of several asymptotic expansions concerning the eulermascheroni constant landau and lebesgue constants glaisherkinkelin constant and choisrivastava constants

moments of random matrices and hypergeometric orthogonal polynomials we establish a new connection between moments of n times n random matrices xn and hypergeometric orthogonal polynomials specifically we consider moments mathbb e rm tr x n s as a function of the complex variable s in mathbb c whose analytic structure we describe completely we discover several remarkable features including a reflection symmetry or functional equation zeros on a critical line in the complex plane and orthogonality relations an application of the theory resolves part of an integrality conjecture of cunden et al j math phys 57 111901 2016 on the time delay matrix of chaotic cavities in each of the classical ensembles of random matrix theory gaussian laguerre jacobi we characterise the moments in terms of the askey scheme of hypergeometric orthogonal polynomials we also calculate the leading order n rightarrow infty asymptotics of the moments and discuss their symmetries and zeroes we discuss aspects of these phenomena beyond the random matrix setting including the mellin transform of products and wronskians of pairs of classical orthogonal polynomials when the random matrix model has orthogonal or symplectic symmetry we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials

on mutual information in multipartite quantum states and equality in strong subadditivity of entropy the challenge of equality in the strong subadditivity inequality of entropy is approached via a general additivity of correlation information in terms of nonoverlapping clusters of subsystems in multipartite states density operators a family of tripartite states satisfying equality is derived

determinants of laplacians the determinant of the laplacian on spinor fields on a riemann surface is evaluated in terms of the value of the selberg zeta function at the middle of the critical strip a key role in deriving this relation is played by the barnes double gamma function

functional determinants in quantum field theory functional determinants of differential operators play a prominent role in theoretical and mathematical physics and in particular in quantum field theory they are however difficult to compute in non trivial cases for one dimensional problems a classical result of gel fand and yaglom dramatically simplifies the problem so that the functional determinant can be computed without computing the spectrum of eigenvalues here i report recent progress in extending this approach to higher dimensions i e functional determinants of partial differential operators with applications in quantum field theory

geometric zeta functions of locally symmetric spaces the theory of geometric zeta functions for locally symmetric spaces is generalized to the case of higher rank spaces we show that the zeta functions can be continued to meromorphic functions on the plane describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalizes a conjecture of patterson we also extend the range of zeta functions in considering higher dimensional flats

on state dependent implication in quantum mechanics this paper studies a generalization of the a priori implication e or f meaning of course ef e in the quantum logic p h of all projectors the new definition can be applied also to new pairs of events being based on the indistinguishability of some events when considered from the standpoint of a fixed state rho the mathematical theory of factor boolean algebras on which this extension is based is clarified

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

the periodic nature of the positive solutions of a nonlinear fuzzy max difference equation we investigate the periodic nature of the positive solutions of the fuzzy difference equation x n maxa 0x n k a 1x n m where k m are positive integers a 0 a 1 are positive fuzzy numbers and the initial values x i i d d 1 1 d max k m are positive fuzzy numbers in addition we give conditions so that the solutions of this equation are unbounded

airy function exact wkb results for potentials of odd degree an exact wkb wentzel kramers brillouin treatment of 1d homogeneous schrodinger operators with the confining potentials n even is extended to odd degrees n the resulting formalism is first illustrated theoretically and numerically upon the spectrum of the cubic oscillator potential concerning the linear potential n 1 the theory exhibits a duality in which the airy functions ai become paired with the spectral determinants of the quartic oscillator n 4 classic identities for the airy function as well as some less familiar ones appear in this new perspective as special cases in a general setting

applications of the mellin barnes integral representation we apply the mellin barnes integral representation to several situations of interest in mathematical physics at the purely mathematical level we derive useful asymptotic expansions of different zeta functions and partition functions these results are then employed in different topics in quantum field theory which include the high temperature expansion of the free energy of a scalar field in ultrastatic curved spacetime the asymptotics of the p brane density of states and an explicit approach to the asymptotics of the determinants that appear in string theory

master equation vs partition function canonical statistics of ideal bose einstein condensates abstract within the canonical ensemble a partially condensed ideal bose gas with arbitrary single particle energies is equivalent to a system of uncoupled harmonic oscillators we exploit this equivalence for deriving a formula which expresses all cumulants of the canonical distribution governing the number of condensate particles in terms of the poles of a generalized zeta function provided by the single particle spectrum this formula lends itself to systematic asymptotic expansions which capture the non gaussian character of the condensate fluctuations with utmost precision even for relatively small finite systems as confirmed by comparison with exact numerical calculations we use these results for assessing the accuracy of a recently developed master equation approach to the canonical condensate statistics this approach turns out to be quite accurate even when the master equation is solved within a simple quasithermal approximation as a further application of the cumulant formula we show that and explain why all cumulants of a homogeneous bose einstein condensate in a box higher than the first retain a dependence on the boundary conditions in the thermodynamic limit

embedding problem of noncompact fuzzy number space e abstract in this paper we use the embedding j wu and zhang from noncompact fuzzy number space e into the concrete frechet space c 0 1 c 0 1 to discuss some properties of the e valued functions such as measurability integrability etc

on the fuzzy difference equation x n 1 a x n x n m in this paper we study the existence the boundedness and the asymptotic behavior of the positive solutions of the fuzzy equation xn 1 a xn xn m n 0 1 where xn is a sequence of fuzzy numbers a is a fuzzy number and m 1 2

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

meyer vietoris type formula for determinants of elliptic differential operators abstract for a closed codimension one submanifold of a compact manifold m let m be the manifold with boundary obtained by cutting m along let a be an elliptic differential operator on m and b and c be two complementary boundary conditions on if a b is an elliptic boundary valued problem on m then one defines an elliptic pseudodifferential operator r of neumann type on and prove the following factorization formula for the regularized determinants det a det a b k det r with k a local quantity depending only on the jets of the symbols of a b and c along the particular case when m has dimension 2 a is the laplace beltrami operator and b resp c is the dirichlet resp neumann boundary condition is considered

sums associated with the zeta function abstract we evaluate the sums of certain classes of series involving the riemann zeta function by using the theory of the double gamma function which has recently been revived in the study of determinants of laplacians relevant connections with various known results are also pointed out

asymptotic formulas for the triple gamma function γ3 by means of its integral representation abstract there is an abundant literature on inequalities for the gamma function and its various related functions as well as their approximations only very recently several authors began to investigate various inequalities for the double gamma function 2 and its approximation here in this sequel to some of these recent works we aim at presenting an integral representation of the triple gamma function 3 which is then used to derive an asymptotic formula for 3 as a by product of the results presented here integral representations and asymptotic formulas for the gamma function and the double gamma function 2 are also given the methods and techniques used in this paper can easily be extended to derive the corresponding integral representations and asymptotic formulas for the multiple gamma functions n n 4

integrals involving a function associated with the euler maclaurin summation formula the authors first evaluate sums of certain series involving the riemann zeta function by using the theory of the double gamma function with or without the use of these results the authors then establish seberal integrals involving a certain periodic function which is associated for example with the euler maclaurin summation formula finally by introducing some new mathematical constants several families of definite integrals involving logarithms of gamma functions are considered relevant connections of the results presented here with those given in earlier works are also pointed out

zeta invariants for sequences of spectral type special functions and the lerch formula we discuss the spectral properties of a class of sequences of what we call spectral type we introduce an effective method to calculate the zeta invariants for this type of sequence such invariants are given in terms of some new and old special functions and we consider a number of examples in which we study the properties of these special functions

local l factors of motives and regularized determinants for a smooth projective variety x over a global field k consider the completed l function of h w x it is defined as the euler product over all places v of k of the local l factors l hw x s introduced in se in the function field case the generalized lefschetz trace formula in adic cohomology gives a global cohomological expression for the l function this has been used on the following purposes

boundedness and asymptotic behavior of the solutions of a fuzzy difference equation in this paper we study the existence the uniqueness the boundedness and the asymptotic behavior of the positive solutions of the fuzzy difference equation xn 1 i 0kai xn ipi where k 1 2 ai i 0 1 k are positive fuzzy numbers pi i 0 1 k are positive constants and xi i k k 1 0 are positive fuzzy numbers

determinants of laplace like operators on riemann surfaces we calculate determinants of second order partial differential operators defined on riemann surfaces of genus greater than one using a relation between selberg s zeta function and functional determinants in addition we perform a calculation of these determinants directly using selberg s trace formula and compare our results with previous computations which followed the latter route

some mathematical constants the theory of multiple gamma functions was studied in about 1900 and has been revived in the study of determinants of laplacians in the middle or so of 1980s there are some classes of mathematical constants involved naturally in the gamma and multiple gamma functions here we introduce those known mathematical constants associated with the gamma and multiple gamma functions and will show how they are involved if possible

the burghelea friedlander kappeler gluing formula for zeta determinants on a warped product manifold and a product manifold the burghelea friedlander kappeler bfk gluing formula for the regularized zeta determinants of laplacians contains a constant which is expressed by the constant term in the asymptotic expansion of the regularized zeta determinants of a one parameter family of the dirichlet to neumann operators when the dimension of a cutting hypersurface is odd or the metric is a product one near a cutting hypersurface this constant is well known in this paper we discuss this constant in two cases one is when a warped product metric is given near a cutting hypersurface and the other is when a manifold is a product manifold especially in the first case we use the result of fucci and kirsten commun math phys 317 635 665 2013 in which the regularized zeta determinant of the laplacian defined on a warped product manifold is computed

on certain definite integrals which arise in automorphic lie theory calculates in closed form a family of definite integrals in b a c n 0 1 2 3 which arise in the calculation of regularized functional determinants associated with a compact space form x of a rank 1 riemannian symmetric space in the special case when n 0 b 1 2 a pi c 1 and x is a riemann surface the integral i0 1 2 pi 1 is known for example in the context of polyakov string theory

the role of coherence entropy of physical twin observables in entanglement the concept of physical twin observables pto for bipartite quantum states introduced and proved relevant for quantum information theory in recent work is substantially simplified the relation of observable and state is studied in detail from the point of view of coherence entropy properties of this quantity are further explored it is shown that besides for pure states also for a class of mixed states quantum discord measure of entanglement can be expressed through the coherence entropy of a pto complete in relation to the state

a family of log gamma integrals and associated results abstract the authors present a systematic investigation of the following log gamma integral 0 z log t 1 d t and of its several related integral formulas relevant connections among the various mathematical constants involved naturally in the evaluation of the proposed integral are pointed out some approximate numerical values of the derivative 1 a of the hurwitz zeta function are also considered importance of such derivatives as 1 a lies in their usefulness in the effective lagrangian theory of quark confinement

certain classes of series involving the zeta function the authors apply the theory of the double gamma function which was recently revived in the study of determinants of laplacians to evaluate some families of series involving the riemann zeta function introducing a presumably new mathematical constant in the theory of the double gamma function they also systematically evaluate a definite integral of the double gamma function and various classes of series associated with zeta functions some of these definite integrals are expressed in terms of quotients of double gamma functions

unstable periodic orbits and semiclassical quantisation the bohr sommerfeld quantisation condition has a meaningful extension to classically chaotic systems whose periodic unstable orbits are isolated it provides a semiclassical euler factorisation for the functional determinant of the quantal hamiltonian in contrast to the hadamard infinite product over the eigenvalues by which the exact determinant is defined

quantum discord a measure of the quantumness of correlations two classically identical expressions for the mutual information generally differ when the systems involved are quantum this difference defines the quantum discord it can be used as a measure of the quantumness of correlations separability of the density matrix describing a pair of systems does not guarantee vanishing of the discord thus showing that absence of entanglement does not imply classicality we relate this to the quantum superposition principle and consider the vanishing of discord as a criterion for the preferred effectively classical states of a system i e the pointer states

on the fuzzy difference equation xn f xn 1 xn k abstract in this paper we study the global asymptotic behavior of the positive solutions of the following fuzzy difference equation x n f x n 1 x n k n 0 1 under appropriate assumptions where k 1 2 and the initial values x k x k 1 x 1 are positive fuzzy numbers we give sufficient conditions under which every positive solution of the above equation converges to a positive equilibrium

regularized products and determinants zeta regularized products are used to define determinants of operators in infinite dimensional spaces this article provides a general theory of regularized products and determinants which delivers a better approach to their existence and explicit determination

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

on the system of two difference equations xn 1 i 0kai yn ipi yn 1 i 0kbi xn iqi abstract in this paper we study the system of two difference equations of the form x n 1 i 0 k a i y n i p i y n 1 i 0 k b i x n i q i where a i b i i 0 1 k x i y i i k k 1 0 are positive numbers and p i q i i 0 k are positive constants more precisely we investigate the boundedness the persistence of the positive solutions the existence of a unique positive equilibrium and the global asymptotic stability of the positive equilibrium of the above system finally we find solutions of the system which do not oscillate about the positive equilibrium

turing s method for the selberg zeta function in one of his final research papers alan turing introduced a method to certify the completeness of a purported list of zeros of the riemann zeta function in this paper we consider turing s method in the analogous setting of selberg zeta functions and we demonstrate that it can be carried out rigorously in the prototypical case of the modular surface

on the new solutions to the fuzzy difference equation xn 1 a bxn abstract in this paper using a generalization of division for fuzzy numbers we study the existence and global behavior of the fuzzy difference equation x n 1 a b x n n 0 1 where a and b are positive fuzzy numbers some examples are presented to illustrate the applicability of our results

on first order fuzzy ricatti difference equation this paper deals with the existence the boundedness and the asymptotic behavior of the positive solutions to a first order fuzzy ricatti difference equation xn 1 a xnb xn n 0 1 where xn is a sequence of positive fuzzy numbers a b and the initial value x0 are positive fuzzy numbers moreover an example is given to demonstrate the effectiveness of the results obtained

on transfer operators for continued fractions with restricted digits for any non empty subset i of the natural numbers let lambda i denote those numbers in the unit interval whose continued fraction digits all lie in i define the corresponding transfer operator formula for formula where re rs theta i is the abscissa of convergence of the series formula when acting on a certain hilbert space hi rs we show that the operator li rs is conjugate to an integral operator ki rs if furthermore rs is real then ki rs is selfadjoint so that li rs hi rs hi rs has purely real spectrum it is proved that li rs also has purely real spectrum when acting on various hilbert or banach spaces of holomorphic functions on the nuclear space c omega 0 1 and on the frechet space c infty 0 1 the analytic properties of the map rs m li rs are investigated for certain alphabets i of an arithmetic nature for example i primes i squares i an arithmetic progression i the set of sums of two squares it is shown that rs m li rs admits an analytic continuation beyond the half plane re rs theta i

krull s theory for the double gamma function abstract the barnes g function g x 1 2 satisfies the functional equation log g x 1 log g x log x we complement w krull s work in bemerkungen zur differenzengleichung g x 1 g x x math nachrichten 1 1948 365 376 with additional results that yield a different characterization of the function g new expansions and sharp bounds for g on x 0 in terms of gamma and digamma functions a new expansion for the gamma function and summation formulae with polygamma functions

landau singularities and singularities of holonomic integrals of the ising class we consider families of multiple and simple integrals of the ising class and the linear ordinary differential equations with polynomial coefficients they are solutions of we compare the full set of singularities given by the roots of the head polynomial of these linear odes and the subset of singularities occurring in the integrals with the singularities obtained from the landau conditions for these ising class integrals we show that the landau conditions can be worked out either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral the singular behaviour of these integrals is obtained in the self dual variable w s 2 1 s2 with s sinh 2k where k j kt is the usual ising model coupling constant switching to the variable s we show that the singularities of the analytic continuation of series expansions of these integrals actually break the kramers wannier duality we revisit the singular behaviour zenine et al 2005 j phys a math gen 38 9439 74 of the third contribution to the magnetic susceptibility of ising model 3 at the points 1 3w 4w2 0 and show that 3 s is not singular at the corresponding points inside the unit circle s 1 while its analytical continuation in the variable s is actually singular at the corresponding points 2 s s2 0 outside the unit circle s 1

asymptotics of tracy widom distributions and the total integral of a painleve ii function the tracy widom distribution functions involve integrals of a painleve ii function starting from positive infinity in this paper we express the tracy widom distribution functions in terms of integrals starting from minus infinity there are two consequences of these new representations the first is the evaluation of the total integral of the hastings mcleod solution of the painleve ii equation the second is the evaluation of the constant term of the asymptotic expansions of the tracy widom distribution functions as the distribution parameter approaches minus infinity for the gue tracy widom distribution function this gives an alternative proof of the recent work of deift its and krasovsky the constant terms for the goe and gse tracy widom distribution functions are new

the genus and a regularization of an s1 equivariant euler class we show that a new multiplicative genus in the sense of hirzebruch can be obtained by generalizing a calculation due to atiyah and witten we introduce this as the genus compute its value for some examples and highlight some of its interesting properties we also indicate a connection with the study of multiple zeta values which gives an algebraic interpretation for our proposed regularization procedure

the asymptotics of the arakelov geen s function and faltings delta invariant we study the behavior of the arakelov green s function and faltings delta invariant on degenerating riemann surfaces

new solutions for first order linear fuzzy difference equations abstract in this paper we study the different formulations of fuzzy difference equation x n w x n 1 q where w q are positive fuzzy numbers using a generalization of division for fuzzy numbers we investigate the existence uniqueness and global behavior of the solution

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

quantum eigenfunctions in terms of periodic orbits of chaotic systems a resummed formula for the wigner function corresponding to an eigenfunction of a chaotic system in terms of periodic orbits is developed the infinite sum over periodic orbits is effectively truncated with the help of an extension of a method that was applied to the spectral determinant by berry and keating 1992 in principle the formula enables the computation of eigenstates and the probability density of wavefunctions from classical periodic orbits the conditions for appearance of scars are discussed

riemann hilbert problems from donaldson thomas theory we study a class of riemann hilbert problems arising naturally in donaldson thomas theory in certain special cases we show that these problems have unique solutions which can be written explicitly as products of gamma functions we briefly explain connections with gromov witten theory and exact wkb analysis

determinants of laplacians and isopolar metrics on surfaces of infinite area we construct a determinant of the laplacian for infinite area surfaces which are hyperbolic near infinity and without cusps in the case of a convex co compact hyperbolic metric the determinant can be related to the selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the laplacian in the hyperbolic near infinity case the determinant is analyzed through the zeta regularized relative determinant for a conformal metric perturbation we establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics these results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances

a quantum measure of coherence and incompatibility the well known two slit interference is understood as a special relation between an observable localization at the slits and a state being on both slits the relation between an observable and a quantum state is investigated in the general case it is assumed that the amount of coherence equals that of the incompatibility between an observable and a state on these grounds an argument is presented that leads to a natural quantum measure of coherence called coherence or incompatibility information its properties are studied in detail making use of the mixing property of relative entropy derived in this paper a precise relation between the measure of coherence of an observable and that of its coarsening is obtained and discussed from the intuitive point of view convexity of the measure is proved and thus the fact that it is an information entity is established a few more detailed properties of the coherence information are derived with a view to investigating the final state entanglement in general repeatable measurement and more importantly general bipartite entanglement in follow ups of this study

relation between primes and nontrivial zeros in the riemann hypothesis legendre polynomials modified zeta function and schrodinger equation we study the dependence between prime numbers and the real and imaginary parts of the nontrivial zeros of the riemann zeta function the legendre polynomials and the partial derivatives of the riemann zeta function are used to investigate the above dependence along with the riemann hypothesis with physical interpretations a modified zeta function with finite terms is defined as a new implement for the study of the zeta function and its zeros

certain classes of series associated with the zeta function and multiple gamma functions abstract the authors apply the theory of multiple gamma functions which was recently revived in the study of the determinants of the laplacians in order to evaluate some families of series involving the riemann zeta function by introducing a certain mathematical constant they also systematically evaluate this constant and some definite integrals of the triple gamma function various classes of series associated with the zeta function are expressed in closed forms many of these results are also used here to compute the determinant of the laplacian on the four dimensional unit sphere s 4 explicitly

asymptotic expansions of the double zeta function abstract the double zeta function of barnes 2 v z w is considered for large and small values of z and w with w 0 arg z r v 2 and r z 0 to the whole complex z plane and complex v plane with arg z

semiclassical quantization of sawtooth map with inclusion of next leading order and its failure the semiclassical quantization of sawtooth maps is reconsidered from the viewpoint of the higher order terms in the asymptotic series we carry out the semiclassical quantization in the leading and next leading orders however we fail in both cases the reasons are discussed in detail

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

series involving the zeta function and multiple gamma functions the theory of multiple gamma functions which was recently revived in the study of the determinants of the laplacians was applied in several earlier works in order to evaluate some families of series involving the riemann zeta function as well as to compute the determinants of the laplacians here in the present paper the authors address the converse problem and apply various known or new formulas for series associated with the zeta and related functions with a view to developing the corresponding theory of multiple gamma functions and then using these series to compute the determinants of the laplacians on the n dimensional unit sphere s n n 5 6 7 explicitly

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

spectral functions for regular sturm liouville problems in this paper we provide a detailed analysis of the analytic continuation of the spectral zeta function associated with one dimensional regular sturm liouville problems endowed with self adjoint separated and coupled boundary conditions the spectral zeta function is represented in terms of a complex integral and the analytic continuation in the entire complex plane is achieved by using the well known liouville green or wkb asymptotic expansion of the eigenfunctions associated with the problem the analytically continued expression of the spectral zeta function is then used to compute the functional determinant of the sturm liouville operator and the coefficients of the asymptotic expansion of the associated heat kernel

quantumness of bipartite states in terms of conditional entropies quantum discord as defined by olliver and zurek 2002 phys rev lett 88 017901 as the difference of two natural quantum extensions of the classical mutual information plays an interesting role in characterizing quantumness of correlations inspired by this idea we will study quantumness of bipartite states arising from different quantum analogs of the classical conditional entropy our approach is intrinsic in contrast to the olliver zurek method that involves extrinsic local measurements for this purpose we introduce two alternative variants of quantum conditional entropies via conditional density operators which in turn are intuitive quantum extensions of equivalent classical expressions for the conditional probability the significance of these quantum conditional entropies in characterizing quantumness of bipartite states is illustrated through several examples

general considerations on the finite size corrections for coulomb systems in the debye huckel regime we study the statistical mechanics of classical coulomb systems in a low coupling regime debye huckel regime in a confined geometry with dirichlet boundary conditions for the electric potential we use a method recently developed by the authors which relates the grand partition function of a coulomb system in a confined geometry with a certain regularization of the determinant of the laplacian on that geometry with dirichlet boundary conditions we study several examples of fully confining geometry in two and three dimensions and semi confined geometries where the system is confined only in one or two directions of the space we also generalize the method to study systems confined in arbitrary geometries with smooth boundary we find a relation between the expansion for small argument of the heat kernel of the laplacian and the large size expansion of the grand potential of the coulomb system this allow us to find the finite size expansion of the grand potential of the system in general we recover known results for the bulk grand potential in two and three dimensions and the surface tension for two dimensional systems we find the surface tension for three dimensional systems for two dimensional systems our general calculation of the finite size expansion gives a proof of the existence a universal logarithmic finite size correction predicted some time ago at least in the low coupling regime for three dimensional systems we obtain a prediction for the curvature correction to the grand potential of a confined system

last passage time for linear diffusions and application to the emptying time of a box we study the statistics of last passage time for linear diffusions first we present an elementary derivation of the laplace transform of the probability density of the last passage time thus recovering known results from the mathematical literature we then illustrate them on several explicit examples in a second step we study the spectral properties of the schrodinger operator associated to such diffusions in an even potential u x u x unveiling the role played by the so called weyl coefficient indeed in this case our approach allows us to relate the last passage times for dual diffusions i e diffusions driven by opposite force fields and to obtain new explicit formulae for the mean last passage time we further show that for such even potentials the small time t expansion of the mean last passage time on the interval 0 t involves the korteveg de vries invariants which are well known in the theory of schrodinger operators finally we apply these results to study the emptying time of a one dimensional box of size l containing n independent brownian particles subjected to a constant drift in the scaling limit where both n rightarrow infty and l rightarrow infty keeping the density rho n l fixed we show that the limiting density of the emptying time is given by a gumbel distribution our analysis provides a new example of the applications of extreme value statistics to out of equilibrium systems

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

a rule for quantizing chaos the authors find a real function delta e whose zeros approximate the quantum energy levels of a system with chaotic classical trajectories delta e is a finite sum over combinations of classical periodic orbits it is obtained from gutzwiller s infinite and divergent sum 1982 representing the spectral density in terms of periodic orbits by means of a resummation conjectured by analogy with a derivation of the riemann siegel formula for the riemann zeros they assess the practicality of the quantization condition

analytic torsion and closed geodesics on hyperbolic manifolds pour une variete hyperbolique orientee fermee x et une representation orthogonale de 1 x on definit une fonction zeta de type ruelle r s pour re s grand comme le produit sur les geodesiques closes premieres des facteurs det i exp sl ou l est la longueur de on continue analytiquement r s et on calcule son terme principal en s 0

some two sided inequalities for multiple gamma functions and related results there is an abundant literature on inequalities for the euler s gamma function c and its various related functions yet only very recently several authors began to study inequalities for the barnes double gamma function c 2 here in this paper we aim at presenting several two sided inequalities for the multiple gamma functions c n n 2 3 4 5 in our investigation of these two sided inequalities for the multiple gamma functions c n n 2 3 4 5 we employ and extend a method based upon taylor s formula and express email protected n 1 x as series involving the zeta functions we also give a more convenient explicit form of the multiple gamma functions c n email protected n n being the set of positive integers the main two sided inequalities for the multiple gamma functions c n n 2 3 4 5 which we have presented in this paper are presumably new and their derivations provide a fruitful insight into the corresponding problem for the multiple gamma functions c n when email protected 6

a comment on the gelfand yaglom theorem zeta functions and heat kernels for pt symmetric hamiltonians we propose a numerical method for computing zeta functions for general classes of hamiltonians based on the gelfand yaglom theorem which computes the determinant without needing to compute the eigenvalues we verify that this agrees with known analytic results for certain hermitean hamiltonians and we present numerical evidence that it may extend also to non hermitean pt symmetric hamiltonians

some integral and asymptotic formulas associated with the hurwitz zeta function the main object of this paper is to present an easily accessible integral representation of the partial sum l u x a 0 n x n a u of the hurwitz zeta function z u a which entails a number of important implications for l u x a and z u a and for their derivatives furthermore a new proof is given for a fundamental summation formula contained in theorem 3 below which is shown to possess numerous further consequences and applications relevant connections of the results obtained here with those considered in many other earlier and recent works are also discussed

trichotomy of a system of two difference equations abstract in this paper we study the boundedness and the asymptotic behavior of the positive solutions of the system of difference equations x n 1 a i 1 k a i x n p i j 1 m b j y n q j y n 1 b i 1 k c i y n p i j 1 m d j x n q j where k m 1 2 a b ai ci bj dj i 1 k j 1 m are positive constants pi qj i 1 k j 1 m are positive integers such that p1

zeta functions of nearly circular domains we study the zeta functions of negative integer powers of eigenvalues of the laplacian on two dimensional domains which are close to the unit disc d for closed form expressions are obtained which describe when the technique developed is applied to derive an asymptotic expansion for zeta functions of regular n sided polygons in the limit

multiple gamma and related functions the authors give several new and potentially useful relationships between the multiple gamma functions and other mathematical functions and constants as by products of some of these relationships a classical definite integral due to euler and other definite integrals are also considered together with closed form evaluations of some series involving the riemann and hurwitz zeta functions

on the fuzzy difference equations of finance in this paper our intention is to demonstrate the applicability of fuzzy difference equations in the field of finance in order to achieve that we follow the work done by deeba et al and papashinopoulos et al and we study the existence of positive solutions to fuzzy financial difference equations initially we appose a method realized by buckley that deals with fuzzy analogues of the elementary compound interest problem afterwards we approach this case via the fuzzy difference equations method after making a comparison between the two methods we notice that they end up to the same general formula the main contribution of this paper is the simplicity of the method presented which gives us the opportunity to create formulas for cases which are specialized and also more realistic than the very basic one of the compound interest moreover by using difference equations we can study the factors which cause the change in the value of money in different time periods as a source for these cases of finance we use the work of kwapisz and we study a well known problem of financial mathematics this problem deals with the time value of money and as a direct application of it we study the balance of bank deposits finally a numerical example is presented for better understanding

some q extensions of the apostol bernoulli and the apostol euler polynomials of order n and the multiple hurwitz zeta function abstract in this paper we first investigate several further interesting properties of the multiple hurwitz lerch zeta function n z s a which was introduced recently by choi et al j choi d s jang h m srivastava a generalization of the hurwitz lerch zeta function integral transform spec funct 19 2008 we then introduce and investigate some q extensions of the multiple hurwitz lerch zeta function n z s a the apostol bernoulli polynomials b k n x of order n and the apostol euler polynomials e k n x of order n relevant connections of the results presented here with those obtained in earlier works are also indicated precisely

on determinants of laplacians on riemann surfaces determinants of laplacians on tensors and spinors of arbitrary weights on compact hyperbolic riemann surfaces are computed in terms of values of selberg zeta functions at half integer points