series representations for some mathematical constants abstract the authors apply a classical series identity involving the psi or digamma function with a view to deriving series representations for a number of known mathematical constants several closely related consequences and results are also considered

on generalized harmonic number sums we evaluate generalized harmonic number sums with parameter in terms of values of polylogarithm functions and several examples are given instances of such sums occur in diverse areas including analytic number theory and in calculations of high energy nuclear and atomic physics

some classes of infinite series associated with the riemann zeta and polygamma functions and generalized harmonic numbers the authors aim at presenting a systematic investigation of several families of infinite series which are associated with the riemann zeta function the digamma and polygamma functions the harmonic and generalized harmonic numbers and the stirling numbers of the first kind relevant connections of the results derived here with those considered in many earlier works on this subject are also indicated

on some slowly convergent series involving the hurwitz zeta function we shall extract the essence of the adamchik srivastava generating function method analysis munich 18 1998 131 by proving the most far reaching ramanujan yoshimoto formula and by showing that some of the results stated in srivastava and choi series associated with the zeta and related functions kluwer academic publishers dordrecht 2001 are simple consequences of the above mentioned formula

on some series containing p x p y and 9 p x p y 2 for certain values of x and y abstract in this article we discuss series containing the factor k 1 k 1 2 1 2 k 1 2 and 1 2 k 1 case1 2 k with k n here x d dxlog x and x is the gammafunction

the wick ordered exponential of the dipole field as a field of type s we construct the wick ordered exponential of the dipole field an infrared singular field we show that the field operator is an object much more singular than its own vacuum expectation values the actual construction involves the spaces of type s by gelfand and shilov and seems to provide the most general distributional scheme for the choice of test function in gauge quantum field theories

conformal deformation to constant negative scalar curvature on noncompact riemannian manifolds si m g est une variete de riemann complete de courbure scalaire non positive s satisfaisant s x e 0 pour x m m 0 ou m 0 est un ensemble compact alors il y a une metrique conforme complete g de courbure scalaire s 1

a certain family of infinite series associated with digamma functions abstract the sums of several interesting infinite series were recently expressed in terms of the psi or digamma functions the object of this paper is to present a systematic account of these and of numerous similar or more general series whose sums can be found in the literature in various equivalent forms some relevant unifications and further generalizations are also indicated

on the structure of degenerate solutions of the einstein conformally invariant scalar system we address the question of the existence and construction of nontrivial regular solutions of the einstein conformally invariant massless scalar field equations i e solutions g satisfying 1 2 r 4 2 g 0 and additionally geometry and the scalar field are regular across the degeneracy region defined as the zeros of 1 2 under the assumptions 1 the solution g is minimally of class c3 and admits a hypersurface orthogonal timelike killing vector field and 2 relative to the three spacelike hypersurfaces perpendicular to the killing field the degeneracy region constitute regular two surfaces and the induced positive definite three metric possesses a degenerate ricci we show that the conformal system admits nontrivial regular across the degeneracy region solutions and we demonstrate that any such solution necessarily admits an additional local g 3 group of isometries possessing two dimensional orbits of constant gaussian curvature coinciding with the

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

static spherically symmetric solutions of the einstein yang mills equations we study the global behaviour of static spherically symmetric solutions of the einstein yang mills equations with gauge groupsu 2 our analysis results in three disjoint classes of solutions with a regular origin or a horizon the 3 spaces t const of the first generic class are compact and singular the second class consists of an infinite family of globally regular resp black hole solutions the third type is an oscillating solution which although regular is not asymptotically flat

on existence and behaviour of asymptotically flat solutions to the stationary einstein equations we show existence and uniqueness of asymptotically flat solutions to the stationary einstein equations ins 3 b r whereb r is a ball of radiousr 0 when a small enough continuous complex function u on s is given regularity and decay estimates imply that these solutions are analytic in the interior ofs and also at infinity when suitably conformally rescaled

a general phase transition model for vehicular traffic an extension of the colombo phase transition model is proposed the congestion phase is described by a two dimensional zone defined around an equilibrium flux known as the classical fundamental diagram general criteria to build such a set valued fundamental diagram are enumerated and instantiated on several equilibrium fluxes with different concavity properties the solution of the riemann problem in the presence of phase transitions is obtained through the construction of a riemann solver which enables the definition of the solution of the cauchy problem using wavefront tracking the free flow phase is described using a newell daganzo fundamental diagram which allows for a more tractable definition of phase transition compared to the original colombo phase transition model the accuracy of the numerical solution obtained by a modified godunov scheme is assessed on benchmark scenarios for the different flux functions constructed

existence of global bounded weak solutions to nonsymmetric systems of keyfitz kranzer type in this paper we study the global l solutions for the cauchy problem of nonsymmetric system 1 1 of keyfitz kranzer type when n 1 1 1 is the aw rascle traffic flow model first we introduce a new flux approximation to obtain a lower bound e 0 for the parabolic system generated by adding artificial viscosity to the aw rascle system then using the compensated compactness method with the help of l1 estimate of we t x we prove the pointwise convergence of the viscosity solutions under the general conditions on the function p which includes prototype function p 1 a where 1 0 0 a is a constant second by means of bv estimates on the riemann invariants and the compensated compactness method we prove the global existence of bounded entropy weak solutions for the cauchy problem of general nonsymmetric systems 1 1

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

on the analyticity of stationary gravitational fields at spatial infinity it is proved that all stationary vacuum solutions of einstein s equations which satisfy certain weak differentiability conditions characterizing asymptotic flatness possess an analytic structure near spatial infinity this analyticity theorem implies the existence of a multipole expansion whose coefficients can be expressed in terms of the geroch hansen multipole moments defined at the point at infinity on the conformal manifold this proves a longstanding conjecture that these moments uniquely determine the local structure of a stationary asymptotically flat vacuum metric

derivation of continuum traffic flow models from microscopic follow the leader models in this paper we establish a connection between a microscopic follow the leader model based on ordinary differential equations and a semidiscretization of a macroscopic continuum model based on a c

analytic continuation of the single eigenvalue probability density function an exact expression for the single eigenvalue probability density function for any dimension of a gaussian orthogonal ensemble is found by analytic continuation it is shown that the form can be used to derive a number of exact results in a simple way in odd dimensions it is shown that the single eigenvalue probability density function can be written as a sum of two densities

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

further reducible cases of certain kampe de fe riet functions associated with incomplete integrals of cylindrical functions abstract new reduction formulas are derived for the kampe de feriet functions f q 1 0 p 2 1 and f 1 2 1 1 1 0 which are closely connected with representations for incomplete lipschitz hankel type integrals of cylindrical functions some general series identities and related reduction formulas relevant to the present investigation are also considered

on one dimensional digamma and polygamma series related to the evaluation of feynman diagrams we consider summations over digamma and polygamma functions often with summands of the form 1 n j n p q n r and 1 n j m n p q n r where m p q and r are positive integers we develop novel general integral representations and present explicit examples special cases of the sums reduce to known linear euler sums the sums of interest find application in quantum field theory including evaluation of feynman amplitudes

a 2 phase traffic model based on a speed bound we extend the classical lighthill whitham richards lwr traffic model allowing different maximal speeds for different vehicles then we add a uniform bound on the traffic speed the result presented in this paper is a new macroscopic model displaying two phases based on a nonsmooth 2 times2 system of conservation laws this model is compared with other models of the same type in the current literature as well as with a kinetic one moreover we establish a rigorous connection between a microscopic follow the leader model based on ordinary differential equations and this macroscopic continuum model

values of the polygamma functions at rational arguments gauss in 1812 in his celebrated memoir on the hypergeometric series presented a remarkable formula for the psi or digamma function z at rational arguments z which can be expressed in terms of elementary functions davis in 1935 extended gauss s result to the polygamma functions by using a known series representation of n z in an elementary yet technical way k lbig in 1996 in his cern technical report also gave two extensions to n z by using the series definition of polylogarithm function and the above known series representation here we aim at deriving general formulae expressing n z as rational arguments in terms of other functions which will be obtained in two ways in addition several special cases are also considered and as a by product of our main results we derive in a simple and unified manner all formulae given by gauss davis and k lbig finally it should be noted that all our results in view of the relationship between n z and the hurwitz zeta function s a could be rewritten in the representation of s a

on hyperboloidal cauchy data for vacuum einstein equations and obstructions to smoothness of scri the relationship between the geometric properties of hyperboloidal cauchy data for vacuum einstein equations at the conformal boundary of the initial data surface and between the space time geometry is analyzed in detail we prove that a necessary condition for existence of a smooth or a polyhomogeneous scri i e a scri around which the metric is expandable in terms ofr j log i r rather than in terms ofr j is the vanishing of the shear of the conformal boundary of the initial data surface we derive the boundary constraints which have to be satisfied by an initial data set for compatibility with friedrich s conformal framework we show that a sufficient condition for existence of a smooth scri not necessarily complete is the vanishing of the shear of the conformal boundary of the initial data surface and smoothness up to boundary of the conformally rescaled initial data we also show that the occurrence of some log terms in an asymptotic expansion at the conformal boundary of solutions of the constraint equations is related to the non vanishing of the weyl tensor at the conformal boundary

on the regularity of solutions to the yamabe equation and the existence of smooth hyperboloidal initial data for einstein s field equations the regularity of the solutions to the yamabe problem is considered in the case of conformally compact manifolds and negative scalar curvature the existence of smooth hyperboloidal initial data for einstein s field equations is demonstrated

closed form summations of certain hypergeometric type series containing the digamma function recently interesting novel summation formulae for hypergeometric type series containing a digamma function as a factor have been established by miller 2006 j phys a math gen 39 3011 20 mainly by exploiting already known results and using certain transformation and reduction formulae in the theory of the kampe de feriet double hypergeometric function it is shown here that these and several other series of this type can be closed form summed by simpler and more direct arguments based only on a derivative formula for the pochhammer symbol and the theory of the digamma or function and generalized hypergeometric function in addition a new reduction formula for the kampe de feriet function is obtained

a class of multi phase traffic theories for microscopic kinetic and continuum traffic models in the present paper a review and numerical comparison of a special class of multi phase traffic theories based on microscopic kinetic and macroscopic traffic models is given macroscopic traffic equations with multi valued fundamental diagrams are derived from different microscopic and kinetic models numerical experiments show similarities and differences of the models in particular for the appearance and structure of stop and go waves for highway traffic in dense situations for all models but one phase transitions can appear near bottlenecks depending on the local density and velocity of the flow

fractional calculus and the sums of certain families of infinite series abstract although the concept of fractional calculus can be traced back to l hopital s naive three centuries old curiosity about the meaning of leibniz s nth derivative notation dny dxn when n 1 2 and many monumental works on the subject of fractional calculus have since appeared and are still appearing in the mathematical literature the five volume work published recently by k nishimoto fractional calculus vols i iv 1984 1987 1989 1991 an essence of nishimoto s fractional calculus calculus in the 21st century integrations and differentiations of arbitrary order 1991 contains an interesting account of the theory and applications of fractional calculus in a number of areas of mathematical analysis such as ordinary and partial differential equations summation of series et cetera the main object of the present paper is to examine rather systematically some of the most recent contributions by nishimoto et al cf e g j college engrg nihon univ ser b 32 1991 7 13 15 21 23 30 j fractional calculus1 1992 7 16 see also fractional calculus vol iv 1991 j college engrg nihon univ ser b33 1992 1 8 l galue et al j fractional calculus 1 1992 17 21 and others on the use of fractional calculus in finding the sums of numerous interesting families of infinite series various other classes of infinite sums found in the literature without using fractional calculus and their hitherto unnoticed connections with certain known results are also considered

some summation formulae for spherical spinors two families of series with terms involving spherical spinors spinor spherical harmonics for a spin one half particle and are summed to closed forms involving the legendre function of the first kind and its derivatives with respect to an argument and or an index in particular it is shown that the closed forms of the series 0 n n are remarkably simple and with i denoting the 2 2 unit matrix and standing for the vector of pauli matrices integral eigenvalue equations solved by the spherical spinors with either n n or as kernels are provided it is pointed out that n n is the green function of the operator 1 i 1 2 while is the generalized reduced green function of the operator k 1 i k 1 2 with ir

hyperbolic phase transitions in traffic flow this paper provides a mathematical model of the phenomenon of phase transitions in traffic flow the model consists of a scalar conservation law coupled with a 2 2 system of conservation laws the coupling is achieved via a free boundary where the phase transition takes place for this model the riemann problem is stated and globally solved the cauchy problem is proved to admit a solution defined globally in time without any assumption about the smallness of the initial data or the number of phase boundaries qualitative properties of real traffic flow are shown to agree with properties of the solutions of the model

on the existence of analytic null asymptotically flat solutions of einstein s vacuum field equations this paper proves the existence of analytic solutions of the asymptotic characteristic initial value problem for einstein s field equations for analytic data on past null infinity and on an incoming null hypersurface

the multipole structure of stationary space times a definition of multipole moments for stationary asymptotically flat solutions of einstein s equations is proposed it is shown that these moments characterize a given space time uniquely conversely they can be arbitrarily prescribed i e they generate power series for the field variables which satisfy the field equations to all orders despite their apparently rather different origin they are shown to be identical with the geroch hansen ones

optimal rate of convergence for sequences of a prescribed form abstract we provide a highly convergent version of the generalized euler sequence and we determine the element with the optimal rate of convergence in several classes of sequences of a prescribed form the new approach extends and unifies many previous efforts in this direction in addition we answer an open problem on the asymptotic expansion of the harmonic number sequence

on some series representations of the hurwitz zeta function a variety of infinite series representations for the hurwitz zeta function are obtained particular cases recover known results while others are new specialization of the series representations apply to the riemann zeta function leading to additional results the method is briefly extended to the lerch zeta function most of the series representations exhibit fast convergence making them attractive for the computation of special functions and fundamental constants

on the dirichlet problem for stationary and axisymmetric einstein equations in suitable coordinates einstein s field equations for a rigidly rotating perfect fluid in equilibrium can be written as a semilinear system of purely elliptic partial differential equations of second order therefore the formulation of a boundary value problem is appropriate in this situation it is shown that the dirichlet problem for the vacuum region outside a ball and for a ball inside the matter region has a unique regular solution if the boundary data are in a characteristic way limited by the diameter of the ball this restriction seems to be closely connected with stability limits for rotating stars furthermore the used mathematical methods are directly related to a numerical solution technique for such physical systems

entropy solutions for a traffic model with phase transitions abstract we introduce an appropriate notion of entropy solution for a generalization of the two phase macroscopic traffic model proposed in goatin 2006 we first apply the wave front tracking method to prove existence and a priori bounds for weak solutions then in the case the characteristic field of the free phase is linearly degenerate we prove that the obtained weak solutions are in fact entropy solutions the case of solutions attaining values at the vacuum is considered we also present an explicit numerical example to describe some qualitative features of the solutions

on existence of static metric extensions in general relativity motivated by problems related to quasi local mass in general relativity we study the static metric extension conjecture proposed by r bartnik 4 we show that for any metric on b1 that is close enough to the euclidean metric and has reflection invariant boundary data there always exists an asymptotically flat and scalar flat static metric extension in m 3 b1 such that it satisfies bartnik s geometric boundary condition 4 on b1

on the stationary implies axisymmetric theorem for extremal black holes in higher dimensions all known stationary black hole solutions in higher dimensions possess additional rotational symmetries in addition to the stationary killing field also for all known stationary solutions the event horizon is a killing horizon and the surface gravity is constant in the case of non degenerate horizons non extremal black holes a general theorem was previously established 24 proving that these statements are in fact generally true under the assumption that the spacetime is analytic and that the metric satisfies einstein s equation here we extend the analysis to the case of degenerate extremal black holes it is shown that the theorem still holds true if the vector of angular velocities of the horizon satisfies a certain diophantine condition which holds except for a set of measure zero

characteristic initial data and wavefront singularities in general relativity the structure of singularities caustics self intersections of wavefronts null hypersurfaces and wavefront families null coordinates in arbitrary space times is discussed in detail and illustrated by explicit examples of stable wavefront singularities in minkowski space it is shown how characteristic initial data determine the caustics and the self intersections of the characteristics of einstein s field equations

the polygamma function p k x for x 1 4 and x 3 4 expressions for the polygamma function j k x for the arguments x 14 and x 14 are given in terms of bernoulli numbers euler numbers the riemann zeta function for odd integer arguments and the related series of reciprocal powers of integers b m

series representations in the spirit of ramanujan abstract using an integral transform with a mild singularity we obtain series representations valid for specific regions in the complex plane involving trigonometric functions and the central binomial coefficient which are analogues of the types of series representations first studied by ramanujan over certain intervals on the real line we then study an exponential type series rapidly converging to the special values of l functions and the riemann zeta function in this way a new series converging to catalan s constant with geometric rate of convergence less than a quarter is deduced further evaluations of some series involving hyperbolic functions are also given

fredholm hodge and liouville theorems on noncompact manifolds fredholm liouville hodge and l2 cohomology theorems are proved for laplacians associated with a class of metrics defined on manifolds that have finitely many ends the metrics are conformal to ones that are asymptotically translation invariant they are not necessarily complete the fredholm results are of necessity with respect to weighted sobolev spaces embedding and compact embedding theorems are also proved for these spaces two of the most useful facts in analysis on a compact riemannian manifold are that the laplacian is fredholm and its kernel consists of closed and coclosed forms that provide unique representatives for all the de rham cohomology classes naturally one would like to extend these results to noncompact manifolds the first such result in this direction is due to kodaira 9 12 p 165 it is that l2 a m g is the orthogonal direct sum of dco aq 1m dg co aq lm and k2 awm g ae l2 am g ida dga 0 unfortunately if no restriction is made on the manifold or the metric one cannot improve on this the laplacian need not be fredholrn l2 harmonic forms need not be closed or coclosed even if they are closed and coclosed the space of harmonic l2 forms need not be finite dimensional see 7 and even if the laplacian is fredholm and l2 harmonic forms are closed and coclosed those forms need not provide unique or total representation of de rham cohomology thus one of the main questions in analysis on noncompact manifolds is what conditions on m and g allow one to carry over the fredholm and hodge theorems for compact manifolds and if they cannot be carried over completely to what extent can they be in the case of hodge s theorem i e properties of the space of l2 harmonic forms this question has been quite actively investigated recently see 2 8 11 and 13 for instance one of the results of atiyah patodi and singer in 2 is that if a manifold has cylindrical ends then k 2 a m g is naturally isomorphic to the image of hcomp m in hfdr m see 2 proposition 4 9 in 11 muller investigates the spectrum of the laplacian on manifolds that outside a compact set are q x r with received by the editors august 30 1985 and in revised form april 10 1986 1980 mathematics subject classification 1985 revision primary 58gi0 35j05 1987 american mathematical society 0002 9947 87 1 00 25 per page this content downloaded from 157 55 39 45 on tue 19 jul 2016 04 20 24 utc all use subject to http about jstor org terms

closed form summations of dowker s and related trigonometric sums through a unified and relatively simple approach which uses complex contour integrals particularly convenient integration contours and calculus of residues closed form summation formulas for 12 very general families of trigonometric sums are deduced one of them is a family of cosecant sums which was first summed in closed form in a series of papers by dowker 1987 phys rev d 36 3095 101 1989 j math phys 30 770 3 1992 j phys a math gen 25 2641 8 whose method has inspired our work in this area all of the formulas derived here involve the higher order bernoulli polynomials this article is part of a special issue of journal of physics a mathematical and theoretical in honour of stuart dowker s 75th birthday devoted to applications of zeta functions and other spectral functions in mathematics and physics

future stability of the einstein non linear scalar field system we consider the question of future global non linear stability in the case of einstein s equations coupled to a non linear scalar field the class of potentials v to which our results apply is defined by the conditions v 0 0 v 0 0 and v 0 0 thus einstein s equations with a positive cosmological constant represents a special case obtained by demanding that the scalar field be zero in that context there are stability results due to helmut friedrich the methods of which are however not so easy to adapt to the presence of matter the goal of the present paper is to develop methods that are more easily adaptable due to the extreme nature of the causal structure in models of this type it is possible to prove a stability result which only makes local assumptions concerning the initial data and yields global conclusions in time to be more specific we make assumptions in a set of the form b 4r 0 p for some r 0 0 on the initial hypersurface and obtain the conclusion that all causal geodesics in the maximal globally hyperbolic development that start in b r 0 p are future complete furthermore we derive expansions for the unknowns in a set that contains the future of b r 0 p the advantage of such a result is that it can be applied regardless of the global topology of the initial hypersurface as an application we prove future global non linear stability of a large class of spatially locally homogeneous spacetimes with compact spatial topology

conformal structures of static vacuum data in the cauchy problem for asymptotically flat vacuum data the solution jets along the cylinder at space like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future past null infinity the tendency of these singularities to spread along the null generators of null infinity obstructs the development of a smooth conformal structure at null infinity for the solution jets arising from time reflection symmetric data to extend smoothly to the critical sets it is necessary that the cotton tensor of the initial three metric h satisfies a certain conformally invariant condition at space like infinity it is sufficient that h be asymptotically static at space like infinity the purpose of this article is to characterize the gap between these conditions we show that with the class of metrics which satisfy condition on the cotton tensor and a certain non degeneracy requirement is associated a one form with conformally invariant differential d we provide two criteria if h is real analytic is closed and one of its integrals satisfies a certain equation then h is conformal to static data near space like infinity if h is smooth is asymptotically closed and one of its integrals satisfies a certain equation asymptotically then h is asymptotically conformal to static data at space like infinity

the asymptotic characteristic initial value problem for einstein s vacuum field equations as an initial value problem for a first order quasilinear symmetric hyperbolic system the asymptotic characteristic initial value problem for einstein s vacuum field equations where data are given on an incoming null hypersurface and on part of past null infinity is reduced to a characteristic initial value problem for a first order quasilinear symmetric hyperbolic system of differential equations for which existence and uniqueness of solutions can be shown it is delineated how the same method can be applied to the standard cauchy problems for einstein s vacuum and conformal vacuum equations

summations for certain series containing the digamma function apparently new summations in terms of well known special functions are deduced for hypergeometric type series containing a digamma function as a factor as a by product of this investigation new reduction formulae for the kampe de feriet function f0 2 12 1 0 x x are obtained

certain summation formulas involving harmonic numbers and generalized harmonic numbers abstract harmonic numbers and generalized harmonic numbers have been studied since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science various branches of number theory elementary particle physics and theoretical physics here we aim at presenting further interesting identities about certain finite or infinite series involving harmonic numbers and generalized harmonic numbers by applying an algorithmic method to a known summation formula for the hypergeometric function 5 f 4 1