a new bartholdi zeta function of a digraph we introduce a new type of the bartholdi zeta function of a digraph d and give a determinant expression of it furthermore we define a new type of the bartholdi l function of d and give a determinant expression of it as a corollary we obtain a decomposition formula for a new type of the bartholdi zeta function of a group covering of d by new bartholdi l functions of d

some expansions related to the hubbell rectangular source integral abstract four series expansions are presented and examined for the hubbell rectangular source integral f a b b0 arctan a 1 x 2 1 x 2 1 dx for the extremal situations when one or both of the parameters a b tend to zero or infinity

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

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

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

special values of multiple polylogarithms historically the polylogarithm has attracted specialists and non specialists alike with its lovely evaluations much the same can be said for euler sums or multiple harmonic sums which within the past decade have arisen in combinatorics knot theory and high energy physics more recently we have been forced to consider multidimensional extensions encompassing the classical polylogarithm euler sums and the riemann zeta function here we provide a general framework within which previously isolated results can now be properly understood applying the theory developed herein we prove several previously conjectured evaluations including an intriguing conjecture of don zagier

the algebra of multiple harmonic series abstract recently there has been much interest in multiple harmonic series i 1 i 2 i k n 1 n 2 n k 1 1 n 1 i 1 n 2 i 2 n k i k which converge when the exponentsijare positive integers andi1 1 also known as multiple zeta values or euler zagier sums starting with the noncommutative polynomial algebraq x y we define a second multiplication which is commutative and associative and call the resulting structure the harmonic algebra h as a graded commutative algebra h turns out to be a free polynomial algebra with the number of generators in degreengiven by the witt formula for the numbern n of basis elements of degreenin the free lie algebra on two generators multiple harmonic series can be thought of as images under a map h 0 rwhich is a homomorphism with respect to the commutative multiplication where h 0is an appropriate subalgebra of h if we calli1 ikthe weight of the series i1 ik then forn 1 there are at mostn n series of weightnthat are irreducible in the sense that they are not sums of rational multiples of products of multiple harmonic series of lower weight in fact our approach gives an explicit set of algebraically irreducible multiple harmonic series we also show that there is a subalgebra h 0 h 1 h related to the shuffle algebra which contains the algebra s of symmetric functions in fact maps the elements of s h 0to algebraic combinations of zeta values i for integeri 2 the map is not injective we show how several results about multiple harmonic series can be recast as statements about the kernel of and propose some conjectures on the structure of the algebra h 0 ker

on the rapid computation of various polylogarithmic constants we give algorithms for the computation of the d th digit of certain transcendental numbers in various bases these algorithms can be easily implemented multiple precision arithmetic is not needed require virtually no memory and feature run times that scale nearly linearly with the order of the digit desired they make it feasible to compute for example the billionth binary digit of log 2 or on a modest work station in a few hours run time

the coefficients of the reduced bartholdi zeta function abstract in this paper we establish a new zeta function based on the bartholdi zeta function for an undirected graph g called the reduced bartholdi zeta function we study the relation between its coefficients and the structure of the graph and demonstrate that the coefficients count the star subgraphs in the symmetric digraph d g moreover we investigate the properties of semi principle minors extracted from the adjacency matrix of the oriented line graph of g we also present a general formula for calculating all the coefficients of the reduced bartholdi zeta function

a p adic regulator problem in algebraic k theory and group cohomology is injective where p is a fixed rational prime and n 1 is odd observe that each p is clearly injective for n 1 because ki 0 0 and k op op a much harder problem is whether zp is injective for n 1 and f totally real abelian injectively of zp on the subgroup of 0 consisting of those elements congruent to lmodp for each p p is equivalent to nonvanishing of the p adic regulator br c as an example of 1 let f be quadratic imaginary then is

estimates for the norms of products of sines and cosines abstract in this paper we prove asymptotic formulas for the l p norms of p n k 1 n 1 e i k and q n k 1 n 1 e i k these products can be expressed using k 1 n sin k 2 and k 1 n cos k 2 respectively we prove an estimate for p n at a point near where its maximum occurs finally we give an asymptotic formula for the maximum of the fourier coefficients of q n

trigonometry of spacetimes a new self dual approach to a curvature signature in dependent trigonometry a new method to obtain trigonometry for the real spaces of constant curvature and metric of any even degenerate signature is presented the method could be described as curvature signature in dependent trigonometry and encapsulates trigonometry for all these spaces into a single basic trigonometric group equation this brings to its logical end the idea of an absolute trigonometry and provides equations which hold true for the nine two dimensional spaces of constant curvature and any signature this family of spaces includes both relativistic and non relativistic spacetimes therefore a complete discussion of trigonometry in the six de sitter minkowskian newton hooke and galilean spacetimes follow as particular instances of the general approach distinctive traits of the method are universality and self duality every equation is meaningful for the nine spaces at once and displays invariance explicitly under a duality transformation relating the nine spaces amongst themselves these basic structural properties allow a complete study of trigonometry and in fact any equation previously known for the three classical riemannian spaces also has a version for the remaining six spacetimes in most cases these equations are new

evaluation of a ln tan integral arising in quantum field theory we analytically evaluate a dilogarithmic integral that is prototypical of volumes of ideal tetrahedra in hyperbolic geometry we additionally obtain new representations of the clausen function cl2 and the catalan constant g cl2 2 as well as new relations between sine and clausen function values

degenerations of generalized krichever novikov algebras on tori degenerations of lie algebras of meromorphic vector fields on elliptic curves i e complex tori which are holomorphic outside a certain set of points markings are studied by an algebraic geometric degeneration process certain subalgebras of lie algebras of meromorphic vector fields on p1 the riemann sphere are obtained in case of some natural choices of the markings these subalgebras are explicitly determined it is shown that the number of markings can change

quantum chaos on discrete graphs adapting a method developed for the study of quantum chaos on quantum metric graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76 spectral functions and trace formulae for discrete laplacians on graphs are derived this is achieved by expressing the spectral secular equation in terms of the periodic orbits of the graph and obtaining functions which belong to the class of functions proposed originally by ihara 1966 j mat soc japan 18 219 and expanded by subsequent authors stark and terras 1996 adv math 121 124 kotani and sunada 2000 j math sci univ tokyo 7 7 finally a model of classical dynamics on the discrete graph is proposed it is analogous to the corresponding classical dynamics derived for quantum graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76

integral forms of sums associated with harmonic numbers we use integral identities to establish a relationship with sums that include harmonic numbers more over we obtain some closed forms of finite binomial sums in particular cases we establish some identities for harmonic numbers

an alternative proof of a theorem of zagier abstract we give a new proof of zagier s theorem on evaluation of multiple zeta star values at 2 a 3 2 b the novelty of the proof consists in employing a hypergeometric identity of andrews which expresses a very well poised hypergeometric series in terms of a multiple series and the residue theorem

a geometrical angle on feynman integrals a direct link between a one loop n point feynman diagram and a geometrical representation based on the n dimensional simplex is established by relating the feynman parametric representations to the integrals over contents of n 1 dimensional simplices in non euclidean geometry of constant curvature in particular the four point function in four dimensions is proportional to the volume of a three dimensional spherical or hyperbolic tetrahedron which can be calculated by splitting into birectangular ones it is also shown that the known formula of reduction of the n point function in n 1 dimensions corresponds to splitting the related n dimensional simplex into n rectangular ones

a variational principle for weighted delaunay triangulations and hyperideal polyhedra we use a variational principle to prove an existence and uniqueness theorem for planar weighted delaunay triangulations with non intersecting site circles with prescribed combinatorial type and circle intersection angles such weighted delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space thus the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles more generally we consider weighted delaunay triangulations in piecewise flat surfaces allowing cone singularities with prescribed cone angles in the vertices the material presented here extends work by rivin on delaunay triangulations and ideal polyhedra

quadratic and cubic harmonic number sums abstract this paper develops an approach to evaluation of euler related sums the approach is based on simple integral computations by the approach we can obtain some closed form representations of sums of quadratic and cubic harmonic numbers and reciprocal binomial coefficients the given representations are new

structure of the lie algebra of polynomial vector fields on the riemann sphere with three punctures the lie algebra of polynomial vector fields on the sphere with two punctures also known as the witt algebra or the centerless virasoro algebra can be generalized in different ways in this paper the lie algebra obtained by allowing poles at three points rather than only two is considered using some elementary representation theory of the symmetric group on three letters a convenient basis for this lie algebra is determined such that the commutator of two basis elements is a linear combination of at most three other basis elements the expressions for the basis elements show that this lie algebra is related to modular forms for a level 2 congruence subgroup of sl2 z a conclusion is given with some suggestions for further investigation

the poincare metric and isoperimetric inequalities for hyperbolic polygons we prove several isoperimetric inequalities for the conformal radius or equivalently for the poincare density of polygons on the hyperbolic plane our results include as limit cases the isoperimetric inequality for the conformal radius of euclidean n gons conjectured by g polya and g szego in 1951 and a similar inequality for the hyperbolic n gons of the maximal hyperbolic area conjectured by j hersch both conjectures have been proved in previous papers by the third author our approach uses the method based on a special triangulation of polygons and weighted inequalities for the reduced modules of trilaterals developed by a yu solynin we also employ the dissymmetrization transformation of v n dubinin as an important part of our proofs we obtain monotonicity and convexity results for special combinations of the euler gamma functions which appear to have a significant interest in their own right

on a three dimensional symmetric ising tetrahedron and contributions to the theory of the dilogarithm and clausen functions perturbative quantum field theory for the ising model at the three loop level yields a tetrahedral feynman diagram c a b with masses a and b and four other lines with unit mass the completely symmetric tetrahedron ctet c 1 1 has been of interest from many points of view with several representations and conjectures having been given in the literature we prove a conjectured exponentially fast convergent sum for c 1 1 as well as a previously empirical relation for c 1 1 as a remarkable difference of clausen function values our presentation includes propositions extending the theory of the dilogarithm li2 and clausen cl2 functions as well as their relation to other special functions of mathematical physics the results strengthen connections between feynman diagram integrals volumes in hyperbolic space number theory and special functions and numbers specifically including dilogarithms clausen function values and harmonic numbers

universal central extensions of elliptic affine lie algebras let g be a simple complex finite dimensional lie algebra and let r be the ring of regular functions on a compact complex algebraic curve with a finite number of points removed lie algebras of the form g cr are considered these generalize kac moody loop algebras since for a curve of genus zero with two punctures r c t t 1 the universal central extension of g r is analogous to an untwisted affine kac moody algebra by kassel s theorem the kernel of the universal central extension is linearly isomorphic to the kahler differentials of r modulo exact differentials the dimension of the kernel for any r is determined first restricting to hyperelliptic curves with 2 3 or 4 special points removed a basis for the kernel is determined restricting further to an elliptic curve with punctures at two points of orders one and two in the group law we explicitly determine the cocycles which give the commutation relations for the universal central extension the results involve pollaczek polynomials which ar

on harmonic numbers and nonlinear euler sums abstract in this paper we are interested in euler type sums with products of harmonic numbers stirling numbers and bell numbers we discuss the analytic representations of euler sums through values of polylogarithm function and riemann zeta function moreover we provide explicit evaluations for almost all euler sums with weight 5 which can be expressed in terms of zeta values and polylogarithms furthermore we give explicit formula for several classes of euler related sums in terms of zeta values and harmonic numbers and several examples are given the given representations are new

trace formulae and spectral statistics for discrete laplacians on regular graphs i following the derivation of the trace formulas in the first paper in this series we establish here a connection between the spectral statistics of random regular graphs and the predictions of random matrix theory rmt this follows from the known poisson distribution of cycle counts in regular graphs in the limit that the cycle periods are kept constant and the number of vertices increases indefinitely the result is analogous to the so called diagonal approximation in quantum chaos we also show that by assuming that the spectral correlations are given by rmt to all orders we can compute the leading deviations from the poisson distribution for cycle counts we provide numerical evidence which supports this conjecture

lyndon words and shuffle algebras for generating the coloured multiple zeta values relations tables we study here the coloured multiple zeta values obtained by extending the usual notion of the multiple zeta values by adding roots of unity we state their main combinatorial properties and we give as a result a maple algorithm which generates the relations table between these values

krichever novikov algebras on riemann surfaces of genus zero and one with n punctures bases of meromorphic vector fields on compact riemann surfaces of genus zero and one with an arbitrary number of punctures are given the structure constants are expressed in terms exclusively of algebraic constants generalized graded modules are also considered centrally extended algebras of vector fields as well as generalized heisenberg algebras are constructed the most simple examples are worked out in some detail

integrals of polylogarithmic functions recurrence relations and associated euler sums we show that integrals of the form 0 1 x m li p x li q x dx m 2 p q 1 and 0 1 log r x li p x li q x x dx p q r 1 satisfy certain recurrence relations which allow us to write them in terms of euler sums from this we prove that in the first case for all m p q and in the second case when p q r is even these integrals are reducible to zeta values in the case of odd p q r we combine the known results for euler sums with the information obtained from the problem in this form to give an estimate on the number of new constants which are needed to express the above integrals for a given weight p q r the proofs are constructive giving a method for the evaluation of these and other similar integrals and we present a selection of explicit evaluations in the last section

formulae concerning the computation of the clausen integral cl2 θ the main problem under study concerns the expression of the clausen integral cl2 in closed form in terms of known constants and special functions when is equal to a rational multiple of belonging to 0 2 a general formula giving cl2 p q in terms of the derivative of the di gamma function and the sine function is deduced from an appropriate fourier series expansion some variants of this formula are obtained in further sections the formulae expressing cl2 2 and more generally cl2 m m 2 3 4 as linear combinations of terms of the form cl2 const are established the various results are illustrated by means of typical examples of practical application the last section contains two simple approximations enabling the computation of cl for any in 0 with a relative error smaller than 0 63 and 0 003 resp the paper ends with an appendix in which among other things a peculiar trigonometric identity is established as a by product

the invariant of hyperbolic 3 manifolds in this paper we use the formula in the above theorem to study t m for hyperbolic 3 manifolds and calculate the right hand side of the formula explicitly in a special case our main task is to represent the integral of 1 by some more tractible ones in sect 1 we deal with general compact oriented riemannian manifolds m of dimension 3 let f m be the so 3 oriented frame bundle of m let l be a

evaluation of the multiple zeta values zeta 2 dots 2 3 2 dots 2 a formula is given for the special multiple zeta values occurring in the title as rational linear combinations of products m p 2n with m odd the existence of such a formula had been proved using motivic arguments by francis brown but the explicit formula more precisely certain 2 adic properties of its coefficients were needed for his proof in cite1 of the conjecture that all periods of mixed tate motives over z are q 2pi 1 linear combinations of multiple zeta values the formula is proved indirectly by computing the generating functions of both sides in closed form one as the product of a sine function and a 3 f 2 hypergeometric function and one as a sum of 14 products of sine functions and digamma functions and then showing that both are entire functions of exponential growth and that they agree at sufficiently many points to force their equality we also show that the space spanned by the multiple zeta values in question coincides with the space of double zeta values of odd weight and find a relation between this space and the space of cusp forms on the full modular group

bases on multipunctured riemann surfaces and interacting strings amplitudes the krichever novikov bases are studied on riemann surfaces with more than two punctures the bases are presented and the completness theorem is proven for the case of integer up to a common constant momenta then the interacting strings are considered the amplitudes and partition functions are obtained comparable with that of path integral approach for the amplitudes the simple geometric implication is proposed

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

new properties of multiple harmonic sums modulo p and p analogues of leshchiner s series in this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences 1 a c 1 b 2 a c 2 b and prove a number of congruences for these sums modulo a prime p the congruences obtained allow us to find nice p analogues of leshchiner s series for zeta values and to refine a result due to m hoffman and j zhao about the set of generators of the multiple harmonic sums of weight 7 and 9 modulo p moreover we are also able to provide a new proof of zagier s formula for zeta 2 a 3 2 b based on a finite identity for partial sums of the zeta star series

killing spinors on spheres and hyperbolic manifolds properties of killing spinors on spheres and hyperbolic manifolds are investigated with an emphasis on the relations to killing vectors conformal killing vectors and solutions of maxwell s equations

relations of multiple zeta values and their algebraic expression abstract we establish a new class of relations which we call the cyclic sum identities among the multiple zeta values k1 kl n1 nl 11 n1k1 nkkl these identities have an elementary proof and imply the sum theorem for multiple zeta values they also have a succinct statement in terms of cyclic derivations as introduced by rota sagan and stein in addition we discuss the expression of other relations of multiple zeta values via the shuffle and harmonic products on the underlying vector space h of the noncommutative polynomial ring q x y and also using an action of the hopf algebra of quasi symmetric functions on q x y

the 3 point virasoro algebra and its action on a fock space we define the 3 point virasoro algebra and construct a representation of it on a previously defined fock space for the 3 point affine algebra

some weighted bartholdi zeta function of a digraph related to the weighted bond scattering matrix of a graph we define some weighted bartholdi zeta function of a digraph d and give a determinant expression for it we present a decomposition formula for the weighted zeta function of a group covering of d furthermore we define an l function of d and give a determinant expression for it as a corollary we express the weighted bartholdi zeta function of a group covering of d by means of its l functions

new results concerning power series expansions of the riemann xi function and the li keiper constants the riemann hypothesis is equivalent to the li criterion governing a sequence of real constants that are certain logarithmic derivatives of the riemann xi function evaluated at unity a new represe

superconformal algebra of meromorphic vector fields with multipoles on a super riemann sphere the superconformal algebras of meromorphic vector fields with multipoles and the relevant abelian differential of the third kind on a super riemann sphere were constructed the algebra includes two ramond sectors as subalgebras

on some log cosine integrals related to ζ 3 ζ 4 and ζ 6 we evaluate a variety of log cosine integrals one of which has value a rational multiple of 4 while others evaluate to the form a 6 b 3 4 2 3 where a 0 is rational and b is 1 or 2 and is the riemann zeta function we show that these results may be obtained starting from a known tabulated result the derivation yields results for certain classes of logarithm trigonometric integrals and indicates how further analytic results on digamma series may be obtained

quasi periodic functions on the torus and sl n elliptic lie algebra we investigate properties of the sl n automorphic elliptic algebra e sl n we prove it to be z quasi graded lie algebra which could be viewed as a deformation of a graded loop algebra we show that it admits the decomposition into the direct sum of two subalgebras e sl n e sl n e sl n consistent with the described quasi grading we prove that e sl n e sl n i e lie algebras e sl n e sl n and e sl n constitute the manin triple we explicitly construct a central extension of e sl n we find its algebra of differentiations and its central extension which coincide with the quasi graded deformation of the virasoro algebra

closed form summation of some trigonometric series the problem of numerical evaluation of the classical trigonometric series formule where 1 in the case of s 2n and c 2n 1 with n 1 2 3 has been recently addressed by dempsey liu and dempsey boersma and dempsey and by gautschi we show that when is equal to a rational multiple of 2 these series can in the general case be summed in closed form in terms of known constants and special functions general formulae giving c and s in terms of the generalized riemann zeta function and the cosine and sine functions respectively are derived some simpler variants of these formulae are obtained and various special results are established

presentation of the constant r term krichever novikov type algebras for each constant r term krichever novikov type algebra a minimal set of defining generators is given from which all the operators of the algebra can be constructed by sequential definitions a finite system of polynomial conditions on the defining generators is given guaranteeing the commutation relations of the full algebra

the dilogarithm function for complex argument this paper summarizes the basic properties of the euler dilogarithm function often referred to as the spence function these include integral representations series expansions linear and quadrat

chebyshev coefficients for the clausen function cl 2 x abstract twenty digit chebyshev coefficients are given for the clausen function cl 2 x allowing a fast computation of this function for real values of the argument x

some special cases of the generalized hypergeometric function q 1 f q abstract some special cases of the generalized hypergeometric function q 1 f q with rational numbers as parameters are given in tabular form these results complement existing tables some analytical aspects are discussed and a derivation is given for those cases which correct existing table entries or replace numerical values by analytic expressions

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

some evaluation of parametric euler sums abstract in this paper by using the method of contour integral representations and the theorem of residues and integral representations of series we discuss the analytic representations of parametric euler sums that involve harmonic numbers through zeta values and rational function series either linearly or nonlinearly furthermore we give explicit formulae for several parametric quadratic and cubic sums in terms of zeta values and rational series moreover some interesting new consequences and illustrative examples are considered

on a lie algebra of vector fields on a complex torus this paper discusses a modification of the krichever novikov lie algebra of vector fields on the torus in which the two specified points where singularities can occur are points of order 1 and 2 rather than points of infinite order

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

on the clausen integral cl2 θ and a related integral abstract in this article the clausen integral cl 2 0 ln 2 sin t 2 d t whereby is equal to a rational multiple of belonging to 0 2 is developed into a finite series of derivatives of the di gamma function for the same values of a method is presented to calculate the related integral 1 0 ln 2 2 cos 1 d

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

four parametric linear euler sums abstract in this paper firstly we aim to investigate analytic continuations of altogether four types of parametric linear euler sums by using the euler maclaurin summation formula and the euler boole summation formula secondly we show how nicely three shuffle relations among the four parametric linear euler sums can be derived by using an l summing formula due to hassani and rahimpour thirdly using one of the shuffle relations we present certain series representations of product of riemann zeta functions some special cases and relevant connections of the results presented here with those involving known identities are also demonstrated and indicated

global geometric deformations of current algebras as krichever novikov type algebras we construct algebraic geometric families of genus one i e elliptic current and affine lie algebras of krichever novikov type these families deform the classical current respectively affine kac moody lie algebras the construction is induced by the geometric process of degenerating the elliptic curve to singular cubics if the finite dimensional lie algebra defining the infinite dimensional current algebra is simple then even if restricted to local families the constructed families are non equivalent to the trivial family in particular we show that the current algebra is geometrically not rigid despite its formal rigidity this shows that in the infinite dimensional lie algebra case the relations between geometric deformations formal deformations and lie algebra two cohomology are not that close as in the finite dimensional case the constructed families are e g of relevance in the global operator approach to the wess zumino witten novikov models appearing in the quantization of conformal field theory the algebras are explicitly given by generators and structure equations and yield new examples of infinite dimensional algebras of current and affine lie algebra type

some evaluation of cubic euler sums abstract p flajolet and b salvy 15 prove the famous theorem that a nonlinear euler sum s i 1 i 2 i r q reduces to a combination of sums of lower orders whenever the weight i 1 i 2 i r q and the order r are of the same parity in this article we develop an approach to evaluate the cubic sums s 1 2 m p and s 1 l 1 l 2 l 3 by using the approach we establish some relations involving cubic quadratic and linear euler sums specially we prove the cubic sums s 1 2 m m and s 1 2 l 1 2 2 l 1 are reducible to zeta values quadratic and linear sums moreover we prove that the two combined sums involving multiple zeta values of depth four i j 1 2 i j m i m j 1 1 and i j k 1 2 3 i j k m i m j m k 1 can be expressed in terms of multiple zeta values of depth 3 here 2 m 1 m 2 m 3 n finally we evaluate the alternating cubic euler sums s 1 3 2 r 1 and show that it are reducible to alternating quadratic and linear euler sums the approach is based on tornheim type series computations

l2 torsion invariants we discuss a generalisation of reidemeister franz torsion which applies to infinite dimensional representations of the fundamental group by utilising the theory of finite von neumann algebras our results apply to a class of closed oriented manifolds which are l2 acyclic in the sense of having trivial l2 cohomology evidence is presented for equality of the essentially combinatorial torsion introduced here with an analytic l2 ray singer torsion defined by the second named author