an algebraic proof of the cyclic sum formula for multiple zeta values abstract we introduce an algebraic formulation of the cyclic sum formulas for multiple zeta values and for multiple zeta star values we also present an algebraic proof of cyclic sum formulas for multiple zeta values and for multiple zeta star values by reducing them to kawashima s relation

conical zeta values and their double subdivision relations abstract we introduce the concept of a conical zeta value as a geometric generalization of a multiple zeta value in the context of convex cones the quasi shuffle and shuffle relations of multiple zeta values are generalized to open cone subdivision and closed cone subdivision relations respectively for conical zeta values in order to achieve the closed cone subdivision relation we also interpret linear relations among fractions as subdivisions of decorated closed cones as a generalization of the double shuffle relation of multiple zeta values we give the double subdivision relation of conical zeta values and formulate the extended double subdivision relation conjecture for conical zeta values

on efficient weighted newton methods for solving systems of nonlinear equations in this study we present iterative methods of convergence order four and six for solving systems of nonlinear equations the fourth order scheme is composed of two steps namely newton iteration as the first step and weighted newton iteration as the second step the sixth order scheme is composed of three steps the first two steps are same as that of fourth order scheme whereas the third step is again based on weighted newton iteration computational efficiency in its general form is discussed comparison between the efficiencies of proposed techniques and existing techniques is made it is proved that for large systems the new methods are more efficient numerical tests are performed which confirm the theoretical results moreover theoretical order of convergence is verified in the examples

quasilinearization method and its verification on exactly solvable models in quantum mechanics the proof of the convergence of the quasilinearization method of bellman and kalaba whose origin lies in the theory of linear programming is extended to large and infinite domains and to singular functionals in order to enable the application of the method to physical problems this powerful method approximates solution of nonlinear differential equations by treating the nonlinear terms as a perturbation about the linear ones and is not based unlike perturbation theories on existence of some kind of small parameter the general properties of the method particularly its uniform and quadratic convergence which often also is monotonic are analyzed and verified on exactly solvable models in quantum mechanics namely application of the method to scattering length calculations in the variable phase method shows that each approximation of the method sums many orders of the perturbation theory and that the method reproduces properly the singular structure of the exact solutions the method provides final and reasonable answers for infinite values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist

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

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

kawashima s relations for interpolated multiple zeta values abstract recently yamamoto introduced polynomials in one variable t which interpolates multiple zeta and zeta star values t mzvs for short provided new prospects on two one conjecture of ohno and zudilin and proved the cyclic sum formula for t mzvs in this paper we establish a generalization of kawashima s relations t kawashima relations for t mzvs we prove the cyclic sum formula for t mzvs using a type of derivation operator together with the t kawashima relations

an efficient multi step iterative method for computing the numerical solution of systems of nonlinear equations associated with odes an efficient multi step iterative method for computing the numerical solution of systems of nonlinear equations associated with odes

multivariable hoffman ihara operators and the operad of formal power series abstract we introduce a family of multilinear operators on a quasi shuffle algebra h which we call hoffman ihara operators it generalizes the linear operators constructed by hoffman and further studied by hoffman and ihara we prove the formula on the composition of these operators and show that this formula provides a unified and transparent understanding of some results on shuffle and quasi shuffle products an interpretation as an action of the operad of formal power series is also given

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

higher order multi step jarratt like method for solving systems of nonlinear equations this paper proposes a multi step iterative method for solving systems of nonlinear equations with a local convergence order of 3m4 where m 2 is the number of steps the multi step iterative method includes two parts the base method and the multi step part the base method involves two function evaluations two jacobian evaluations one lu decomposition of a jacobian and two matrixvector multiplications every stage of the multi step part involves the solution of two triangular linear systems and one matrixvector multiplication the computational efficiency of the new method is better than those of previously proposed methods the method is applied to several nonlinear problems resulting from discretizing nonlinear ordinary differential equations and nonlinear partial differential equations

multiple q zeta values we introduce a q analog of the multiple harmonic series commonly referred to as multiple zeta values the multiple q zeta values satisfy a q stuffle multiplica tion rule analogous to the stuffle multiplication rule arising from the series representation of ordinary multiple zeta values additionally multiple q zeta values can be viewed as special values of the multiple q polylogarithm which admits a multiple jackson q integral representation whose limiting case is the drinfel d simplex integral for the ordinary multiple polylogarithm when q 1 the multiple jackson q integral representation for multiple q zeta values leads to a second multiplication rule satisfied by them referred to as a q shuffle despite this it appears that many numerical relations satisfied by ordinary multiple zeta values have no interesting q extension for example a suitable q analog of broadhurst s formula for 3 1 n if one exists is likely to be rather complicated nevertheless we show that a number of infinite classes of relations inc luding hoffman s partition identities ohno s cyclic sum identities granville s sum formula euler s convolution formula ohno s generalized duality relation and the derivation relations of ihara and kaneko extend to multiple q zeta values

double shuffle relation for associators it is proved that drinfel 0 d s pentagon equation implies the generalized double shue relation as a corollary an embedding from the grothen

special geometry cubic polynomials and homogeneous quaternionic spaces the existing classification of homogeneous quaternionic spaces is not complete we study these spaces in the context of certainn 2 supergravity theories where dimensional reduction induces a mapping betweenspecial real kahler and quaternionic spaces the geometry of the real spaces is encoded in cubic polynomials those of the kahler and quaternionic manifolds in homogeneous holomorphic functions of second degree we classify all cubic polynomials that have an invariance group that acts transitively on the real manifold the corresponding kahler and quaternionic manifolds are then homogeneous we find that they lead to a well defined subset of the normal quaternionic spaces classified by alekseevskii and the corresponding special kahler spaces given by cecotti but there is a new class of rank 3 spaces of quaternionic dimension larger than 3 we also point out that some of the rank 4 alekseevskii spaces were not fully specified and correspond to a finite variety of inequivalent spaces a simpler version of the equation that underlies the classification of this paper also emerges in the context ofw3 algebras

generalized log sine integrals and the mordell tornheim zeta values we introduce certain integrals of a product of the bernoulli polynomials and logarithms of milnor s multiple sine functions it is shown that all the integrals are expressed by the mordell tornheim zeta values at positive integers and that the converse is also true moreover we apply the theory of the integral to obtain various new results for the mordell tornheim zeta values

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

the depth structure of motivic multiple zeta values in this paper we construct some maps related to the motivic galois action on depth graded motivic multiple zeta values from these maps we give some short exact sequences about depth graded motivic multiple zeta values in depth two and three in higher depth we conjecture that there are exact sequences of the same type we will show from three conjectures about depth graded motivic lie algebra we can nearly deduce the exact sequences conjectures in higher depth at last we give a new proof of the result that the modulo zeta mathfrak m 2 version motivic double zeta values are generated by the totally odd part we reduce the well known conjecture that the modulo zeta mathfrak m 2 version motivic triple zeta values are generated by the totally odd part to an isomorphism conjecture in linear algebra

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

algebraic differential formulas for the shuffle stuffle and duality relations of iterated integrals abstract in this paper we prove certain algebraic identities which correspond to differentiation of the shuffle relation the stuffle relation and the relations which arise from mobius transformations of iterated integrals these formulas provide fundamental and useful tools in the study of iterated integrals on a punctured projective line

the weil petersson geometry of the moduli space of su n 3 calabi yau manifolds i the weil petersson metric is defined on the moduli space of calabi yau manifolds the curvature of this weil petersson metrics is computed and its potential is explicitely defined it is proved that the moduli space of calabi yau manifolds is unobstructed see tian

on hyper kahler manifolds associated to lagrangian kahler submanifolds of t ast mathbb c n for any lagrangian kahler submanifold m c t cn there exists a canonical hyper kahler metric on t m a kaihler potential for this metric is given by the generalized calabi ansatz of the theoretical physicists cecotti ferrara and girardello this correspondence provides a method for the construction of pseudo hyper kahler manifolds with large automorphism group using it an interesting class of pseudo hyper kaihler manifolds of complex signature 2 2n is constructed for any manifold n in this class a group of automorphisms with a codimension one orbit on n is specified finally it is shown that the bundle of intermediate jacobians over the moduli space of gauged calabi yau 3 folds admits a natural pseudo hyper kahler mnetric of complex signature 2 2n

an iterative finite difference method for solving bratu s problem in this paper we propose a new iterative finite difference ifd scheme based on the newton raphson kantorovich approximation method in function space to solve the classical one dimensional bratu s problem this new numerical method produces accurate solutions with low computational cost the effectiveness and accuracy of the ifd method are confirmed through several numerical examples and compared to some existing numerical solvers

shuffle relations for regularised integrals of symbols we prove shuffle relations which relate a product of regularised integrals of classical symbols int reg sigma i d xi i i 1 ldots k to regularised nested iterated integrals prod i 1 k int reg sigma i d xi i sum tau in sigma k int reg d xi 1 int vert xi 2 vert leq vert xi 1 vert d xi 2 cdots int vert xi l vert leq vert xi k 1 vert d xi k otimes i 1 k sigma tau i where k is the group of permutations over k elements we show that these shuffle relations hold if all the symbols i have vanishing residue this is true of non integer order symbols on which the regularised integrals have all the expected properties such as stokes property mmp in general the shuffle relations hold up to finite parts of corrective terms arising from a renormalisation on tensor products of classical symbols a procedure adapted from renormalisation methods to compute feynman diagrams familiar to physicists we relate the shuffle relations for regularised integrals of symbols with shuffle relations for multiple zeta functions adapting the above constructions to the case of a symbol on the unit circle

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

double shuffle relations for arborified zeta values abstract arborified zeta values are defined as iterated series and integrals using the universal properties of rooted trees this approach allows to study their convergence domain and to relate them to multiple zeta values generalisations to rooted trees of the stuffle and shuffle products are defined and studied it is further shown that arborified zeta values are algebra morphisms for these new products on trees

multiple zeta values vs multiple zeta star values abstract we discuss an algebraic connection between two kinds of multiple zeta values or their q analogues the q multiple zeta values and the q multiple zeta star values these two classes of values generate the same algebra but in this paper we show that the translation map between these two classes has a quite interesting algebraic property in a general setting for example the compatibility with the harmonic product we also study several applications of the result

quantum quasi shuffle algebras ii we establish some properties of quantum quasi shuffle algebras they include the necessary and sufficient condition for the construction of the quantum quasi shuffle product the universal property and the commutativity condition as an application we use the quantum quasi shuffle product to construct a linear basis of t v for a special kind of yang baxter algebras v m

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

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

p adic multiple zeta values i our main aim in this paper is to give a foundation of the theory of p adic multiple zeta values we introduce one variable p adic multiple polylogarithms by coleman s p adic iterated integration theory we define p adic multiple zeta values to be special values of p adic multiple polylogarithms we consider the formal p adic kz equation and introduce the p adic drinfel d associator by using certain two fundamental solutions of the p adic kz equation we show that our p adic multiple polylogarithms appear as coefficients of a certain fundamental solution of the p adic kz equation and our p adic multiple zeta values appear as coefficients of the p adic drinfel d associator we show various properties of p adic multiple zeta values which are sometimes analogous to the complex case and are sometimes peculiar to the p adic case via the p adic kz equation

multiple series connected to hoffman s conjecture on multiple zeta values abstract recent results of zlobin and cresson fischler rivoal allow one to decompose any suitable p uple series of hypergeometric type into a linear combination over the rationals of multiple zeta values of depth at most p in some cases only the multiple zeta values with 2 s and 3 s are involved as in hoffman s conjecture in this text we study the depth p part of this linear combination namely the contribution of the multiple zeta values of depth exactly p we prove that it satisfies some symmetry property as soon as the p uple series does and make some conjectures on the depth p 1 part of the linear combination when p 3 our result generalizes the property that very well poised univariate hypergeometric series involve only zeta values of a given parity which is crucial in the proof by rivoal and ball rivoal that 2 n 1 is irrational for infinitely many n 1 the main feature of the proof is an algebraic approach based on representations of z 2 z p s p

duality between quasi symmetrical functions and the solomon descent algebra abstract the ring qsym of quasi symmetric functions is naturally the dual of the solomon descent algebra the product and the two coproducts of the first extending those of the symmetric functions correspond to a coproduct and two products of the second which are defined by restriction from the symmetric group algebra a consequence is that qsym is a free commutative algebra

shuffle products for multiple zeta values and partial fraction decompositions of zeta functions of root systems the shuffle product plays an important role in the study of multiple zeta values mzvs this is expressed in terms of multiple integrals and also as a product in a certain non commutative polynomial algebra over the rationals in two indeterminates in this paper we give a new interpretation of the shuffle product in fact we prove that the procedure of shuffle products essentially coincides with that of partial fraction decompositions of mzvs of root systems as an application we give a proof of extended double shuffle relations without using drinfel d integral expressions for mzvs furthermore our argument enables us to give some functional relations which include double shuffle relations

on certain kahler quotients of quaternionic kahler manifolds we prove that given a certain isometric action of a two dimensional abelian group a on a quaternionic kahler manifold m which preserves a submanifold n m the quotient m n a has a natural kahler structure we verify that the assumptions on the group action and on the submanifold n m are satisfied for a large class of examples obtained from the supergravity c map in particular we find that all quaternionic kahler manifolds m in the image of the c map admit an integrable complex structure compatible with the quaternionic structure such that n m is a complex submanifold finally we discuss how the existence of the kahler structure on m is required by the consistency of spontaneous mathcal n 2 to mathcal n 1 supersymmetry breaking

duality and q multiple zeta values abstract following bachmann s recent work on bi brackets and multiple eisenstein series zudilin introduced the notion of multiple q zeta brackets which provides a q analog of multiple zeta values possessing both shuffle as well as quasi shuffle relations the corresponding products are related in terms of duality in this work we study zudilin s duality construction in the context of classical multiple zeta values as well as various q analogs of multiple zeta values regarding the former we identify the derivation relation of order two with a hoffman ohno type relation then we describe relations between the ohno okuda zudilin q multiple zeta values and the schlesinger zudilin q multiple zeta values

an efficient fifth order method for solving systems of nonlinear equations in this paper we present a three step iterative method of convergence order five for solving systems of nonlinear equations the methodology is based on the two step homeier s method with cubic convergence homeier 2004 computational efficiency in its general form is discussed and a comparison between the efficiency of proposed technique and existing ones is made the performance is tested through numerical examples moreover theoretical results concerning order of convergence and computational efficiency are verified in the examples it is shown that the present method has an edge over existing methods particularly when applied to large systems of equations

frobenius manifolds projective special geometry and hitchin systems we consider the construction of frobenius manifolds associated to projective special geometry and analyse the dependence on choices involved in particular we prove that the underlying f manifold is canonical we then apply this construction to integrable systems of hitchin type

solving systems of nonlinear equations when the nonlinearity is expensive construction of multi step iterative method for solving system of nonlinear equations is considered when the nonlinearity is expensive the proposed method is divided into a base method and multi step part the convergence order of the base method is five and each step of multi step part adds additive factor of five in the convergence order of the base method the general formula of convergence order is 5 m 2 where m 3 is the step number for a single instance of the iterative method we only compute two jacobian and inversion of one jacobian is required the direct inversion of jacobian is avoided by computing lu factors the computed lu factors are used in the multi step part for solving five systems of linear equations that make the method computational efficient the distinctive feature of the underlying multi step iterative method is the single call to the computationally expensive nonlinear function and thus offers an increment of additive factor of five in the convergence order per single call the numerical simulations reveal that our proposed iterative method clearly shows better performance where the computational cost of the involved nonlinear function is higher than the computational cost for solving five lower and upper triangular systems

renormalization of multiple zeta values abstract multiple zeta values mzvs in the usual sense are the special values of multiple variable zeta functions at positive integers their extensive studies are important in both mathematics and physics with broad connections and applications in contrast very little is known about the special values of multiple zeta functions at non positive integers since the values are usually undefined we define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory and following the hopf algebra approach of connes and kreimer this definition of renormalized mzvs agrees with the convergent mzvs and extends the work of ihara kaneko zagier on renormalization of mzvs with positive arguments we further show that the important quasi shuffle stuffle relation for usual mzvs remains true for the renormalized mzvs

homogeneous kahler manifolds and t algebras in n 2 supergravity and superstrings motivated by the problem of the moduli space of superconformal theories we classify all the normal homogeneous kahler spaces which are allowed in the coupling of vector multiplets ton 2 sugra such homogeneous spaces are in one to one correspondence with the homogeneous quaternionic spaces h n found by alekseevskii there are two infinite families of homogeneous non symmetric spaces each labelled by two integers we construct explicitly the corresponding supergravity models they are described by acubic functionf as in flat potential models they are kahler einstein if and only if they are symmetric we describe in detail the geometry of the relevant manifolds they are siegel bounded domains of the first type we discuss the physical relevance of this class of bounded domains for string theory and the moduli geometry finally we introduce thet algebraic formalism of vinberg to describe in an efficient way the geometry of these manifolds the homogeneous spaces allowed inn 2 sugra are associated to rank 3t algebras in exactly the same way as the symmetric spaces are related to jordan algebras we characterize thet algebras allowed inn 2 supergravity they are those for which theungraded determinant is a polynomial in the matrix entries the kahler potential is simply minus the logarithm of this naive determinant

the algebra of multitangent functions abstract multizeta values are numbers appearing in many different contexts unfortunately their arithmetics remains mostly out of reach in this article we define a functional analogue of the algebra of multizeta values namely the algebra of multitangent functions which are 1 periodic functions defined by a process formally similar to multizeta values we introduce here the fundamental notions of reduction into monotangent functions projection onto multitangent functions and that of trifactorization giving a way of writing a multitangent function in terms of hurwitz multizeta functions this explains why the multitangent algebra is a functional analogue of the algebra of multizeta values we then discuss the most important algebraic and analytic properties of these functions and their consequences on multizeta values as well as their regularization in the divergent case each property of multitangents has a pendant on the side of multizeta values this allows us to propose new conjectures which have been checked up to the weight 18

completeness in supergravity constructions we prove that the supergravity r and c maps preserve completeness as a consequence any component mathcal h of a hypersurface h 1 defined by a homogeneous cubic polynomial h such that partial 2h is a complete riemannian metric on mathcal h defines a complete projective special kahler manifold and any complete projective special kahler manifold defines a complete quaternionic kahler manifold of negative scalar curvature we classify all complete quaternionic kahler manifolds of dimension less or equal to 12 which are obtained in this way and describe some complete examples in 16 dimensions

mixed tate motives over z we prove that the category of mixed tate motives over z is spanned by the motivic fundamental group of p 1 minus three points we prove a conjecture by m hoffman which states that every multiple zeta value is a q linear combination of n 1 n r where n i 2 3

explicit double shuffle relations and a generalization of eulerʼs decomposition formula abstract we give an explicit formula for the shuffle relation in a general double shuffle framework that specializes to double shuffle relations of multiple zeta values and multiple polylogarithms as an application we generalize the well known decomposition formula of euler that expresses the product of two riemann zeta values as a sum of double zeta values to a formula that expresses the product of two multiple polylogarithm values as a sum of other multiple polylogarithm values