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

assumed strain stabilization of the eight node hexahedral element an accurate and robust stabilization based on the assumed strain method and an operator orthogonal to constant strain fields is presented for an eight node hexahedral element with uniform reduced integration the stabilization forces depend only on the element geometry and material properties user specified parameters are not needed yet the assumed strain stabilization is nearly as fast as perturbation type hourglass control in an explicit code it has excellent coarse mesh accuracy for linear elastic bending and is easily incorporated into a nonlinear program the assumed strain field is also used with four point integration which does not require stabilization in addition two forms of the b matrix are studied and it is shown that the mean form is more efficient since it passes the patch test in a simplified form

modules of the 0 hecke algebra and quasisymmetric schur functions abstract we begin by deriving an action of the 0 hecke algebra on standard reverse composition tableaux and use it to discover 0 hecke modules whose quasisymmetric characteristics are the natural refinements of schur functions known as quasisymmetric schur functions furthermore we classify combinatorially which of these 0 hecke modules are indecomposable from here we establish that the natural equivalence relation arising from our 0 hecke action has equivalence classes that are isomorphic to subintervals of the weak bruhat order on the symmetric group focussing on the equivalence classes containing a canonical tableau we discover a new basis for the hopf algebra of quasisymmetric functions and use the cardinality of these equivalence classes to establish new enumerative results on truncated shifted reverse tableau studied by panova and adin king roichman generalizing our 0 hecke action to one on skew standard reverse composition tableaux we derive 0 hecke modules whose quasisymmetric characteristics are the skew quasisymmetric schur functions of bessenrodt et al this enables us to prove a restriction rule that reflects the coproduct formula for quasisymmetric schur functions which in turn yields a quasisymmetric branching rule analogous to the classical branching rule for schur functions

on posets and hopf algebras abstract we generalize the notion of the rank generating function of a graded poset namely by enumerating different chains in a poset we can assign a quasi symmetric function to the poset this map is a hopf algebra homomorphism between the reduced incidence hopf algebra of posets and the hopf algebra of quasi symmetric functions this work implies that the zeta polynomial of a poset may be viewed in terms hopf algebras in the last sections of the paper we generalize the reduced incidence hopf algebra of posets to the hopf algebra of hierarchical simplicial complexes

weak quasi symmetric functions rota baxter algebras and hopf algebras abstract we introduce the hopf algebra of quasi symmetric functions with semigroup exponents generalizing the hopf algebra qsym of quasi symmetric functions as a special case we obtain the hopf algebra qsym n of weak quasi symmetric functions which provides a framework for the study of a question proposed by g c rota relating symmetric type functions and rota baxter algebras we provide the transformation formulas between the weak monomial and fundamental quasi symmetric functions which extends the corresponding results for quasi symmetric functions moreover we show that qsym is a hopf subalgebra and a hopf quotient algebra of qsym n rota s question is addressed by identifying qsym n with the free commutative unitary rota baxter algebra x of weight 1 on one generator x which also allows us to equip x with a hopf algebra structure

eulerian quasisymmetric functions we introduce a family of quasisymmetric functions called eulerian quasisymmetric functions which specialize to enu merators for the joint distribution of the permutation statistics major index and excedance number on permutations of fixed cycle type this family is analogous to a family of quasisymmetric func tions that gessel and reutenauer used to study the joint distribu tion of major index and descent number on permutations of fixed cycle type our central result is a formula for the generating func tion for the eulerian quasisymmetric functions which specializes to a new and surprising q analog of a classical formula of euler for the exponential generating function of the eulerian polynomials this q analog computes the joint distribution of excedance number and major index the only of the four important euler mahonian distributions that had not yet been computed our study of the eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of gessel and reutenauer we also obtain q analogs q p analogs and quasisymmetric func tion analogs of classical results on the symmetry and unimodality of the eulerian polynomials our eulerian quasisymmetric functions refine symmetric functions that have occurred in various represen tation theoretic and enumerative contexts including macmahon s study of multiset derangements work of procesi and stanley on toric varieties of coxeter complexes stanley s work on chromatic symmetric functions and the work of the authors on the homology of a certain poset introduced by bjorner and welker

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

combinatorial models for coalgebraic structures abstract we introduce a convenient category of combinatorial objects known as cell sets on which we study the properties of the appropriate free abelian group functor we obtain a versatile generalization of the notion of incidence coalgebra giving rise to an abundance of coalgebras hopf algebras and comodules all of whose structure constants are positive integers with respect to certain preferred bases our category unifies and extends existing constructions in algebraic combinatorics providing proper functorial descriptions it is inspired in part by the notion of cw complex and is also geared to future applications in algebraic topology and the theory of formal group laws

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

fifteen node tetrahedral elements for explicit methods in nonlinear solid dynamics abstract despite the ease in meshing and benefits for modeling flexure curved shapes etc second order tetrahedral elements are not contained in typical explicit solid dynamic programs this is primarily due to the lack of both a satisfactory consistent nodal loading distribution and mass lumping technique row summation lumping for instance produces negative vertex node masses for the popular ten node serendipity tetrahedron which also has zero vertex node loads resulting from a constant pressure on an element face this has led to piecewise composites of four node tetrahedrons to represent a ten node one in explicit codes in this paper truly second order fifteen node formulations for compressible and for nearly incompressible materials are presented and evaluated in addition to producing all positive nodal loads from a uniform traction row summation mass lumping for the fifteen node element is shown to produce all positive nodal masses performance is assessed in standard benchmark problems and practical applications using various elastic and elastic plastic material models and involving very large strains deformations severe distortions and contact impact comparisons are also made with several first order elements and second order hexahedral formulations the offered elements performed satisfactorily in all examples as recently found for second order hexahedral elements it is shown that the inclusion of face and centroidal nodes is vital for robust performance with row summation lumping and high order quadrature rules are crucial with explicit methods these second order elements are shown to be viable for practical applications especially using today s parallel computers whereas the reliable performance is generally attained at significant computational expense compared with first order and brick types these elements can be more computationally competitive in flexure and have the desirable trait that they are amenable to automatic tetrahedral meshing software

peak quasisymmetric functions and eulerian enumeration abstract via duality of hopf algebras there is a direct association between peak quasisymmetric functions and enumeration of chains in eulerian posets we study this association explicitly showing that the notion of cd index long studied in the context of convex polytopes and eulerian posets arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by stembridge thus eulerian posets having a nonnegative cd index for example face lattices of convex polytopes correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis we diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions and study the g polynomial for eulerian posets as an algebra homomorphism

galois connections for incidence hopf algebras of partially ordered sets abstract an important well known result of rota describes the relationship between the mobius functions of two posets related by a galois connection we present an analogous result relating the antipodes of the corresponding incidence hopf algebras from which the classical formula can be deduced to motivate the derivation of this more general result we first observe that a simple conceptual proof of rota s classical formula can be obtained by interpreting it in terms of bimodules over the incidence algebras bimodules correct the apparent lack of functoriality of incidence algebras with respect to monotone maps the theory of incidence hopf algebras is reviewed from scratch and centered around the notion of cartesian posets also the universal multiplicative function on a poset is constructed and an analog for antipodes of the classical mobius inversion formula is presented

cyclic sum formula for multiple l values abstract the cyclic sum formula for multiple l values which can be viewed as a generalization of the cyclic sum formula for multiple zeta values proved by hoffman and ohno or ohno and wakabayashi is shown an algebraic formulation of the cyclic sum formula is also presented

some counting questions for matrices with restricted entries we obtain upper bounds for the number of arbitrary and symmetric matrices with integer entries in a given box in an arbitrary location and a given determinant we then apply these bounds to estimate the number of matrices in such boxes which have an integer eigenvalues finally we outline some open questions

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

a least squares approach for uniform strain triangular and tetrahedral finite elements a least squares approach is presented for implementing uniform strain triangular and tetrahedral finite elements the basis for the method is a weighted least squares formulation in which a linear displacement field is fit to an element s nodal displacements by including a greater number of nodes on the element boundary than is required to define the linear displacement field it is possible to eliminate volumetric locking common to fully integrated lower order elements such results can also be obtained using selective or reduced integration schemes but the present approach is fundamentally different from those the method is computationally efficient and can be used to distribute surface loads on an element edge or face in a continuously varying manner between vertex mid edge and mid face nodes example problems in two and three dimensional linear elasticity are presented element types considered in the examples include a six node triangle eight node tetrahedron and ten node tetrahedron 1998 john wiley sons ltd

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

large deformation triangular and tetrahedral element formulations for unstructured meshes abstract the increasing use of automatic mesh generators and remeshers has spurred the need for accurate and efficient triangular and tetrahedral elements in unstructured meshes in large deformation analysis of solids accuracy and efficiency require the treatment of volumetric locking and element distortions respectively in this paper a mixed formulations is proposed to treat these issues and applied to triangular and tetrahedral elements the formulations enforces in the weak sense the expression for the volumetric strain in terms of the deformation gradient the mixed formulation is combined with an irreducible formulation to provide a balance between the treatment of volumetric locking and the maintenance of an adequate critical timestep three techniques are proposed to determine such a balance at the local element level example computations in two and three dimensions of an iron cylinder impacting a rigid surface are used to demonstrate the formulations and to compare them to the irreducible displacement based formulation

skew quasisymmetric schur functions and noncommutative schur functions abstract recently a new basis for the hopf algebra of quasisymmetric functions qsym called quasisymmetric schur functions has been introduced by haglund luoto mason van willigenburg in this paper we extend the definition of quasisymmetric schur functions to introduce skew quasisymmetric schur functions these functions include both classical skew schur functions and quasisymmetric schur functions as examples and give rise to a new poset l c that is analogous to young s lattice we also introduce a new basis for the hopf algebra of noncommutative symmetric functions nsym this basis of nsym is dual to the basis of quasisymmetric schur functions and its elements are the pre image of the schur functions under the forgetful map nsym sym we prove that the multiplicative structure constants of the noncommutative schur functions equivalently the coefficients of the skew quasisymmetric schur functions when expanded in the quasisymmetric schur basis are nonnegative integers satisfying a littlewood richardson rule analogue that reduces to the classical littlewood richardson rule under as an application we show that the morphism of algebras from the algebra of poirier reutenauer to sym factors through nsym we also extend the definition of schur functions in noncommuting variables of rosas sagan in the algebra ncsym to define quasisymmetric schur functions in the algebra ncqsym we prove these latter functions refine the former and their properties and project onto quasisymmetric schur functions under the forgetful map lastly we show that by suitably labeling l c skew quasisymmetric schur functions arise in the theory of pieri operators on posets

a stabilized nodally integrated tetrahedral a stabilized nodally integrated linear tetrahedral is formulated and analysed it is well known that linear tetrahedral elements perform poorly in problems with plasticity nearly incompressible materials and acute bending for a variety of reasons low order tetrahedral elements are preferable to quadratic tetrahedral elements particularly for nonlinear problems but the severe locking problems of tetrahedrals have forced analysts to employ hexahedral formulations for most nonlinear problems on the other hand automatic mesh generation is often not feasible for building many 3d hexahedral meshes a stabilized nodally integrated linear tetrahedral is developed and shown to perform very well in problems with plasticity nearly incompressible materials and acute bending the formulation is analytically and numerically shown to be stable and optimally convergent for the compressible case provided sufficient smoothness of the exact solution u c2 h1 3 future work may extend the formulation to the incompressible regime and relax the regularity requirements nonetheless the results demonstrate that the method is not susceptible to locking and performs quite well in several standard linear and nonlinear benchmarks published in 2006 by john wiley sons ltd

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

random walks on quasisymmetric functions abstract conditions are provided under which an endomorphism on quasisymmetric functions gives rise to a left random walk on the descent algebra which is also a lumping of a left random walk on permutations spectral results are also obtained several important random walks are now realized this way stanley s qs distribution results from endomorphisms given by evaluation maps a shuffles result from the a th convolution power of the universal character and the tchebyshev operator of the second kind introduced recently by ehrenborg and readdy yields traditional riffle shuffles a conjecture of ehrenborg regarding the spectra for a family of random walks on ab words is proven a theorem of stembridge from the theory of enriched p partitions is also recovered as a special case

kondo effect in single atom contacts the importance of the atomic geometry co single atom junctions on copper surfaces are studied by scanning tunneling microscopy and ab initio calculations the kondo temperature of single cobalt atoms on the cu 111 surface has been measured at various tip sample distances ranging from tunneling to the point contact regime the experiments show a constant kondo temperature for a whole range of tip substrate distances consistently with the predicted energy position of the spin polarized d levels of co this is in striking difference to experiments on co cu 100 junctions where a substantial increase of the kondo temperature has been found our calculations reveal that the different behavior of the co adatoms on the two cu surfaces originates from the interplay between the structural relaxations and the electronic properties in the near contact regime

the hopf algebra approach to feynman diagram calculations the hopf algebra structure underlying feynman diagrams which governs the process of renormalization in perturbative quantum field theory is reviewed recent progress is briefly summarized with an emphasis on further directions of research

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

structure of the malvenuto reutenauer hopf algebra of permutations we analyze the structure of the malvenuto reutenauer hopf algebra ssym of permutations in detail we give explicit formulas for its antipode prove that it is a cofree coalgebra determine its primitive elements and its coradical filtration and show that it decomposes as a crossed product over the hopf algebra of quasi symmetric functions in addition we describe the structure constants of the multiplication as a certain number of facets of the permutahedron as a consequence we obtain a new interpretation of the product of monomial quasi symmetric functions in terms of the facial structure of the cube the hopf algebra of malvenuto and reutenauer has a linear basis indexed by permutations our results are obtained from a combinatorial description of the hopf algebraic structure with respect to a new basis for this algebra related to the original one via mobius inversion on the weak order on the symmetric groups this is in analogy with the relationship between the monomial and funda mental bases of the algebra of quasi symmetric functions our results reveal a close relationship between the structure of the malvenuto reutenauer hopf algebra and the weak order on the symmetric groups

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

the use of the local truncation error to improve arbitrary order finite elements for the linear wave and heat equations abstract the local truncation error in space and time can be efficiently used for the analysis and the increase in accuracy of the linear and high order finite elements in the 1 d 2 d and 3 d cases on uniform and non uniform meshes several applications of the local truncation error are considered in the paper it is proven that for the 1 d wave equation with a piece wise constant wave velocity the local truncation error is zero if the linear finite elements with the element size proportional to the wave velocity the lumped mass matrix and the central difference method with the time increments equal to the stability limit are used it is shown in the 1 d and multidimensional cases that the optimal lumped mass matrix can be calculated by the minimization of the order of the local truncation error and yields the maximum possible order of accuracy the minimization of the order of the local truncation error allows us to develop the linear finite elements and the isogeometric high order elements with improved accuracy i e accuracy is improved from order 2 p the conventional elements to order 4 p the new elements where p is the order of the polynomial approximations new high order boundary conditions are developed in order to keep a high order accuracy of the developed technique the new elements can be equally applied to linear wave propagation and heat transfer problems it is also shown that non uniform meshes may lead to inaccurate results due to the increase in the local truncation error the difference in accuracy between the quadrilateral and triangular linear elements is analyzed with the suggested approach the presented numerical examples are in good agreement with the theoretical results the approach considered in the paper can be easily applied to the analysis of different aspects of finite elements techniques as well as other numerical approaches

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

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

a uniform strain hexahedron and quadrilateral with orthogonal hourglass control the treatment of zero energy modes which arise due to one point integration of first order isoparametric finite elements is addressed a method for precisely isolating these modes for arbitrary geometry is developed two hourglass control schemes viscous and elastic are presented and examined in addition a convenient one point integration scheme which analytically integrates the element volume and uniform strain modes is presented

irreducible representations of semisimple algebraic groups and supersolvable lattices abstract let be the cross section lattice of an irreducible representation of a semisimple algebraic group certain combinatorial properties of are studied supersolvable s are determined in terms of dynkin diagrams the characteristic polynomial of a supersolvable is computed

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

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

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

interpolation of multiple zeta and zeta star values we define polynomials of one variable t whose values at t 0and1 are the multiple zeta values and the multiple zeta star values respectively we give an application to the two one conjecture of ohno zudilin and also prove the sum formula and the cyclic sum formula for these polynomials

hopf algebras and the logarithm of the s transform in free probability let k be a positive integer and let g k denote the set of all joint distributions of k tuples a 1 a k in a noncommutative probability space a such that a 1 a k 1 g k is a group under the operation of the free multiplicative convolution we identify g k as the group of characters of a certain hopf algebra y k then by using the log map from characters to infinitesimal characters of y k we introduce a transform ls for distributions g k ls is a power series in k noncommuting indeterminates z 1 z k its coefficients can be computed from the coefficients of the r transform of by using summations over chains in the lattices nc n of noncrossing partitions the ls transform has the linearizing property that ls ls ls g k such that in the particular case k 1 one has that y 1 is naturally isomorphic to the hopf algebra sym of symmetric functions and that the ls transform is very closely related to the logarithm of the s transform of voiculescu by the formula ls z z log s z g 1 in this case the group g 1 can be identified as the group of characters of sym in such a way that the s transform its reciprocal 1 s and its logarithm log s relate in a natural sense to the sequences of complete elementary and respectively power sum symmetric functions

symmetric gaussian quadrature formulae for tetrahedronal regions abstract quadrature formulae of degrees 2 to 6 are presented for the numerical integration of a function over tetrahedronal regions the formulae presented are of gaussian type and fully symmetric with respect to the four vertices of the tetrahedron

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

hourglass control in linear and nonlinear problems abstract mesh stabilization techniques for controlling the hourglass modes in under integrated hexahedral and quadrilateral elements are described it is shown that the orthogonal hourglass techniques previously developed can be obtained from simple requirements that insure the consistency of the finite element equations in the sense that the gradients of linear fields are evaluated correctly it is also shown that this leads to an hourglass control that satisfies the patch test the nature of the parameters which relate the generalized stresses and strains for controlling hourglass modes is examined by means of a mixed variational principle and some guidelines for their selection are discussed finally effective means of implementing these hourglass procedure in computer codes are described applications to both the laplace equation and the equations of solid mechanics in 2 and 3 dimensions are considered

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

tetrahedral composite finite elements we develop and analyse a composite ct3d tetrahedral element consisting of an ensemble of 12 four node linear tetrahedral elements coupled to a linear assumed deformation defined over the entire domain of the composite element the element is designed to have well defined lumped masses and contact tractions in dynamic contact problems while at the same time minimizing the number of volume constraints per element the relation between displacements and deformations is enforced weakly by recourse to the hu washizu principle the element arrays are formulated in accordance with the assumed strain prescription the formulation of the element accounts for fully non linear kinematics integrals over the domain of the element are computed by a five point quadrature rule the element passes the patch test in arbitrarily distorted configurations our numerical tests demonstrate that ct element has been found to possess a convergence rate comparable to those of linear simplicial elements and that these convergence rates are maintained as the near incompressible limit is approached we have also verified that the element satisfies the babuska brezzi condition for a regular mesh configuration these tests suggest that the ct3d element can indeed be used reliably in calculations involving near incompressible behaviour which arises e g in the presence of unconfined plastic flow

hopf algebras and edge labeled posets abstract given a finite graded poset with labeled hasse diagram we construct a quasi symmetric generating function for chains whose labels have fixed descents this is a common generalization of a generating function for the flag f vector defined by ehrenborg and of a symmetric function associated with certain edge labeled posets that arose in the theory of schubert polynomials we show that this construction gives a hopf morphism from an incidence hopf algebra of edge labeled posets to the hopf algebra of quasi symmetric functions

correlations of eigenvalues on multi dimensional flat tori we show that for almost all k dimensional flat tori the eigenvalues of the laplacian follow a uniform distribution with respect to their 2 3 and k 2 level correlations

triangles and tetrahedra in explicit dynamic codes for solids explicit dynamic codes which are used currently for the study of plastic deformations in impact or with some modification for metal forming suffer two serious limitations first only quadrilateral or hexahedral linear elements can be used thus limiting the possibilities of adaptive refinement and adaptive meshing second even with the use of such elements special devices such as reduced integration must be introduced to avoid locking and reduce costs these necessitate complex hour glass control mending type procedures the main difficulties are those due to the need of treatment of almost incompressible deformation modes recently similar difficulties have been overcome in the context of fluid dynamics soil dynamics and we show here how the processes introduced there can be adopted effectively to the present problem thus allowing an almost unrestricted choice of element interpolations 1998 john wiley sons ltd

square lattice ising model susceptibility series expansion method and differential equation for χ 3 in a previous paper 2004 j phys a math gen 37 9651 68 we have given the fuchsian linear differential equation satisfied by 3 the three particle contribution to the susceptibility of the isotropic square lattice ising model this paper gives the details of the calculations with some useful tricks and tools allowing one to obtain long series in polynomial time the method is based on series expansion in the variables that appear in the n 1 dimensional integrals representing the n particle contribution to the isotropic square lattice ising model susceptibility the integration rules are straightforward due to remarkable formulae we derived for these variables we obtain without any numerical approximation 3 as a fully integrated series in the variable w s 2 1 s2 where s sh 2k with k j kt the conventional ising model coupling constant we also give some perspectives and comments on these results

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