degenerations of pre lie algebras we consider the variety of pre lie algebra structures on a given n dimensional vector space the group gln k acts on it and we study the closure of the orbits with respect to the zariski topology this leads to the definition of pre lie algebra degenerations we give fundamental results on such degenerations including invariants and necessary degeneration criteria we demonstrate the close relationship to lie algebra degenerations finally we classify all orbit closures in the variety of complex two dimensional pre lie algebras

an algebraic approach to invariant preserving integators the case of quadratic and hamiltonian invariants in this article conditions for the preservation of quadratic and hamiltonian invariants by numerical methods which can be written as b series are derived in a purely algebraical way the existence of a modified invariant is also investigated and turns out to be equivalent up to a conjugation to the preservation of the exact invariant a striking corollary is that a symplectic method is formally conjugate to a method that preserves the hamitonian exactly another surprising consequence is that the underlying one step method of a symmetric multistep scheme is formally conjugate to a symplectic p series when applied to newton s equations of motion

computing with groups and their character tables in this survey an attempt is made to give some impression of the capabilities of currently available programs for computations with finitely generated groups and their representations

identifying nonlinear biomechanical models by multicriteria analysis in this study the methodology developed by srdjevic and cveticanin international journal of industrial ergonomics 34 2004 307 318 for the nonbiased objective parameter identification of the linear biomechanical model exposed to vertical vibrations is extended to the identification of n degree of freedom dof nonlinear biomechanical models the dynamic performance of the n dof nonlinear model is described in terms of response functions in the frequency domain such as the driving point mechanical impedance and seat to head transmissibility function for randomly generated parameters of the model nonlinear equations of motion are solved using the runge kutta method the appropriate data transformation from the time to frequency domain is performed by a discrete fourier transformation squared deviations of the response functions from the target values are used as the model performance evaluation criteria thus shifting the problem into the multicriteria framework the objective weights of criteria are obtained by applying the shannon entropy concept the suggested methodology is programmed in pascal and tested on a 4 dof nonlinear lumped parameter biomechanical model the identification process over the 2000 generated sets of parameters lasts less than 20 s the model response obtained with the imbedded identified parameters correlates well with the target values therefore justifying the use of the underlying concept and the mathematical instruments and numerical tools applied it should be noted that the identified nonlinear model has an improved accuracy of the biomechanical response compared to the accuracy of a linear model

a comodule bialgebra structure for word series substitution and mould composition abstract an internal coproduct is described which is compatible with hoffman s quasi shuffle product hoffman s quasi shuffle hopf algebra with deconcatenation coproduct is a comodule hopf algebra over the bialgebra thus defined the relation with ecalle s mould calculus i e mould composition and contracting arborification is precised

computational group theory in crystal physics a review abstract a brief account is given of the role played by three dimensional crystallographic point and space groups and their representations in computer programs in crystal physics it covers their use in specific applications for example energy band programs and lattice dynamics programs together with their use in those programs written to calculate for example irreducible characters and matrix representatives selection rules clebsch gordan coefficients etc which have general application in crystal physics

on the pre lie algebra of specified feynman graphs we study the pre lie algebra of specified feynman graphs v t and we define a pre lie structure on its doubling space f t we prove that f t is pre lie module on v t and we find some relations between the two pre lie structures also we study the enveloping algebras of two pre lie algebras denoted respectively by d t and h t and we prove that d t is a module bialgebra on h t we study the pre lie algebra of specified feynman graphs v t and we define a pre lie structure on its doubling space f t we prove that f t is pre lie module on v t and we find some relations between the two pre lie structures also we study the enveloping algebras of two pre lie algebras denoted respectively by d t and h t and we prove that d t is a module bialgebra on h t

a rough path perspective on renormalization abstract we develop the algebraic theory of rough path translation particular attention is given to the case of branched rough paths whose underlying algebraic structure connes kreimer grossman larson makes it a useful model case of a regularity structure in the sense of hairer pre lie structures are seen to play a fundamental rule which allow a direct understanding of the translated i e renormalized equation under consideration this construction is also novel with regard to the algebraic renormalization theory for regularity structures due to bruned hairer zambotti 2016 the links with which are discussed in detail

algebraic renormalisation of regularity structures we give a systematic description of a canonical renormalisation procedure of stochastic pdes containing nonlinearities involving generalised functions this theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms this subgroup is sufficiently large to be able to implement a version of the bphz renormalisation prescription in this context this is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations our construction is based on bialgebras of decorated coloured forests in cointeraction more precisely we have two hopf algebras in cointeraction coacting jointly on a vector space which represents the generalised functions of the theory two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic birkhoff factorisation that arises naturally in perturbative quantum field theory

homogeneous averaging operators on simple finite conformal lie algebras we describe all homogeneous averaging operators on finite simple conformal lie algebras in the case of current algebras these operators are closely related to rational solutions of the classical yang baxter equation

integrable renormalization i the ladder case in recent years a hopf algebraic structure underlying the process of renormalization in quantum field theory was found it led to a birkhoff factorization for regularized hopf algebra characters i e for feynman rules in this work we would like to show that this birkhoff factorization finds its natural formulation in terms of a classical r matrix coming from a rota baxter structure underlying the target space of the regularized hopf algebra characters working in the rooted tree hopf algebra the simple case of the hopf subalgebra of ladder trees is treated in detail the extension to the general case i e the full hopf algebra of rooted trees or feynman graphs is briefly outlined

construction de certaines operades et bigebres associees aux polytopes de stasheff et hypercubes stasheff polytopes introduced by stasheff in his study of h spaces are linked to associativity the direct sum of their cellular complexes is the underlying complex of the a operad which describes homotopy associative algebras in particular there exists a quasi isomorphism a as here we define on the direct sum of their dual cellular complexes the structure of a differential graded operad this construction extends the dendriform operad of loday which corresponds to the vertices of the polytopes we also define the structure of a differential graded operad on the direct sum of the dual cellular complexes of the hypercubes we define a quasi isomorphism from as to each of these operads we also define non differential variants of the two preceding operads and a morphism from as to each of these operads we show that the free algebras have a coproduct which turns them into bialgebras les polytopes de stasheff introduits pour l etude des h espaces sont lies a l associativite la somme directe de leurs complexes cellulaires forme le complexe sous jacent a l operade a qui decrit les algebres associatives a homotopie pres en particulier il existe un quasi isomorphisme a as ici on munit la somme directe des duaux de leurs complexes cellulaires d une structure d operade differentielle graduee cette construction generalise l operade des algebres dendriformes de loday qui correspond aux sommets des polytopes on munit aussi la somme directe des duaux des complexes cellulaires des hypercubes d une structure d operade differentielle graduee on definit un quasi isomorphisme de as clans chacune de ces deux operades on construit egalement des variantes non differentielles des deux operades precedentes on definit un morphisme de as dans chacune de ces operades et on montre que les algebres libres sont munies d un coproduit coassociatif qui en fait des bigebres

a lie theoretic approach to renormalization motivated by recent work of connes and marcolli based on the connes kreimer approach to renormalization we augment the latter by a combinatorial lie algebraic point of view our results rely both on the properties of the dynkin idempotent one of the fundamental lie idempotents in the theory of free lie algebras and on properties of hopf algebras encapsulated in the notion of associated descent algebras besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures the new techniques give rise to an algebraic approach to the galois theory of renormalization in particular they do not depend on the geometry underlying the case of dimensional regularization and the riemann hilbert correspondence this is illustrated with a discussion of the bphz renormalization scheme

a general method to calculate all irreducible representations of a finite group a so called stochastic method is introduced which can be used to decompose the reducible representations of any finite group into irreducible representations in this way the regular representation of a finite group is reduced completely to obtain all of the irreducible representations of the group it also induces a new approach to calculate the character table of a finite group based on this method these calculations or decompositions can be performed by means of a computer

mathematical models for the apparent masses of standing subjects exposed to vertical whole body vibration linear lumped parameter models of the apparent masses of human subjects in standing positions when exposed to vertical whole body vibration have been developed simple models with a single degree of freedom d o f and with two d o f were considered for practical use model parameters were optimised using both the mean apparent mass of 12 male subjects and the apparent masses of individual subjects measured in a previous study the calculated responses of two d o f models with a massless support structure showed best agreement with the measured apparent mass and phase with errors less than 0 1 in the normalised apparent mass i e corresponding to errors less than 10 of the static mass and errors less than 5 in the phase for a normal standing posture the model parameters obtained with the mean measured apparent masses of the 12 subjects were similar to the means of the 12 sets of parameters obtained when fitting to the individual apparent masses it was found that the effects of vibration magnitude and postural changes on the measured apparent mass could be represented by changes to the stiffness and damping in the two d o f models

hopf algebras of rooted forests cocyles and free rota baxter algebras the hopf algebra and the rota baxter algebra are the two algebraic structures underlying the algebraic approach of connes and kreimer to renormalization of perturbative quantum field theory in particular the hopf algebra of rooted trees serves as the baby model of feynman graphs in their approach and can be characterized by certain universal properties involving a hochschild 1 cocycle decorated rooted trees have also been applied to study feynman graphs we will continue the study of universal properties of various spaces of decorated rooted trees with such a 1 cocycle leading to the concept of a cocycle hopf algebra we further apply the universal properties to equip a free rota baxter algebra with the structure of a cocycle hopf algebra

combinatorics of rooted trees and hopf algebras we begin by considering the graded vector space with a basis consisting of rooted trees with grading given by the count of non root vertices we define two linear operators on this vector space the growth and pruning operators which respectively raise and lower grading their commutator is the operator that multiplies a rooted tree by its number of vertices and each operator naturally associates a multiplicity to each pair of rooted trees by using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint and obtain several results about the associated multiplicities now the symmetric algebra on the vector space of rooted trees after a degree shift can be endowed with a coproduct to make a hopf algebra this was defined by kreimer in connection with renormalization we extend the growth and pruning operators as well as the inner product mentioned above to kreimer s hopf algebra on the other hand the vector space of rooted trees itself can be given a noncommutative multiplication with an appropriate coproduct this leads to the hopf algebra of grossman and larson we show that the inner product on rooted trees leads to an isomorphism of the grossman larson hopf algebra with the graded dual of kreimer s hopf algebra correcting an earlier result of panaite

structure of the loday ronco hopf algebra of trees abstract loday and ronco defined an interesting hopf algebra structure on the linear span of the set of planar binary trees they showed that the inclusion of the hopf algebra of non commutative symmetric functions in the malvenuto reutenauer hopf algebra of permutations factors through their hopf algebra of trees and these maps correspond to natural maps from the weak order on the symmetric group to the tamari order on planar binary trees to the boolean algebra we further study the structure of this hopf algebra of trees using a new basis for it we describe the product coproduct and antipode in terms of this basis and use these results to elucidate its hopf algebraic structure in the dual basis for the graded dual hopf algebra our formula for the coproduct gives an explicit isomorphism with a free associative algebra we also obtain a transparent proof of its isomorphism with the non commutative connes kreimer hopf algebra of foissy and show that this algebra is related to non commutative symmetric functions as the commutative connes kreimer hopf algebra is related to symmetric functions

on overlapping divergences using set theoretic considerations we show that the forest formula for overlapping divergences comes from the hopf algebra of rooted trees

renormalization in quantum field theory and the riemann hilbert problem i the hopf algebra structure of graphs and the main theorem this paper gives a complete selfcontained proof of our result announced in 6 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the riemann hilbert problem we shall first show that for any quantum field theory the combinatorics of feynman graphs gives rise to a hopf algebra which is commutative as an algebra it is the dual hopf algebra of the enveloping algebra of a lie algebra whose basis is labelled by the one particle irreducible feynman graphs the lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa the corresponding lie group g is the group of characters of we shall then show that using dimensional regularization the bare unrenormalized theory gives rise to a loop where c is a small circle of complex dimensions around the integer dimension d of space time our main result is that the renormalized theory is just the evaluation at z d of the holomorphic part of the birkhoff decomposition of we begin to analyse the group g and show that it is a semi direct product of an easily understood abelian group by a highly non trivial group closely tied up with groups of diffeomorphisms the analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title

harmonic polynomials invariant under a finite subgroup of o n in this paper an algorithm is described which allows a systematic computation of harmonic polynomials of a given degree invariant under a finite subgroup of the group o n an application of the algorithm to the octahedral cubic subgroup is given

theoretical foundation methods and criteria for calibrating human vibration models using frequency response functions while simulations of the measured biodynamic responses of the whole human body or body segments to vibration are conventionally interpreted as summaries of biodynamic measurements and the resulting models are considered quantitative this study looked at these simulations from a different angle model calibration the specific aims of this study are to review and clarify the theoretical basis for model calibration to help formulate the criteria for calibration validation and to help appropriately select and apply calibration methods in addition to established vibration theory a novel theorem of mechanical vibration is also used to enhance the understanding of the mathematical and physical principles of the calibration based on this enhanced understanding a set of criteria was proposed and used to systematically examine the calibration methods besides theoretical analyses a numerical testing method is also used in the examination this study identified the basic requirements for each calibration method to obtain a unique calibration solution this study also confirmed that the solution becomes more robust if more than sufficient calibration references are provided practically however as more references are used more inconsistencies can arise among the measured data for representing the biodynamic properties to help account for the relative reliabilities of the references a baseline weighting scheme is proposed the analyses suggest that the best choice of calibration method depends on the modeling purpose the model structure and the availability and reliability of representative reference data

irreducible representations of finite groups abstract the flodmark blokker scheme for finding irreducible representations of finite groups is sufficiently general for treating all little groups of crystallographic space groups this scheme is generalized in two respects a the case when no element has a non degenerate eigenvalue but the group contains a set with commuting matrix representatives for each irreducible representation b the case when any set of commuting irreducible matrix representatives has a common eigenspace with dimension higher than 1 degenerate with respect to the eigenvalues of the set in case b we make linear combinations of group elements to find a complete set of commuting operators the completeness being with respect to a degenerate subspace of an irreducible space of the group we state a theorem to prove the generality of this method the procedure is valid for arbitrary finite groups and it has been programmed for implementation by a computer

a combinatorial hopf algebra for nonlinear output feedback control systems abstract in this work a combinatorial description is provided of a faa di bruno type hopf algebra which naturally appears in the context of fliess operators in nonlinear feedback control theory it is a connected graded commutative and non cocommutative hopf algebra defined on rooted circle trees a cancellation free forest formula for its antipode is given

nonabelian generalized lax pairs the classical yang baxter equation and postlie algebras we generalize the classical study of generalized lax pairs the related mathcal o operators and the modified classical yang baxter equation by introducing the concepts of nonabelian generalized lax pairs extended mathcal o operators and the extended classical yang baxter equation we study in this context the nonabelian generalized r matrix ansatz and the related double lie algebra structures relationship between extended mathcal o operators and the extended classical yang baxter equation is established especially for self dual lie algebras this relationship allows us to obtain an explicit description of the manin triples for a new class of lie bialgebras furthermore we show that a natural structure of postlie algebra is behind mathcal o operators and fits in a setup of the triple lie algebra that produces self dual nonabelian generalized lax pairs

mathematical models for the apparent mass of the seated human body exposed to vertical vibration alternative mathematical models of the vertical apparent mass of the seated human body are developed the optimum parameters of four models two single degree of freedom models and two two degree of freedom models are derived from the mean measured apparent masses of 60 subjects 24 men 24 women 12 children previously reported the best fits were obtained by fitting the phase data with single degree of freedom and two degree of freedom models having rigid support structures for these two models curve fitting was performed on each of the 60 subjects so as to obtain optimum model parameters for each subject for the averages of each of the three groups of subjects and for the entire group of subjects the values obtained are tabulated use of a two degree of freedom model provided a better fit to the phase of the apparent mass at frequencies greater than about 8 hz and an improved fit to the modulus of the apparent mass at frequencies around 5 hz it is concluded that the two degree of freedom model provides an apparent mass similar to that of the human body but this does not imply that the body moves in the same manner as the masses in this optimized two degree of freedom model

algebraic birkhoff factorization and the euler maclaurin formula on cones we equip the space of lattice cones with a coproduct which makes it a cograded coaugmented connnected coalgebra the exponential generating sum and exponential generating integral on lattice cones can be viewed as linear maps on this space with values in the space of meromorphic germs with linear poles at zero we investigate the subdivision properties reminiscent of the inclusion exclusion principle for the cardinal on finite sets of such linear maps and show that these properties are compatible with the convolution quotient of maps on the coalgebra implementing the algebraic birkhoff factorization procedure on the linear maps under consideration we factorize the exponential generating sum as a convolution quotient of two maps with each of the maps in the factorization satisfying a subdivision property a direct computation shows that the polar decomposition of the exponential generating sum on a smooth lattice cone yields an euler maclaurin formula the compatibility with subdivisions of the convolution quotient arising in the algebraic birkhoff factorization then yields the euler maclaurin formula for any lattice cone this provides a simple formula for the interpolating factor by means of a projection formula

the odd origin of gerstenhaber brackets batalin vilkovisky operators and master equations using five basic principles we treat gerstenhaber lie brackets batalin vilkovisky bv operators and master equations appearing in mathematical and physical contexts in a unified way the different contexts for this are given by the different types of feynman graphs that underlie the particular situation two of the maxims we bring forth are 1 that extending to the non connected graphs gives a commutative multiplication forming a part of the bv structure and 2 that there is a universal odd twist that unifies and explains seemingly ad hoc choices of signs and is responsible for the bv operator being a differential our treatment results in uniform general theorems these allow us to prove new results and recover and connect many constructions that have appeared independently throughout the literature the more general point of view also allows us to disentangle the necessary from the circumstantial

normal coordinates and primitive elements in the hopf algebra of renormalization we introduce normal coordinates on the infinite dimensional group g introduced by connes and kreimer in their analysis of the hopf algebra of rooted trees we study the primitive elements of the algebra and show that they are generated by a simple application of the inverse poincare lemma given a closed left invariant 1 form on g for the special case of the ladder primitives we find a second description that relates them to the hopf algebra of functionals on power series with the usual product either approach shows that the ladder primitives are given by the schur polynomials the relevance of the lower central series of the dual lie algebra in the process of renormalization is also discussed leading to a natural concept of k primitiveness which is shown to be equivalent to the one already in the literature

nested sums expansion of transcendental functions and multiscale multiloop integrals expansion of higher transcendental functions in a small parameter are needed in many areas of science for certain classes of functions this can be achieved by algebraic means these algebraic tools are based on nested sums and can be formulated as algorithms suitable for an implementation on a computer examples such as expansions of generalized hypergeometric functions or appell functions are discussed as a further application we give the general solution of a two loop integral the so called c topology in terms of multiple nested sums in addition we discuss some important properties of nested sums in particular we show that they satisfy a hopf algebra

generalizations of the classical yang baxter equation and o operators tensor solutions r matrices of the classical yang baxter equation cybe in a lie algebra obtained as the classical limit of the r matrix solution of the quantum yang baxter equation is an important structure appearing in different areas such as integrable systems symplectic geometry quantum groups and quantum field theory further study of cybe led to its interpretation as certain operators giving rise to the concept of o operators the o operators were in turn interpreted as tensor solutions of cybe by enlarging the lie algebra bai c a unified algebraic approach to the classical yang baxter equation j phys a math theor 40 11073 2007 10 1088 1751 8113 40 36 007 the purpose of this paper is to extend this study to a more general class of operators that were recently introduced bai c guo l and ni x nonabelian generalized lax pairs the classical yang baxter equation and postlie algebras commun math phys 297 553 2010 10 1007 s00220 010 0998 7 in the study of lax pa

a unified algebraic approach to the classical yang baxter equation in this paper the different operator forms of the classical yang baxter equation are given in the tensor expression through a unified algebraic method it is closely related to left symmetric algebras which play an important role in many fields in mathematics and mathematical physics by studying the relations between left symmetric algebras and the classical yang baxter equation we can construct left symmetric algebras from certain classical r matrices and conversely there is a natural classical r matrix constructed from a left symmetric algebra which corresponds to a parakahler structure in geometry moreover the former in a special case gives an algebraic interpretation of the left symmetry as a lie bracket left twisted by a classical r matrix

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

renormalised multiple integrals of symbols with linear constraints given a holomorphic regularisation procedure e g riesz or dimensional regularisation on classical symbols we define renormalised multiple integrals of radial classical symbols with linear constraints to do so we first prove the existence of meromorphic extensions of multiple integrals of holomorphic perturbations of radial symbols with linear constraints and then implement either generalised evaluators or a birkhoff factorisation renormalised multiple integrals are covariant and factorise over independent sets of constraints

the normal distribution is infinitely divisible abstract we prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution we study the voiculescu transform first by giving a survey of its combinatorial implications and then analytically including a proof of free infinite divisibility in fact we prove that a sub family of askey wimp kerov distributions are freely infinitely divisible of which the normal distribution is a special case at the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution the others being the cauchy distribution and the free 1 2 stable distribution

structural phase transitions in crystals broken symmetry isotropy groups a systematic procedure ensuring the determination of all isotropy groups of a given representation of a space group is presented for the first time isotropy groups play an important role in various areas of theoretical solid state physics for example only isotropy groups may occur in a structural phase transition driven by an order parameter belonging to a given representation the method uses the chain criterion directly on the image of the representation employing a labeling of the matrices by space group elements a notion of a substar of a wave vector associated with the representation is central to the method finally as a detailed illustration the method is applied to a structural phase transition in a15 systems driven by an x point order parameter the result agrees with previously reported ones

on the invariance of residues of feynman graphs we use simple iterated one loop graphs in massless yukawa theory and qed to pose the following question what are the symmetries of the residues of a graph under a permutation of places to insert subdivergences the investigation confirms partial invariance of the residue under such permutations the highest weight transcendental is invariant under such a permutation for qed this result is gauge invariant i e the permutation invariance holds for any gauge computations are done making use of the hopf algebra structure of graphs and employing ginac to automate the calculations

su mn su m su n isoscalar factors and s f1 f2 s f1 s f2 isoscalar factors a simple relation is found between the isoscalar factor isf of the unitary group and those of the permutation group i e the su mn su m su n isf is equal to the s f1 f2 s f1 s f2 isf since the values of s f1 f2 s f1 s f2 isf are independent of m and n one arrives at an important conclusion that the values of su mn su m su n isf are also independent of m and n therefore they can be calculated for all m and n by a single stroke instead of one m and one n at a time an eigenfunction metho for evaluating the su mn su m su n isf is given which can be easily translated into a computer program

rota baxter operators on sl 2 c and solutions of the classical yang baxter equation we explicitly determine all rota baxter operators of weight zero on sl 2 c under the cartan weyl basis for the skew symmetric operators we give the corresponding skew symmetric solutions of the classical yang baxter equation in sl 2 c confirming the related study by semenov tian shansky in general these rota baxter operators give a family of solutions of the classical yang baxter equation in the six dimensional lie algebra sl 2 c ad sl 2 c they also give rise to three dimensional pre lie algebras which in turn yield solutions of the classical yang baxter equation in other six dimensional lie algebras

hopf algebras cyclic cohomology and the transverse index theorem in this paper we solve a longstanding internal problem of noncommutative geometry namely the computation of the index of transversally elliptic operators on foliations we show that the computation of the local index formula for transversally hypoelliptic operators can be settled thanks to a very specific hopf algebra associated to each integer codimension this hopf algebra reduces transverse geometry to a universal geometry of affine nature the structure of this hopf algebra its relation with the lie algebra of formal vector fields as well as the computation of its cyclic cohomology are done in the present paper in which we also show that under a suitable unimodularity condition the cosimplicial space underlying the hochschild cohomology of a hopf algebra carries a highly nontrivial cyclic structure

supercharacters symmetric functions in noncommuting variables and related hopf algebras we identify two seemingly disparate structures supercharacters a useful way of doing fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field and the ring of symmetric functions in noncommuting variables each is a hopf algebra and the two are isomorphic as such this allows developments in each to be transferred the identification suggests a rich class of examples for the emerging field of combinatorial hopf algebras

quantification of changes in modal parameters due to the presence of passive people on a slender structure abstract this work proposes a method to predict changes in modal parameters of a generic structure due to the presence of passive people the method requires to accurately know the modal model of the empty structure i e natural frequencies damping ratios and mode shapes no restriction on the number of degrees of freedom of the structure is required each passive subject on the structure is modelled through his apparent mass and is introduced locally on the empty structure to obtain a model of the joint human structure system to include the apparent mass of the subjects in the model both measurements and models available in literature were used the effect of using different models of the apparent mass was investigated and the effect of using apparent masses obtained with different levels of vibration was analysed the proposed model was validated by means of experimental tests carried out on a lightly damped staircase the empty structure was first tested to obtain its modal model in order to apply the proposed approach after performing several tests with passive people standing on the structure the experimental results showed small changes of natural frequencies increase or decrease and a high increase of damping ratios due to people s presence thus the proposed approach was used to predict the above mentioned changes the measured and predicted frequency response functions natural frequencies and damping ratios were compared in all the cases taken into consideration the predicted results fitted with the experimental evidence

generalizing the connes moscovici hopf algebra to contain all rooted trees this paper defines a generalization of the connes moscovici hopf algebra h 1 that contains the entire hopf algebra of rooted trees a relationship between the former a much studied object in non commutative geometry and the latter a much studied object in perturbative quantum field theory has been established by connes and kreimer the results of this paper open the door to study the cohomology of the hopf algebra of rooted trees

development of a two dimensional dynamic model of the foot ankle system exposed to vibration abstract workers in mining mills construction and some types of manufacturing are exposed to vibration that enters the body through the feet exposure to foot transmitted vibration ftv is associated with an increased risk of developing vibration induced white foot viwft viwft is a vascular and neurological condition of the lower limb leading to blanching in the toes and numbness and tingling in the feet which can be disabling for the worker this paper presents a two dimensional dynamic model describing the response of the foot ankle system to vibration using four segments and eight kelvin voigt models the parameters of the model have been obtained by minimizing the quadratic reconstruction error between the experimental and numerical curves of the transmissibility and the apparent mass of participants standing in a neutral position the average transmissibility at five locations on the foot has been optimized by minimizing the difference between experimental data and the model prediction between 10 and 100 hz the same procedure has been repeated to fit the apparent mass measured at the driving point in a frequency range between 2 and 20 hz monte carlo simulations were used to assess how the variability of the mass stiffness and damping matrices affect the overall data dispersion results showed that the 7 of freedom model correctly described the transmissibility the average transmissibility modulus error was 0 1 the error increased when fitting the transmissibility and apparent mass curves the average modulus error was 0 3 however the obtained values were reasonable with respect to the average inter participant variability experimentally estimated at 0 52 for the modulus study results can contribute to the development of materials and equipment to attenuate ftv and consequently lower the risk of developing viwft

commutative post lie algebra structures on nilpotent lie algebras and poisson algebras abstract we give an explicit description of commutative post lie algebra structures on some classes of nilpotent lie algebras for non metabelian filiform nilpotent lie algebras and lie algebras of strictly upper triangular matrices we show that all cpa structures are associative and induce an associated poisson admissible algebra

decomposition spaces incidence algebras and mobius inversion i basic theory abstract this is the first in a series of papers devoted to the theory of decomposition spaces a general framework for incidence algebras and mobius inversion where algebraic identities are realised by taking homotopy cardinality of equivalences of groupoids a decomposition space is a simplicial groupoid satisfying an exactness condition weaker than the segal condition expressed in terms of active and inert maps in just as the segal condition expresses composition the new exactness condition expresses decomposition and there is an abundance of examples in combinatorics after establishing some basic properties of decomposition spaces the main result of this first paper shows that to any decomposition space there is an associated incidence coalgebra spanned by the space of 1 simplices and with coefficients in groupoids we take a functorial viewpoint throughout emphasising conservative ulf functors these induce coalgebra homomorphisms reduction procedures in the classical theory of incidence coalgebras are examples of this notion and many are examples of decalage of decomposition spaces an interesting class of examples of decomposition spaces beyond segal spaces is provided by hall algebras the waldhausen s construction of an abelian or stable infinity category is shown to be a decomposition space in the second paper in this series we impose further conditions on decomposition spaces to obtain a general mobius inversion principle and to ensure that the various constructions and results admit a homotopy cardinality in the third paper we show that the lawvere menni hopf algebra of mobius intervals is the homotopy cardinality of a certain universal decomposition space two further sequel papers deal with numerous examples from combinatorics note the notion of decomposition space was arrived at independently by dyckerhoff and kapranov 17 who call them unital 2 segal spaces our theory is quite orthogonal to theirs the definitions are different in spirit and appearance and the theories differ in terms of motivation examples and directions

rota baxter 3 lie algebras in this paper we introduce the concepts of a rota baxter operator and differential operator with weights on an n algebra we then focus on rota baxter 3 lie algebras and show that they can be derived from rota baxter lie algebras and pre lie algebras and from rota baxter commutative associative algebras with derivations we also establish the inheritance property of rota baxter 3 lie algebras

coalgebras of words and phrases abstract we introduce two constructions of a coassociative comultiplication in the algebra of phrases in a given alphabet as a preliminary step we give two constructions of a pre lie comultiplication in the module generated by words

the geometry of the space of branched rough paths we construct an explicit transitive free action of a banach space of holder functions on the space of branched rough paths which yields in particular a bijection between theses two spaces this endows the space of branched rough paths the structure of a principal homogeneous space over a banach space and allows to characterize its automorphisms the construction is based on the baker campbell hausdorff formula and on the hairer kelly map which allows to describe branched rough paths in terms of anisotropic geometric rough paths

qed hopf algebras on planar binary trees abstract in this paper we describe the hopf algebras on planar binary trees used to renormalize the feynman propagators of quantum electrodynamics and the coaction which describes the renormalization procedure both structures are related to some semi direct coproduct of hopf algebras

etude de l algebre de lie double des arbres enracines decores the double lie algebra ld of rooted trees decorated by a set d is introduced generalising the construction of connes and kreimer it is shown that it is a simple lie algebra its derivations and its automorphisms are described as well as some central extensions finally the category of lowest weight modules is introduced and studied

from rota baxter algebras to pre lie algebras rota baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics rota baxter algebras provide a construction of pre lie algebras from associative algebras in this paper we give all rota baxter operators of weight 1 on complex associative algebras in dimension 3 and their corresponding pre lie algebras

hopf algebras renormalization and noncommutative geometry we explore the relation between the hopf algebra associated to the renormalization of qft and the hopf algebra associated to the ncg computations of tranverse index theory for foliations

on hopf algebra structures over free operads the operad lie can be constructed as the operad of primitives primas from the operad as of associative algebras this is reflected by the theorems of friedrichs poincare birkhoff witt and cartier milnor moore we replace the operad as by families of free operads p which include the operad mag freely generated by a non commutative non associative binary operation and the operad of stasheff polytopes we obtain poincare birkhoff witt type theorems and collect information about the operads primp e g in terms of characteristic functions

a new approach to permutation group representations ii the formula for the yamanouchi matrix elements is rederived by the eigenfunction method in a simple fashion

a bijection between rooted trees and fermionic fock states grafting and growth operators in fock space and fermionic operators for rooted trees we show that fermionic fock states in the occupation number representation can be indexed uniquely by rooted trees our main ingredients in this construction are the matula numbers the fundamental theorem of arithmetic and a relabeling of fermionic quantum states by natural numbers as a byproduct of the correspondence mentioned above we realize the grafting and the growth operators comprising important constructions in the context of hopf algebras in the fermionic fock space also we show how to construct fermionic creation and annihilation operators in the context of rooted trees new representations of the solutions of combinatorial dyson schwinger equations and of the antipode in the connes kreimer hopf algebra of rooted trees related to the occupation number picture are presented

stochastic expansions and hopf algebras we study solutions to nonlinear stochastic differential systems driven by a multi dimensional wiener process a useful algorithm for strongly simulating such stochastic systems is the castell gaines method which is based on the exponential lie series when the diffusion vector fields commute it has been proved that at low orders this method is more accurate in the mean square error than corresponding stochastic taylor methods however it has also been shown that when the diffusion vector fields do not commute this is not true for strong order one methods here we prove that when there is no drift and the diffusion vector fields do not commute the exponential lie series is usurped by the sinh log series in other words the mean square error associated with a numerical method based on the sinh log series is always smaller than the corresponding stochastic taylor error in fact to all orders our proof uses the underlying hopf algebra structure of these series and a two alphabet associative algebra of shuffle and concatenation operations we illustrate the benefits of the proposed series in numerical studies

modified group projector technique induced representations the group projector technique is developed for the representations of the direct product form d d x d in which one of them has been induced from a representation of a subgroup it is shown that the group projectors and symmetry adapted bases are essentially determined in terms of the subgroup representations to illustrate both the technical and conceptual advantages of the method it is shown how the calculation of the normal modes of polymers polyacetylene as an example can be obtained using the symmetry of the monomer only and several results of the induction theory are reconsidered within the new framework

birkhoff type decompositions and the baker campbell hausdorff recursion we describe a unification of several apparently unrelated factorizations arising from quantum field theory vertex operator algebras combinatorics and numerical methods in differential equations the unification is given by a birkhoff type decomposition that was obtained from the baker campbell hausdorff formula in our study of the hopf algebra approach of connes and kreimer to renormalization in perturbative quantum field theory there we showed that the birkhoff decomposition of connes and kreimer can be obtained from a certain baker campbell hausdorff recursion formula in the presence of a rota baxter operator we will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for lie algebras that arose in vertex operator algebra and conformal field theory and the even odd decomposition of combinatorial hopf algebra characters as well as the lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations

general dyson schwinger equations and systems we classify combinatorial dyson schwinger equations giving a hopf subalgebra of the hopf algebra of feynman graphs of the considered quantum field theory we first treat single equations with an arbitrary eventually infinite number of insertion operators we distinguish two cases in the first one the hopf subalgebra generated by the solution is isomorphic to the faa di bruno hopf algebra or to the hopf algebra of symmetric functions in the second case we obtain the dual of the enveloping algebra of a particular associative algebra seen as a lie algebra we also treat systems with an arbitrary finite number of equations with an arbitrary number of insertion operators with at least one of degree 1 in each equation

on the stability of some groups of formal diffeomorphisms by the birkhoff decomposition abstract let g be the group of one parameter identity tangent diffeomorphisms on the line whose coefficients are formal laurent series in the parameter e with a pole of finite order at 0 it is well known that the birkhoff decomposition can be defined in such a group we investigate the stability of the birkhoff decomposition in subgroups of g and give a formula for this decomposition these results are strongly related to renormalization in quantum field theory since it was proved by a connes and d kreimer that after dimensional regularization the unrenormalized effective coupling constants are the image by a formal identity tangent diffeomorphism of the coupling constants of the theory in the massless 6 3 theory this diffeomorphism is in g and its birkhoff decomposition gives directly the bare coupling constants and the renormalized coupling constants

an algebraic birkhoff decomposition for the continuous renormalization group this paper aims at presenting the first steps towards a formulation of the exact renormalization group equation in the hopf algebra setting of connes and kreimer it mostly deals with some algebraic preliminaries allowing us to formulate perturbative renormalization within the theory of differential equations the relation between renormalization formulated as a change of boundary condition for a differential equation and an algebraic birkhoff decomposition for rooted trees is explicited