locality and renormalization universal properties and integrals on trees the purpose of this paper is to build an algebraic framework suited to regularize branched structures emanating from rooted forests and which encodes the locality principle this is achieved by means of the universal properties in the locality framework of properly decorated rooted forests these universal properties are then applied to derive the multivariate regularization of integrals indexed by rooted forests we study their renormalization along the lines of kreimer s toy model for feynman integrals the purpose of this paper is to build an algebraic framework suited to regularize branched structures emanating from rooted forests and which encodes the locality principle this is achieved by means of the universal properties in the locality framework of properly decorated rooted forests these universal properties are then applied to derive the multivariate regularization of integrals indexed by rooted forests we study their renormalization along the lines of kreimer s toy model for feynman integrals

the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

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

polynomial functors and combinatorial dyson schwinger equations we present a general abstract framework for combinatorial dyson schwinger equations in which combinatorial identities are lifted to explicit bijections of sets and more generally equivalences of groupoids key features of combinatorial dyson schwinger equations are revealed to follow from general categorical constructions and universal properties rather than beginning with an equation inside a given hopf algebra and referring to given hochschild 1 cocycles our starting point is an abstract fixpoint equation in groupoids shown canonically to generate all the algebraic structures precisely for any finitary polynomial endofunctor p defined over groupoids the system of combinatorial dyson schwinger equations x 1 p x has a universal solution namely the groupoid of p trees the isoclasses of p trees generate naturally a connes kreimer like bialgebra in which the abstract dyson schwinger equation can be internalised in terms of canonical b operators the solution to this equation is a series the

finite dimensional comodules over the hopf algebra of rooted trees the aim of this paper is an algebraic study of the hopf algebra h r of rooted trees which was introduced in cite kreimer1 connes broadhurst kreimer2 we first construct comodules over h r from finite families of primitive elements furthermore we classify these comodules by restricting the possible families of primitive elements and taking the quotient by the action of certain groups in the next section we give a formula about primitive elements of the subalgebra of ladders and construct a projection operator on the space of primitive elements it allows us to obtain a basis of the primitive elements by an inductive process which answers one of the questions of cite kreimer in the last sections we classify the hopf algebra endomorphisms and the coalgebras endomorphisms using the graded hopf algebra gr h r associated to the filtration by deg p of cite kreimer we then prove that h r is isomorphic to gr h r and deduce a decomposition of the group of the hopf algebra automorphisms of h r as a semi direct product

punctuality and idleness in just in time scheduling abstract in scheduling problems with irregular cost functions such as deviation functions in earliness tardiness scheduling problems optimal solutions usually contain some idleness periods during which no activity is processed then minimizing the penalties for not delivering on time and minimizing the idleness cost are two complementary criteria for a schedule a dynamic programming procedure in which the states are represented by continuous two dimensional piecewise linear functions is proposed to compute the cost of the pareto optimal schedules

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

identities on fractional integrals and various integral transforms in this work we will introduce theorems relating the riemann liouville fractional integral and the weyl fractional integral to some well known integral transforms including laplace transforms stieltjes transforms generalized stieltes transforms hankel transforms and k transforms as applications of the theorems and their results a number of infinite integrals of elementary functions and special functions are evaluated and some illustrative examples are presented

some parseval goldstein type identities involving the fs 2 transform the fc 2 transform and the p4 transform and their applications abstract in the present paper the authors consider several new integral transforms including the f s 2 transform the f c 2 transform and the p 4 transform as generalizations of the classical fourier sine transform the classical fourier cosine transform and the classical stieltjes transform respectively many identities involving these transforms are given by making use of these identities a number of new parseval goldstein type identities are obtained for these and many other well known integral transforms the identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions some examples are also given as illustrations of the results presented here

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

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

classification of systems of dyson schwinger equations in the hopf algebra of decorated rooted trees abstract we consider systems of combinatorial dyson schwinger equations briefly sdse x 1 b 1 f 1 x 1 x n x n b n f n x 1 x n in the connes kreimer hopf algebra h i of rooted trees decorated by i 1 n where b i is the operator of grafting on a root decorated by i and f 1 f n are non constant formal series the unique solution x x 1 x n of this equation generates a graded subalgebra h s of h i we characterise here all the families of formal series f 1 f n such that h s is a hopf subalgebra more precisely we define three operations on sdse change of variables dilatation and extension and give two families of sdse cyclic and fundamental systems and prove that any sdse s such that h s is hopf is the concatenation of several fundamental or cyclic systems after the application of a change of variables a dilatation and iterated extensions

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

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

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

a theorem on widder s potential transform and its applications abstract in the present paper the authors prove a parseval goldstein type theorem involving the classical laplace transform the fourier sine transform and widder s potential transform the theorem is then shown to yield a simple algorithm for evaluating infinite integrals some illustrative examples are also given

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

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

minmax scheduling with job classes and earliness tardiness costs abstract we address scheduling problems with job dependent due dates and general possibly nonlinear and asymmetric earliness and tardiness costs the number of distinct due dates is substantially smaller than the number of jobs thus jobs are partitioned to classes where all jobs of a given class share a common due date we consider the settings of a single machine and parallel identical machines our objective is of a minmax type i e we seek a schedule that minimizes the maximum earliness tardiness cost among all jobs we introduce a nonlinear program that needs to be solved in order to obtain an optimal solution for the single machine case the case of parallel identical machines is np hard even for two machines linear costs and a single common due date we introduce an lpt based largest processing time first heuristic and a simple lower bound on the optimal cost both the heuristic and the lower bound are asymptotically accurate under very general conditions our numerical tests indicate that the heuristic produces very close to optimal schedules in all settings

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

a new method of solving bessel s differential equation using the l 2 transform in the present paper a new method of solving bessel s differential equation is given using the l 2 transform

groupoids and faa di bruno formulae for green functions in bialgebras of trees we prove a faa di bruno formula for the green function in the bialgebra of p trees for any polynomial endofunctor p the formula appears as relative homotopy cardinality of an equivalence of groupoids

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

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

identities for the e2 1 transform and their applications in the present paper the authors introduce the e 2 1 transform with kernel the exponential integral function it is shown that the third iterate of the l 2 transform is the exponential integral transform and some identities involving the new transform the l 2 transform and the widder potential transform are given using the identities parseval goldstein type results involving these transforms are proved some illustrative examples are also given

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

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 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

identities for the exponential integral and the complementary error transforms abstract in the present paper the authors introduce the exponential integral transform and the complementary error transform then it is shown that the third iterate of the laplace transform is the exponential integral transform and a modified third iterate of the laplace transform is the complementary error transform some identities involving these new transforms the laplace transform and the stieltjes transform are given using the identities parseval goldstein type results involving these transforms are proved some illustrative examples are also given

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

identities and parseval type relations for the l2 transform abstract in the present paper the authors introduce identities for the l 2 transform which yield a parseval type relationship for the l 2 transform the parseval type relationship proven in this paper give rise to useful corollaries for evaluating indefinite integrals of special functions some examples are also given as illustrations of the results presented here

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

identities for the hankel transform and their applications abstract in the present paper the authors show that iterations of the hankel transform with k transform is a constant multiple of the widder transform using these iteration identities several parseval goldstein type theorems for these transforms are given by making use of these results a number of new goldstein type exchange identities are obtained for these and the laplace transform the identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions some examples are also given as illustration of the results presented here

towards cohomology of renormalization bigrading the combinatorial hopf algebra of rooted trees the renormalization of quantum field theory twists the antipode of a noncocommutative hopf algebra of rooted trees decorated by an infinite set of primitive divergences the hopf algebra of undecorated rooted trees r generated by a single primitive divergence solves a universal problem in hochschild cohomology it has two nontrivial closed hopf subalgebras the cocommutative subalgebra ladder of pure ladder diagrams and the connes moscovici noncocommutative subalgebra cm of noncommutative geometry these three hopf algebras admit a bigrading by n the number of nodes and an index k that specifies the degree of primitivity in each case we use iterations of the relevant coproduct to compute the dimensions of subspaces with modest values of n and k and infer a simple generating procedure for the remainder the results for ladder are familiar from the theory of partitions while those for cm involve novel transforms of partitions most beautiful is the bigrading of r the largest of the three thanks to sloane s superseeker we discovered that it saturates all possible inequalities we prove this by using the universal hochschild closed one cocycle b which plugs one set of divergences into another and by generalizing the concept of natural growth beyond that entailed by the connes moscovici case we emphasize the yet greater challenge of handling the infinite set of decorations of realistic quantum field theory

mulitgraded dyson schwinger systems we study systems of combinatorial dyson schwinger equations with an arbitrary number n of coupling constants the considered hopf algebra of feynman graphs is mathbb n n graded and we wonder if the graded subalgebra generated by the solution is hopf or not we first introduce a family of pre lie algebras which we classify dually providing systems generating a hopf subalgebra we also describe the associated groups as extensions of groups of formal diffeomorphisms on several variables we then consider systems coming from feynman graphs of a quantum field theory we show that if the number n of independent coupling constants is the number of interactions of the considered qft then the generated subalgebra is hopf for qed varphi 3 and qcd we also prove that this is the minimal value of n all these examples are generalizations of the first family of dyson schwinger systems in the one coupling constant case called fundamental we also give a generalization of the second family called cyclic

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

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

faa di bruno subalgebras of the hopf algebra of planar trees from combinatorial dyson schwinger equations we consider the combinatorial dyson schwinger equation x b p x in the non commutative connes kreimer hopf algebra of planar rooted trees hnck where b is the operator of grafting on a root and p a formal series the unique solution x of this equation generates a graded subalgebra an p of hnck we describe all the formal series p such that an p is a hopf subalgebra we obtain in this way a 2 parameters family of hopf subalgebras of hnck organized into three isomorphism classes a first one restricted to a polynomial ring in one variable a second one restricted to the hopf subalgebra of ladders isomorphic to the hopf algebra of quasi symmetric functions a last infinite one which gives a non commutative version of the faa di bruno hopf algebra by taking the quotient the last class gives an infinite set of embeddings of the faa di bruno algebra into the connes kreimer hopf algebra of rooted trees moreover we give an embedding of the free faa di bruno hopf algebra on d variables into a hopf algebra of decorated rooted trees together with a non commutative version of this embedding

one processor scheduling with symmetric earliness and tardiness penalties we consider one processor scheduling problems having the following form tasks t1 t2 tn are given with each ti having a specified length li and a preferred starting time ai or equivalently a preferred completion time bi the tasks are to be scheduled nonpreemptively i e a task cannot be split on a single processor to begin as close to their preferred starting times as possible we examine two different cost measures for such schedules the sum of the absolute discrepancies from the preferred starting times and the maximum such discrepancy for the first of these we show that the problem of finding minimum cost schedules is np complete however we give an efficient algorithm that finds minimum cost schedules whenever the tasks either all have the same length or are required to be executed in a given fixed sequence for the second cost measure we give an efficient algorithm that finds minimum cost schedules in general with no constraints on the ordering or lengths of the tasks

the epstein glaser approach to perturbative quantum field theory graphs and hopf algebras the paper aims at investigating perturbative quantum field theory in the approach of epstein and glaser eg and in particular its formulation in the language of graphs and hopf algebras has various has are encountered each one associated with a special combination of physical concepts such as normalization localization pseudounitarity causal regularization and renormalization the algebraic structures representing the perturbative expansion of the s matrix are imposed on operator valued distributions equipped with appropriate graph indices translation invariance ensures the algebras to be analytically well defined and graded total symmetry allows to formulate bialgebras the algebraic results are given embedded in the corresponding physical framework covering the two eg versions by fredenhagen and scharf that differ with respect to the concrete recursive implementation of causality besides the ultraviolet divergences occurring in feynman s representation are mathematically reasoned as a final result the change of the renormalization scheme in the context of eg is modeled via a ha and interpreted as the eg analog of kreimer s ha the paper aims at investigating perturbative quantum field theory in the approach of epstein and glaser eg and in particular its formulation in the language of graphs and hopf algebras has various has are encountered each one associated with a special combination of physical concepts such as normalization localization pseudounitarity causal regularization and renormalization the algebraic structures representing the perturbative expansion of the s matrix are imposed on operator valued distributions equipped with appropriate graph indices translation invariance ensures the algebras to be analytically well defined and graded total symmetry allows to formulate bialgebras the algebraic results are given embedded in the corresponding physical framework covering the two eg versions by fredenhagen and scharf that differ with respect to the concrete recursive implementation of causality besides the ultraviolet divergences occurring in feynman s representation are mathematically reasoned as a

a short survey of noncommutative geometry we give a survey of selected topics in noncommutative geometry with some emphasis on those directly related to physics including our recent work with dirk kreimer on renormalization and the riemann hilbert problem we discuss at length two issues the first is the relevance of the paradigm of geometric space based on spectral considerations which is central in the theory as a simple illustration of the spectral formulation of geometry in the ordinary commutative case we give a polynomial equation for geometries on the four sphere with fixed volume the equation involves an idempotent e playing the role of the instanton and the dirac operator d it is of the form e 12 d e 4 5 and determines both the sphere and all its metrics with fixed volume form the expectation x is the projection on the commutant of the algebra of 4 by 4 matrices we also show using the noncommutative analog of the polyakov action how to obtain the noncommutative metric in spectral form on the noncommutative tori fr

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

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

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

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

a theorem on the generalized stieltjes transform and its applications abstract this paper presents a proof of parseval goldstein type theorem involving the generalized stieltjes transform and the classical laplace transform the theorem is then shown to yield a number of results about the widder potential transform and the laguerre transform some illustrative examples are also given

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

hopf algebraic structure of families of trees in this paper we describe hopf algebras which are associated with certain families of trees these hopf algebras originally arose in a natural fashion one of the authors 5 was investigating data structures based on trees which could be used to efficiently compute certain differential operators given data structures such as trees which can be multiplied and which act as higherorder derivations on an algebra one expects to find a hopf algebra of some sort we were pleased to find that not only was there a hopf algebra associated with these data structures but that it could be used to give new proofs of enumerations of such objects as rooted trees and ordered rooted trees previous work applying hopf algebras to combinatorial objects such as 10 13 or 14 has concerned itself with algebraic structures on polynomial algebras and on partially ordered sets rather than on trees themselves

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

a dynamic programming method for single machine scheduling abstract this paper treats the single machine scheduling problem that minimizes the sum of general cost functions defined for jobs e g weighted sums of earliness and tardiness since the straightforward dynamic programming recursion requires 2 n states where n is the number of jobs an ssdp successive sublimation dynamic programming method is developed to keep the number of generated states within manageable level by utilizing several types of dynamic programming recursions representing varied degrees of state space relaxations computational results on problem instances of up to n 35 exhibit a clear superiority of this ssdp approach over the original dynamic programming recursion

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

new identities and parseval type relations for the generalized integral transforms in the present paper the authors consider several new integral transforms including the l 4 n transform the p 4 n transform the f s 2 n transform and the f c 2 n transform as generalizations of the classical laplace transform the classical stieltjes transform the classical fourier sine transform and the classical fourier cosine transform respectively identities involving these transforms are given using this identities a number of new parseval goldstein type identities are obtained some examples are also given as illustrations of the results presented here

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

new identities involving the laplace and the ℒ 2 transforms and their applications in the present paper parseval goldstein type theorems involving the 2 transforms and the laplace transform are proved the theorems are then shown to yield a number of new identities involving several well known integral transforms and special functions using the theorem and its corollaries a number of interesting infinite integrals of elementary and special functions are presented some illustrative examples are also given

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

lie algebras associated to systems of dyson schwinger equations we consider systems of combinatorial dyson schwinger equations in the connes kreimer hopf algebra hi of rooted trees decorated by a set i let h s be the subalgebra of hi generated by the homogeneous components of the unique solution of this system if it is a hopf subalgebra we describe it as the dual of the enveloping algebra of a lie algebra g s of one of the following types 1 g s is an associative algebra of paths associated to a certain oriented graph 2 or g s is an iterated extension of the faa di bruno lie algebra 3 or g s is an iterated extension of an infinite dimensional abelian lie algebra we also describe the character groups of h s

single machine scheduling with symmetric earliness and tardiness penalties abstract this research focuses on scheduling jobs with varying processing times and distinct due dates on a single machine subject to earliness and tardiness penalties hence this work will find application in a just in time jit production environment the scheduling problem is formulated as a 0 1 linear integer program with three sets of constraints where the objective is to minimize the sum of the absolute deviations between job completion times and their respective due dates the first two sets of constraints are equivalent to the supply and demand constraints of an assignment problem the third set which represents the process time non overlap constraints is relaxed to form the lagrangian dual problem the dual problem is then solved using the subgradient algorithm efficient heuristics have also been developed in this work to yield initial primal feasible solutions and to convert primal infeasible solutions to feasibility the computational results show that the relative deviation from optimality obtained by the subgradient algorithm is less than 3 for problem sizes varying from 10 to 100 jobs