lie theory for nilpotent l algebras the deligne groupoid is a functor from nilpotent differential graded lie algebras concentrated in positive degrees to groupoids in the special case of lie algebras over a field of characteristic zero it gives the associated simply connected lie group we generalize the deligne groupoid to a functor from l algebras concentrated in degree n to n groupoids we actually construct the nerve of the n groupoid which is an enriched kan complex the construction of gamma is quite explicit it is based on dupont s proof of the de rham theorem and yields higher dimensional analogues of holonomy and of the campbell hausdorff formula in the case of abelian l algebras i e chain complexes the functor is the dold kan simplicial set

unifying derived deformation theories abstract we develop a framework for derived deformation theory valid in all characteristics this gives a model category reconciling local and global approaches to derived moduli theory in characteristic 0 we use this to show that the homotopy categories of dglas and shlas l algebras considered by kontsevich hinich and manetti are equivalent and are compatible with the derived stacks of toen vezzosi and lurie another application is that the cohomology groups associated to any classical deformation problem in any characteristic admit the same operations as andre quillen cohomology

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

words and forbidden factors given a finite or infinite word v we consider the set m v of minimal forbidden factors of v we show that the set m v is of fundamental importance in determining the structure of the word v in the case of a finite word w we consider two parameters that are related to the size of m w the first counts the minimal forbidden factors of w and the second gives the length of the longest minimal forbidden factor of w we derive sharp upper and lower bounds for both parameters we prove also that the second parameter is related to the minimal period of the word w we are further interested to the algorithmic point of view indeed we design linear time algorithm for the following two problems i given w construct the set m w and conversely ii given m w reconstruct the word w in the case of an infinite word x we consider the following two functions gx that counts for each n the allowed factors of x of length n and fx that counts for each n the minimal forbidden factors of x of length n we address the following general problem what information about the structure of x can be derived from the pair gx fx we prove that these two functions characterize up to the automorphism exchanging the two letters the language of factors of each single infinite sturmian word

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

on matrix differential equations and abstract fg algorithm abstract there is well known relationship between both the isospectral flows and the unitarily self equivalent flows and the abstract fg eigenvalue algorithm namely the values of the isospectral and self equivalent flows at integer times coincide with the iterates of fg algorithm we show that such a relation is characteristic of a whole class of matrix differential equations all these equations are built by means of various representations of an arbitrary lie algebra on k n n

simple finite dimensional double algebras abstract a double algebra is a linear space v equipped with linear map v v v v additional conditions on this map lead to the notions of lie and associative double algebras we prove that simple finite dimensional lie double algebras do not exist over an arbitrary field and all simple finite dimensional associative double algebras over an algebraically closed field are trivial over an arbitrary field every simple finite dimensional associative double algebra is commutative a double algebra structure on a finite dimensional space v is naturally described by a linear operator r on the algebra end v of linear transformations of v double lie algebras correspond in this sense to skew symmetric rota baxter operators double associative algebra structures to left averaging operators

the wess zumino witten term of the m5 brane and differential cohomotopy we combine rational homotopy theory and higher lie theory to describe the wess zumino witten wzw term in the m5 brane sigma model we observe that this term admits a natural interpretation as a twisted 7 cocycle on super minkowski spacetime with coefficients in the rational 4 sphere this exhibits the wzw term as an element in twisted cohomology with the twist given by the cocycle of the m2 brane we consider integration of this rational situation to differential cohomology and differential cohomotopy

subword complexity and power avoidance abstract we begin a systematic study of the relations between subword complexity of infinite words and their power avoidance among other things we show that the thue morse word has the minimum possible subword complexity over all overlap free binary words and all 7 3 power free binary words but not over all 7 3 power free binary words the twisted thue morse word has the maximum possible subword complexity over all overlap free binary words but no word has the maximum subword complexity over all 7 3 power free binary words if some word attains the minimum possible subword complexity over all square free ternary words then one such word is the ternary thue word the recently constructed 1 2 bonacci word has the minimum possible subword complexity over all symmetric square free ternary words

formality and kontsevich duflo type theorems for lie pairs newcommand poly operatorname poly bullet newcommand td operatorname td l a nabla frac 1 2 newcommand cx 1 operatorname tot big gamma lambda bullet a vee otimes r mathcal 1 poly big newcommand cy 1 mathbb h bullet operatorname ce a mathcal 1 poly kontsevich s formality theorem states that there exists an l infty quasi isomorphism from the dgla t poly m of polyvector fields on a smooth manifold m to the dgla d poly m of polydifferential operators on m which extends the classical hochschild kostant rosenberg map in this paper we extend kontsevich s formality theorem to lie pairs a framework which includes a range of diverse geometric contexts such as complex manifolds foliations and mathfrak g manifolds the spaces cx t and cx d associated with a lie pair l a each carry an l infty algebra structure canonical up to l infty isomorphism these two spaces serve as replacements for the spaces of polyvector fields and polydifferential operators respectively their corresponding cohomology groups cy t and cy d admit canonical gerstenhaber algebra structures we establish the following formality theorem for lie pairs there exists an l infty quasi isomorphism from cx t to cx d whose first taylor coefficient is equal to operatorname hkr circ td here td acts on cx t by contraction furthermore we prove a kontsevich duflo type theorem for lie pairs the hochschild kostant rosenberg map twisted by the square root of the todd class of the lie pair l a is an isomorphism of gerstenhaber algebras from cy t to cy d as applications we establish formality theorems and kontsevich duflo type theorems for complex manifolds foliations and mathfrak g manifolds in the case of complex manifolds we recover the kontsevich duflo theorem of complex geometry

generalized lax pairs the modified classical yang baxter equation and affine geometry of lie groups we generalize the usual lax equationd dt l m l byd dt l m l where is an arbitrary representation of a lie algebra g the values ofm in a representation spacev the values ofl the usual classicalr matrix programme for hamiltonian integrable systems is generalized tor matrices taking values in g v ther matrices are then considered as left invariant torsion free covariant derivatives on a lie groupk with lie algebrav the classical yang baxter equation cybe is equivalent to the flatness ofk whereas the modified cybe implies thatk is an affine locally symmetric space an example is discussed

a version of the goldman millson theorem for filtered l algebras abstract in this paper we consider l algebras equipped with complete descending filtrations we prove that under some mild conditions an l quasi isomorphism u l l induces a weak equivalence between the deligne getzler hinich dgh groupoids corresponding to l and l respectively this paper may be considered as a modest addition to foundational paper 10 by ezra getzler

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

palindromic closures using multiple antimorphisms a generalized pseudostandard word u as introduced in 2006 by de luca and de luca is given by a directive sequence of letters from an alphabet a and by a directive sequence of involutory antimorphisms acting on a prefixes of u with increasing length are constructed using a pseudopalindromic closure operator we show that generalized thue morse words tb m with b m n and b m 2 are generalized pseudostandard words if and only if tb m is a periodic word or b m this extends the result of de luca and de luca obtained for the classical thue morse words

presenting higher stacks as simplicial schemes abstract we show that an n geometric stack may be regarded as a special kind of simplicial scheme namely a duskin n hypergroupoid in affine schemes where surjectivity is defined in terms of covering maps yielding artin n stacks deligne mumford n stacks and n schemes as the notion of covering varies this formulation adapts to all hag contexts so in particular works for derived n stacks replacing rings with simplicial rings we exploit this to describe quasi coherent sheaves and complexes on these stacks and to draw comparisons with kontsevich s dg schemes as an application we show how the cotangent complex controls infinitesimal deformations of higher and derived stacks

infinite words with linear subword complexity abstract in section 1 we study the relations among some combinatorial properties of infinite words especially in the case of infinite words with linear subword complexity the main result of section 1 concerns a permutation property of infinite words with linear complexity in section 2 we investigate the special case of the sturmian infinite words the main result is that a sturmian infinite word associated to a real number contains no k th powers iff has a continued fraction expansion with bounded partial quotients in section 3 it is shown that some of the results of sections 1 and 2 are optimal

words and special factors in this paper we consider sets of factors of a given finite word over a finite alphabet which permit us to reconstruct the entire word this analysis is based on the notion of special factor a factor u of a finite word w is called right resp left special if there exist two distinct letters x and y such that ux uy resp xu yu are factors of w a factor is bispecial if it is right and left special a proper box of w is any factor of w of the kind asb with a b letters and s a bispecial factor of w the initial resp terminal box of w is the shortest prefix resp suffix of w which is an unrepeated factor a box is called maximal if it is not a proper factor of another box the main result of the paper is the following theorem maximal box theorem any finite word w is uniquely determined by the initial box the terminal box and the set of maximal boxes a consequence is that a finite word w is uniquely determined by the knowledge of its factors up to the length n max rw kw 1 where kw is the length of the terminal box and rw is the minimal natural number for which there is no right special factor of length rw some structural properties of boxes are studied another important combinatorial notion is that of superbox a superbox is any factor of w of the kind asb with a b letters and such that s is a repeated factor whereas as and sb are unrepeated factors a theorem for superboxes similar to the maximal box theorem is proved some algorithms allowing us to construct boxes and superboxes and conversely to reconstruct the word are given in this combinatorial frame we give an upper and a lower bound to the number of states of a minimal deterministic automaton recognizing the set of the factors of w these bounds are sharper than the known bounds

orlicz norms of sequences of random variables let f i i 1 n be copies of a random variable f and let n be an orlicz function we show that for every x r n the expectation e x i f i n i 1 n is maximal up to an absolute constant if f i i 1 n are independent in that case we show that the expectation e x i f i n i 1 n is equivalent to x m for some orlicz function m depending on n and on distribution of f only we provide applications of this result

properties of free baxter algebras abstract the study of free baxter algebras was started by rota and cartier 30 years ago we continue this study by applying two recent constructions of free baxter algebras we investigate the basic structure of a free baxter algebra and characterize in detail when a free baxter algebra is a domain or a reduced algebra we also describe the nilpotent radical of a free baxter algebra when it is not reduced

conditional entropy of some automatic sequences burrows and sulston 1991 have introduced conditional block entropies hn from information theory in order to give a quantitative measure of disorder for sequences and if possible a characterization of quasicrystalline sequences we give here some properties concerning these block entropies and give an explicit formula for the sequences hn n in n corresponding to the thue morse sequence the rudin shapiro sequence and the paperfolding sequence we deduce from these computations that this measure of disorder does not enable us to distinguish between deterministic sequences even if they have different spectral properties

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

deformations and homotopy theory of relative rota baxter lie algebras we determine the l infty algebra that controls deformations of a relative rota baxter lie algebra and show that it is an extension of the dg lie algebra controlling deformations of the underlying mathsf lie mathsf rep pair by the dg lie algebra controlling deformations of the relative rota baxter operator consequently we define the cohomology of relative rota baxter lie algebras and relate it to their infinitesimal deformations a large class of relative rota baxter lie algebras is obtained from triangular lie bialgebras and we construct a map between the corresponding deformation complexes next the notion of a homotopy relative rota baxter lie algebra is introduced we show that a class of homotopy relative rota baxter lie algebras is intimately related to pre lie infty algebras

on a combinatorial property of sturmian words recall that a semigroup has the property pn if for any sequence of n of its elements two differently permuted products of these n elements are equal let s be an infinite sturmian word on a 2 letter alphabet a we prove that the rees quotient of a by the set of the non factors of s has p4 and that this result is the best possible we prove also that if st is the set of all finite sturmian words then the rees quotient a a st has p8

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

renormalization in quantum field theory and the riemann hilbert problem ii the β function diffeomorphisms and the renormalization group we showed in part i that the hopf algebra of feynman graphs in a given qft is the algebra of coordinates on a complex infinite dimensional lie group g and that the renormalized theory is obtained from the unrenormalized one by evaluating at 0 the holomorphic part of the riemann hilbert decomposition 1 of the loop g provided by dimensional regularization we show in this paper that the group g acts naturally on the complex space x of dimensionless coupling constants of the theory more precisely the formula g 0 gz 1 z 3 3 2 for the effective coupling constant when viewed as a formal power series does define a hopf algebra homomorphism between the hopf algebra of coordinates on the group of formal diffeomorphisms to the hopf algebra this allows first of all to read off directly without using the group g the bare coupling constant and the renormalized one from the riemann hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms this shows that renormalization is intimately related with the theory of non linear complex bundles on the riemann sphere of the dimensional regularization parameter it also allows to lift both the renormalization group and the function as the asymptotic scaling in the group g this exploits the full power of the riemann hilbert decomposition together with the invariance of under a change of unit of mass this not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group g for the full higher pole structure of minimal subtracted counterterms in terms of the residue

deformations of schemes and other bialgebraic structures there has long been a philosophy that every deformation problem in characteristic zero should be governed by a differential graded lie algebra dgla in this paper we show how to construct a simplicial deformation complex sdc governing any bialgebraic deformation problem examples of such problems are deformations of a hopf algebra or of an arbitrary scheme in characteristic zero sdcs and dglas are shown to be equivalent

orlicz sequence spaces spanned by identically distributed independent random variables in lp spaces abstract let 1 p 2 and let l p l p 0 1 be the classical l p space of all classes of p integrable functions on 0 1 it is known that any subspace in l p spanned by a sequence of independent copies of a mean zero random variable f l p is isomorphic to some orlicz sequence space l n we employ methods from interpolation theory to fully describe the class of orlicz functions n which can be obtained by this procedure and to identify the precise connection between n and the distribution of f our methods strengthen and complement classical results of m i kadec as well as those of j bretagnolle and d dacunha castelle

baxter algebras and hopf algebras by applying a recent construction of free baxter algebras we obtain a new class of hopf algebras that generalizes the classical divided power hopf algebra we also study conditions under which these hopf algebras are isomorphic

koszul duality for lie algebroids abstract this paper studies the role of dg lie algebroids in derived deformation theory more precisely we provide an equivalence between the homotopy theories of formal moduli problems and dg lie algebroids over a commutative dg algebra of characteristic zero at the level of linear objects we show that the category of representations of a dg lie algebroid is an extension of the category of quasi coherent sheaves on the corresponding formal moduli problem we describe this extension geometrically in terms of pro coherent sheaves

on the geometry of random convex sets between polytopes and zonotopes abstract in this work we study a class of random convex sets that interpolate between polytopes and zonotopes these sets arise from considering a q t h moment q 1 of an average of order statistics of 1 dimensional marginals of a sequence of n n independent random vectors in r n we consider the random model of isotropic log concave distributions as well as the uniform distribution on an l p n sphere 1 p with respect to the cone probability measure and study the geometry of these sets in terms of the support function and mean width we provide asymptotic formulas for the expectation of these geometric functionals which are sharp up to absolute constants our model includes and generalizes the standard one for random polytopes

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

maurer cartan moduli and models for function spaces we set up a formalism of maurer cartan moduli sets for l algebras and associated twistings based on the closed model category structure on formal differential graded algebras a k a differential graded coalgebras among other things this formalism allows us to give a compact and manifestly homotopy invariant treatment of chevalley eilenberg and harrison cohomology we apply the developed technology to construct rational homotopy models for function spaces

fibonacci morphisms and sturmian words abstract marton kosa 1987 has stated five conjectures devoted to the study of some binary morphisms and the factors of infinite words these morphisms generate positive answers to the first four of these conjectures are given in this paper the main result being the characterization of a class of morphisms whose stationary words are sturmian

dimensional regularization in position space and a forest formula for epstein glaser renormalization we reformulate dimensional regularization as a regularization method in position space and show that it can be used to give a closed expression for the renormalized time ordered products as solutions to the induction scheme of epstein glaser this closed expression which we call the epstein glaser forest formula is analogous to zimmermann s forest formula for bph renormalization for scalar fields the resulting renormalization method is always applicable we compute several examples we also analyze the hopf algebraic aspects of the combinatorics our starting point is the main theorem of renormalization of stora and popineau and the arising renormalization group as originally defined by stuckelberg and petermann

discrepancy chaining and subgaussian processes the research leading to the results presented here has received funding from the european research council under the european community s seventh frame work programme fp7 2007 2013 erc grant agreement 203134 from the israel science foundation grant 666 06 and from the australian research council grant dp0986563

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

renormalization thermodynamic formalism and quasi crystals in subshifts we examine the thermodynamic formalism for a class of renormalizable dynamical systems which in the symbolic space is generated by the thue morse substitution and in complex dynamics by the feigenbaum coullet tresser map the basic question addressed is whether fixed points v of a renormalization operator mathcal r acting on the space of potentials are such that the pressure function beta mapsto mathcal p beta v exhibits phase transitions this extends the work by baraviera leplaideur and lopes on the manneville pomeau map where such phase transitions were indeed detected in this paper however the attractor of renormalization is a cantor set rather than a single fixed point which admits various classes of fixed points of mathcal r some of which do and some of which do not exhibit phase transitions in particular we show it is possible to reach as a ground state a quasi crystal before temperature zero by freezing a dynamical system

the ring of k regular sequences ii in this paper we continue our study of k regular sequences begun in 1992 we prove some new results give many new examples from the literature and state some open problems

an algebraic scheme associated with the noncommutative kp hierarchy and some of its extensions a well known ansatz trace method for soliton solutions turns the equations of the non commutative kp hierarchy and those of certain extensions into families of algebraic sum identities we develop an algebraic formalism in particular involving a mixable shuffle product to explore their structure more precisely we show that the equations of the non commutative kp hierarchy and its extension xnckp in the case of a moyal deformed product as derived in previous work correspond to identities in this algebra furthermore the moyal product is replaced by a more general associative product this leads to a new even more general extension of the non commutative kp hierarchy relations with rota baxter algebras are established

minimal models of quantum homotopy lie algebras via the bv formalism using the batalin vilkovisky formalism of mathematical physics an explicit construction for the minimal model of a quantum l algebra is given as a formal super integral the approach taken herein to these formal integrals is axiomatic and they can be approached using perturbation theory to obtain combinatorial formulae as shown in the appendix additionally there exists a canonical differential graded lie algebra morphism mapping formal functions on homology to formal functions on the whole space an inverse l algebra morphism to this differential graded lie algebra morphism is constructed as a formal super integral using the batalin vilkovisky formalism of mathematical physics an explicit construction for the minimal model of a quantum l algebra is given as a formal super integral the approach taken herein to these formal integrals is axiomatic and they can be approached using perturbation theory to obtain combinatorial formulae as shown in the appendix additionally there exists a canonical differential graded lie algebra morphism mapping formal functions on homology to formal functions on the whole space an inverse l algebra morphism to this differential graded lie algebra morphism is constructed as a formal super integral

some combinatorial properties of the thue morse sequence and a problem in semigroups abstract some combinatorial properties of the special factors of the thue morse sequence in a two letter alphabet are considered an application of this analysis to a problem in semigroups is 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

canonical positions for the factors in paperfolding sequences it has been proved by the first author that the number of factors of length k in any paperfolding sequence is equal to 4k once k 7 we prove here that for every integer k there exists a set of integers pk of cardinality 4k for k 7 such that the factors of length k of any paperfolding sequence are exactly the factors beginning at the places indexed by the integers in pk this gives a bijective proof of the result mentioned above then we give an explicit uniform linear upper bound for the recurrence function of paperfolding sequences finally we study the set of all factors of all paperfolding sequences evaluating their number for a given length and studying the language of these words

on the subword complexity of thue morse polynomial extractions let the subword complexity of a sequence u u n n 0 over a finite set s be the function m p u m where p u m is the number of distinct blocks of length m in u let t t n n 0 denote the thue morse sequence in this paper we study the complexity of the sequences t h t h n n 0 when h n q n is a polynomial with h n n in particular we solve an open problem of allouche and shallit regarding t n 2 n 0 we also study the vector space over z 2z spanned by the sequences t h

what do homotopy algebras form abstract in paper 4 we constructed a symmetric monoidal category s lie mc whose objects are shifted and filtered l algebras here we fix a cooperad c and show that algebras over the operad cobar c naturally form a category enriched over s lie mc following 4 we integrate this s lie mc enriched category to a simplicial category hoalg c whose mapping spaces are kan complexes the simplicial category hoalg c gives us a particularly nice model of an 1 category of cobar c algebras we show that the homotopy category of hoalg c is the localization of the category of cobar c algebras and morphisms with respect to quasi isomorphisms finally we show that the homotopy transfer theorem is a simple consequence of the goldman millson theorem

on weakly bounded empirical processes let f be a class of functions on a probability space and let x 1 x k be independent random variables distributed according to we establish an upper bound that holds with high probability on rm sup f in f i f x i geq t for every t 0 and that depends on a natural geometric parameter associated with f we use this result to analyze the supremum of empirical processes of the form z f left k 1 sum i 1 k f p x i mathbb e f p right for p 1 using the geometry of f we also present some geometric applications of this approach based on properties of the random operator gamma k 1 2 sum i 1 k x i e i where x i i 1 k are sampled according to an isotropic log concave measure on mathbb r n

deformations of associative rota baxter operators abstract rota baxter operators and more generally o operators on associative algebras are important in probability combinatorics associative yang baxter equation and splitting of algebras using a method of uchino we construct an explicit graded lie algebra whose maurer cartan elements are given by o operators this allows us to construct cohomology for an o operator this cohomology can also be seen as the hochschild cohomology of a certain algebra with coefficients in a suitable bimodule next we study linear and formal deformations of an o operator which are governed by the above defined cohomology we introduce nijenhuis elements associated with an o operator which give rise to trivial deformations as applications we conclude deformations of weight zero rota baxter operators associative r matrices and averaging operators

a finiteness condition for semigroups generalizing a theorem of hotzel in 1979 hotzel proved 1 l the following interesting finiteness condition for semigroups let s be a finitely generated semigroup if s verifies the minimal condition on principal left ideals resp right ideals and all subgroups of s have finite orders then s is finite this theorem solved a question posed by coudrain and schiitzenberger 3 who proved a similar result with the only difference that s satisfies the minimal condition on principal bi ideals instead of left ideals moreover two finiteness conditions for semigroups due respectively to simon 16 and to de luca and restivo 4 are based on hotzel s theorem in this paper we prove the following generalization of hotzel s theorem in which one supposes that only finitely generated subgroups are finite

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

on t extensions of the hankel determinants of certain automatic sequences in 1998 allouche peyriere wen and wen considered the thue morse sequence and proved that all the hankel determinants of the period doubling sequence are odd integers we speak of t extension when the entries along the diagonal in the hankel determinant are all multiplied by t we prove that the t extension of each hankel determinant of the period doubling sequence is a polynomial in t whose leading coefficient is the only one to be an odd integer our proof makes use of the combinatorial set up developed by bugeaud and han which appears to be very suitable for this study as the parameter t counts the number of fixed points of a permutation finally we prove that all the t extensions of the hankel determinants of the regular paperfolding sequence are polynomials in t of degree less than or equal to 3

disconnected rational homotopy theory we construct two algebraic versions of homotopy theory of rational disconnected topological spaces one based on differential graded commutative associative algebras and the other one on complete differential graded lie algebras as an application of the developed technology we obtain results on the structure of maurer cartan spaces of complete differential graded lie algebras

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

on the combinatorics of finite words in this paper we consider a combinatorial method for the analysis of finite words recently introduced in colosimo and de luca special factors in biological strings preprint 97 42 dipt matematica univ di roma for the study of biological macromolecules the method is based on the analysis of right special factors of a given word a factor u of a word w is special if there exist at least two occurrences of the factor u in w followed on the right by two distinct letters we show that in the combinatorics of finite words two parameters play an essential role the first denoted by r represents the minimal integer such that there do not exist special factors of w of length r the second that we denote by k is the minimal length of a factor of w which cannot be extended on the right in a factor of w some new results are proved in particular a new characterization in terms of special factors and of r and k is given for the set per of all words w having two periods p and q which are coprimes and such that w p q 2