algebraic and combinatorial structures on pairs of twin binary trees abstract we give a new construction of a hopf algebra defined first by reading 2005 rea05 whose bases are indexed by objects belonging to the baxter combinatorial family i e baxter permutations pairs of twin binary trees etc our construction relies on the definition of the baxter monoid analog of the plactic monoid and the sylvester monoid and on a robinson schensted like correspondence and insertion algorithm indeed the baxter monoid leads to the definition of a lattice structure over pairs of twin binary trees and the definition of a hopf algebra the algebraic properties of this hopf algebra are studied and among other multiplicative bases are provided and freeness and self duality proved

hopf algebra of the planar binary trees is a sub hopf algebra there is a basis qn of soln such that the composite map k qn soln k sn has the following property its linear dual k sn k qn is induced by a set theoretic map sn qn in between sn and qn there is an intermediate set yn which is made of planar binary trees with n vertices for instance for n 3 the number of elements in s3 y3 and q3 is 6 5 and 4 respectively the projection from sn to qn is the composite of two maps sn w yn w qn article no ai981759

hopf co addition for free magma algebras and the non associative hausdorff series abstract generalizations of the series exp and log to noncommutative non associative and other types of algebras were considered by m lazard and recently by v drensky and l gerritzen there is a unique power series exp x in one non associative variable x such that exp x exp x exp 2 x exp 0 1 we call the unique series h h x y in two non associative variables satisfying exp h exp x exp y the non associative hausdorff series and we show that the homogeneous components h n of h are primitive elements with respect to the co addition for non associative variables we describe the space of primitive elements for the co addition in non associative variables using taylor expansion and a projector onto the algebra a 0 of constants for the partial derivations by a theorem of kurosh a 0 is a free algebra we describe a procedure to construct a free algebra basis consisting of primitive elements

a categorification of biclosed sets of strings abstract we consider a closure space known as biclosed sets of strings of a gentle algebra of finite representation type palu pilaud and plamondon proved that the collection of all biclosed sets of strings forms a lattice and moreover that this lattice is congruence uniform many interesting examples of finite congruence uniform lattices may be represented as the lattice of torsion classes of an associative algebra we introduce a generalization the lattice of torsion shadows and we prove that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows when every indecomposable module over the gentle algebra is a brick finite congruence uniform lattices admit an alternate partial order known as the core label order in many cases the core label order of a congruence uniform lattice is isomorphic to a lattice of wide subcategories of an associative algebra analogous to torsion shadows we introduce wide shadows and prove that the core label order of the lattice of biclosed sets is isomorphic to a lattice of wide shadows

random martingales and localization of maximal inequalities abstract let x d be a metric measure space for r 0 consider the hardy littlewood maximal operator m r f x def sup r r 1 b x r b x r f d we show that if there is an n 1 such that one has the microdoubling condition b x 1 1 n r b x r for all x x and r 0 then the weak 1 1 norm of m r has the following localization property m r l 1 x l 1 x sup r 0 m r r n r l 1 x l 1 x an immediate consequence is that if x d is ahlfors david n regular then the weak 1 1 norm of m r is n log n generalizing a result of stein and stromberg 1983 47 we show that this bound is sharp by constructing a metric measure space x d that is ahlfors david n regular for which the weak 1 1 norm of m 0 is n log n the localization property of m r is proved by assigning to each f l 1 x a distribution over random martingales for which the associated random doob maximal inequality controls the weak 1 1 inequality for m r

intervals of balanced binary trees in the tamari lattice we show that the set of balanced binary trees is closed by interval in the tamari lattice we establish that the intervals t t where t and t are balanced binary trees are isomorphic as posets to a hypercube we introduce synchronous grammars that allow to generate tree like structures and obtain fixed point functional equations to enumerate these we also introduce imbalance tree patterns and show that they can be used to describe some sets of balanced binary trees that play a particular role in the tamari lattice finally we investigate other families of binary trees that are also closed by interval in the tamari lattice

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

periodic solutions of some fourth order nonlinear differential equations abstract in this paper using mawhin s continuation theorem we obtain existence results concerning t periodic solutions of fourth order nonlinear differential equations of the type u p u g t u e t where p r g r 2 r is continuous t periodic in t and e r r is continuous t periodic with mean value zero applications are given for the extended fisher kolmogorov and swift hohenberg equations with forcing term

weak bruhat order on the set of faces of the permutohedron and the associahedron abstract we define a partial order on the set of faces of the permutohedron which extends the weak bruhat order of the symmetric group the restriction of this order gives a partially ordered set structure to the set of faces of the associahedron which extends the tamari order defined on the set of planar binary rooted trees these orders are used to describe associative algebra structures on the vector spaces spanned by the set of faces of the permutohedra on one hand and on the vector space spanned by the set of planar rooted trees on the other hand our results extend the description of the dendriform algebra structure on the spaces spanned by permutations and by planar binary trees in terms of the weak bruhat order and the tamari order respectively obtained by j l loday and the second author in j l loday m ronco order structure and the algebra of permutations and of planar binary trees j algebraic combin 15 2002 253 270

wavelet galerkin solutions for one dimensional partial differential equations in this paper we describe how wavelets may be used to solve partial differential equations these problems are currently solved by techniques such as finite differences finite elements and multi grid the wavelet method however offers several advantages over traditional methods wavelets have the ability to represent functions at different levels of resolution thereby providing a logical means of developing a hierarchy of solutions furthermore compactly supported wavelets such as those due to daubechies1 are localized in space which means that the solution can be refined in regions of high gradient e g stress concentrations without having to regenerate the mesh for the entire problem in order to demonstrate the wavelet technique we consider the one dimensional counterpart of helmholtz s equation by comparison with a simple finite difference solution to this problem with periodic boundary conditions we show how a wavelet technique may be efficiently developed dirichlet boundary conditions are then imposed using the capacitance matrix method described by proskurowski and widlund2 and others the convergence rates of the wavelet solutions are examined and they are found to compare extremely favourably to the finite difference solutions preliminary investigations also indicate that the wavelet technique is a strong contender to the finite element method at least for problems with simple geometries

ideals of quasi symmetric functions and super covariant polynomials for sn the aim of this work is to study the quotient ring r n of the ring q x 1 x n over the ideal j n generated by non constant homogeneous quasi symmetric functions we prove here that the dimension of r n is given by c n the n th catalan number this is also the dimension of the space sh n of super covariant polynomials that is defined as the orthogonal complement of j n with respect to a given scalar product we construct a basis for r n whose elements are naturally indexed by dyck paths this allows us to understand the hilbert series of sh n in terms of number of dyck paths with a given number of factors

wavelets and the numerical solution of partial differential equations we present a new numerical method for the solution of partial differential equations in nonseparable domains the method uses a wavelet galerkin solver with a nontrivial adaptation of the standard capacitance matrix method the numerical solutions exhibit spectral convergence with regard to the order of the compactly supported daubechies wavelet basis furthermore the rate of convergence is found to be independent of the geometry we solve the helmholtz equation since for variations in the parameter the solutions have qualitative properties that well illustrate the applications of our method

crystals and trees quasi kashiwara operators monoids of binary trees and robinson schensted type correspondences abstract kashiwara s crystal graphs have a natural monoid structure that arises by identifying words labelling vertices that appear in the same position of isomorphic components the celebrated plactic monoid the monoid of young tableaux arises in this way from the crystal graph for the q analogue of the general linear lie algebra gl n and the so called kashiwara operators interact beautifully with the combinatorics of young tableaux and with the robinson schensted knuth correspondence the authors previously constructed an analogous quasi crystal structure for the related hypoplactic monoid the monoid of quasi ribbon tableaux which has similarly neat combinatorial properties this paper constructs an analogous crystal type structure for the sylvester and baxter monoids the monoids of binary search trees and pairs of twin binary search trees respectively both monoids are shown to arise from this structure just as the plactic monoid does from the usual crystal graph the interaction of the structure with the sylvester and baxter versions of the robinson schensted knuth correspondence is studied the structure is then applied to prove results on the number of factorizations of elements of these monoids and to prove that both monoids satisfy non trivial identities

combinatorics of cyclic shifts in plactic hypoplactic sylvester baxter and related monoids abstract the cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift this paper examines the cyclic shift graphs of plactic like monoids whose elements can be viewed as combinatorial objects of some type aside from the plactic monoid itself the monoid of young tableaux examples include the hypoplactic monoid quasi ribbon tableaux the sylvester monoid binary search trees the stalactic monoid stalactic tableaux the taiga monoid binary search trees with multiplicities and the baxter monoid pairs of twin binary search trees it was already known that for many of these monoids connected components of the cyclic shift graph consist of elements that have the same evaluation that is contain the same number of each generating symbol this paper focuses on the maximum diameter of a connected component of the cyclic shift graph of these monoids in the rank n case for the hypoplactic monoid this is n 1 for the sylvester and taiga monoids at least n 1 and at most n for the stalactic monoid 3 except for ranks 1 and 2 when it is respectively 0 and 1 for the plactic monoid at least n 1 and at most 2 n 3 the current state of knowledge including new and previously known results is summarized in a table

a set operad of formal fractions and dendriform like sub operads abstract we introduce an operad of formal fractions abstracted from the mould operads and containing both the dendriform and the tridendriform operads we consider the smallest set operad contained in this operad and containing four specific elements of arity two corresponding to the generators and the associative elements of the dendriform and tridendriform operads we obtain a presentation of this operad by binary generators and quadratic relations and an explicit combinatorial description using a new kind of bi coloured trees similar results are also presented for related symmetric operads

the algebra of binary search trees we introduce a monoid structure on the set of binary search trees by a process very similar to the construction of the plactic monoid the robinson schensted insertion being replaced by the binary search tree insertion this leads to a new construction of the algebra of planar binary trees of loday ronco defining it in the same way as non commutative symmetric functions and free symmetric functions we briefly explain how the main known properties of the loday ronco algebra can be described and proved with this combinatorial point of view and then discuss it from a representation theoretical point of view which in turns leads to new combinatorial properties of binary trees

jordan trialgebras and post jordan algebras abstract we compute minimal sets of generators for the s n modules n 4 of multilinear polynomial identities of arity n satisfied by the jordan product and the jordan diproduct resp pre jordan product in every triassociative resp tridendriform algebra these identities define jordan trialgebras and post jordan algebras jordan analogues of the lie trialgebras and post lie algebras introduced by dotsenko et al pei et al vallette loday we include an extensive review of analogous structures existing in the literature and their interrelations in order to identify the gaps filled by our two new varieties of algebras we use computer algebra linear algebra over finite fields representation theory of symmetric groups to verify in both cases that every polynomial identity of arity 6 is a consequence of those of arity 4 we conjecture that in both cases the next independent identities have arity 8 imitating the glennie identities for jordan algebras we formulate our results as a commutative square of operad morphisms which leads to the conjecture that the squares in a much more general class are also commutative

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

the weak type 1 1 bounds for the maximal function associated to cubes grow to infinity with the dimension let md be the centered hardy littlewood maximal function associated to cubes in r d with lebesgue measure and let cd denote the lowest constant appearing in the weak type 1 1 inequality satised by md we show that cd 1 as d 1 thus answering for the case of cubes a longstanding

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

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

wavelet galerkin method for fourth order linear and nonlinear differential equations abstract in this paper we propose a wavelet galerkin method for fourth order linear and nonlinear differential equations using compactly supported daubechies wavelets 2 term connection coefficients have been effectively used for a computationally economical evaluation of higher order derivatives the orthogonality and compact support properties of basis functions lead to highly sparse linear systems the quasilinearization strategy is effectively employed in dealing with wavelet coefficients of nonlinear problems the stability and the convergence analysis in the form of error analysis have been carried out an efficient compression algorithm is proposed to reduce the computational cost of the method finally the method is tested on several examples and found to be in good agreement with exact solution

tree expansion in time dependent perturbation theory the computational complexity of time dependent perturbation theory is well known to be largely combinatorial whatever the chosen expansion method and family of parameters combinatorial sequences goldstone and other feynman type diagrams etc we show that a very efficient perturbative expansion both for theoretical and numerical purposes can be obtained through an original parametrization by trees and generalized iterated integrals we emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics these tools are applied to the adiabatic approximation and the effective hamiltonian we prove perturbatively and nonperturbatively the convergence of morita s generalization of the gell mann and low wave function we show that summing all the terms associated with the same tree leads to an utter simplification where the sum is simpler than any of its terms finally we recover the rayleigh schrodinger time independent equation for the wave operator and we give an explicit nonrecursive expression for the term corresponding to an arbitrary tree

cambrian hopf algebras abstract cambrian trees are oriented and labeled trees which fulfill local conditions around each node generalizing the classical conditions for binary search trees similar to binary trees for the tamari lattice cambrian trees provide convenient combinatorial models for n reading s cambrian lattices of type a based on a natural surjection from signed permutations to cambrian trees we define the cambrian hopf algebra extending j l loday and m ronco s algebra on binary trees we describe combinatorially the products and coproducts of both the cambrian algebra and its dual in terms of operations on cambrian trees we then construct the baxter cambrian algebra which extends s law and n reading s baxter hopf algebra on rectangulations and s giraudo s equivalent hopf algebra on twin binary trees