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

minor identities for quasi determinants and quantum determinants we present several identities involving quasi minors of noncommutative generic matrices these identities are specialized to quantum matrices yieldingq analogues of various classical determinantal formulas

matroid representations and free arrangements we show that terao s conjecture freeness of the module of logarithmic forms at a hyperplane arrangement is determined by its abstract matroid holds over fields with at most four elements however an example demonstrates that the field characteristic has to be fixed for this

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

superconvergence and a posteriori error estimates of splitting positive definite mixed finite element methods for elliptic optimal control problems in this paper we investigate superconvergence and a posteriori error estimates of splitting positive definite mixed finite element methods for elliptic optimal control problems the presented scheme is independent symmetric and positive definite for the state variables and the adjoint state variables moreover the matching relation i e lbb condition between the mixed element spaces v h and wh is not necessary thus we can choose the approximation spaces more flexibly in order to derive the superconvergence we will use classical mixed finite element spaces the state and co state are approximated by the lowest order raviart thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions at first we derive some superconvergence properties for the control variable the state variables and the adjoint state variables then using a recovery operator we obtain a superconvergence result for the control variable next combining energy approach with postprocessing method we derive a posteriori error estimates for optimal control problems we will show that the method does not involve the jump residuals which make the analysis simpler finally a numerical example is given to demonstrate the theoretical results on superconvergence

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

noncommutative rational functions invariant under the action of a finite solvable group abstract this paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in d generators these invariant skew fields are always finitely generated which contrasts with the free algebra case for abelian groups or solvable groups g with a well behaved representation theory it is shown that the invariant skew fields are free on g d 1 1 generators finally positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented

whittaker coinvariants for gl m n abstract let w m n be the finite w algebra attached to the principal nilpotent orbit in the general linear lie superalgebra gl m n c in this paper we study the whittaker coinvariants functor which is an exact functor from category o for gl m n c to a certain category of finite dimensional modules over w m n we show that this functor has properties similar to soergel s functor v in the setting of category o for a semisimple lie algebra we also use it to compute the center of w m n explicitly and deduce consequences for the classification of blocks of o up to morita derived equivalence

a formula for the characteristic polynomial of an arrangement on etablit une formule permettant de calculer le polynome caracteristique d un arrangement constitue d un ensemble fini d hyperplans en termes de serie de poincarre de modules gradues

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

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

vertex operators arising from jacobi trudi identities we give an interpretation of the boson fermion correspondence as a direct consequence of the jacobi trudi identity this viewpoint enables us to construct from a generalized version of the jacobi trudi identity the action of a clifford algebra on the polynomial algebras that arrive as analogues of the algebra of symmetric functions a generalized giambelli identity is also proved to follow from that identity as applications we obtain explicit formulas for vertex operators corresponding to characters of the classical lie algebras shifted schur functions and generalized schur symmetric functions associated to linear recurrence relations

inversion des matrices de hankel abstract hankel determinants can be viewed as special schur symmetric functions this provides without computations of determinants or matrices most of the algebra of hankel determinants and explains their links with other classical fields

malcev neumann series on the free group and questions of rationality let g be an ordered free group b a basis contained in g the sub monoid spanned by b and k a field we construct a family q n g n n of sub division rings in k m g the division ring consisting of the so called malcev neumann series describe their intersections with k and give some applications concerning rational series in k

finite element approximation of parabolic time optimal control problems the numerical approximation of a parabolic time optimal control problem via piecewise linear splines is considered at each stage of the approximation a bang bang approximate control is selected by solving for its switching times as the solution of a constrained nonlinear least squares optimization problem the well posedness of the approximation scheme is shown and the rate of convergence to the exact solution investigated numerical results for some one and two dimensional parabolic control problems are given

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

adaptive finite element approximation for distributed elliptic optimal control problems in this paper sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems these error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach our numerical results indicate that the sharp error estimators work satisfactorily in guiding the mesh adjustments and can save substantial computational work

the campbell baker hausdorff dynkin formula and solutions of differential equations the campbell baker hausdorff dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems specifically suppose u t takes values in a c manifold and satisfies the initial value problem u t a t u t u 0 a where a t is a vector field on the manifold depending continuously on t then u t exp z t a here exp z a means the solution at s 1 to v s z v s v 0 a for z t r 1 pr e r2r 1e tr t x a s 1 a s 2 a s r ds as t 0 where tr t s rr 0 s1 s2 sr t pr is the set of permutations on 1 r e is the number of errors in ordering consecutive terms in 1 r and is the usual commutator of vector fields under appropriate analyticity assumptions the series for z t is convergent for small t this formula gives an explicit formulation of results of k t chen published in 1957 applications are given to problems in sub riemannian geometry and to improving convergence estimates for the campbell baker hausdorff dynkin formula in the context of banach algebras our formula can be thought of as a noncommutative generalization of the familiar formula u t a exp 0t a s ds in the scalar linear case in the same way that the campbell baker hausdorff dynkin formula is a noncommutative generalization of the familiar formula exey ex y

three dimensional gaussian fluctuations of non commutative random surfaces along time like paths abstract we construct a continuous time non commutative markov operator on u gl n with corresponding morphisms u gl n end l 2 u n at any fixed times 0 t 1 t 2 this is an analog of a continuous time non commutative markov operator on the group von neumann algebra v n u n constructed in 16 and is a variant of discrete time non commutative random walks on u gl n 2 9 it is also shown that when restricting to the gelfand tsetlin subalgebra of u gl n the non commutative markov operator matches a 2 1 dimensional random surface model introduced in 7 as an application it is then proved that the moments converge to an explicit gaussian field along time like paths combining with 7 which showed convergence to the gaussian free field along space like paths this computes the entire three dimensional gaussian field in particular it matches a gaussian field from eigenvalues of random matrices 5

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

analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls an optimal control problem for impressed cathodic systems in electrochemistry is studied the control in this problem is the current density on the anode a matching objective functional is considered we first demonstrate the existence and uniqueness of solutions for the governing partial differential equation with a nonlinear boundary condition we then prove the existence of an optimal solution next we derive a necessary condition of optimality and establish an optimality system of equations finally we define a finite element algorithm and derive optimal error estimates

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

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

a priori error estimates for space time finite element discretization of parabolic optimal control problems part i problems without control constraints in this paper we develop a priori error analysis for galerkin finite element discretizations of optimal control problems governed by linear parabolic equations the space discretization of the state variable is done using usual conforming finite elements whereas the time discretization is based on discontinuous galerkin methods for different types of control discretizations we provide error estimates of optimal order with respect to both space and time discretization parameters the paper is divided into two parts in the first part we develop some stability and error estimates for space time discretization of the state equation and provide error estimates for optimal control problems without control constraints in the second part of the paper the techniques and results of the first part are used to develop a priori error analysis for optimal control problems with pointwise inequality constraints on the control variable

a splitting positive definite mixed finite element method for elliptic optimal control problem in this paper we propose a splitting positive definite mixed finite element method mfem for the approximation of convex optimal control problem governed by elliptic equations with control constraints by selecting the variation functional properly the presented procedure can be split into two independent symmetric and positive definite weak formula for the unknown state variable y and for the unknown flux variable s it then follows from the first order necessary and sufficient optimality condition we deduce another two corresponding adjoint state equations for z and w which are also independent symmetric and positive definite also a variational inequality for the control variable u is involved convergence analysis shows that the method yields the approximate solutions with optimal accuracy in l 2 w u norm for the control u l 2 w norm for the original state y and adjoint state z h div w norm for the flux state s and adjoint state w without requiring the lbb consistency condition finally some numerical examples are presented to confirm our theoretical results

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

a decomposition of solomon s descent algebra abstract a descent class in the symmetric group sn is the collection of permutations with a given descent set it was shown by l solomon j algebra41 1976 255 264 that the product in the group algebra q sn of two descent classes is a linear combination of descent classes thus descent classes generate a subalgebra of q sn we refer to it here as solomon s descent algebra and denote it by n this algebra is not semisimple but it has a faithul representation in terms of upper triangular matrices the main goal of this paper is a decomposition of its multiplicative structure it develops that n acts in a natural way on the so called lie monomials this action has a purely combinatorial description and is a crucial tool in the construction of a complete set of indecomposable representations of n in particular we obtain a natural basis of irreducible orthogonal idempotents indexed by partitions of n for the quotient n n natural bases of nilpotents and idempotents for the subspaces e ne for two arbitrary partitions and are also constructed and the dimensions of these spaces are given a combinatorial interpretation in terms of the so called decreasing factorization of an arbitary word into a product of lyndon words

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

an extension of pontryagin s principle for state constrained optimal control of semilinear elliptic equations and variational inequalities this paper deals with state constrained optimal control problems governed by semilinear elliptic equations or variational inequalities by using ekeland s principle a minimum principle of pontryagin s type under some stability conditions of the optimal cost with respect to the state constraints is derived

a hopf algebra approach to inner plethysm abstract we use the hopf algebra structure of the algebra of symmetric functions to study the adams operators of the complex representation rings of symmetric groups and we give new proofs of all of littlewood s formulas for inner plethysm we also study the adams operations for orthogonal and symplectic group characters

superconvergence of mixed finite element approximations over quadrilaterals a superconvergence result is established in this article for approximate solutions of second order elliptic equations by mixed finite element methods over quadrilaterals the superconvergence indicates an accuracy of cal o h k 2 for the mixed finite element approximation if the raviart thomas or brezzi douglas fortin marini elements of order k are employed with optimal error estimate of cal o h k 1 numerical experiments are presented to illustrate the theoretical result

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

superconvergence of mixed finite element methods for optimal control problems t in this paper we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods the state and co state variables are approximated by the lowest order raviart thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique we derive l 2 superconvergence properties for the flux functions along the gauss lines and for the scalar functions at the gauss points via mixed projections moreover global l 2 superconvergence results are obtained by virtue of an interpolation postprocessing technique thus based on these superconvergence estimates some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods finally some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence

a generating set of solomon s descent algebra abstract solomon s descent algebra is generated by sums of descent classes corresponding to certain hook shapes this particularly implies that the ring of class functions of any finite symmetric group s n is generated by the irreducible characters corresponding to certain hook partitions of n as another consequence a second generating set of solomon s descent algebra and of the ring of class functions of s n is obtained related to the major index of permutations

l infty estimates for approximated optimal control problems an optimal control problem for a two dimensional elliptic equation is investigated with pointwise control constraints this paper is concerned with discretization of the control by piecewise linear functions the state and the adjoint state are discretized by linear finite elements approximation of order h in the l infty norm is proved in the main result

the characteristic polynomial of a multiarrangement abstract given a multiarrangement of hyperplanes we define a series by sums of the hilbert series of the derivation modules of the multiarrangement this series turns out to be a polynomial using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arrangement the characteristic polynomial of an arrangement is a combinatorial invariant but this generalized characteristic polynomial is not however when the multiarrangement is free we are able to prove the factorization theorem for the characteristic polynomial the main result is a formula that relates global data to local data of a multiarrangement given by the coefficients of the respective characteristic polynomials this result gives a new necessary condition for a multiarrangement to be free consequently it provides a simple method to show that a given multiarrangement is not free

superconvergence of quadratic optimal control problems by triangular mixed finite element methods the aim of this work is to investigate the discretization of a quadratic convex optimal control problem using the mixed finite element method the state and co state are approximated by the order k 1 raviart thomas mixed finite element spaces and the control is approximated by piecewise constant functions we construct an interpolation of the exact control and a projection of the discrete scalar co state to be the approximated solution of the control variable for the continuous optimal control problem as a result it can be proved that the difference between the interpolation and the piecewise constant approximation has superconvergence property for the control of order h 3 2 for k 0 and of order h 2 for k 1 moreover only for the order k 1 raviart thomas mixed finite element approximation does the postprocessing technique possess the superconvergence property of order h 2 finally some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence

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

generalized exponents of a free arrangement of hyperplanes and shepherd todd brieskorn formula we define an n arrangement as a finite family of hyperplanes through the origin in c 1 in 11 and 12 we studied the free arrangement and defined its structure sequence their definitions will be given again in sect 2 in this article we say the generalized exponents instead of the structure sequence let d o d 1 d be the generalized exponents of a free n arrangement x let xj u h our main result is hex

vandermonde and wronskian matrices over division rings in l the first author initiated the study of vandermonde matrices with entries over a division ring k even in the case when k is a field the definition of a vandermonde matrix in l is already more general than that in the classical sense as it takes into account a given endomorphism s of k under the epilogue of l it was further pointed out that the results of that paper remain valid if in the definition of a vandermonde matrix one allows for a given s derivation d and uses the appropriate definition of the power functions in the s d setting such a generalization is worthwhile because in this setting the study of vandermonde matrices over k also encompasses the study of wronskian matrices with respect to d so results on vandermonde matrices may be used to study linear differential equations arising from the operator d more generally vandermonde matrices seem to be an important tool in studying the skew polynomial ring k t s d so a deeper understanding of vandermonde matrices should be important to the study of the arithmetic of a division ring the purpose of this paper is threefold first we shall develop under sec tion 2 the basic facts on skew polynomials in the s d setting define the evaluation of such polynomials and prove the all important product theorem 2 7 second we shall give a general computation of the rank of a vandermonde matrix with respect to s d as in l the computation is first reduced to the case when the elements a a used to build the vandermonde matrix are pairwise s d conjugate in this case the rank of the vandermonde matrix is computed by the dimension of a vector space over a certain division subring of k cf theorem 4 5 last under

finite dimensional approximation of a class of constrained nonlinear optimal control problems an abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed nonlinearities occur in both the objective functional and the constraints the framework includes an abstract nonlinear optimization problem posed on infinite dimensional spaces an approximate problem posed on finite dimensional spaces together with a number of hypotheses concerning the two problems the framework is used to show that optimal solutions exist to show that lagrange multipliers may be used to enforce the constraints to derive an optimality system from which optimal states and controls may be deduced and to derive existence results and error estimates for solutions of the approximate problem the abstract framework and the results derived from that framework are then applied to three concrete control or optimization problems and their approximation by finite element methods the first involves the von k arm an plate equations of nonlinear elasticity the second the ginzburg landau equations of superconductivity and the third the navier stokes equations for incompressible viscous flows

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

superconvergence properties of optimal control problems an optimal control problem for a two dimensional 2 d elliptic equation is investigated with pointwise control constraints this paper is concerned with discretization of the control by piecewise constant functions the state and the adjoint state are discretized by linear finite elements approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state it is proved that these approximations have convergence order h2