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

twisted descent algebras and the solomon tits algebra abstract the notion of descent algebra of a bialgebra is lifted to the barratt joyal setting of twisted bialgebras the new twisted descent algebras share many properties with their classical counterparts for example there are twisted analogs of classical lie idempotents and of the peak algebra moreover the universal twisted descent algebra is equipped with two products and a coproduct and there is a fundamental rule linking all three this algebra is shown to be naturally related to the geometry of the coxeter complex of type a

symplectic integration of constrained hamiltonian systems by composition methods recent work reported in the literature suggests that for the long term integration of hamiltonian dynamical systems one should use methods that preserve the symplectic structure of the flow in this paper the symplecticity of numerical integrators is investigated for constrained hamiltonian systems with holonomic constraints the following two results will be derived i it is shown that any first or second order symplectic integrator for unconstrained problems can be generalized to constrained systems such that the resulting scheme is symplectic and preserves the constraints based on this higher order methods can be derived by the same composition methods used for unconstrained problems ii leimkuhler and reich math comp 63 1994 pp 589 605 derived symplectic integrators based on dirac s reformulation of the constrained problem as an unconstrained hamiltonian system however although the unconstrained reformulation can be handled by direct application of any symplectic implicit runge kutta m

conservative integration of rigid body motion by quaternion parameters with implicit constraints summary an angular momentum and energy conserving time integration algorithm for rigid body rotation is formulated in terms of the quaternion parameters and the corresponding four component conjugate momentum vector via hamilton s equations the introduction of an extended mass matrix leads to a symmetric set of eight state space equations of motion the extra inertial parameter serves as a multiplier on the kinematic constraint and it is demonstrated that convergence characteristics are improved by selecting this parameter somewhat larger than the inertial moments external loads enter these equations via the set of momentum equations initially the normalization of the quaternion array is introduced via a lagrange multiplier however this lagrange multiplier can be expressed explicitly in terms of the gradient of the external load potential and elimination of the lagrange multiplier from the final format leaves only an explicit projection applied to the external load potential gradient an algorithm is developed by forming a finite increment of the hamiltonian this procedure identifies the proper selection of increments and mean values and leads to an algorithm with conservation of momentum and energy implementation conservation properties and accuracy of the algorithm are illustrated by examples with a flying box and a spinning top copyright 2012 john wiley sons ltd

explicit lie poisson integration and the euler equations we give a wide class of lie poisson systems for which explicit lie poisson integrators preserving all casimirs can be constructed the integrators are extremely simple and have widespread applicability examples are the free rigid body a moment truncation and a new fast algorithm for the sine bracket truncation of the 2d euler equations

simple and efficient integration of rigid rotations suitable for constraint solvers simple and efficient way of integrating rigid rotations is presented the algorithm is stable second order accurate and in its explicit version involves evaluation of only two exponential maps per time step the semi explicit version of the proposed scheme improves upon the long term stability while it retains the explicitness in the force evaluation the algebraic structure of both schemes makes them suitable forthe analysis of constrained multi body systems the explicit algorithm is specifically aimed at the analysis involving small incremental rotations where its modest computational cost becomes the major advantage the semi explicit scheme naturally broadens the scope of possible applications copyright 2009 john wiley sons ltd

shuffle compatible permutation statistics abstract since the early work of richard stanley it has been observed that several permutation statistics have a remarkable property with respect to shuffles of permutations we formalize this notion of a shuffle compatible permutation statistic and introduce the shuffle algebra of a shuffle compatible permutation statistic which encodes the distribution of the statistic over shuffles of permutations this paper develops a theory of shuffle compatibility for descent statistics statistics that depend only on the descent set and length which has close connections to the theory of p partitions quasisymmetric functions and noncommutative symmetric functions we use our framework to prove that many descent statistics are shuffle compatible and to give explicit descriptions of their shuffle algebras thus unifying past results of stanley gessel stembridge aguiar bergeron nyman and petersen

explicit momentum conserving integrator for dynamics of rigid bodies approximating the midpoint lie algorithm we reformulate the midpoint lie algorithm which is implicit in the torque calculation to achieve explicitness in the torque evaluation this is effected by approximating the impulse imparted over the time step with discrete impulses delivered at either the beginning of the time step or at the end of the time step thus we obtain two related variants both of which are explicit in the torque calculation but only first order in the time step both variants are momentum conserving and both are symplectic consequently drawing on the properties of the composition of maps we introduce another algorithm that combines the two variants in a single time step the resulting algorithm is explicit momentum conserving symplectic and second order its accuracy is outstanding and consistently outperforms currently known implicit and explicit integrators

a new algorithm for computing liquid crystal stable configurations the harmonic mapping case in this article we propose a new algorithm for minimizing the energy of a nematic liquid crystal based on the equal elastic constants oseen frank model the problem reduces to finding harmonic minimizing maps that take values into the unit sphere of bbb r 3 the convergence of this algorithm is proved in a continuous setting then numerous numerical results that show its efficiency are given

peak quasisymmetric functions and eulerian enumeration abstract via duality of hopf algebras there is a direct association between peak quasisymmetric functions and enumeration of chains in eulerian posets we study this association explicitly showing that the notion of cd index long studied in the context of convex polytopes and eulerian posets arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by stembridge thus eulerian posets having a nonnegative cd index for example face lattices of convex polytopes correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis we diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions and study the g polynomial for eulerian posets as an algebra homomorphism

lie idempotent algebras abstract besides the inner product inherited from each symmetric group algebra ksn on the direct sum ks nksn a second convolution product arises from the hopf algebra structure on ks for any subalgebra a of ks for both the inner and the convolution product it is shown that the lie idempotent algebra l a of a generated by all lie idempotents in a with respect to the convolution product is also closed under inner products its nth homogeneous component l n a is thus a subalgebra of ksn for all n provided that a contains a lie idempotent in ksk for all k n solomon s descent algebra d n is a subalgebra of l n a in this case a number of results on the structure of l n a is derived including a description of its jacobson radical its principal indecomposables and its cartan invariants furthermore solomon s epimorphism from d n onto the ring of class functions clk sn of sn extends to an epimorphism of algebras c n l n a cl k s n additional results are obtained concerning a bialgebra structure of l a in the particular case of a n d n this amounts to a new approach to several well known results on d n

enriched p partitions and peak algebras abstract we develop a more general view of stembridge s enriched p partitions and use this theory to outline the structure of peak algebras for the symmetric group and the hyperoctahedral group initially we focus on commutative peak algebras spanned by sums of permutations with the same number of peaks where we consider several variations on the definition of peak whereas stembridge s enriched p partitions are related to quasisymmetric functions the dual coalgebra of solomon s type a descent algebra our generalized enriched p partitions are related to type b quasisymmetric functions the dual coalgebra of solomon s type b descent algebra using these functions we move on to explore non commutative peak algebras spanned by sums of permutations with the same set of peaks while some of these algebras have been studied before our approach gives explicit structure constants with a combinatorial description

representation theory of 0 hecke clifford algebras abstract the representation theory of 0 hecke clifford algebras as a degenerate case is not semisimple and also with rich combinatorial meaning bergeron hivert and thibon have proved that the grothendieck ring of the category of finitely generated supermodules of 0 hecke clifford algebras is isomorphic to the algebra of peak quasisymmetric functions defined by stembridge in this paper we further study the category of finitely generated projective supermodules and clarify the correspondence between it and the peak algebra of symmetric groups in particular two kinds of restriction rules for induced projective supermodules are obtained after that we consider the corresponding heisenberg double and its fock representation to prove that the ring of peak quasisymmetric functions is free over its subring of symmetric functions spanned by schur s q functions

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

accurate discretization of a non linear micromagnetic problem abstract in this paper we propose a finite element discretization of the maxwell landau lifchitz gilbert equations governing the electromagnetic field in a ferromagnetic material our point of view is that it is desirable for the discrete problem to possess conservation properties similar to the continuous system we first prove the existence of a new class of liapunov functions for the continuous problem and then for a variational formulation of the continuous problem we also show a special continuous dependence result then we propose a family of mass lumped finite element schemes for the problem for the resulting semi discrete problem we show that magnetization is conserved and that semi discrete liapunov functions exist finally we show the results of some computations that show the behavior of the fully discrete liapunov functions

semi implicit reversible algorithms for rigid body rotational dynamics this paper presents two semi implicit algorithms based on splitting methodology for rigid body rotational dynamics the first algorithm is a variation of partitioned runge kutta prk methodology that can be formulated as a splitting method the second algorithm is akin to a multiple time stepping scheme and is based on modified crouch grossman mcg methodology which can also be expressed as a splitting algorithm these algorithms are second order accurate and time reversible however they are not poisson integrators i e non symplectic these algorithms conserve some of the first integrals of motion but some others are not conserved however the fluctuations in these invariants are bounded over exponentially long time intervals these algorithms exhibit excellent long term behavior because of their reversibility property and their approximate poisson structure preserving property the numerical results indicate that the proposed algorithms exhibit superior performance compared to some of the currently well known algorithms such as the simo wong algorithm newmark algorithm discrete moser veselov algorithm lewis simo algorithm and the liemid ea algorithm

an algorithmic proof theory for hypergeometric ordinary and q multisum integral identities it is shown that every proper hypergeometric multisum integral identity orq identity with a fixed number of summations and or integration signs possesses a short computer constructible proof we give a fast algorithm for finding such proofs most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here we give many examples of the method including computer generated proofs of identities of mehta dyson selberg hille hardy q saalschutz and others the prospect of using the method for proving multivariate identities that involve an arbitrary number of summations integrations is discussed

numerical integration of lie poisson systems while preserving coadjoint orbits and energy in this paper we apply geometric integrators of the rkmk type to the problem of integrating lie poisson systems numerically by using the coadjoint action of the lie group g on the dual lie algebra mbox normalsize mathfrak g to advance the numerical flow we devise methods of arbitrary order that automatically stay on the coadjoint orbits first integrals known as casimirs are retained to machine accuracy by the numerical algorithm within the proposed class of methods we find integrators that also conserve the energy these schemes are implicit and of second order nonlinear iteration in the lie algebra and linear error growth of the global error are discussed numerical experiments with the rigid body and a finite dimensional truncation of the euler equations for a two dimensional 2d incompressible fluid are used to illustrate the properties of the algorithm

random walks on quasisymmetric functions abstract conditions are provided under which an endomorphism on quasisymmetric functions gives rise to a left random walk on the descent algebra which is also a lumping of a left random walk on permutations spectral results are also obtained several important random walks are now realized this way stanley s qs distribution results from endomorphisms given by evaluation maps a shuffles result from the a th convolution power of the universal character and the tchebyshev operator of the second kind introduced recently by ehrenborg and readdy yields traditional riffle shuffles a conjecture of ehrenborg regarding the spectra for a family of random walks on ab words is proven a theorem of stembridge from the theory of enriched p partitions is also recovered as a special case

symplectic integration of constrained hamiltonian systems a hamiltonian system in potential form h q p ptmp 2 f q subject to smooth constraints on q can be viewed as a hamiltonian system on a manifold but numerical computations must be performed in rn in this paper methods which reduce hamiltonian differential algebraic equations to odes in euclidean space are examined the authors study the construction of canonical parametrizations or local charts as well as methods based on the construction of ode systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold in each case a hamiltonian system of ordinary differential equations is produced the stability of the constraint invariants and the behavior of the original hamiltonian along solutions are investigated both numerically and analytically

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

a numerical test of long time stability for rigid body integrators summary in the context of hamiltonian odes a necessary condition for an integrator to be symplectic or conjugate symplectic is that it nearly preserves the exact hamiltonian this paper introduces a numerical test of this necessity for rigid body methods it turns out that several rigid body integrators proposed in literature fail this test hence these integrators should be used with caution for long time simulation copyright 2011 john wiley sons ltd

a gauss seidel projection method for micromagnetics simulations abstract one of the main difficulties in micromagnetics simulation is the severe time step constraint introduced by the exchange field using standard explicit integrators leads to a physical time step of sub pico seconds which is often two orders of magnitude smaller than the fastest physical time scales direct implicit integrators require solving complicated coupled systems in this paper we introduce an implicit method whose complexity is comparable to solving the scalar heat equation implicitly this method is based on a combination of a gauss seidel implementation of a fractional step implicit solver for the gyromagnetic term and the projection method for the heat flow of harmonic maps this method allows us to carry out fully resolved calculations for the switching of the magnetization in micron sized elements

the peak algebra and the descent algebras of types b and d we show the existence of a unital subalgebra n of the symmetric group algebra linearly spanned by sums of permutations with a common peak set which we call the peak algebra we show that n is the image of the descent algebra of type b under the map to the descent algebra of type a which forgets the signs and also the image of the descent algebra of type d the algebra n contains a two sided ideal n which is defined in terms of interior peaks this object was introduced in previous work by nyman 2003 we find that it is the image of certain ideals of the descent algebras of types b and d we derive an exact sequence of the form 0 n n n 2 0 we obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs in particular we construct two new commutative semisimple subalgebras of the descent algebra of dimensions n and n 2 1 by grouping permutations according to their number of peaks or interior peaks we discuss the hopf algebraic structures that exist on the direct sums of the spaces n and n over n 0 and explain the connection with previous work of stembridge 1997 we also obtain new properties of his descents to peaks map and construct a type b analog

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 campbell hausdorff near ring over n a topological view abstract the famous baker campbell hausdorff series defines a group composition on the set of the so called grouplike elements of a completed free lie algebra as we can find in p cartier s paper on the campbell hausdorff formula in 1956 according to a baider and r c churchill we call this group the campbell hausdorff group over the alphabet x of the free lie algebra by a certain coproduct and certain bialgebra endomorphisms of the free associative algebra over a set x which should be at most countably infinite we get an alternative realisation of this campbell hausdorff group this realisation is much more comfortable to deal with than the classical one the usual composition of endomorphisms gives a near ring structure to this group in this paper we consider a metric on the campbell hausdorff near ring over the alphabet n and the compatibility of this metric with the near ring compositions and we make some topological remarks

a short elementary and easy wz proof of the askey gasper inequality that was used by de branges in his proof of the bieberbach conjecture

the shifted poirier reutenauer algebra based on the shifted schensted correspondence and the shifted knuth equivalence a shifted analog of the poirier reutenauer algebra is introduced it is a right coideal subalgebra of the poirier reutenauer algebra and turns out to be a higher lift of schur s p functions its close relations with the peak subalgebra and the stembridge algebra of peak functions are also uncovered

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 peak algebra of the symmetric group revisited abstract the linear span p n of the sums of all permutations in the symmetric group sn with a given set of peaks is a sub algebra of the symmetric group algebra due to nyman this peak algebra is a left ideal of the descent algebra d n and the direct sum p of all p n is a hopf sub algebra of the direct sum d of all d n dual to the stembridge algebra of peak functions in our self contained approach peak counterparts of several results on the descent algebra are established including a simple combinatorial characterization of the algebra p n an algebraic characterization of p n based on the action on the poincare birkhoff witt basis of the free associative algebra the display of peak variants of the classical lie idempotents an eulerian type sub algebra of p n a description of the jacobson radical of p n and its nil potency index of the principal indecomposable and irreducible p n modules and of the cartan matrix of p n furthermore it is shown that the primitive lie algebra of p is free and that p is its enveloping algebra

explicit newmark verlet algorithm for time integration of the rotational dynamics of rigid bodies we reformulate the traditional velocity based vector space newmark algorithm for the rotational dynamics of rigid bodies that is for the setting of the so 3 lie group we show that the most naive re write of the vector space algorithm possesses the properties of symplecticity and almost momentum conservation thus we obtain an explicit algorithm for rigid body dynamics that matches or exceeds performance of existing algorithms but which curiously does not seem to have been considered in the open literature so far copyright 2005 john wiley sons ltd

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

geometric integration on spheres and some interesting applications geometric integration theory can be employed when numerically solving odes or pdes with constraints in this paper we present several one step algorithms of various orders for odes on a collection of spheres to demonstrate the versatility of these algorithms we present representative calculations for reduced free rigid body motion a conservative ode and a discretization of micromagnetics a dissipative pde we emphasize the role of isotropy in geometric integration and link numerical integration schemes to modern differential geometry through the use of partial connection forms this theoretical framework generalizes moving frames and connections on principal bundles to manifolds with nonfree actions

momentum conserving symplectic integrators abstract in this paper we show that symplectic partitioned runge kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of hamiltonian differential equations by extended point transformation we also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of lie poisson systems on certain submanifolds of the general matrix group gl n

mechanical integrators derived from a discrete variational principle many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion in this paper we present a procedure to construct time stepping algorithms that approximate the flow of continuous odes for mechanical systems by discretizing hamilton s principle rather than the equations of motion the discrete equations share similarities to the continuous equations by preserving invariants including the symplectic form and the momentum map we first present a formulation of discrete mechanics along with a discrete variational principle we then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of noether s theorem we then use the discrete mechanics formulation to develop a procedure for constructing symplectic momentum mechanical integrators for lagrangian systems with holonomic constraints we apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators

a proof of solomon s rule abstract we put forward a proof of solomon s rule in terms of matrices for multiplication in the descent algebra of the symmetric group our proof exploits the graphs that we can obtain from all the subsets of the set of transpositions i i 1 n 1i 1

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells