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

radial spline collocation method for static analysis of beams conventional spline collocation method scm uses equally spaced knots in the computation of approximation however when load distribution geometry material property of structure are discontinuous it will greatly reduce accuracy the goal of this paper is mainly to develop unequally spaced knots to improve the disadvantage of using equally spaced knots applying the concept of radial basis functions rbf the spline function is transformed into radial spline function rsf consequently rsf can conveniently calculate the values for unequally spaced knots so that the complicated beam problems with discontinuous characteristics can be solved accurately and efficiently by the proposed radial spline collocation method rscm in this paper the proposed rscm also can be extended to solve other structural problems such as frame plate shell etc

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

angle variable holonomy in adiabatic excursion of an integrable hamiltonian if an integrable classical hamiltonian h describing bound motion depends on parameters which are changed very slowly then the adiabatic theorem states that the action variables i of the motion are conserved here the fate of the angle variables is analysed because of the unavoidable arbitrariness in their definition angle variables belonging to distinct initial and final hamiltonians cannot generally be compared however they can be compared if the hamiltonian is taken on a closed excursion in parameter space so that initial and final hamiltonians are the same the result shows that the angle variable change arising from such an excursion is not merely the time integral of the instantaneous frequency omega dh di but differs from it by a definite extra angle which depends only on the circuit in parameter space not on the duration of the process the 2 form which describes this angle variable holonomy is calculated

enforcement of essential boundary conditions in meshless approximations using finite elements abstract a technique for easily treating essential boundary conditions for approximations which are not interpolants such as moving least square approximations in the element free galerkin method is presented the technique employs a string of elements along the essential boundaries and combines the finite element shape functions with the approximation in the resulting approximation essential boundary conditions can be treated exactly as in finite elements it is shown that the resulting approximation can exactly reproduce linear polynomials so that it satisfies the patch test numerical studies show that the method retains the high rate of convergence associated with moving least square approximations

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

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

a renormalized rough path over fractional brownian motion we construct in this article a rough path over fractional brownian motion with arbitrary hurst index by i using the fourier normal ordering algorithm introduced in unterberger commun math phy 298 1 1 36 2010 to reduce the problem to that of regularizing tree iterated integrals and ii applying the bogolioubov parasiuk hepp zimmermann bphz renormalization algorithm to feynman diagrams representing tree iterated integrals

non commutative hopf algebra of formal diffeomorphisms abstract this paper deals with two hopf algebras which are the non commutative analogues of two different groups of formal power series the first group is the set of invertible series with the group law being multiplication of series while the second is the set of formal diffeomorphisms with the group law being a composition of series the motivation to introduce these hopf algebras comes from the study of formal series with non commutative coefficients invertible series with non commutative coefficients still form a group and we interpret the corresponding new non commutative hopf algebra as an alternative to the natural hopf algebra given by the co ordinate ring of the group which has the advantage of being functorial in the algebra of coefficients for the formal diffeomorphisms with non commutative coefficients this interpretation fails because in this case the composition is not associative anymore however we show that for the dual non commutative algebra there exists a natural co associative co product defining a non commutative hopf algebra moreover we give an explicit formula for the antipode which represents a non commutative version of the lagrange inversion formula and we show that its coefficients are related to planar binary trees then we extend these results to the semi direct co product of the previous hopf algebras and to series in several variables finally we show how the non commutative hopf algebras of formal series are related to some renormalization hopf algebras which are combinatorial hopf algebras motivated by the renormalization procedure in quantum field theory and to the renormalization functor given by the double tensor algebra on a bi algebra

holder continuous rough paths by fourier normal ordering we construct in this article an explicit geometric rough path over arbitrary d dimensional paths with finite 1 variation for any alpha in 0 1 the method may be coined as fourier normal ordering since it consists in a regularization obtained after permuting the order of integration in iterated integrals so that innermost integrals have highest fourier frequencies in doing so there appear non trivial tree combinatorics which are best understood by using the structure of the hopf algebra of decorated rooted trees in connection with the chen or multiplicative property and of the hopf shuffle algebra in connection with the shuffle or geometric property holder continuity is proved by using besov norms the method is well suited in particular in view of applications to probability theory see the companion article 34 for the construction of a rough path over multidimensional fractional brownian motion with hurst index 1 4 or 35 for a short survey in that case

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

the normal distribution is infinitely divisible abstract we prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution we study the voiculescu transform first by giving a survey of its combinatorial implications and then analytically including a proof of free infinite divisibility in fact we prove that a sub family of askey wimp kerov distributions are freely infinitely divisible of which the normal distribution is a special case at the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution the others being the cauchy distribution and the free 1 2 stable distribution

groups of tree expanded series abstract in ch brouder a frabetti renormalization of qed with planar binary trees eur phys j c 19 2001 715 741 ch brouder a frabetti qed hopf algebras on planar binary trees j algebra 267 2003 298 322 we introduced three hopf algebras on planar binary trees related to the renormalization of quantum electrodynamics one of them the algebra h is commutative and is therefore the ring of coordinate functions of a proalgebraic group g the other two algebras h e and h are free non commutative therefore their abelian quotients are the coordinate rings of two proalgebraic groups g e and g in this paper we describe explicitly these groups using two monoidal structures and a set operad structure on planar binary trees we show that these groups can be realized on formal series expanded over trees and that the group laws are generalizations of the multiplication and the composition of usual series in one variable therefore we obtain some new groups of invertible tree expanded series and of tree expanded formal diffeomorphisms respectively the hopf algebra describing the renormalization of the electric charge corresponds to the subgroup of tree expanded formal diffeomorphisms formed of the translations which fix the zero by some particular tree expanded series which remind the proper correlation functions in quantum field theory in turn the group of tree expanded formal diffeomorphisms and some of its subgroups give rise to new hopf algebras on trees all the constructions are done in a general operad theoretic setting and then applied to the specific duplicial operad on trees

loop of formal diffeomorphisms and faa di bruno coloop bialgebra abstract we consider a generalization of pro algebraic loops defined on general categories of algebras and the dual notion of a coloop bialgebra suitable to represent them as functors we prove that the natural loop of formal diffeomorphisms with associative coefficients is proalgebraic and we give the closed formulas of the codivisions on its coloop bialgebra this result provides a generalization of the lagrange inversion formula to series with non commutative coefficients and a loop theoretic explanation to the existence of the non commutative faa di bruno hopf algebra

an example of phase holonomy in wkb theory discusses the application of wkb theory to harper s equation psi n 1 psi n 1 2 alpha cos 2 pi beta n delta psi n e psi m in the case in which beta is very close to a rational number p q the wkb wavefunction for this system is a vector valued quantity proportional to an eigenvector u of a matrix h x p which is parametrised by the phase space coordinates x and p the complex phase of u is determined by a non holonomic connection rule when transported around a cycle and in phase space u is multiplied by a phase factor ei gamma c this phase change manifests itself as a modification of the bohr sommerfeld quantisation conditions

a finite point method in computational mechanics applications to convective transport and fluid flow the paper presents a fully meshless procedure fo solving partial differential equations the approach termed generically the finite point method is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals some examples showing the accuracy of the method for solution of adjoint and non self adjoint equations typical of convective diffusive transport and also to the analysis of compressible fluid mechanics problem are presented

qed hopf algebras on planar binary trees abstract in this paper we describe the hopf algebras on planar binary trees used to renormalize the feynman propagators of quantum electrodynamics and the coaction which describes the renormalization procedure both structures are related to some semi direct coproduct of hopf algebras

new concepts in meshless methods meshless methods have been extensively popularized in literature in recent years due to their flexibility in solving boundary value problems two kinds of truly meshless methods the meshless local boundary integral equation mlbie method and the meshless local petrov galerkin mlpg approach are presented and discussed both methods use the moving least squares approximation to interpolate the solution variables while the mlbie method uses a local boundary integral equation formulation and the mlpg employs a local symmetric weak form the two methods are truly meshless ones as both of them do not need a finite element or boundary element mesh either for purposes of interpolation of the solution variables or for the integration of the energy all integrals can be easily evaluated over regularly shaped domains in general spheres in three dimensional problems and their boundaries numerical examples presented in the paper show that high rates of convergence with mesh refinement are achievable in essence the present meshless method based on the lswf is found to be a simple efficient and attractive method with a great potential in engineering applications copyright 2000 john wiley sons ltd

hopf algebras renormalization and noncommutative geometry we explore the relation between the hopf algebra associated to the renormalization of qft and the hopf algebra associated to the ncg computations of tranverse index theory for foliations

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

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

the structure of hopf algebras with a projection the smash product algebra and the smash coproduct coalgebra are well known in the context of hopf algebras 1 5 and these notions can be viewed as being motivated by the semidirect product construction in the theory of groups and in the theory of afiine group schemes respectively let

solving partial differential equations by collocation using radial basis functions after a series of application papers have proven the approach to be numerically effective this paper gives the first theoretical foundation for methods solving partial differential equations by collocation with possibly radial basis functions

application of the multiquadric method for numerical solution of elliptic partial differential equations abstract we have used the multiquadric mq approximation scheme for the solution of elliptic partial differential equations with dirichlet and or neumann boundary conditions the scheme has the advantage of using the data points in arbitrary locations with an arbitrary ordering two dimensional laplace poisson and biharmonic equations describing the various physical processes have been taken as the test examples the agreement is found to be very good between the computed and exact solutions the method also provides an excellent approximation with a curved boundary

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h