wavevector dependent susceptibility in z invariant pentagrid ising model we study the q dependent susceptibility q of a z invariant ferromagnetic ising model on a penrose tiling as first introduced by korepin using de bruijn s pentagrid for the rapidity lines the pair correlation function for this model can be calculated exactly using the quadratic difference equations from our previous papers its fourier transform q is studied using a novel way to calculate the joint probability for the pentagrid neighborhoods of the two spins reducing this calculation to linear programming since the lattice is quasiperiodic we find that q is aperiodic and has everywhere dense peaks which are not all visible at very low or high temperatures more and more peaks become visible as the correlation length increases that is as the temperature approaches the critical temperature

characteristic cut finite element methods for convection diffusion problems on time dependent surfaces we develop a finite element method for convection diffusion problems on a given time dependent surface for instance modeling the evolution of a surfactant the method is based on a characteristic

on the convergence of multistep methods for nonlinear stiff differential equations convergence estimates are given fora stable multistep methods applied to singularly perturbed differential equations and nonlinear parabolic problems the approach taken here combines perturbation arguments with frequency domain techniques

a theory of pictures for quasi posets abstract the theory of pictures between posets is known to encode much of the combinatorics of symmetric group representations and related topics such as young diagrams and tableaux many reasons combinatorial e g since semi standard tableaux can be viewed as double quasi posets and topological quasi posets identify with finite topologies lead to extend the theory to quasi posets this is the object of the present article

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

self similarity of quasilattices in two dimensions ii the non bravais type n gonal quasilattice for pt i see ibid vol 22 p 193 1989 presents a systematic method of dividing an n gonal lattice into identical sublattices which are also n gonal lattices a non bravais type n gonal quasilattice in two dimensions is constructed with the projection method by assigning windows with different shapes sizes and or orientations to the sublattices the symmetries and the orientations of the windows are determined by the relative point symmetries of the sublattices to the n gonal lattice and windows assigned to equivalent sublattices differ from one another only in their orientations the sublattices are transformed among themselves by a volume conserving linear transformation induced by a complex pv unit in the n cyclotomic field a necessary and sufficient condition for a pv unit to be a complex self similarity ratio of a non bravais type quasilattice is presented it is shown that every non bravais type n gonal quasilattice has a self similarity characterised by pv unit

flows and stochastic taylor series in ito calculus for general stochastic systems driven by continuous semimartingales an explicit formula for the logarithm of the ito flow map is given the computation relies on the lift to quasi shuffle algebras of formulas involving products of ito integrals of semimartingales whereas the chen strichartz formula computing the logarithm of the stratonovich flow map is classically expanded as a formal sum indexed by permutations the analogous formula in ito calculus is naturally indexed by surjections this reflects the change of algebraic background involved in the transition between the two integration theories lastly we extend our formula for the quasi shuffle chen strichartz series for the logarithm of the flow map to the non commutative case for linear matrix valued sdes driven by arbitrary semimartingales we obtain a similar formula

a fully discrete evolving surface finite element method in this paper we consider a time discrete evolving surface finite element method for the advection and diffusion of a conserved scalar quantity on a moving surface in earlier papers using a suitable variational formulation in time dependent sobolev space we proposed and analyzed a finite element method using surface finite elements on evolving triangulated surfaces ima j numer anal 25 2007 pp 385 407 math comp to appear optimal order l 2 gamma t and h 1 gamma t error bounds were proved for linear finite elements in this work we prove optimal order error bounds for a backward euler time discretization

higher order finite element methods and pointwise error estimates for elliptic problems on surfaces we define higher order analogues to the piecewise linear surface finite element method studied in g dziuk finite elements for the beltrami operator on arbitrary surfaces in partial differential equations and calculus of variations springer verlag berlin 1988 pp 142 155 and prove error estimates in both pointwise and l 2 based norms using the laplace beltrami problem on an implicitly defined surface gamma as a model pde we define lagrange finite element methods of arbitrary degree on polynomial approximations to gamma which likewise are of arbitrary degree then we prove a priori error estimates in the l 2 h 1 and corresponding pointwise norms that demonstrate the interaction between the pde error that arises from employing a finite dimensional finite element space and the geometric error that results from approximating gamma we also consider parametric finite element approximations that are defined on gamma and thus induce no geometric error computational examples confirm the sharpness of our error estimates

gauss runge kutta time discretization of wave equations on evolving surfaces this paper is concerned with an error analysis for a full discretization of the linear wave equation on a moving surface the equation is discretized in space by the evolving surface finite element method discretization in time is done by gau runge kutta grk methods aiming for higher order accuracy in time and unconditional stability of the fully discrete scheme the latter is established in the natural time dependent norms by using the algebraic stability and the coercivity property of the grk methods together with the properties of the spatial semi discretization under sufficient regularity conditions optimal order error estimates for this class of fully discrete methods are shown numerical experiments are presented to confirm some of the theoretical results

a second order stabilization method for linearizing and decoupling nonlinear parabolic systems a new time discretization method for strongly nonlinear parabolic systems is constructed by combining the fully explicit two step backward difference formula and a second order stabilization of wav

transport and diffusion of material quantities on propagating interfaces via level set methods we develop theory and numerical algorithms to apply level set methods to problems involving the transport and diffusion of material quantities in a level set framework level set methods are computational techniques for tracking moving interfaces they work by embedding the propagating interface as the zero level set of a higher dimensional function and then approximate the solution of the resulting initial value partial differential equation using upwind finite difference schemes the traditional level set method works in the trace space of the evolving interface and hence disregards any parameterization in the interface description consequently material quantities on the interface which themselves are transported under the interface motion are not easily handled in this framework we develop model equations and algorithmic techniques to extend the level set method to include these problems we demonstrate the accuracy of our approach through a series of test examples and convergence studies

l 2 discretization error bounds for maps into riemannian manifolds we study the approximation of functions that map a euclidean domain omega subset mathbb r d into an n dimensional riemannian manifold m g minimizing an elliptic semilinear energy in a function set h subset w 1 2 omega m the approximation is given by a restriction of the energy minimization problem to a family of conforming finite dimensional approximations s h subset h we provide a set of conditions on s h such that we can prove a priori w 1 2 and l 2 approximation error estimates comparable to standard euclidean finite elements this is done in an intrinsic framework independently of embeddings of the manifold or the choice of coordinates

runge kutta approximation of quasi linear parabolic equations we study the convergence properties of implicit runge kutta meth ods applied to time discretization of parabolic equations with time or solution dependent operator error bounds are derived in the energy norm the con vergence analysis uses two different approaches the first technically simpler approach relies on energy estimates and requires algebraic stability of the runge kutta method the second one is based on estimates for linear time invariant equations and uses fourier and perturbation techniques it applies to a 9 stable runge kutta methods and yields the precise temporal order of conver gence this order is noninteger in general and depends on the type of boundary conditions

plane posets special posets and permutations abstract we study the self dual hopf algebra h sp of special posets introduced by malvenuto and reutenauer and the hopf algebra morphism from h sp to the hopf algebra of free quasi symmetric functions fqsym given by linear extensions in particular we construct two hopf subalgebras both isomorphic to fqsym the first one is based on plane posets the second one on heap ordered forests an explicit isomorphism between these two hopf subalgebras is also defined with the help of two combinatorial transformations on special posets the restriction of the hopf pairing of h sp to these hopf subalgebras and others is also studied as well as certain isometries between them these problems are solved using duplicial and dendriform structures

local rules for quasicrystals the relationship of local ordering and long range order is studied for quasicrystalline tilings of plane and space two versions of the concept of local rules are introduced strong and weak necessary conditions of the existence of strong local rules are found they are mainly reduced to the constraints for irrational numbers related to incommensurabilities of the quasicrystals for planar quasicrystals the quadratic irrationalities a b sqrt d left a b in mathbb q d in mathbb z right play an important role for three dimensional quasicrystals not only quadratic but also cubic irrationalities a b 3 sqrt d c 3 sqrt d 2 left a b c in mathbb q d in mathbb z right are allowed the existence of weak local rules is established for almost all two dimensional quasicrystals based on quadratic irrationalities and for the three dimensional quasicrystal having icosahedral symmetry

algebraic construction of one dimensional quasiperiodic tilings for an arbitrary one dimensional quasiperiodic tiling constructed via the grid method with periodically spaced grids an algebraic equation is derived for the positions of the vertices dependent on a single variable the cardinal position of the vertices

numerical analysis for a system coupling curve evolution to reaction diffusion on the curve we consider a finite element approximation for a system consisting of the evolution of a closed planar curve by forced curve shortening flow coupled to a reaction diffusion equation on the evolving curve the scheme for the curve evolution is based on a parametric description allowing for tangential motion whereas the discretization for the pde on the curve uses an idea from g dziuk and c m elliott ima j numer anal 27 2007 pp 262 292 we prove optimal error bounds for the resulting fully discrete approximation and present numerical experiments these confirm our estimates and also illustrate the advantage of the tangential motion of the mesh points in practice

diffusion on a quasiperiodic chain an exact solution to the problem of diffusion on a quasiperiodic chain is provided by a decimation method involving scaling by an irrational length factor

heptagonal quasicrystal tilings four colour aperiodic selfsimilar patterns in two dimensions with seven fold symmetry and bragg spectrum are introduced and described with the help of substitutional sequences

sharply localized pointwise and w 1 estimates for finite element methods for quasilinear problems we establish pointwise and w 1 estimates for finite element methods for a class of second order quasilinear elliptic problems defined on domains in r these estimates are localized in that they indicate that the pointwise dependence of the error on global norms of the solution is of higher order our pointwise estimates are similar to and rely on results and analysis techniques of schatz for linear problems we also extend estimates of schatz and wahlbin for pointwise differences e x 1 e x 2 in pointwise errors to quasilinear problems finally we establish estimates for the error in w 1 d where d is a subdomain these negative norm estimates are novel for linear as well as for nonlinear problems our analysis heavily exploits the fact that galerkin error relationships for quasilinear problems may be viewed as perturbed linear error relationships thus allowing easy application of properly formulated results for linear problems

overlapping unit cells in 3d quasicrystal structure a three dimensional quasiperiodic lattice with overlapping unit cells and periodic in one direction is constructed using grid and projection methods pioneered by de bruijn each unit cell consists of 26 points of which 22 are the vertices of a convex polytope and 4 are interior points also shared with other neighbouring unit cells using kronecker s theorem the frequencies of all possible types of overlapping are found

an adaptive finite element method for the laplace beltrami operator on implicitly defined surfaces we present an adaptive finite element method for approximating solutions to the laplace beltrami equation on surfaces in mathbb r 3 which may be implicitly represented as level sets of smooth functions residual type a posteriori error bounds which show that the error may be split into a residual part and a geometric part are established in addition implementation issues are discussed and several computational examples are given

a convergent evolving finite element algorithm for mean curvature flow of closed surfaces a proof of convergence is given for semi and full discretizations of mean curvature flow of closed two dimensional surfaces the numerical method proposed and studied here combines evolving finite elements whose nodes determine the discrete surface like in dziuk s method and linearly implicit backward difference formulae for time integration the proposed method differs from dziuk s approach in that it discretizes huisken s evolution equations for the normal vector and mean curvature and uses these evolving geometric quantities in the velocity law projected to the finite element space this numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five the error analysis combines stability estimates and consistency estimates to yield optimal order h 1 norm error bounds for the computed surface position velocity normal vector and mean curvature the stability analysis is based on the matrix vector formulation of the finite element method and does not use geometric arguments the geometry enters only into the consistency estimates numerical experiments illustrate and complement the theoretical results

a θ stable approximations of abstract cauchy problems we study the approximation of linear parabolic cauchy problems by means of galerkin methods in space anda stable multistep schemes of arbitrary order in time the error is evaluated in the norm ofl t 2 h x 1 l t l x 2

stability and error analysis for a diffuse interface approach to an advection diffusion equation on a moving surface in this paper we analyze a fully discrete numerical scheme for solving a parabolic pde on a moving surface the method is based on a diffuse interface approach that involves a level set description of the moving surface under suitable conditions on the spatial grid size the time step and the interface width we obtain stability and error bounds with respect to natural norms furthermore we present test calculations that confirm our analysis

ground states of two dimensional quasicrystals hamiltonians for nonperiodic tilings are considered it is shown that the quasicrystalline tiling obtained by the cut and strip method from ad dimensional cubic lattice may bs a ground state only if the tiling possesses a high orientational symmetry the 2 d quasicrystal should haved fold symmetry ifd is even and 2d fold symmetry ifd is odd for interactions of a finite range the restrictions are stronger only a 2 5 quasicrystal penrose tiling may be a stable ground state

non periodic tessellations with unit and non unit pisot inflation factors non periodic planar patterns generated by deterministic inflation rules are described several types of patterns with both unit and non unit pisot numbers as inflation factors are formed by triangular prototiles when the inflation factor is the silver mean the patterns can be obtained by superimposing two subpatterns and some of them can be converted into face to face patterns the geometric constructions corresponding to 12 fold symmetry are also studied the fourier transform of point masses placed in vertex positions of the transformed subpatterns exhibit two fold four fold and six fold symmetries although the peaks with highest intensities are situated in rings with apparent octagonal and dodecagonal symmetries

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

modeling and computation of two phase geometric biomembranes using surface finite elements biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions the modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces this leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints in this paper we propose a new computational tool for computing equilibria based on an l 2 relaxation flow for the total energy in which the line energy is approximated by a surface ginzburg landau phase field functional the relaxation dynamics couple a nonlinear fourth order geometric evolution equation of willmore flow type for the membrane with a surface allen cahn equation describing the lateral decomposition a novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface the mean curvature vector and the surface phase field function the resulting variational formulation uses h 1 spaces and we employ triangulated surfaces and h 1 conforming quadratic surface finite elements for approximating solutions together with a semi implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells discocytes and starfishes with lateral phase separation

a finite volume method for solving parabolic equations on logically cartesian curved surface meshes we present a second order finite volume scheme for the constant coefficient diffusion equation on curved parametric surfaces while our scheme is applicable to general quadrilateral surface meshes based on smooth or piecewise smooth coordinate transformations our primary motivation for developing the present scheme is to solve diffusion problems on a particular set of circular and spherical meshes introduced in d a calhoun c helzel and r j leveque siam rev 50 2008 pp 723 752 for the discretization of hyperbolic problems these grids are generated from mappings of a single cartesian grid and were designed to have nearly uniform cells sizes and avoid the pole singularity associated with polar or spherical grid mappings the present method for parabolic equations offers several advantages it does not require analytic metric terms shows second order accuracy on our disk and sphere grids can be easily coupled to existing finite volume solvers for logically cartesian meshes and handles general mixed boundary conditions our parabolic scheme should appeal to researchers in the fields of geophysical fluid dynamics computational biology and any other discipline that requires the solution of parabolic equations on quadrilateral surface meshes in this article we present several numerical examples demonstrating the accuracy of the scheme and then use the scheme to solve advection reaction diffusion equations modeling biological pattern formation on surfaces

a finite volume method on general surfaces and its error estimates abstract in this paper we study a finite volume method and its error estimates for the numerical solution of some model second order elliptic partial differential equations defined on a smooth surface the discretization is defined via a surface mesh consisting of piecewise planar triangles and piecewise polygons the optimal error estimates of the approximate solution are proved in both the h 1 and l 2 norms which are of first order and second order respectively under mesh regularity assumptions some numerical tests are also carried out to experimentally verify our theoretical analysis

theory of self similarity of periodic approximants to a quasilattice iii the case of a nonbravais type quasilattice for pt ii see ibid vol 25 p 2147 1992 the mother lattice l of a non bravais type quasilattice nbtql is a non bravais type periodic lattice with higher dimensionality and has an associated bravais lattice l0 the main problems on periodic approximants pas to the nbtql are reduced to similar problems in the case of the relevant bravais type quasilattice derived from l0 and the previous theories on the space groups and self similarity of the pas apply to the nbtql the present theory includes a general prescription of obtaining pas to a nbtql the author applies the theory to several nbtqls with an octagonal decagonal or dodecagonal point symmetry the second important bravais lattice lh called the host lattice is associated with l and he has clarified the difference in roles between l0 and lh in the theory of pas to the nbtql

error analysis of a space time finite element method for solving pdes on evolving surfaces in this paper we present an error analysis of an eulerian finite element method for solving parabolic partial differential equations pdes posed on evolving hypersurfaces in rd d 2 3 the method employs discontinuous piecewise linear in time continuous piecewise linear in space finite elements and is based on a space time weak formulation of a surface pde problem trial and test surface finite element spaces consist of traces of standard volumetric elements on a space time manifold resulting from the evolution of a surface we prove first order convergence in space and time of the method in an energy norm and second order convergence in a weaker norm furthermore we derive regularity results for solutions of parabolic pdes on an evolving surface which we need in a duality argument used in the proof of the second order convergence estimate

two dimensional dodecagonal quasilattices two dimensional quasiperiodic lattices with 12 fold rotational symmetry are constructed by projecting a four dimensional hyperhexagonal lattice onto a two dimensional subspace the projection method in the present case is greatly simplified by representing the four dimensional lattice as a complex two dimensional lattice by choosing windows with different forms one obtains several quasiperiodic tilings which have a self similarity one of the tilings is quite similar to the pattern of the dodecagonal quasicrystal of a nicr alloy

l 2 estimates for the evolving surface finite element method in this paper we consider the evolving surface nite element method for the advection and di usion of a conserved scalar quantity on a moving surface in an earlier paper using a suitable variational formulation in time dependent sobolev space we proposed and analysed a nite element method using surface nite elements on evolving triangulated surfaces an optimal order h error bound was proved for linear nite elements in this work we prove the optimal error bound in l t uniformly in time

weak matching rules for quasicrystals weak matching rules for a quasicrystalline tiling are local rules that ensure that fluctuations in perp space are uniformly bounded it is shown here that weak matching rules exist forn fold symmetric tilings wheren is any integer not divisible by four the result suggests that contrary to previous indications quasicrystalline ground states are not confined to those symmetries for which the incommensurate ratios of wavevectors are quadratic irrationals an explicit method of constructing weak matching rules forn fold symmetric tilings in two dimensions is presented it is shown that the generalization of the construction yields weak matching rules in the case of icosahedral symmetry as well

eulerian finite element methods for parabolic equations on moving surfaces three new eulerian finite element methods for parabolic pdes on a moving surface gamma t are presented and compared in numerical experiments these are space time galerkin methods which are derived from a weak formulation in space and time the trial and test spaces contain the traces on the space time manifold of an outer prismatic finite element space the numerical experiments show that two of the methods converge with second order with respect to both the time step size and the spatial mesh width

an n gonal quasilattice in two dimensions as a dual lattice to a multiple periodic grid an n gonal quasilattice in two dimensions is constructed for the case of n 6m m or 2 as a dual to an m tuple periodic grid gm g0 union rg0 union union rm 1g0 where g0 is a periodic grid with hexagonal point symmetry and r the rotation operation of the plane by angle 2 pi n the author investigates the case where g0 is a honeycomb triangular kagome or diced grid the resulting quasilattice is not of bravais type unless g0 is the honeycomb grid it is shown also that the same quasilattice is obtained with the projection method from the 2m dimensional lattice l l0 l0 l0 where l0 is a triangular honeycomb diced or kagome lattice being dual to g0 in the projection method the 2m dimensional euclidean space is divided into cells with a 2m dimensional grid g which is dual to l it is shown that gm is identical to a two dimensional section of g

variational problems and partial differential equations on implicit surfaces a novel framework for solving variational problems and partial differential equations for scalar and vector valued data defined on surfaces is introduced in this paper the key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed cartesian coordinate system using this new embedding function the equations are then both intrinsic to the surface and defined in the embedding space this approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces as is commonly done in the literature we describe the framework and present examples in computer graphics and image processing applications including texture synthesis flow field visualization and image and vector field intrinsic regularization for data defined on 3d surfaces

absence of weak local rules for the planar quasicrystalline tiling with the 8 fold rotational symmetry levitov s theory of local rules the preceding paper of this issue gives no prediction for the planar quasicrystalline tiling having 8 fold rotational symmetry absence of weak local rules for this particular tiling is proven

on diffraction by aperiodic structures this paper gives a rigorous treatment of some aspects of diffraction by aperiodic structures such as quasicrystals it analyses diffraction in the limit of the infinite system through an appropriately defined autocorrelation the main results are a justification of the standard way of calculating the diffraction spectrum of tilings obtained by the projection method and a proof of a variation on a conjecture by bombieri and taylor

convergence of finite elements on an evolving surface driven by diffusion on the surface for a parabolic surface partial differential equation coupled to surface evolution convergence of the spatial semidiscretization is studied in this paper the velocity of the evolving surface is not given explicitly but depends on the solution of the parabolic equation on the surface various velocity laws are considered elliptic regularization of a direct pointwise coupling a regularized mean curvature flow and a dynamic velocity law a novel stability and convergence analysis for evolving surface finite elements for the coupled problem of surface diffusion and surface evolution is developed the stability analysis works with the matrix vector formulation of the method and does not use geometric arguments the geometry enters only into the consistency estimates numerical experiments complement the theoretical results

a classification of two dimensional quasiperiodic tilings obtained with the grid method two dimensional quasiperiodic tilings qpt obtained from n grids are classified into local isomorphism li classes by using the invariants of the grids the number of the invariants is given by n phi n if n is odd or n phi 2n if n is even where phi is euler s function in number theory all the qpt obtained from n grids with n 2k k or 2 belong to a single li class whose point symmetry is d2n when n is not a power of 2 the qpt are classified into several li classes of the li classes two have the highest point symmetry d2n one of the two classes is associated with n grids in which all the invariants vanish and the other with n grids in which all of them take 1 2 in the case of an odd n there is also one continuous series of qpt with point symmetry dn the author also presents a general formula of the tile statistics of the quasiperiodic tilings obtained with the grid method

linearly implicit full discretization of surface evolution stability and convergence of full discretizations of various surface evolution equations are studied in this paper the proposed discretization combines a higher order evolving surface finite element method for space discretization with higher order linearly implicit backward difference formulae for time discretization the stability of the full discretization is studied in the matrix vector formulation of the numerical method the geometry of the problem enters into the bounds of the consistency errors but does not enter into the proof of stability numerical examples illustrate the convergence behaviour of the full discretization

the implicit closest point method for the numerical solution of partial differential equations on surfaces many applications in the natural and applied sciences require the solutions of partial differential equations pdes on surfaces or more general manifolds the closest point method is a simple and accurate embedding method for numerically approximating pdes on rather general smooth surfaces however the original formulation is designed to use explicit time stepping this may lead to a strict time step restriction for some important pdes such as those involving the laplace beltrami operator or higher order derivative operators to achieve improved stability and efficiency we introduce a new implicit closest point method for surface pdes the method allows for large stable time steps while retaining the principal benefits of the original method in particular it maintains the order of accuracy of the discretization of the underlying embedding pde it works on sharply defined bands without degrading the accuracy of the method and it applies to general smooth surfaces it also is very simple and may be applied to a rather general class of surface pdes convergence studies for the in surface heat equation and a fourth order biharmonic problem are given to illustrate the accuracy of the method we demonstrate the flexibility and generality of the method by treating flows involving diffusion reaction diffusion and fourth order spatial derivatives on a variety of interesting surfaces including surfaces of mixed codimension

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 hopf algebra of finite topologies and t partitions a noncommutative and noncocommutative hopf algebra on finite topologies h t is introduced and studied freeness cofreeness self duality generalizing stanley s definition of p partitions associated to a special poset we define the notion of t partitions associated to a finite topology and deduce a hopf algebra morphism from h t to the hopf algebra of packed words wqsym generalizing stanley s decomposition by linear extensions we deduce a factorization of this morphism which induces a combinatorial isomorphism from the shuffle product to the quasi shuffle product of wqsym it is strongly related to a partial order on packed words here described and studied

monte carlo approach to dendritic growth the connection between diffusion limited aggregation and the equations of dendritic growth is critically examined a different type of monte carlo simulation is proposed and used to construct two dimensional dendrite like patterns the wavelength occurring for short times is in good agreement with the linear stability analysis the time dependence of the characteristic wavelengths is also determined

a formula for the number of tilings of an octagon by rhombi we propose the first algebraic determinantal formula to enumerate tilings of a centrosymmetric octagon of any size by rhombi this result uses the gessel viennot technique and generalizes to any octagon a formula given by elnitsky in a special case

stability of implicit explicit backward difference formulas for nonlinear parabolic equations we analyze stability properties of backward difference formulas bdf methods of order up to 5 for linear parabolic equations as well as of implicit explicit bdf methods for nonlinear parabolic equations by energy techniques time dependent norms also play a key role in the analysis

error analysis of implicit euler methods for quasilinear hyperbolic evolution equations in this paper we study the convergence of the semi implicit and the implicit euler methods for the time integration of abstract quasilinear hyperbolic evolution equations the analytical framework considered here includes certain quasilinear maxwell s and wave equations as special cases our analysis shows that the euler approximations are well posed and convergent of order one the techniques will be the basis for the future investigation of higher order time integration methods and full discretizations of certain quasilinear hyperbolic problems

local rules for quasiperiodic tilings of quadratic 2 planes in r 4 we prove that quasiperiodic tilings of the plane appearing in the strip pro jection method always admit local rules when the linear embedding of i z in 4 has quadratic coefficients these local rules are constructed and studied the connection between novikov quasicrystallographic groups and the quasiperiodic tilings of eu clidean space is explained all the point groups in novikov s sense compatible with these local rules are enlisted the two dimensional quasicrystals with infinite fold rotational symmetry are constructed and studied

evolving surface finite element method for the cahn hilliard equation we use the evolving surface finite element method to solve a cahn hilliard equation on an evolving surface with prescribed velocity we start by deriving the equation using a conservation law and appropriate transport formulae and provide the necessary functional analytic setting the finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity we go on to show a rigorous well posedness result for the continuous equations by showing convergence along a subsequence of the finite element scheme we conclude the paper by deriving error estimates and present various numerical examples

implicit explicit multistep methods for nonlinear parabolic equations implicit explicit multistep methods for nonlinear parabolic equations were recently analyzed in 2 3 1 in these papers the linear operator of the equation is assumed time independent self adjoint and positive definite then the linear part is discretized implicitly and the remaining part explicitly here we slightly relax the hypotheses on the linear operator by allowing part of it to be time dependent or nonself adjoint we establish optimal order a priori error estimates

space group theory for a nonperiodic icosahedral quasilattice the hypercubic lattice y is considered in euclidean space e6 embedding of the icosahedral group a 5 determines a well known noncrystallographic action of a 5 on an irreducible subspace e3 from boundaries of y and its metrical dual y new convex polytopes called klotze are constructed in e6 the klotze tile e6 periodically but not face to face and provide a new fundamental domain under the hypercubic translation group all boundaries of a klotz are parallel or perpendicular to the subspace e3 the intersection of the lattice y with e3 determine a nonperiodic face to face rhombohedral tiling of e3 a quasicrystal model is introduced in e6 and e3 the model has stable density supported on the rhombohedral tiles its fourier transform yields sharp diffraction maxima and reduces to integrals in e3 over the representative tiles and their boundaries

meyer s concept of quasicrystal and quasiregular sets this paper relates two mathematical concepts of long range order of a set of atoms each of which is based on restrictions on the set of interatomic distances a set in n is ameyer set if is a delone set and there is a finite setf such that lambda lambda subseteq lambda f text y meyer proposed that such sets include the possible structures of quasicrystals he obtained a structure theory for such sets which reformulates results that he obtained in harmonic analysis around 1970 and which relates these sets to cut and project sets in geometric crystallography v i galiulin introduced the concept ofquasiregular set which is a set such that both and are delone sets this paper shows that these two concepts are identical

equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings the two main techniques for the generation of quasiperiodic tilings de bruijn s grid method 1981 and the projection formalism are generalised a very broad class of quasiperiodic tilings is obtained in this way the two generalised methods are shown to be equivalent the standard calculation of fourier spectra is extended to the whole general class of tilings various applications are discussed

a simple derivation of quasi crystalline spectra a quasi crystalline solid is an aperiodic array of atoms which gives rise to bragg like diffraction peaks using the projection method for generating such structures the authors provide a simple derivation of this paradoxical behaviour

fully implicit linearly implicit and implicit explicit backward difference formulae for quasi linear parabolic equations quasi linear parabolic equations are discretised in time by fully implicit backward difference formulae bdf as well as by implicit explicit and linearly implicit bdf methods up to order five under appropriate stability conditions for the various methods considered we establish optimal order a priori error bounds by energy estimates which become applicable via the nevanlinna odeh multiplier technique

a self similar dodecagonal quasiperiodic tiling of the plane in terms of squares regular hexagons and thin rhombi the author presents a dodecagonal quasiperiodic tiling of the plane in terms of three kinds of tiles a square a regular hexagon and a thin rhombus whose acute inner angles are 30 degrees the dodecagonal tiling is self similar with respect to a dilatation by 2 square root 3 it is associated with a dodecagonal quasiperiodic lattice which is obtained by a projection from a hyperhoneycomb lattice in four dimensions the hyperhoneycomb lattice is a non bravais lattice composed of four bravais sublattices and the author must assign windows with different orientations to different sublattices the author discusses in detail the relationships of the present method of obtaining a dodecagonal tiling to other methods

lumped finite elements for reaction cross diffusion systems on stationary surfaces all the authors am is cv mf thank the isaac newton institute for mathematical sciences for its hospitality during the programme coupling geometric pdes with physics for cell morphology motility and pattern formation epsrc ep k032208 1 this work am has received funding from the european union s horizon 2020 research and innovation programme under the marie sklodowska curie grant agreement no 642866 am and cv acknowledge support from the engineering and physical sciences research council ep j016780 1 on modelling analysis and simulation of spatial patterning on evolving biological surfaces and the leverhulme trust research project grant rpg 2014 149 on unravelling new mathematics for 3d cell migration am was partially supported by a fellowship from the simons foundation am is a royal society wolfson research merit award holder generously funded by the wolfson foundation

a finite element method for surface pdes matrix properties we consider a recently introduced new finite element approach for the discretization of elliptic partial differential equations on surfaces the main idea of this method is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface the method is particularly suitable for problems in which there is a coupling with a problem in an outer domain that contains the surface for example two phase flow problems it has been proved that the method has optimal order of convergence both in the h 1 and in the l 2 norm in this paper we address linear algebra aspects of this new finite element method in particular the conditioning of the mass and stiffness matrix is investigated for the two dimensional case we present an analysis which proves that the effective spectral condition number of the diagonally scaled mass matrix and the diagonally scaled stiffness matrix behaves like h 3 ln h and h 2 ln h respectively where h is the mesh size of the outer triangulation

random tilings of high symmetry i mean field theory we study random tiling models in the limit of high rotational symmetry in this limit a mean field theory yields reasonable predictions for the configurational entropy of free boundary rhombus tilings in two dimensions we base our mean field theory on an iterative tiling construction inspired by the work of de bruijn in addition to the entropy we consider correlation functions phason elasticity and the thermodynamic limit tilings of dimension other than two are considered briefly

natural vorticity boundary conditions on solid walls abstract we derive a new kind of boundary conditions for the vorticity equation with solid wall boundaries for fluid flow problems the formulation uses a dirichlet condition for the normal component of vorticity and neumann type conditions for the tangential components in a galerkin integral formulation the tangential condition is natural i e it is enforced by a right hand side functional and does not impose a boundary constraint on trial and test spaces the functional involves the pressure variable and we discuss several velocity vorticity formulations where the proposed condition is appropriate several numerical experiments are given that illustrate the validity of the new approach

fixed boundary octagonal random tilings a combinatorial approach some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied a geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems thus generalizing previous results on the more restricted class of codimension one tilings in particular this method gives access to counting formulas which are directly related to questions of entropy in these statistical systems methods and tools from the field of enumerative combinatorics are used

implicit explicit multistep methods for quasilinear parabolic equations efficient combinations of implicit and explicit multistep methods for nonlinear parabolic equations were recently studied in 1 in this note we present a refined analysis to allow more general nonlinearities the abstract theory is applied to a quasilinear parabolic equation

structure factor of 1d systems superlattices based on two letter substitution rules i delta bragg peaks the recent generalization to the case of arbitrary tile lengths and arbitrary scattering factors of the calculation of the structure factor of 1d substitutional systems is studied in detail this method makes a easy to find all the peaks in the diffraction spectrum of a system the well known periodic and quasiperiodic spectra with delta peaks at integer multiples of a single number and integer linear combinations of two incommensurate frequencies respectively were found to be the l 0 subsets of two more general types of spectra infinite periodic or limit periodic and infinite quasiperiodic or limit quasiperiodic characterized by rational numbers of the type m nl l 0 infinity in place of the above integers substitution rules that produce quasicrystalline quasiperiodic and infinite quasiperiodic spectra give the same type of spectrum for all values of the ratio rho rho a rho b of the two tile lengths rho a and rho b this is not the case for the other rules thus the same substitution rule such as the copper mean rule can give an infinite periodic spectrum for a single rational ratio rho rho a rho b of the two tile lengths rho a and rho b a periodic like spectrum for other rational rho and a spectrum in some aspects similar to that of a random system when rho is an irrational number on the other hand a thue morse system diffracts as a periodic crystal when p not 1 but has no non trivial delta peaks when rho 1 other thue morse like systems can have infinite periodic spectra for all rho

numerical simulation of dealloying by surface dissolution via the evolving surface finite element method in this article we formulate a numerical method for the simulation of dealloying of a binary alloy by the selective removal of one component via electrochemical dissolution such that there is phase separation of the other component at the solid electrolyte interface the evolution of the interface is modelled by a forced mean curvature flow strongly coupled to the solution of a cahn hilliard equation modelling surface phase separation the method is based on a triangulated hypersurface whose evolution is calculated as well as the solution of the cahn hilliard equation by the evolving surface finite element method esfem the numerical experiments simulate complex morphology and concentration patterns providing evidence that the mathematical model may describe the formation of nanoporosity

a finite element method for elliptic equations on surfaces in this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated the main idea is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface the method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface we give an analysis that shows that the method has optimal order of convergence both in the h 1 and in the l 2 norm results of numerical experiments are included that confirm this optimality

the exponential lie series for continuous semimartingales we consider stochastic differential systems driven by continuous semimartingales and governed by non commuting vector fields we prove that the logarithm of the flowmap is an exponential lie series this relies on a natural change of basis to vector fields for the associated quadratic covariation processes analogous to stratonovich corrections the flowmap can then be expanded as a series in compositional powers of vector fields and the logaritm of the flowmap can thus be expanded in the lie algebra of vector fields further we give a direct self contained proof of the corresponding chen strichartz formula which provides an explicit formula for the lie series coefficients such exponential lie series are important in the development of strong lie group integration schemes that ensure approximate solutions themselves lie in any homogeneous manifold on which the solution evolves

dualisation of voronoi domains and klotz construction a general method for the generation of proper space fillings the authors present a general method for the generation of proper space fillings of euclidean spaces with convex polytopes by projection from higher dimensional spaces as a special case quasiperiodic tilings are obtained by the projection of periodic structures the concept is general enough for the treatment of defects in quasiperiodic tilings by placing defects in high dimensional translation lattices

a posteriori error estimates for finite volume approximations of elliptic equations on general surfaces abstract in this paper we present a residual based a posteriori error estimate for the finite volume discretization of steady convection diffusion reaction equations defined on surfaces in r 3 which are often implicitly represented as level sets of smooth functions reliability and efficiency of the proposed a posteriori error estimator are rigorously proved numerical experiments are also conducted to verify the theoretical results and demonstrate the robustness of the error estimator