convergence in measure with respect to a matrix valued measure and some matrix completion problems for a matrix valued measure m we introduce a notion of convergence in measure m which generalizes the notion of convergence in measure with respect to a scalar measure and takes into account the matrix structure of m let s be a subset of the set of matrices of given size it is easy to see that the set of s valued measurable functions is closed under convergence in measure with respect to a matrix valued measure if and only if s is a closed set i e if and only if sp is closed for any orthoprojector p we discuss the behaviour of closed sets under operations of linear algebra and the closedness of particular classes of matrices

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

density of polynomials in some l2 spaces on radial rays in the complex plane in this paper we study the necessary and sufficient conditions for the density of the linear space of matrix polynomials in a linear space of square integrable functions with respect to a matrix of measures supported on a set of radial rays of the complex plane the connection with a completely indeterminate hamburger matrix moment problem is stated vector valued functions associated in a natural way with a function defined in the union of the radial rays are used thus our first aim is the construction of a linear space of square integrable functions with respect to a matrix of measures supported on a set of radial rays and a positive semi definite matrix acting on the discrete part of the corresponding inner product an isometric transformation which allows to reduce the problem of density to the case of the real line is introduced finally some examples of such spaces are shown and its completeness is studied in detail

imex hdg dg a coupled implicit hybridized discontinuous galerkin and explicit discontinuous galerkin approach for shallow water systems abstract we propose imex hdg dg schemes for planar and spherical shallow water systems of interest is subcritical flow where the speed of the gravity wave is faster than that of nonlinear advection in order to simulate these flows efficiently we split the governing system into a stiff part describing the gravity wave and a non stiff part associated with nonlinear advection the former is discretized implicitly with the hdg method while an explicit runge kutta dg discretization is employed for the latter the proposed imex hdg dg framework 1 facilitates high order solutions both in time and space 2 avoids overly small time step sizes 3 requires only one linear system solve per time stage 4 relative to dg generates smaller and sparser linear systems while promoting further parallelism and 5 suppresses the fast modes in the system with a large time step size numerical results for various test cases demonstrate that our methods are beneficial for applications where non stiff terms are accurately treated while stiff terms are less accurately handled

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

a synchronous and iterative flux correction formalism for coupled transport equations many problems of fluid dynamics involve the coupled transport of several density like dependent variables for instance densities of mass and momenta in elastic flows in this paper a conservative and synchronous flux corrected transport fct formalism is developed which aims at a consistent transport of such variables the technique differs from traditional fct algorithms in two respects first the limiting of transportive fluxes of the primary variables e g mass and momentum does not derive from smooth estimates of the variables but it derives from analytic constraints implied by the lagrangian form of the governing continuity equations which are imposed on the specific mixing ratios of the variables e g velocity components second the traditional fct limiting based on sufficiency conditions is augmented by an iterative procedure which approaches the necessity requirements this procedure can also be used in the framework of traditional fct schemes and a demonstration is provided that it can significantly reduce some of the pathological behaviors of fct algorithms although the approach derived is applicable to the transport of arbitrary conserved quantities it is particularly useful for the synchronous transport of mass and momenta in elastic flows where it assures intrinsic stability of the algorithm regardless of the magnitude of the mass density variable this latter property becomes especially important in fluids with large density variations or in models with a material vertical coordinate e g geophysical hydrostatic stratified flows in isopycnic isentropic coordinates where material surfaces can collapse to zero mass layers admitting therefore arbitrarily large local courant numbers

total variation diminishing implicit explicit runge kutta methods for the simulation of double diffusive convection in astrophysics we put forward the use of total variation diminishing or more generally strong stability preserving implicit explicit runge kutta methods for the time integration of the equations of motion associated with the semiconvection problem in the simulation of stellar convection the fully compressible navier stokes equation augmented by continuity and total energy equations and an equation of state describing the relation between the thermodynamic quantities is semi discretized in space by essentially non oscillatory schemes and dissipative finite difference methods it is subsequently integrated in time by runge kutta methods which are constructed such as to preserve the total variation diminishing or strong stability property satisfied by the spatial discretization coupled with the forward euler method we analyse the stability accuracy and dissipativity of the time integrators and demonstrate that the most successful methods yield a substantial gain in computational efficiency as compared to classical explicit runge kutta methods

a standard test set for numerical approximations to the shallow water equations in spherical geometry a suite of seven test cases is proposed for the evaluation of numerical methods intended for the solution of the shallow water equations in spherical geometry the shallow water equations exhibit the major difficulties associated with the horizontal dynamical aspects of atmospheric modeling on the spherical earth these cases are designed for use in the evaluation of numerical methods proposed for climate modeling and to identify the potential trade offs which must always be made in numerical modeling before a proposed scheme is applied to a full baroclinic atmospheric model it must perform well on these problems in comparison with other currently accepted numerical methods the cases are presented in order of complexity they consist of advection across the poles steady state geostrophically balanced flow of both global and local scales forced nonlinear advection of an isolated low zonal flow impinging on an isolated mountain rossby haurwitz waves and observed atmospheric states one of the cases is also identified as a computer performance algorithm efficiency benchmark for assessing the performance of algorithms adapted to massively parallel computers

on a class of matrix orthogonal polynomials on the real line abstract it is shown how to construct matrix orthogonal polynomials on the real line provided we have a sequence of scalar orthogonal polynomials over an algebraic harmonic curve we consider the moment matrix associated with a distribution on and related to a basis of the linear space of polynomials obtained through the polynomial which defines the curve this particular moment matrix can be regarded as a hankel block matrix and thus matrix orthogonal polynomials appear in a natural way

free akivis algebras primitive elements and hyperalgebras free akivis algebras and primitive elements in their universal enveloping algebras are investigated it is proved that subalgebras of free akivis algebras are free and that finitely generated subalgebras are finitely residual decidability of the word problem for the variety of akivis algebras is also proved the conjecture of k h hofmann and k strambach problem 6 15 in topological and analytic loops in quasigroups and loops theory and applications series in pure mathematics o chein h o pflugfelder and j d h smith eds vol 8 pp 205 262 heldermann verlag berlin 1990 on the structure of primitive elements is proved to be not valid and a full system of primitive elements in free nonassociative algebra is constructed finally it is proved that every algebra b can be considered as a hyperalgebra that is a system with a series of multilinear operations that plays a role of a tangent algebra for a local analytic loop where the hyperalgebra operations on b are interpreted by certain primitive elements

eulerian idempotents and milnor moore theorem for certain non cocommutative hopf algebras abstract the purpose of this paper is to prove a milnor moore style theorem for a particular kind of non cocommutative hopf algebras the dendriform algebras a dendriform hopf algebra is a hopf algebra such that the product is the sum of two operations and verifying certain conditions between them and with the coproduct the role of lie algebras is played by brace algebras which are defined by n ary operations one for each n 2 satisfying some relations we show that a dendriform hopf algebra is isomorphic to the enveloping algebra of its brace algebra of primitive elements one of the ingredients of the proof is the construction of eulerian idempotents in this context

orthogonal matrix polynomials and higher order recurrence relations abstract it is well known that orthogonal polynomials on the real line satisfy a three term recurrence relation and conversely every system of polynomials satisfying a three term recurrence relation is orthogonal with respect to some positive borel measure on the real line we extend this result and show that every system of polynomials satisfying some 2 n 1 term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are n n matrices we apply this result to polynomials orthogonal with respect to a discrete sobolev inner product and other inner products in the linear space of polynomials as an application we give a short proof of krein s characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points

an all scale anelastic model for geophysical flows dynamic grid deformation we have developed an adaptive grid refinement approach for simulating geophysical flows on scales from micro to planetary our model is nonoscillatory forward in time nft nonhydrostatic and anelastic the major focus in this effort to date has been the design of a generalized mathematical framework for the implementation of deformable coordinates and its efficient numerical coding in a generic eulerian semi lagrangian nft format the key prerequisite of the adaptive grid is a time dependent coordinate transformation implemented rigorously throughout the governing equations of the model the transformation enables mesh refinement indirectly via dynamic change of the metric coefficients while retaining advantages of cartesian mesh calculations speed low memory requirements and accuracy conducted fully in the computational domain diverse test results presented in this paper simulations of a traveling stratospheric inertio gravity wave packet with numerically advected dense mesh region and an idealized climate of the earth with analytically prescribed adaptive transformations demonstrate the potential and the efficacy of the new deformable grid model for tracing targeted flow features and dynamically adjusting to prescribed undulations of model boundaries

a parallel adaptive barotropic model of the atmosphere the parallel adaptive model plasma has been developed for modeling a barotropic atmosphere this model adapts the computational grid at every time step according to a physical error indicator thus compared to uniform grid experiments the number of grid points is reduced significantly at the same time the error increases only slightly when comparing with uniform grid solutions for the discretization of the underlying spherical shallow water equations a lagrange galerkin method is used the unstructured triangular grid is maintained by the grid generator amatos and the large linear systems are solved by the parallel solver interface fossi experimental convergence is shown by means of steady state and unsteady analytical solutions plasma yields satisfactory results for quasi standard experiments that is the rossby haurwitz wave and zonal flows over an isolated mountain

an edge based unstructured mesh discretisation in geospherical framework an arbitrary finite volume approach is developed for discretising partial differential equations governing fluid flows on the sphere unconventionally for unstructured mesh global models the governing equations are cast in the anholonomic geospherical framework established in computational meteorology the resulting discretisation retains proven properties of the geospherical formulation while it offers the flexibility of unstructured meshes in enabling irregular spatial resolution the latter allows for a global enhancement of the spatial resolution away from the polar regions as well as for a local mesh refinement a class of non oscillatory forward in time edge based solvers is developed and applied to numerical examples of three dimensional hydrostatic flows including shallow water benchmarks on a rotating sphere

double shuffle relations for arborified zeta values abstract arborified zeta values are defined as iterated series and integrals using the universal properties of rooted trees this approach allows to study their convergence domain and to relate them to multiple zeta values generalisations to rooted trees of the stuffle and shuffle products are defined and studied it is further shown that arborified zeta values are algebra morphisms for these new products on trees

iterated upwind schemes for gas dynamics a class of high resolution schemes established in integration of anelastic equations is extended to fully compressible flows and documented for unsteady and steady problems through a span of mach numbers from zero to supersonic the schemes stem from iterated upwind technology of the multidimensional positive definite advection transport algorithm mpdata the derived algorithms employ standard and modified forms of the equations of gas dynamics for conservation of mass momentum and either total or internal energy as well as potential temperature numerical examples from elementary wave propagation through computational aerodynamics benchmarks to atmospheric small and large amplitude acoustics with intricate wave flow interactions verify the approach for both structured and unstructured meshes and demonstrate its flexibility and robustness

a discontinuous galerkin method for the shallow water equations in spherical triangular coordinates a global model of the atmosphere is presented governed by the shallow water equations and discretized by a runge kutta discontinuous galerkin method on an unstructured triangular grid the shallow water equations on the sphere a two dimensional surface in r 3 are locally represented in terms of spherical triangular coordinates the appropriate local coordinate mappings on triangles on every triangular grid element this leads to a two dimensional representation of tangential momentum and therefore only two discrete momentum equations the discontinuous galerkin method consists of an integral formulation which requires both area elements and line element faces integrals here we use a rusanov numerical flux to resolve the discontinuous fluxes at the element faces a strong stability preserving third order runge kutta method is applied for the time discretization the polynomial space of order k on each curved triangle of the grid is characterized by a lagrange basis and requires high order quadature rules for the integration over elements and element faces for the presented method no mass matrix inversion is necessary except in a preprocessing step the validation of the atmospheric model has been done considering standard tests from williamson et al d l williamson j b drake j j hack r jakob p n swarztrauber a standard test set for numerical approximations to the shallow water equations in spherical geometry j comput phys 102 1992 211 224 unsteady analytical solutions of the nonlinear shallow water equations and a barotropic instability caused by an initial perturbation of a jet stream a convergence rate of o dx k 1 was observed in the model experiments furthermore a numerical experiment is presented for which the third order time integration method limits the model error thus the time step dt is restricted by both the cfl condition and accuracy demands conservation of mass was shown up to machine precision and energy conservation converges for both increasing grid resolution and increasing polynomial order k

n extremal matrices of measures for an indeterminate matrix moment problem in this paper we study the n extremal matrices of measures associated to a completely indeterminate matrix moment problem i e those matrices of measure w solutions of a completely indeterminate matrix moment problem for which the linear space of matrix polynomials is dense in the corresponding l2 w

the spectral element method for the shallow water equations on the sphere the spectral element method is implemented for the shallow water equations in spherical geometry and its performance is compared with other models this is the first step in evaluating the suitability of spectral elements for climate modeling the potential advantages and disadvantages of spectral elements over more conventional models used for climate studies are discussed the method requires that the sphere be tiled with rectangles for which we make use of the gnomonic projection to map the sphere onto the cube to measure the performance of the method relative to other models results are presented from a standard suite of shallow water test cases for the sphere these results confirm the spectral accuracy of the method

mpdata an edge based unstructured grid formulation we present an advancement in the evolution of mpdata multidimensional positive definite advection transport algorithm over the last two decades mpdata has proven successful in applications using single block structured cuboidal meshes viz cartesian meshes while employing homeomorphic mappings to accommodate time dependent curvilinear domains motivated by the strengths of the cartesian mesh mpdata we develop a new formulation in an arbitrary finite volume framework with a fully unstructured polyhedral hybrid mesh in mpdata as in any taylor series based integration method for pde the choice of data structure has a pronounced impact on the technical details of the algorithm aiming at a broad range of applications with a large number of control volume cells we select a general compact and computationally efficient edge based data structure this facilitates the use of mpdata for problems involving complex geometries and or inhomogeneous anisotropic flows where mesh adaptivity is advantageous in this paper we describe the theory and implementation of the basic finite volume mpdata and document extensions important for applications a fully monotone scheme diffusion scheme and generalization to complete flow solvers theoretical discussions are illustrated with benchmark results in two and three spatial dimensions

implicit explicit formulations of a three dimensional nonhydrostatic unified model of the atmosphere numa we derive an implicit explicit imex formalism for the three dimensional 3d euler equations that allow a unified representation of various nonhydrostatic flow regimes including cloud resolving and mesoscale flow in a 3d cartesian domain as well as global regimes flow in spherical geometries this general imex formalism admits numerous types of methods including single stage multistep methods e g adams methods and backward difference formulas and multistage single step methods e g additive runge kutta methods the significance of this result is that it allows a numerical model to reuse the same machinery for all classes of time integration methods described in this work we also derive two classes of imex methods one dimensional and 3d and show that they achieve their expected theoretical rates of convergence regardless of the geometry e g 3d box or sphere and introduce a new second order imex runge kutta method that performs better than the other second order methods considered we

a pseudo analyzer approach to formal group laws not of operad type abstract formal group schemes associated to affine group schemes or lie groups by completion can be described by classical formal group laws more generally cogroup objects in categories of complete algebras e g associative are described by group laws for operads or analyzers m lazard has introduced analyzers to study formal group laws and group law chunks truncated formal power series a main example of a type of generalized formal group laws not given by an operad or analyzer are group laws corresponding to noncommutative complete hopf algebras to cover this case and other types of group laws pseudo analyzers are introduced we point out differences to the quadratic operad case e g there is no classification of group laws by koszul duality on the other hand we show how pseudo analyzer cohomology can be used to describe extension of group law chunks

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

on hopf algebra structures over free operads the operad lie can be constructed as the operad of primitives primas from the operad as of associative algebras this is reflected by the theorems of friedrichs poincare birkhoff witt and cartier milnor moore we replace the operad as by families of free operads p which include the operad mag freely generated by a non commutative non associative binary operation and the operad of stasheff polytopes we obtain poincare birkhoff witt type theorems and collect information about the operads primp e g in terms of characteristic functions

high order discontinuous galerkin schemes on general 2d manifolds applied to the shallow water equations an innovating approach is proposed to solve vectorial conservation laws on curved manifolds using the discontinuous galerkin method this new approach combines the advantages of the usual approaches described in the literature the vectorial fields are expressed in a unit non orthogonal local tangent basis derived from the polynomial mapping of curvilinear triangle elements while the convective flux functions are written is the usual 3d cartesian coordinate system the number of vectorial components is therefore minimum and the tangency constraint is naturally ensured while the method remains robust and general since not relying on a particular parametrization of the manifold the discontinuous galerkin method is particularly well suited for this approach since there is no continuity requirement between elements for the tangent basis definition the possible discontinuities of this basis are then taken into account in the riemann solver on inter element interfaces the approach is validated on the sphere using the shallow water equations for computing standard atmospheric benchmarks in particular the williamson test cases are used to analyze the impact of the geometry on the convergence rates for discretization error the propagation of gravity waves is eventually computed on non conventional irregular curved manifolds to illustrate the robustness and generality of the method

numerical solution of the reaction advection diffusion equation on the sphere a finite volume algorithm for the solution of the reaction advection diffusion equation on the sphere is derived and evaluated using analytical solutions the proposed approach is based on the principle of semidiscretization the convective and diffusive fluxes are approximated first and then the resulting set of the ordinary differential equations odes is solved using the appropriate time stepping algorithm in the first part of the paper solutions to both the linear advection and the advection diffusion problems for a single conservative scalar are discussed the monotonicity of the scheme is achieved with the explicit adaptive dissipation the development as well as the selected applications of the method are illustrated using a finite volume mesh constructed on the basis of geodesic icosahedral grid which in the past 40 years has been frequently applied in different models of geophysical fluid dynamics the performance of the solver is assessed using a suite of standard tests based on solid body rotation for different initial conditions after analysis of the advection diffusion problem the extension of the method for the equations with reactive terms is presented the performance of the solver is assessed by comparing the results to the analytical solution of the linearized reaction diffusion system in the final part of the paper the application of the solver for studies of nonlinear reactions on the sphere is illustrated the main intended application of the proposed method includes the simulation of transport of chemical constituents in the earth s atmosphere as well as the forecasting of moisture and cloud water fields in numerical weather prediction and climate models

runge kutta imex schemes for the horizontally explicit vertically implicit hevi solution of wave equations many operational weather forecasting centres use semi implicit time stepping schemes because of their good efficiency however as computers become ever more parallel horizontally explicit solutions of the equations of atmospheric motion might become an attractive alternative due to the additional inter processor communication of implicit methods implicit and explicit imex time stepping schemes have long been combined in models of the atmosphere using semi implicit split explicit or hevi splitting however most studies of the accuracy and stability of imex schemes have been limited to the parabolic case of advection diffusion equations we demonstrate how a number of runge kutta imex schemes can be used to solve hyperbolic wave equations either semi implicitly or hevi a new form of hevi splitting is proposed ufpreb which dramatically improves accuracy and stability of simulations of gravity waves in stratified flow as a consequence it is found that there are hevi schemes that do not lose accuracy in comparison to semi implicit ones the stability limits of a number of variations of trapezoidal implicit and some runge kutta imex schemes are found and the schemes are tested on two vertical slice cases using the compressible boussinesq equations split into various combinations of implicit and explicit terms some of the runge kutta schemes are found to be beneficial over trapezoidal especially since they damp high frequencies without dropping to first order accuracy we test schemes that are not formally accurate for stiff systems but in stiff limits nearly incompressible and find that they can perform well the scheme ark2 2 3 2 performs the best in the tests

aspects of the dynamical core of a nonhydrostatic deep atmosphere unified weather and climate prediction model the dynamical core which governs the evolution of resolved fluid dynamical processes is a critical element of any atmospheric model its governing equations must include all relevant dynamical terms and the numerical formulae used to approximate them must be accurate stable and efficient this is particularly so in a unified modeling environment in which the same dynamical core is used for both operational weather prediction and long term climate simulations recent research at the met office on unified dynamical core issues is reviewed aspects covered include properties of various equation sets vertical coordinates semi lagrangian advection and conservation trajectory computation and dynamical equivalence horizontal and vertical discretization and coupling of physical parameterizations to a dynamical core

algebras hyperalgebras nonassociative bialgebras and loops abstract sabinin algebras are a broad generalization of lie algebras that include lie malcev and bol algebras as very particular examples we present a construction of a universal enveloping algebra for sabinin algebras and the corresponding poincare birkhoff witt theorem a nonassociative counterpart of hopf algebras is also introduced and a version of the milnor moore theorem is proved loop algebras and universal enveloping algebras of sabinin algebras are natural examples of these nonassociative hopf algebras identities of loops move to identities of nonassociative hopf algebras by a linearizing process in this way nonassociative algebras and hopf algebras interlace smoothly