coupling fluid flow with porous media flow the transport of substances back and forth between surface water and groundwater is a very serious problem we study herein the mathematical model of this setting consisting of the stokes equations in the fluid region coupled with the darcy equations in the porous medium coupled across the interface by the beavers joseph saffman conditions we prove existence of weak solutions and give a complete analysis of a finite element scheme which allows a simulation of the coupled problem to be uncoupled into steps involving porous media and fluid flow subproblems this is important because there are many legacy codes available which have been optimized for uncoupled porous media and fluid flow

pbw bases of coideal subalgebras and a freeness theorem let h be a character hopf algebra every right coideal subalgebra u that contains the coradical has a pbw basis which can be extended up to a pbw basis of h if additionally u is a bosonization of an invariant with respect to the left adjoint action subalgebra then h is a free left and right u module with a free pbw basis over u these results remain valid if h is a braided hopf algebra generated by a categorically ordered subset of primitive elements if the ground field is algebraically closed the results are still true provided that h is a pointed hopf algebra with commutative coradical and is generated over the coradical by a direct sum of finite dimensional yetter drinfeld submodules of skew primitive elements

hopf algebras of low dimension the main aim of this paper is to classify all types of hopf algebras of dimension less thn or equal to 11 over an algebraically closed field of characteristic 0 if a is such a hopf algebra that is not semisimple then we shall prove that a or a is pointed this property will result from the fact that under some assumptions any hopf algebra that is generated as an algebra by a four dimensional simple subcoalgebra is a hopf quotient of the coordinate ring of quantum sl 2 k the first result allows us to reduce the classification to the case of pointed hopf algebras of dimension 8 we shall describe their types in the last part of the paper

hopf algebras of dimension pq ii abstract let h be a hopf algebra of dimension pq over an algebraically closed field of characteristic zero where p q are odd primes with p q 4 p 11 we prove that h is semisimple and thus isomorphic to a group algebra or the dual of a group algebra

convergence of ipdg for coupled time dependent navierstokes and darcy equations a numerical method is proposed and analyzed for the coupled time dependent navierstokes equations and darcy equations existence and uniqueness of the solution are obtained under a small data condition a priori error estimates are derived numerical examples confirm the theoretical convergence rates

on the even powers of the antipode of a finite dimensional hopf algebra abstract the trace of powers of the square of the antipode s 2 of a finite dimensional hopf algebra a over a field k is studied it is shown in many cases that the trace function vanishes on s 2 m when s 2 m 1 a finer properties of the antipode are related to this phenomenon

on semisimple hopf algebras of dimension pq2 let p and q be distinct prime numbers we prove a result on the existence of nontrivial group like elements in a certain class of semisimple hopf algebras of dimension pqr we conclude the classification of semisimple hopf algebras a of dimension pq2 over an algebraically closed field k of characteristic zero such that both a and a are of frobenius type we also complete the classification of semisimple hopf algebras of dimension pq2 100

an approach to quasi hopf algebras via frobenius coordinates abstract we study quasi hopf algebras and their subobjects over certain commutative rings from the point of view of frobenius algebras we introduce a type of radford formula involving an anti automorphism and the nakayama automorphism of a frobenius algebra then view several results in quantum algebras from this vantage point in addition separability and strong separability of quasi hopf algebras are studied as frobenius algebras

a freeness theorem for nichols algebras abstract we prove a freeness result for nichols algebras over certain nichols subalgebras this result allows us in particular to classify pointed hopf algebras of prime and prime squared index

on cocommutative hopf algebras of finite representation type abstract let g be a finite algebraic group defined over an algebraically closed field k of characteristic p 0 such a group decomposes into a semidirect product g g 0 g red with a constant group g red and a normal infinitesimal subgroup g 0 if the principal block b 0 g of the group algebra h g has finite representation type then both constituents have the same property with at least one of them being semisimple we determine the structure of the infinitesimal constituent g 0 up to the classification of v uniserial groups

long time stability and convergence rate of maccormack rapid solver method for nonstationary stokes darcy problem abstract we propose and study a combination of two second order implicit explicit imex methods for the coupled stokes darcy system that governs flows in karst aquifers the first is a second order explicit two step maccormack scheme and the second is a second order implicit crank nicolson method both algorithms only require the solution of two decoupled problems at each time step one stokes and the other darcy this combination so called the maccormack rapid solver method is very efficient faster at least of second order accuracy in time and space and can be easily implemented using legacy codes under time step limitation of the form t c h where h t are mesh size and time step respectively and c is a physical parameter we prove both long time stability and the rate of convergence of the method some numerical experiments are presented and discussed

accelerating the convergence of spectral deferred correction methods in the recent paper by dutt greengard and rokhlin a variant of deferred or defect correction methods is presented which couples gaussian quadrature with the picard integral equation formulation of the initial value ordinary differential equation the resulting spectral deferred correction sdc methods have been shown to possess favorable accuracy and stability properties even for versions with very high order of accuracy in this paper we show that for linear problems the iterations in the sdc algorithm are equivalent to constructing a preconditioned neumann series expansion for the solution of the standard collocation discretization of the ode this observation is used to accelerate the convergence of sdc using the gmres krylov subspace method for nonlinear problems the gmres acceleration is coupled with a linear implicit approach stability and accuracy analyses show the accelerated scheme provides an improvement in the accuracy efficiency and stability of the original sdc approach furthermore preliminary numerical experiments show that accelerating the convergence of sdc methods can effectively eliminate the order reduction previously observed for stiff ode systems

on the solution of the coupled navier stokes and darcy equations abstract this paper introduces and analyzes two mathematical models for coupling the incompressible navier stokes equations with the porous media flow equations a numerical method that uses continuous finite elements in the incompressible flow region and discontinuous finite elements in the porous medium is proposed existence and uniqueness results under small data condition of the numerical solution are proved optimal a priori error estimates are derived numerical examples comparing the two models under varying physical parameters are provided

quantum groups acting on 4 points we classify the compact quantum groups acting on 4 points these are the quantum subgroups of the quantum permutation group q4 our main tool is a new presentation for the algebra c q4 corresponding to an isomorphism of type q4 so 1 3 the quantum subgroups of q4 are subject to a mckay type correspondence that we describe at the level of algebraic invariants

numerical analysis of a second order algorithm for a non stationary navier stokes darcy model abstract in this paper we construct a second order algorithm based on the semi implicit spectral deferred correction method for the non stationary coupled navier stokes darcy equations with the beavers joseph saffman s interface condition we present a complete theoretical analysis to prove that this algorithm is stable and possesses second order accuracy in time numerical examples confirm the convergence rate and the effectiveness of our algorithm

on fusion categories using a variety of methods developed in the literature in particular the theory of weak hopf algebras we prove a number of general results about fusion categories in characteristic zero we show that the global dimension of a fusion category is always positive and that the s matrix of any not necessarily hermitian modular category is unitary we also show that the category of module functors between two module categories over a fusion category is semisimple and that fusion categories and tensor functors between them are undeformable generalized ocneanu rigidity in particular the number of such categories functors realizing a given fusion datum is finite finally we develop the theory of frobenius perron dimensions in an arbitrary fusion category at the end of the paper we generalize some of these results to positive characteristic

numerical solution to a mixed navier stokes darcy model by the two grid approach we study numerical methods for a coupled navier stokes darcy model which describes a fluid flow filtrating through porous media a decoupled and linearized two grid algorithm is proposed numerical analysis and experiments are presented to show the efficiency and effectiveness of the decoupled and linearized algorithm

decoupled schemes for a non stationary mixed stokes darcy model we study numerical methods for solving a non stationary mixed stokes darcy problem that models coupled fluid flow and porous media flow a decoupling approach based on interface approximation via temporal extrapolation is proposed for devising decoupled marching algorithms for the mixed model error estimates are derived and numerical experiments are conducted to demonstrate the computational effectiveness of the decoupling approach

on hopf algebras of dimension 4p abstract in this paper we prove that a non semisimple hopf algebra h of dimension 4 p with p an odd prime over an algebraically closed field of characteristic zero is pointed provided h contains more than two group like elements in particular we prove that non semisimple hopf algebras of dimensions 20 28 and 44 are pointed or their duals are pointed and this completes the classification of hopf algebras in these dimensions

a two grid method of a mixed stokes darcy model for coupling fluid flow with porous media flow we study numerical methods for solving a coupled stokes darcy problem in porous media flow applications a two grid method is proposed for decoupling the mixed model by a coarse grid approximation to the interface coupling conditions error estimates are derived for the proposed method both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two grid approach for solving multimodeling problems potential extensions and future directions are discussed

noetherian pi hopf algebras are gorenstein we prove that every noetherian affine pi hopf algebra has finite injective dimension which answers a question of brown 1998

on the flatness and the projectivity over hopf subalgebras of hopf algebras over dedekind rings abstract we study the flatness and the projectivity of hopf algebras defined over a dedekind ring over their hopf subalgebras we give a criterion for the faithful flatness and use it to show the faithful flatness of an arbitrary flat hopf algebra upon its finite normal hopf subalgebras for the projectivity over hopf subalgebras of a hopf algebra we need some finiteness condition in terms of the module of integrals in particular we show that the module of integrals is projective of rank one

voter models on heterogeneous networks we study simple interacting particle systems on heterogeneous networks including the voter model and the invasion process these are both two state models in which in an update event an individual changes state to agree with a neighbor for the voter model an individual imports its state from a randomly chosen neighbor here the average time t n to reach consensus for a network of n nodes with an uncorrelated degree distribution scales as n ensuremath mu 1 2 ensuremath mu 2 where ensuremath mu k is the k text th moment of the degree distribution quick consensus thus arises on networks with broad degree distributions we also identify the conservation law that characterizes the route by which consensus is reached parallel results are derived for the invasion process in which the state of an agent is exported to a random neighbor we further generalize to biased dynamics in which one state is favored the probability for a single fitter mutant located at a node of degree k to overspread the population char22 the fixation probability char22 is proportional to k for the voter model and to 1 k for the invasion process

frobenius extensions of subalgebras of hopf algebras we consider when extensions s r of subalgebras of a hopf algebra are frobenius that is frobenius of the second kind given a hopf algebra h we show that when s r are hopf algebras in the yetter drinfeld category for h the extension is frobenius provided r is finite over s and the extension of biproducts s h r h is cleft more generally we give conditions for an extension to be frobenius in particular we study extensions of integral type and consider when the frobenius property is inherited by the subalgebras of coinvariants we apply our results to extensions of enveloping algebras of lie coloralgebras thus extending a result of bell and farnsteiner for lie superalgebras 0 introduction in this paper we consider when various extensions s r of subalgebras of a hopf algebra are frobenius such extensions generalize the usual notion of frobenius extensions by having the module action on one side twisted by an automorphism of s it was already known that any extension of finite dimensional hopf algebras is frobenius a result of the third author sch 92 as is any finite extension u k u l of enveloping algebras of lie superalgebras a result of bell and farnsteiner bf note that u l is not an ordinary hopf algebra but rather a hopf algebra in the category of z2 graded modules these results were an important motivation for this paper and raised the question as to when an extension of hopf algebras in a category was frobenius one of the main results of this paper is that an extension s r of hopf algebras of finite index in the yetter drinfeld category hyd for a given hopf algebra h is frobenius provided that the associated extension of hopf algebras s h r h the biproducts of s and r with h has a normal basis theorem 5 6 this will happen whenever r and h are finite dimensional or when r h is pointed as an application we are able to generalize the bf result to lie coloralgebras if u k u l is a finite extension of enveloping algebras of lie coloralgebras then it is frobenius moreover we give an explicit description of the automorphism of u k and of the frobenius homomorphism f u l u k corollary 6 3 along the way we prove a number of other results about frobenius extensions and conditions that ensure that an extension is frobenius in section 1 we give a short direct proof of the fact that an extension b a of finite dimensional hopf algebras is always frobenius moreover we give explicit received by the editors december 10 1995 1991 mathematics subject classification primary 16w30 secondary 17b35 17b37 c 1997 american mathematical society

semi implicit spectral deferred correction methods for highly nonlinear partial differential equations abstract the goal of this paper is to develop a novel semi implicit spectral deferred correction sdc time marching method the method can be used in a large class of problems especially for highly nonlinear ordinary differential equations odes without easily separating of stiff and non stiff components which is more general and efficient comparing with traditional semi implicit sdc methods the proposed semi implicit sdc method is based on low order time integration methods and corrected iteratively the order of accuracy is increased for each additional iteration and we also explore its local truncation error analytically this sdc method is intended to be combined with the method of lines which provides a flexible framework to develop high order semi implicit time marching methods for nonlinear partial differential equations pdes in this paper we mainly focus on the applications of the nonlinear pdes with higher order spatial derivatives e g convection diffusion equation the surface diffusion and willmore flow of graphs the cahn hilliard equation the cahn hilliard brinkman system and the phase field crystal equation coupled with the local discontinuous galerkin ldg spatial discretization the fully discrete schemes are all high order accurate in both space and time and stable numerically with the time step proportional to the spatial mesh size numerical experiments are carried out to illustrate the accuracy and capability of the proposed semi implicit sdc method

locally conservative coupling of stokes and darcy flows a locally conservative numerical method for solving the coupled stokes and darcy flows problem is formulated and analyzed the approach employs the mixed finite element method for the darcy region and the discontinuous galerkin method for the stokes region a discrete inf sup condition and optimal error estimates are derived

high order multi implicit spectral deferred correction methods for problems of reactive flow models for reacting flow are typically based on advection diffusion reaction a d r partial differential equations many practical cases correspond to situations where the relevant time scales associated with each of the three sub processes can be widely different leading to disparate time step requirements for robust and accurate time integration in particular interesting regimes in combustion correspond to systems in which diffusion and reaction are much faster processes than advection the numerical strategy introduced in this paper is a general procedure to account for this time scale disparity the proposed methods are high order multi implicit generalizations of spectral deferred correction methods misdc methods constructed for the temporal integration of a d r equations spectral deferred correction methods compute a high order approximation to the solution of a differential equation by using a simple low order numerical method to solve a series of correction equations each of which increases the order of accuracy of the approximation the key feature of misdc methods is their flexibility in handling several sub processes implicitly but independently while avoiding the splitting errors present in traditional operator splitting methods and also allowing for different time steps for each process the stability accuracy and efficiency of misdc methods are first analyzed using a linear model problem and the results are compared to semi implicit spectral deferred correction methods furthermore numerical tests on simplified reacting flows demonstrate the expected convergence rates for misdc methods of orders three four and five the gain in efficiency by independently controlling the sub process time steps is illustrated for nonlinear problems where reaction and diffusion are much stiffer than advection although the paper focuses on this specific time scales ordering the generalization to any ordering combination is straightforward