algebraic splitting for incompressible navier stokes equations fully discretized incompressible navier stokes equations are solved by splitting the algebraic system with an approximate factorization this splitting affects the temporal convergence order of velocity and pressure the splitting error is proportional to the pressure variable and a simple analysis shows that the original convergence order of the time integration scheme can be retained by solving for incremental pressure the combination of splitting and incremental pressure is shown to be equivalent to an error correcting method using the full pressure in numerical experiments employing a third order time integration scheme and various orders for the pressure increment the splitting error is shown to control the convergence order and the full order of the scheme is recaptured for both velocity and pressure the difference between perturbing the momentum or the continuity equation is also explored

the goldie theorem for h semiprime algebras abstract the main result states that under certain assumptions about a hopf algebra h every h semiprime right noetherian h module algebra has a quasi frobenius classical right quotient ring another question treated in the paper is concerned with the extension of h module structures to quotient rings these results have an application to the semiprimeness problem for smash product algebras a h

a mackey imprimitivity theory for algebraic groups let g be an affine algebraic group over an algebraically closed field k and let h be a closed subgroup of g if v is a rational h module a comodule for the coordinate ring of h there is a now well known notion of an induced module vi g for g defined as the space morph g v of all h equivariant morphisms from g to a finite dimensional subspace of v with obvious g action the question arises given a rational g module m how can one recognize m as an induced module vig for a finite group g the answer is the mackey imprimitivity theorem the module m is induced if and only if it is a direct sum of subspaces permuted transitively by g with h the stabilizer of one of these subspaces called v one uses this result for example in proving the famous mackey decomposition theorem which describes the restriction of any induced module to a second subgroup l as a direct sum of suitable induced modules given the imprimitivity theorem the proof is just a matter of grouping the summands permuted by g into their orbits under l in the case of algebraic groups the situation is quite different for some subgroups h all g modules are induced this occurs for example if g is connected and k g h k e g if h is parabolic also if g is a connected unipotent group then a rational g module m is induced from some proper subgroup if and only if its endomorphism ring contains a two dimensional submodule e for the conjugation action of g with e k 1 and such that k l is precisely the subspace annihilated by the action of the lie algebra of g on e 27 cf also 5 5 below the latter is an application of a general criterion in case g h is affine a rational g module m is induced if and only if there is an action of the coordinate ring a of g h compatible with the action of g on both a and m note that this generalizes the imprimitivity theorem in the case of finite groups since then a has a k basis of g h orthogonal idempotents permuted transitively by g

coring stabilizers for a hopf algebra coaction abstract the coring stabilizer stab p is introduced for any prime ideal p of a right h comodule algebra a such that the factor ring a p is either right or left goldie this notion is used to obtain hopf algebraic analogs of two category equivalences associated with a homogeneous space the category of linearized quasicoherent sheaves on a noncommutative homogeneous space is interpreted as a suitable quotient category of the category of hopf modules birational invariance of such quotient categories is proved it is shown that for a birational h coequivariant extension b of a properly defined subsets of prime ideals of a and b correspond to each other bijectively

congruence subgroups and generalized frobenius schur indicators we introduce generalized frobenius schur indicators for pivotal categories in a spherical fusion category mathcal c an equivariant indicator of an object in mathcal c is defined as a functional on the grothendieck algebra of the quantum double z mathcal c via generalized frobenius schur indicators the set of all equivariant indicators admits a natural action of the modular group using the properties of equivariant indicators we prove a congruence subgroup theorem for modular categories as a consequence all modular representations of a modular category have finite images and they satisfy a conjecture of eholzer in addition we obtain two formulae for the generalized indicators one of them a generalization of bantay s second indicator formula for a rational conformal field theory this formula implies a conjecture of pradisi sagnotti stanev as well as a conjecture of borisov halpern schweigert

algebraic splitting methods for the steady incompressible navier stokes equations at moderate reynolds numbers abstract the unsteady incompressible navier stokes equations in primitive variables are often numerically solved by segregating the computation of velocity and pressure according to either functional analysis arguments following the pioneering work of a j chorin and r temam std split then discretize paradigm or linear algebra arguments based on the inexact block factorization of the discretized problem dts discretize then split paradigm the presence of the time derivative allows for the calibration of an appropriate approximation of the pseudo differential operator of the pressure problem and excellent results in terms of both accuracy and efficiency have been obtained as witnessed by the abundant literature the extension of the same segregated approach to the steady navier stokes equations is unclear unless a pseudo time advancing formulation is undertaken in this paper we present a methodology for a segregated computation of the primitive variables in a genuinely steady formulation so to avoid iterations to get to the steady limit the approach is largely inspired by the algebraic factorization of the unsteady problem dts approach yet we detail specific settings required by the absence of the velocity time derivative the basic idea relies on the introduction of some parameters in a modified picard linearization we discuss stability bounds and the convergence of the segregated method to the unsplit solution several numerical results on different test cases confirm the efficiency of the procedure

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

on kaplansky s fifth conjecture abstract we prove that the antipode of a semisimple hopf algebra is an involution if the characteristic of the base field is very large

on hopf algebras with cocommutative coradicals 1 zf k is a right coideal subalgebra of h such that s k k h is faithfully flat as a right or left k module 2 zf x h n h is a quotient left h module coalgebra h is faithfully coflat as a right or left x h comodule 3 k h nk and 7 h h establish a 1 1 correspondence between the right coideal subalgebras k of h such that s k ko and the quotient left h module coalgebras n h n h 4 suppose k x in 3 there is a left x h comodule and right k module isomorphism h n x h q k if either a h is pointed or b k is finite dimensional over k

the character table for a hopf algebra arising from the drinfel d double abstract in this paper we give a definition of the character table for a hopf algebra h over a commutative field k of characteristic 0 when h is a finite dimensional semisimple hopf algebra over an algebraically closed field k of characteristic 0 we obtain a formula for the character table which extends the notion of character table for a finite group

adaptive time stepping for incompressible flow part i scalar advection diffusion even the simplest advection diffusion problems can exhibit multiple time scales this means that robust variable step time integrators are a prerequisite if such problems are to be efficiently solved computationally the performance of the second order trapezoid rule using an explicit adams bashforth method for error control is assessed in this work this combination is particularly well suited to long time integration of advection dominated problems herein it is shown that a stabilized implementation of the trapezoid rule leads to a very effective integrator in other situations specifically diffusion problems with rough initial data and general advection diffusion problems with different physical time scales governing the system evolution

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

module categories over pointed hopf algebras we develop some techniques for studying exact module categories over some families of pointed finite dimensional hopf algebras as an application we classify exact module categories over the tensor category of representations of the small quantum groups u q mathfrak sl 2

invariants of spin three manifolds from chern simons theory and finite dimensional hopf algebras abstract a version of kirby calculus for spin and framed three manifolds is given and is used to construct invariants of spin and framed three manifolds in two situations the first is ribbon categories which possess odd degenerate objects this case includes the quantum group situations corresponding to the half integer level chern simons theories conjectured to give spin tqfts by dijkgraaf and witten 1990 commun math phys 129 393 429 in particular the spin invariants constructed by kirby and melvin 1991 invent math 105 473 545 are shown to be identical to the invariants associated to so 3 second an invariant of spin manifolds analogous to the hennings invariant is constructed beginning with an arbitrary factorizable unimodular quasitriangular hopf algebra in particular a framed manifold invariant is associated to every finite dimensional hopf algebra via its quantum double and is conjectured to be identical to kuperberg s noninvolutory invariant of framed manifolds associated to that hopf algebra

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

a method of moments approach for the efficient and accurate modeling of moderately thick cylindrical wire antennas this paper introduces a moment method formulation which is capable of accurately modeling moderately thick cylindrical wire antennas new algorithms are presented for the efficient computation of the cylindrical wire kernel and related impedance matrix integrals these algorithms make use of exact series representations as well as efficient numerical procedures and lead to a significant reduction in overall computation time for thicker wires another major advantage of this moment method technique is that it is no longer restricted by the segment length to radius ratio limitations inherent in past formulations thereby making it possible to achieve solution convergence for a much wider class of wire antenna structures several examples illustrating the superior convergence properties of this new moment method formulation are presented and discussed

higher genus mapping class group invariants from factorizable hopf algebras abstract lyubashenko s construction associates representations of mapping class groups map g n of riemann surfaces of any genus g with any number n of holes to a factorizable ribbon category we consider this construction as applied to the category of bimodules over a finite dimensional factorizable ribbon hopf algebra h for any such hopf algebra we find an invariant of map g n for all values of g and n more generally we obtain such invariants for any pair h where is a ribbon automorphism of h our results are motivated by the quest to understand higher genus correlation functions of bulk fields in two dimensional conformal field theories with chiral algebras that are not necessarily semisimple so called logarithmic conformal field theories

algebraic fractional step schemes with spectral methods for the incompressible navier stokes equations the numerical investigation of a recent family of algebraic fractional step methods for the solution of the incompressible time dependent navier stokes equations is presented these methods are improved versions of the yosida method proposed in a quarteroni f saleri a veneziani factorization methods for the numerical approximation of navier stokes equations comput methods appl mech engrg 188 1 3 2000 505 526 a quarteroni f saleri a veneziani j math pures appl 9 78 5 1999 473 503 and one of them the yosida4 method is proposed in this paper for the first time they rely on an approximate lu block factorization of the matrix obtained after the discretization in time and space of the navier stokes system yielding a splitting in the velocity and pressure computation in this paper we analyze the numerical performances of these schemes when the space discretization is carried out with a spectral element method with the aim of investigating the impact of the splitting on the global accuracy of the computation

pressure correction algebraic splitting methods for the incompressible navier stokes equations in this paper we present a new family of methods for the effective numerical solution of the incompressible unsteady navier stokes equations these methods resort to an algebraic splitting of the discretized problem based on inexact lu block factorizations of the corresponding matrix following a quarteroni f saleri and a veneziani comput methods appl mech engrg 188 2000 pp 505 526 in particular we will start from inexact algebraic factorizations of algebraic chorin temam and yosida type and introduce a pressure correction step aimed at improving the time accuracy one of the schemes obtained in this way the algebraic chorin temam pressure correction method resembles a method previously introduced in the framework of differential projection schemes see l timmermans p minev and f v de vosse internat j numer methods fluids 22 1996 pp 673 688 a prohl projection and quasi compressibility methods for solving the incompressible navier stokes equations teubner stuttgart 1997 the stability and the dependence of splitting error on the time step of the new methods is investigated and tested on several numerical cases

comments on fundamentals of thin wire integral equations with the finite gap generator in papers 1 2 the authors extend the so called effective current method to the hallen s and pocklington s equations with the finite gap generator fgg the indicated method considered earlier in 3 with a reference to 4 is associated with application of the method of auxiliary sources to the analysis of cylindrical dipole antennas 5 7 the purpose of these comments is to draw the authors and readers attention to the existence ofpaper 8 where the pocklington s equation with the fgg is solved for the axial current considered there to be auxiliary the equation is solved by the galerkin method using the piecewise sinusoidal basis and testing functions the auxiliary axial current is used further for calculation of the surface current which is referred to as the effective current in 1 4 and the surface current itself is used for calculation of the input impedance so in some respects papers 1 7 just repeat the main idea and technique of paper 8 although of course papers 1 7 also contain a lot of their own new interesting and useful analytical and numerical results

a note on the consistency and stability properties of yosida fractional step schemes for the unsteady stokes equations in this short note we give some improvements to the stability and consistency analysis of algebraic splitting methods for solving the incompressible fluid equations carried out in p gervasio siam j numer anal 46 2008 pp 1682 1703 in particular we generalize to any value of the approximation index p in mathbb n proposition 3 1 and lemma 3 2 of p gervasio siam j numer anal 46 2008 pp 1682 1703 on the consistency error associated with the algebraic splitting originally proved for p 0 1 2 moreover we remove an assumption of lemma 3 5 of p gervasio siam j numer anal 46 2008 pp 1682 1703

on the oscillations appearing in numerical solutions of solvable and nonsolvable integral equations for thin wire antennas differences between certain solvable and nonsolvable ill posed integral equations with the same nonsingular kernel are discussed the main results come from constructing a solvable equation in the context of straight thin wire antennas the kernel of this equation is the usual approximate also called reduced kernel while its exact solution is the familiar sinusoidal current numerical solutions to this solvable equation are compared to corresponding numerical solutions of the usual halle n and pocklington equations with the approximate kernel it is known from previous publications that these last two equations are nonsolvable and that their numerical solutions present severe oscillations when the number of basis functions is sufficiently large it is found that the difficulties encountered in the former solvable equation are much less important compared to those of the nonsolvable ones the same conclusion is brought out from other integral equations arising in different contexts thin wire circular loop antenna method of auxiliary sources and straight wire antenna of infinite length we discuss the consistency of our results with picard s theorem the results in this paper supplement previous publications regarding the difficulties of numerically solving thin wire integral equations with the approximate kernel

convergence analysis of high order algebraic fractional step schemes for time dependent stokes equations in this paper we analyze the family of yosida algebraic fractional step schemes proposed in a quarteroni f saleri and a veneziani comput methods appl mech engrg 188 2000 pp 505 526 f saleri and a veneziani siam j numer anal 43 2005 pp 174 194 and p gervasio f saleri and a veneziani j comput phys 214 2006 pp 347 365 when applied to time dependent stokes equations under suitable regularity assumptions on the data splitting error estimates both for velocity and pressure are established in particular we analyze the first three methods of this family providing respectively convergence of the fractional step solution towards the numerical solution achieved without any operator splitting of orders 3 2 5 2 7 2 for the velocity and 1 2 3 for the pressure moreover a general way to set up higher order schemes is proposed the present analysis is carried out when spectral element methods are employed for space discretization

a time adaptive space time finite element method for incompressible lagrangian flows with free surfaces computational issues abstract in order to further enhance the performance of the space time galerkin least squares method for solving incompressible navier stokes problems involving free surfaces issues related to the solution strategy and time adaptivity are addressed due to the a priori unknown boundary positions a nonlinear system of equations normally arises at each space time slab which is solved by the newton raphson approach in addition a linear system of equations for velocity and pressure that provides an alternative approach to the solution of the problem is also derived in this paper this linear solution scheme can significantly reduce the computational costs in terms of computer cpu time and memory requirements without the sacrifice of the solution accuracy if the time step size is sufficiently small furthermore the possibility of adaptively adjusting time step size is fully exploited by choosing the volume loss rate as an error indicator a simple adaptive time stepping scheme is presented finally several numerical examples are provided to assess the performances of the proposed schemes

on the structure of irreducible yetter drinfeld modules over quasi triangular hopf algebras abstract let h r be a finite dimensional semisimple and cosemisimple quasi triangular hopf algebra over a field k in this paper we give the structure of irreducible objects of the yetter drinfeld module category yd h h let h r be the majid s transmuted braided group of h r we show that h r is cosemisimple as a coalgebra let h r d 1 d r be the sum of minimal h adjoint stable subcoalgebras for each i 1 i r we choose a minimal left coideal w i of d i and we can define the r adjoint stable algebra n w i of w i using ostrik s theorem on characterizing module categories over monoidal categories we prove that v h h yd is irreducible if and only if there exists an i 1 i r and an irreducible right n w i module u i such that v u i n w i h w i our structure theorem generalizes the results of dijkgraaf pasquier roche and gould on yetter drinfeld modules over finite group algebras if k is an algebraically closed field of characteristic 0 we stress that the r adjoint stable algebra n w i is an algebra over which the dimension of each irreducible right module divides its dimension

semisimple hopf algebras of dimension 60 abstract we determine the isomorphism classes of semisimple hopf algebras of dimension 60 which are simple as hopf algebras

on module categories over finite dimensional hopf algebras abstract we show that indecomposable exact module categories over the category reph of representations of a finite dimensional hopf algebra h are classified by left comodule algebras h simple from the right and with trivial coinvariants up to equivariant morita equivalence specifically any indecomposable exact module category is equivalent to the category of finite dimensional modules over a left comodule algebra this is an alternative approach to the results of etingof and ostrik for this we study the stabilizer introduced by yan and zhu and show that it coincides with the internal hom we also describe the correspondence of module categories between reph and rep h

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

relative fourier transforms and expectations on coideal subalgebras abstract for an algebraic compact quantum group h we establish a bijection between the set of right coideal subalgebras a h and that of left module quotient coalgebras h c it turns out that the inclusion a h always splits as a map of right a modules and right h comodules and the resulting expectation e h a is positive and lifts to a positive map on the full c completion on h if and only if a is invariant under the squared antipode of h the proof proceeds by tannaka reconstructing the coalgebra c corresponding to a h by means of a fiber functor from h equivariant a modules to hilbert spaces while the characterization of those a h which admit positive expectations makes use of a fourier transform turning elements of h into functions on c

on the application of numerical methods to hallen s equation the so called hallen integral equation for the current on a finite linear antenna center driven by a delta function generator takes two forms depending on the choice of kernel the two kernels are usually referred to as the exact and the approximate or reduced kernel with the approximate kernel the integral equation has no solution nevertheless the same numerical method is often applied to both forms of the integral equation in this paper the behavior of the numerical solutions thus obtained is investigated and the similarities and differences between the two numerical solutions are discussed the numerical method is galerkin s method with pulse functions we first apply this method to the two corresponding forms of the integral equation for the current on a linear antenna of infinite length in this case the method yields an infinite toeplitz system of algebraic equations in which the width of the pulse basis functions enters as a parameter the infinite system is solved exactly for nonzero pulse width the exact solution is then developed asymptotically for the case where the pulse width is small when the asymptotic expressions for the case of the infinite antenna are used as a guide for the behavior of the solutions of the finite antenna the latter problem is greatly facilitated for the approximate kernel the main results of this paper carry over to a certain numerical method applied to the corresponding equation of the pocklington type

some interrelations between hopf algebras and their duals abstract in this paper we study certain interrelations between subsets of hopf algebras h and their duals which stem from various morphisms t defined on h we define and study t cocommutative elements cocommutative elements generators of h as an h module and invariants under various actions and coactions are all examples an important role is played by nakayama automorphisms these ideas are applied to co quasitriangular hopf algebras in particular when they are factorizable then the drinfeld element gives rise to a nakayama automorphism and to a special element in the dual explicit calculations are carried out for the drinfeld double

radford drinfeld and cardy boundary states in the 1 p logarithmic conformal field models we introduce p 1 pseudocharacters in the space of 1 p model vacuum torus amplitudes to complete the distinguished basis in the 2p dimensional fusion algebra to a basis in the whole 3p 1 dimensional space of torus amplitudes and the structure constants in this basis are not necessarily non negative integer numbers we obtain a generalized verlinde formula that gives these structure constants in the context of theories with boundaries we identify the space of vacuum torus amplitudes with the space of ishibashi states then we propose 3p 1 boundary states satisfying the cardy condition

invariants of 3 manifolds and projective representations of mapping class groups via quantum groups at roots of unity an example of a finite dimensional factorizable ribbon hopf algebra is given by a quotienth u q g of the quantized universal enveloping algebrau q g at a root of unityq of odd degree the mapping class groupm g 1 of a surface of genusg with one hole projectively acts by automorphisms in theh moduleh g ifh is endowed with the coadjointh module structure there exists a projective representation of the mapping class groupm g n of a surface of genusg withn holes labeled by finite dimensionalh modulesx 1 x n in the vector space hom h x 1 x n h g an invariant of closed oriented 3 manifolds is constructed modifications of these constructions for a class of ribbon hopf algebras satisfying weaker conditions than factorizability including most ofu q g at roots of unityq of even degree are described

the use of the frill generator in thin wire integral equations for a frill generator feed certain fundamental mathematical properties of hallen s and pocklington s equations with the approximate kernel are stated and derived for a particular moment method procedure galerkin s method with pulse functions applied to hallen s equation the consequences to the numerical solutions are carefully examined generalizations to other numerical methods with subsectional basis functions are discussed many of our results come from studying the simpler problem of the infinite dipole analytically and applying the understanding thus obtained to the case of the finite dipole

segregated runge kutta methods for the incompressible navier stokes equations in this work we propose runge kutta time integration schemes for the incompressible navier stokes equations with two salient properties first velocity and pressure computations are segregated at the time integration level without the need to perform additional fractional step techniques that spoil high orders of accuracy second the proposed methods keep the same order of accuracy for both velocities and pressures the segregated runge kutta methods are motivated as an implicit explicit runge kutta time integration of the projected navier stokes system onto the discrete divergence free space and its re statement in a velocity pressure setting using a discrete pressure poisson equation we have analysed the preservation of the discrete divergence constraint for segregated runge kutta methods and their relation in their fully explicit version with existing half explicit methods we have performed a detailed numerical experimentation for a wide set of schemes from first to third order including implicit and imex integration of viscous and convective terms for incompressible laminar and turbulent flows further segregated runge kutta schemes with adaptive time stepping are proposed

on twisting of finite dimensional hopf algebras in this paper we study the properties of drinfeld s twisting for finite dimensional hopf algebras we determine how the integral of the dual to a unimodular hopf algebra h changes under twisting of h we show that the classes of cosemisimple unimodular cosemisimple involutive cosemisimple quasitriangular finite dimensional hopf algebras are stable under twisting we also prove the cosemisimplicity of a coalgebra obtained by twisting of a cosemisimple unimodular hopf algebra by two different twists on two sides such twists are closely related to bi galois extensions and describe the representation theory of its dual next we define the notion of a non degenerate twist for a hopf algebra h and set up a bijection between such twists for h and h this bijection is based on miyashita ulbrich actions of hopf algebras on simple algebras it generalizes to the non commutative case the procedure of inverting a non degenerate skew symmetric bilinear form on a vector space finally we apply these results to classification of twists in group algebras and of cosemisimple triangular finite dimensional hopf algebras in positive characteristic generalizing the previously known classification in characteristic zero

rigid dualizing complexes over quantum homogeneous spaces abstract a quantum homogeneous space of a hopf algebra is a right coideal subalgebra over which the hopf algebra is faithfully flat it is shown that the auslander gorenstein property of a hopf algebra is inherited by its quantum homogeneous spaces if the quantum homogeneous space b of a pointed hopf algebra h is as gorenstein of dimension d then b has a rigid dualizing complex b d the nakayama automorphism is given by ad g s 2 where ad g is the inner automorphism associated to some group like element g h and is the algebra map determined by the left integral of b the quantum homogeneous spaces of u q sl 2 are classified and all of them are proved to be auslander regular as regular and cohen macaulay

structure constants related to symmetric hopf algebras abstract let k be an algebraically closed field of characteristic 0 in this paper we continue our study of structure constants for semisimple hopf algebras h whose character algebra is commutative and for non semisimple factorizable ribbon hopf algebras this is done from the point of view of symmetric algebras such as group algebras in particular we consider general fusion rules which are structure constants associated to products of irreducible characters and structure constants associated to generalizations of class sums and conjugacy classes our methods are reminiscent on one hand of the methods employed when analyzing tilting modules for certain quantum groups and on the other hand of splitting modules for certain drinfeld doubles the family of irreducible characters is divided in two according to the vanishing of their quantum dimension the fusion rules on the character algebra are computed with respect to this division

solvability for semisimple hopf algebras via integrals abstract we use integrals of left coideal subalgebras to develop harmonic analysis for semisimple hopf algebras we show how n the space of functional on n is embedded in h we define a bilinear form on n and show that irreducible n characters are orthogonal with respect to that form we then give an explicit formula for induced characters of n and show how the induced characters are embedded in r h in the second part we give an intrinsic definition for solvable semisimple hopf algebras via left coideal subalgebras and their integrals we show how this definition generalizes solvability for finite groups in particular commutative and nilpotent hopf algebras are solvable we finally prove an analogue of burnside theorem a semisimple quasitriangular hopf algebras of dimension p a q b is solvable

on two invariants of three manifolds from hopf algebras abstract we prove a 20 year old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional hopf algebras namely the kuperberg invariant and the hennings kauffman radford invariant the two invariants can be viewed as a non semisimple generalization of the turaev viro barrett westbury tvbw invariant and the witten reshetikhin turaev wrt invariant respectively by a classical result relating tvbw and wrt it follows that the kuperberg invariant for a semisimple hopf algebra is equal to the hennings kauffman radford invariant for the drinfeld double of the hopf algebra however whether the relation holds for non semisimple hopf algebras has remained open partly because the introduction of framings in this case makes the kuperberg invariant significantly more complicated to handle we give an affirmative answer to this question an important ingredient in the proof involves using a special heegaard diagram in which one family of circles gives the surgery link of the three manifold represented by the heegaard diagram

the static electric field distribution between two semi infinite circular cylinders a model for the feed gap field of a dipole antenna the static electric field distribution in the gap between two solid perfectly conducting semi infinite cylinders is obtained in terms of a fourier bessel eigenfunction series for dipole antennas whose cylinder diameter 2a and gap length 2d are both much less than the operating wavelength lambda this field can serve as the quasistatic excitation field in the gap of the dipole however the fourier bessel series is slowly convergent it is transformed into a rapidly convergent series of ultrasphetical polynomials whose weighting function explicitly satisfies the meixner edge condition numerical results are presented graphically for both the axial electric field on the gap surface and the associated potential distribution gap ratios of d a from 0 01 to 10 0 are considered and it is shown that as d a rightarrow 0 the solution approaches the two dimensional solution obtainable by conformal mapping

an enhanced gap source model an enhanced gap source model is introduced by combination with rao wilton glisson rwg basis functions for decades the stability of the delta gap source model remains a big problem the rwg basis with triangulation and the feeding edge model have been widely used in numerical analysis of antennas since the feeding edge model uses a single rwg edge as the driving source and neglects the gap width effect the resulting current undergoes sharp transition across the gap and thus the model is highly sensitive to the choice of the feeding edge consequently the feeding edge model suffers instability in calculations of current and input impedance to circumvent these pitfalls this enhanced gap source model uses multiple rwg edges and impulsive function to formulate the feeding behavior this new gap source model exhibits significant improvements on both accuracy and stability numerical results are included to illustrate improved accuracy and stability of this new model by virtue of several typical antennas

on the nature of oscillations in discretizations of the extended integral equation using analytical methods previous studies have shown that it is possible for oscillations to occur in the auxiliary surface current determined by applying the method of auxiliary sources mas to problems of scattering by perfect conductors of a very simple shape such oscillations are inherent to mas and would occur even in a hypothetical computer with ideal hardware and software because the integral equation relevant to mas very much resembles the extended integral equation in which the unknown is the actual surface current on the conductor one might surmise that similar oscillations also occur in discretizations of the latter equation in this communication we use analytical means to show that this is not the case therefore any oscillations that do occur in discretizations of the extended integral equation are at least for sufficiently simple problems likely to be due to matrix ill conditioning which magnifies errors that would otherwise be unimportant

weakly group theoretical and solvable fusion categories we introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups weakly group theoretical categories and solvable categories these are fusion categories that are morita equivalent to iterated extensions in the world of fusion categories of arbitrary respectively solvable finite groups weakly group theoretical categories have integer dimension and all known fusion categories of integer dimension are weakly group theoretical our main results are that a weakly group theoretical category c has the strong frobenius property i e the dimension of any simple object in an indecomposable c module category divides the dimension of c and that any fusion category whose dimension has at most two prime divisors is solvable a categorical analog of burnside s theorem for finite groups this has powerful applications to classification of fusion categories and semsisimple hopf algebras of a given dimension in particular we show that any fusion category of integer dimension 84 is weakly group theoretical i e comes from finite group theory and give a full classification of semisimple hopf algebras of dimensions pqr and pq2 where p q r are distinct primes

adaptive time stepping for incompressible flow part ii navier stokes equations we outline a new class of robust and efficient methods for solving the navier stokes equations we describe a general solution strategy that has two basic building blocks an implicit time integrator using a stabilized trapezoid rule with an explicit adams bashforth method for error control and a robust krylov subspace solver for the spatially discretized system we present numerical experiments illustrating the potential of our approach

nilpotent fusion categories abstract in this paper we extend categorically the notion of a finite nilpotent group to fusion categories to this end we first analyze the trivial component of the universal grading of a fusion category c and then introduce the upper central series of c for fusion categories with commutative grothendieck rings e g braided fusion categories we also introduce the lower central series we study arithmetic and structural properties of nilpotent fusion categories and apply our theory to modular categories and to semisimple hopf algebras in particular we show that in the modular case the two central series are centralizers of each other in the sense of m muger

solution of hallen s integral equations by means of nystrom method to solve hallen s integral equations accurately nystrom method is applied in this paper to obtain an exact and finite solution both the exact kernel and a finite gap width must be taken into consideration in the case of a gap feed a modification function that converts the approximate kernel to the exact kernel is derived as the modification function is singular a one point correction method is proposed as a singularity subtraction method using a step function source we can use various quadrature rules to solve the problem the results show small difference between them the validity of this theory is confirmed by comparing the measured results and the calculated results the effect of the feed system is shown to be strong

aladins an algebraic splitting time adaptive solver for the incompressible navier stokes equations we address a time adaptive solver specifically devised for the incompressible navier stokes ins equations one of the challenging issues in this context is the identification of a reliable a posteriori error estimator typical strategies are based on the combination of the solutions computed either with two different time steps or two schemes with different accuracy in this paper we move from the pressure correction algebraic factorizations formerly proposed by saleri veneziani 2005 these schemes feature an intrinsic hierarchical nature such that an accurate solution for the pressure is obtained by computing intermediate low order guesses the difference between the two estimates provide a natural a posteriori estimator after introducing the incremental formulation of the pressure correction schemes we address the properties of this approach including extensive implementation details numerical results presented refer to 2d and 3d unstructured problems with a particular emphasis on cardiovascular problems which are expected to largely benefit from time adaptive solvers in memory of f saleri 1965 2007

are we counting or measuring something let h be a semisimple hopf algebras over an algebraically closed field k of characteristic 0 we define hopf algebraic analogues of commutators and their generalizations and show how they are related to h the hopf algebraic analogue of the commutator subgroup we introduce a family of central elements of h which on one hand generate h and on the other hand give rise to a family of functionals on h when h kg g a finite group these functionals are counting functions on g it is not clear yet to what extent they measure any specific invariant of the hopf algebra however when h is quasitriangular they are at least characters on h

word maps conjugacy classes and a noncommutative waring type theorem let w w x xd 1 be a nontrivial group word we show that if g is a sufficiently large finite simple group then every element g e g can be expressed as a product of three values of w in g this improves many known results for powers commutators as well as a theorem on general words obtained in 19 the proof relies on probabilistic ideas algebraic geometry and character theory our methods which apply the zeta function g s x eirrg xo s giye rise to various additional results of independent interest including applications to conjectures of ore and thompson

factorization methods for the numerical approximation of navier stokes equations we investigate a general approach for the numerical approximation of incompressible navier stokes equations based on splitting the original problem into successive subproblems which are cheaper to solve the splitting is obtained through an algebraic approximate factorization of the matrix arising from space and time discretization of the original equations several schemes based on approximate factorization are investigated for some of these methods a formal analogy with well known time advancing schemes such as the projection chorin temam s can be pointed out features and limits of this analogy that was earlier introduced in b perot j comp phys 108 1993 51 8 are addressed other new methods can also be formulated starting from this approach in particular we introduce the so called yosida method which can be investigated in the framework of quasi compressibility schemes numerical results illustrating the different performances of the different methods addressed are presented for a couple of test cases

fundamentals of thin wire integral equations with the finite gap generator part ii we discuss certain fundamental properties of hallen s and pocklington s equations with the finite gap generator fgg which has long been used as a feed model for cylindrical antennas for the case of the exact kernel we focus on whether important quantities are finite or infinite and compare to corresponding quantities obtained using the delta function generator dfg for the approximate kernel we focus on the unsolvability of the two equations and the relation of unsolvability to the important phenomenon of unphysical oscillations we also extend to the fgg the so called effective current method this is an easy to apply remedy for the oscillatory solutions that allows one to obtain reasonable answers for important quantities such as current distributions and the input impedance it is shown here that these answers are close to those obtained with the exact kernel we show that in a certain sense the method is more advantageous to apply in the present fgg case than in the dfg case our discussions especially in part ii of this paper bring out similarities to 1 the phenomenon of superdirectivity and 2 recently discussed numerical phenomena occurring within the method of auxiliary sources

eliminating unphysical oscillations arising in galerkin solutions to classical integral equations of antenna theory an asymptotic study previous works have discussed in detail the difficulties occurring when one applies numerical methods to hallen s and pocklington s integral equations for the current distribution along a linear antenna when the so called approximate kernel is used the main difficulty is the appearance of unphysical oscillations near the driving point and or near the ends of the antenna another work has proposed an easy to apply possible remedy to overcome these unnatural oscillations the basic idea is to define a new current from the near magnetic field produced by the original oscillating current in the present paper for the case of an antenna center driven by a delta function generator we place this remedy on a much firmer basis by means of an asymptotic study for the case of the linear antenna of infinite length we provide specific connections of our study to actual finite length antennas

models of quasiprojective homogeneous spaces for hopf algebras given an affine group scheme g of finite type over a field k a homogeneous space for g is a scheme x over k containing a rational point x such that g operates transitively on x assuming that g operates on the right we may identify x with the quotient k g of g by the left action of the stabilizer k of x in g the representation theoretic significance ofk g is that the induction functor indk from k modules to g modules factors as a category equivalence

divisibility relations for the dimensions and hilbert series of nichols algebras of non abelian group type abstract we present a divisibility relation for the dimensions and hilbert series of certain classes of nichols algebras of non abelian group type which generalizes nichols algebras over coxeter groups with constant cocycle 1 for this we introduce three groups of isomorphisms acting on nichols algebras which generalizes the exchange operator introduced by milinski and schneider 2000 18 for coxeter groups

solvable hopf algebras and their twists abstract we show that for any solvable group g and a drinfel d twist j k g j is solvable in the sense of the intrinsic definition of solvability given in 2 more generally if a hopf algebra h has a normal solvable series so does h j furthermore while solvable groups are defined as having certain commutative quotients quasitriangular normally solvable hopf algebras have appropriate quantum commutative quotients we end with a detailed example

involutory hopf algebras in 1975 kaplansky conjectured that a finite dimensional semisimple hopf algebra is necessarily involutory twelve years later larson and radford proved the conjecture in characterisitic 0 and obtained significant partial results in positive characteristics the goal of this paper is to offer an efficient proof of these results using rather minimal prerequisites no harpoons and gratifyingly few hits let h be a finite dimensional hopf algebra over a field k and let s h h denote its antipode then h is said to be involutory if s2 idh or equivalently if s is an involution on h in k kaplansky conjectured that a semisimple hopf algebra is necessarily involutory and in a fundamental paper lr2 larson and radford proved this assertion at least for fields of characteristic 0 furthermore they showed that if h is semisimple and cosemisimple and if chark p dimk h 2 then h must again be involutory a nice exposition of this material can be found in n while lr2 is reasonably brief it depelids heavily on certain earlier results on characters of hopf algebras as contained in l and lr 1 and the combined proof is substantial the goal of this paper is to present an efficient proof of the main results of lr2 requiring minimal prerequisites as will be apparent our proof follows the same basic outline as that of larson and radford and therefore we offer nothing new in this regard however by computing traces on certain tensor modules we are able to obtain the necessary character formulas quite quickly and this is where the simplifications occur in some sense the hero of this approach is the linear transformation t described below if v is a left h module then h 0 v is well known to be h isomorphic to h 0 vj where vj is equal to v but with the trivial action determined by the counit e this is proved quite simply by considering the map t h v h v given by t h ov hihi h2v for all h e h v e v h indeed one shows that t is invertible with inverse t 1 h x v e hz x s h2 v h received by the editors july 13 1993 and in revised form june 8 1994 originally communicaated to the proceedings of the ams by lance w small 1991 mathematics subject classification primary 1 6w30 research supported in part by nsf grants dms 9224662 and dms 9003374 1995 american mathematical society

kirby elements and quantum invariants we define the notion of a kirby element of a ribbon category mathcal c not necessarily semisimple kirby elements lead to 3 manifold invariants we characterize a class of kirby elements the algebraic kirby elements in terms of the structure maps of a hopf algebra in mathcal c this class is sufficiently large to recover the quantum invariants of 3 manifolds of reshetikhin and turaev of hennings kauffman and radford and of lyubashenko when these are well defined the cases of a semisimple ribbon category and of a category of representations are explored in detail

projectivity and freeness over comodule algebras let h be a hopf algebra and a an h simple right h comodule algebra it is shown that under certain hypotheses every h a hopf module is either projective or free as an a module and a is either a quasi frobenius or a semisimple ring as an application it is proved that every weakly finite in particular every finite dimensional hopf algebra is free both as a left and a right module over its finite dimensional right coideal subalgebras and the latter are frobenius algebras similar results are obtained for h simple h module algebras

ribbon transformations integrals and triangular decompositions abstract we study the theory of integrals in yetter drinfel d hopf algebras and use the results to determine the integrals of hopf algebras with triangular decomposition

radford s s 4 formula for co frobenius hopf algebras abstract this note extends radford s formula for the fourth power of the antipode of a finite dimensional hopf algebra to co frobenius hopf algebras and studies equivalent conditions to a hopf algebra being involutory for finite dimensional and co frobenius hopf algebras

finite quantum groups and quantum permutation groups abstract we give examples of finite quantum permutation groups which arise from the twisting construction or as bicrossed products associated to exact factorizations in finite groups we also give examples of finite quantum groups which are not quantum permutation groups one such example occurs as a split abelian extension associated to the exact factorization s 4 z 4 s 3 and has dimension 24 we show that in fact this is the smallest possible dimension that a non quantum permutation group can have

on the zero stability of the variable order variable stepsize bdf formulas this paper contains bounds for the mesh size ratiosh j 1 h j of nonequidistant grids such that any combination of the widely used backward differentiation formulas with ordersl 1 l m m 3 4 5 performs zerostable

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

reply to comments on fundamentals of thin wire integral equations with the finite gap generator in papers 1 2 the authors extend the so called effective current method to the hallen s and pocklington s equations with the finite gap generator fgg the indicated method considered earlier in 3 with a reference to 4 is associated with application of the method of auxiliary sources to the analysis of cylindrical dipole antennas 5 7 the purpose of these comments is to draw the authors and readers attention to the existence ofpaper 8 where the pocklington s equation with the fgg is solved for the axial current considered there to be auxiliary the equation is solved by the galerkin method using the piecewise sinusoidal basis and testing functions the auxiliary axial current is used further for calculation of the surface current which is referred to as the effective current in 1 4 and the surface current itself is used for calculation of the input impedance so in some respects papers 1 7 just repeat the main idea and technique of paper 8 although of course papers 1 7 also contain a lot of their own new interesting and useful analytical and numerical results

the trace space and kauffman s knot invariants the traces in the construction of kauffman s knot invariants are studied the trace space is determined for a semisimple finite dimensional quantum hopf algebra and the best lower bound of the dimension of the trace space is given for a unimodular finite dimensional quantum hopf algebra

right coideal subalgebras in u _q so_ 2n 1 we offer a complete classification of right coideal subalgebras which contain all group like elements for the multiparameter version of the quantum group uq sln 1 provided that the main parameter q is not a root of 1 as a consequence we determine that for each subgroup of the group g of all group like elements the quantum borel subalgebra uq sln 1 contains n 1 different homogeneous right coideal subalgebras u such that u g if q has a finite multiplicative order t 2 the classification remains valid for homogeneous right coideal subalgebras of the multiparameter version of the lusztig quantum group uq sln 1 in the paper we consider the quantifications of kac moody algebras as character hopf algebras v k kharchenko a combinatorial approach to the quantifications of lie algebras pacific j math 203 1 2002 191 233

semisimple hopf actions on commutative domains abstract let h be a semisimple so finite dimensional hopf algebra over an algebraically closed field k of characteristic zero and let a be a commutative domain over k we show that if a arises as an h module algebra via an inner faithful h action then h must be a group algebra this answers a question of e kirkman and j kuzmanovich and partially answers a question of m cohen the main results of this article extend to working over k of positive characteristic on the other hand we obtain results on hopf actions on weyl algebras as a consequence of the main theorem

the left and right dimensions of a skew field over the subfield of invariants abstract if h is a hopf algebra and a an h module algebra without nontrivial h stable left or right ideals then the subalgebra of h invariant elements a h is a skew field and a may be regarded as a vector space over a h with respect to either left or right multiplications it is proved in the paper that the left dimension of a over a h is equal to the right dimension under the assumptions that a is semiprimary and dim h in the case when a is itself a skew field this answers a question raised by j bergen m cohen and d fischman