mesoscopic simulation of ostwald ripening the self organization of particles in a two phase system in the coexistence region through a diffusive mechanism is known as ostwald ripening this phenomenon is an example of a multiscale problem in that the microscopic level interaction of the particles can greatly impact the macroscale or observable morphology of the system ostwald ripening is studied here through the use of a mesoscopic model which is a stochastic partial integrodifferential equation that is derived from a spin exchange ising model this model is studied through the use of recently developed and benchmarked spectral schemes for the simulation of solutions to stochastic partial differential equations the typical cluster size is observed to grow like t1 3 over range of times with faster growth at later times the results included here also demonstrate the effect of adjusting the interparticle interaction on the morphological evolution of the system at the macroscopic level

finite element approximation of a sixth order nonlinear degenerate parabolic equation we consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation equation here where generically equation here for any given equation here in addition to showing well posedness of our approximation we prove convergence in space dimensions d leq 3 furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed finally some numerical experiments in one and two space dimensions are presented

numerical analysis of the cahn hilliard equation with a logarithmic free energy a fully discrete finite element method for the cahn hilliard equation with a logarithmic free energy based on the backward euler method is analysed existence and uniqueness of the numerical solution and its convergence to the solution of the continuous problem are proved two iterative schemes to solve the resulting algebraic problem are proposed and some numerical results in one space dimension are presented

error control for the approximation of allen cahn and cahn hilliard equations with a logarithmic potential a fully computable upper bound for the finite element approximation error of allen cahn and cahn hilliard equations with logarithmic potentials is derived numerical experiments show that for the sharp interface limit this bound is robust past topological changes modifications of the abstract results to derive quasi optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold

an energy stable fourth order finite difference scheme for the cahn hilliard equation abstract in this paper we propose and analyze an energy stable numerical scheme for the cahn hilliard equation with second order accuracy in time and the fourth order finite difference approximation in space in particular the truncation error for the long stencil fourth order finite difference approximation over a uniform numerical grid with a periodic boundary condition is analyzed via the help of discrete fourier analysis instead of the standard taylor expansion this in turn results in a reduced regularity requirement for the test function in the temporal approximation we apply a second order bdf stencil combined with a second order extrapolation formula applied to the concave diffusion term as well as a second order artificial douglas dupont regularization term for the sake of energy stability as a result the unique solvability energy stability are established for the proposed numerical scheme and an optimal rate convergence analysis is derived in the l 0 t l 2 l 2 0 t h h 2 norm a few numerical experiments are presented which confirm the robustness and accuracy of the proposed scheme

exotic subfactors of finite depth with jones indices 5 13 2 and 5 17 2 we prove existence of subfactors of finite depth of the hyperfinite ii1 factor with indices and the existence of the former was announced by the second named author in 1993 and that of the latter has been conjectured since then these are the only known subfactors with finite depth which do not arise from classical groups quantum groups or rational conformal field theory

finite element approximation of the cahn hilliard equation with degenerate mobility we consider a fully practical finite element approximation of the cahn hilliard equation with degenerate mobility textstyle frac partial u partial t del b u del gamma lap u psi u where b cdot geq 0 is a diffusional mobility and psi cdot is a homogeneous free energy in addition to showing well posedness and stability bounds for our approximation we prove convergence in one space dimension furthermore an iterative scheme for solving the resulting nonlinear discrete system is analyzed we also discuss how our approximation has to be modified in order to be applicable to a logarithmic homogeneous free energy finally some numerical experiments are presented

non cyclotomic fusion categories etingof nikshych and ostrik ask in 7 2 if every fusion category can be completely defined over a cyclotomic field we show that this is not the case in particular one of the fusion categories coming from the haagerup subfactor 1 and one coming from the newly constructed extended haagerup subfactor 2 can not be completely defined over a cyclotomic field on the other hand we show that the drinfel d center of the even part of the haagerup subfactor is completely defined over a cyclotomic field we identify the minimal field of definition for each of these fusion categories compute the galois groups and identify their galois conjugates ams classification 18d10 46l37

a second order accurate non linear difference scheme for the n component cahn hilliard system a b s t r a c t we consider a second order conservative nonlinear numerical scheme for the ncomponent cahn hilliard system modeling the phase separation of a n component mixture the scheme is based on a crank nicolson finite difference method and is solved by an efficient and accurate nonlinear multigrid method we numerically demonstrate the second order accuracy of the numerical scheme we observe that our numerical solutions are consistent with the exact solutions of linear stability analysis results we also describe numerical experiments such as the evolution of triple junctions and the spinodal decomposition in a quaternary mixture we investigate the effects of a concentration dependent mobility on phase separation

the set of types ofn dimensional semisimple and cosemisimple hopf algebras is finite in this paper we give a partial answer to the above statement proving that the set of types of semisimple and cosemisimple hopf algebras of a given dimension is finite in any characteristic the proof uses the formalism of deformation theory of bialgebras which is concerned from a geometrical point of view with the study of the affine algebraic set b v of all bialgebra structures which can be defined on a vector space v there are two basic facts regarding b v first of all the transport of structures defines an action of gl v on b v such that the orbit of a bialgebra a contains all structures isomorphic to a so the set of types of bialgebras on v is in a one to one correspondence with the set of orbits second we are led in a natural way to a cohomology theory for bialgebras studying the tangent spaces t o and t b v where o is the closure a a a a of the orbit through a bialgebra a one of the most important results of the deformation theory of bialgebras is that the orbit of a bialgebra is open if the cohomology vanishes therefore the number of orbits which correspond to bialgebras having trivial cohomology is finite we obtain our result by proving that the cohomology of a semisimple hopf algebra a vanishes if the characteristic of k is 0 or more generally if a is

an efficient two grid method for the cahn hilliard equation with the concentration dependent mobility and the logarithmic flory huggins bulk potential abstract in this paper we consider the two grid method for solving cahn hilliard equation with the concentration dependent mobility and the logarithmic flory huggins bulk potential this two grid method consists of two steps first step solve the cahn hilliard equation with full implicit mixed finite element method and newton iteration method on a coarse grid second step solve two linear equations on a fine grid the stability and convergence of the proposed scheme are established finally some numerical experiments are presented

numerical solution of distributed order fractional differential equations in this paper a method for the numerical solution of distributed order fdes fractional differential equations of a general form is presented the method applies to both linear and nonlinear equations the caputo type fractional derivative is employed the distributed order fde is approximated with a multi term fde which is then solved by adjusting appropriately the numerical method developed for multi term fdes by katsikadelis several example equations are solved and the response of mechanical systems described by such equations is studied the convergence and the accuracy of the method for linear and nonlinear equations are demonstrated through well corroborated numerical results

existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations a nonlinear parabolic equation of the fourth order is analyzed the equation is characterized by a mobility coefficient that degenerates at 0 existence of at least one weak solution is proved by using a regularization procedure and deducing suitable a priori estimates if a viscosity term is added and additional conditions on the nonlinear terms are assumed then it is proved that any weak solution becomes instantaneously strictly positive this in particular implies uniqueness for strictly positive times and further time regularization properties the long time behavior of the problem is also investigated and the existence of trajectory attractors and under more restrictive conditions of strong global attractors is shown

nonconforming tetrahedral finite elements for fourth order elliptic equations this paper is devoted to the construction of nonconforming finite elements for the discretization of fourth order elliptic partial differential operators in three spatial dimensions the newly constructed elements include two nonconforming tetrahedral finite elements and one quasi conforming tetrahedral element these elements are proved to be convergent for a model bi harmonic equation in three dimensions in particular the quasi conforming tetrahedron element is a modified zienkiewicz element while the nonmodified zienkiewicz element a tetrahedral element of hermite type is proved to be divergent on a special grid

an alternating direction implicit galerkin finite element method for the distributed order time fractional mobile immobile equation in two dimensions abstract in this paper we shall present the alternating direction implicit adi galerkin finite element method fem for solving the distributed order time fractional mobile immobile equation in two dimensions in the time direction the backward euler method is used to deal with the temporal first order derivative and the weighted and shifted grunwald formula is employed to discretize the distributed order time fractional derivative galerkin fem is used for discretization of the spatial direction and then an adi galerkin finite element scheme is constructed the stability and l 2 norm convergence are proved several numerical experiments are provided to verify our theoretical analysis

subfactors of index less than 5 part 3 quadruple points one major obstacle in extending the classification of small index subfactors beyond 3 sqrt 3 is the appearance of infinite families of candidate principal graphs with 4 valent vertices in particular the weeds mathcal q and mathcal q from part 1 morrison and snyder in commun math phys doi 10 1007 s00220 012 1426 y 2012 thus instead of using triple point obstructions to eliminate candidate graphs we need to develop new quadruple point obstructions in this paper we prove two quadruple point obstructions the first uses quadratic tangles techniques and eliminates the weed mathcal q immediately the second uses connections and when combined with an additional number theoretic argument it eliminates both weeds mathcal q and mathcal q finally we prove the uniqueness up to taking duals of the 3311 goodman de la harpe jones subfactor using a combination of planar algebra techniques and connections

a parallel domain decomposition based implicit method for the cahn hilliard cook phase field equation in 3d we present a numerical algorithm for simulating the spinodal decomposition described by the three dimensional cahn hilliard cook chc equation which is a fourth order stochastic partial differential equation with a noise term the equation is discretized in space and time based on a fully implicit cell centered finite difference scheme with an adaptive time stepping strategy designed to accelerate the progress to equilibrium at each time step a parallel newton krylov schwarz algorithm is used to solve the nonlinear system we discuss various numerical and computational challenges associated with the method the numerical scheme is validated by a comparison with an explicit scheme of high accuracy and unreasonably high cost we present steady state solutions of the chc equation in two and three dimensions the effect of the thermal fluctuation on the spinodal decomposition process is studied we show that the existence of the thermal fluctuation accelerates the spinodal decomposition process and that the final steady morphology is sensitive to the stochastic noise we also show the evolution of the energies and statistical moments in terms of the parallel performance it is found that the implicit domain decomposition approach scales well on supercomputers with a large number of processors

tensor categories with fusion rules of self duality for finite abelian groups abstract semisimple tensor categories with fusion rules of self duality for finite abelian groups are classified as an application we prove that the tannaka duals of the dihedral and the quaternion groups of order 8 and the eight dimensional hopf algebra of kac and paljutkin are not isomorphic as abstract tensor categories whereas they have the same fusion rule

diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix a model for phase separation in multicomponent systems is studied in particular the possibility of a concentration dependence of the mobility matrix is taken into account this leads to a system of fourth order degenerate parabolic partial differential equations we derive these equations from balance laws show some properties of the model and prove a global existence result for the degenerate system

on the structure of weak hopf algebras we study the group of group like elements of a weak hopf algebra and derive an analogue of radford s formula for the fourth power of the antipode s which implies that the antipode has a finite order modulo a trivial automorphism we find a sufficient condition in terms of tr s2 for a weak hopf algebra to be semisimple discuss relation between semisimplicity and cosemisimplicity and apply our results to show that a dynamical twisting deformation of a semisimple hopf algebra is cosemisimple

spectral method study of domain coarsening the self organization of particles in a two phase system in the coexistence region through a diffusive mechanism is known as ostwald ripening the late stage of ostwald ripening is studied here through the use of a mesoscopic model in conjunction with a special configuration that allows for the direct measurement of system characteristics such as domain size the mesoscopic model is a stochastic partial integrodifferential equation and is studied through the use of recently developed and benchmarked spectral schemes the size of the growing region is not observed to increase as a power law in this model during the late stages of self organization in contrast to the power law growth observed in simulations of the earlier stages of self organization the results included here also demonstrate the effect of adjusting the interparticle interaction on the morphological evolution of the system

weak hopf algebras i integral theory and c structure we give an introduction to the theory of weak hopf algebras proposed as a coassociative alternative of weak quasi hopf algebras we follow an axiomatic approach keeping as close as possible to the classical theory of hopf algebras the emphasis is put on the new structure related to the presence of canonical subalgebras al and ar in any weak hopf algebra a that play the role of non commutative numbers in many respects a theory of integrals is developed in which we show how the algebraic properties of a such as the frobenius property or semisimplicity or innerness of the square of the antipode are related to the existence of non degenerate normalized or haar integrals in case of c weak hopf algebras we prove the existence of a unique haar measure h a and of a canonical grouplike element g a implementing the square of the antipode and factorizing into left and right elements g glg 1r gl al gr ar further discussion of the c case will be presented in part ii

a fourth order parabolic equation with nonlinear principal part in this paper we study a fourth order parabolic equation with nonlinear principal part modeling epitaxial thin film growth in two space dimensions on the basis of the schauder type estimates and campanato spaces we prove the global existence of classical solutions

non unitary fusion categories and their doubles via endomorphisms we realise non unitary fusion categories using subfactor like methods and compute their quantum doubles and modular data for concreteness we focus on generalising the haagerup izumi family of q systems for example we construct endomorphism realisations of the non unitary yang lee model and non unitary analogues of one of the even subsystems of the haagerup subfactor and of the grossman snyder system we supplement izumi s equations for identifying the half braidings which were incomplete even in his q system setting we conjecture a remarkably simple form for the modular s and t matrices of the doubles of these fusion categories we would expect all of these doubles to be realised as the category of modules of a rational voa and conformal net of factors we expect our approach will also suffice to realise the non semisimple tensor categories arising in logarithmic conformal field theories

numerical solution of non linear fourth order fractional sub diffusion wave equation with time delay abstract in this paper we constructed a linearized compact difference scheme for fourth order non linear fractional sub diffusion equation with time delay and variable coefficients the primary purpose of our work is to use the idea of the l 2 1 formula for temporal dimension and compact linear operator for spatial dimension the proposed method is unconditionally stable and convergent to the analytical solution with the order of convergence o 2 h 4 where and h are temporal and spatial lengths respectively numerical experimentation is carried out to show the efficiency and accuracy of the proposed scheme

a finite element model for the time dependent joule heating problem we study a spatially semidiscrete and a completely discrete finite element model for a nonlinear system consisting of an elliptic and a parabolic partial differential equation describing the electric heating of a conducting body we prove error bounds of optimal order under minimal regularity assumptions when the number of spatial variables d 3 we establish the existence of solutions with the required regularity over arbitrarily long intervals of time when d 2

reflection fusion categories abstract we introduce the notion of a reflection fusion category which is a type of a g crossed category generated by objects of frobenius perron dimension 1 and p where p is an odd prime we show that such categories correspond to orthogonal reflection groups over f p this allows us to use the known classification of irreducible reflection groups over finite fields to classify irreducible reflection fusion categories

the center functor is fully faithful abstract we prove that the notion of drinfeld center defines a functor from the category of indecomposable multi tensor categories with morphisms given by bimodules to that of braided tensor categories with morphisms given by monoidal bimodules moreover we apply some ideas from the physics of topological orders to prove that the center functor restricted to indecom posable multi fusion categories with additional conditions on the target category is fully faithful as byproducts we provide new proofs to some important known results in fusion categories in physics this fully faithful functor gives the precise mathematical description of the boundary bulk relation for 2 1d anomaly free topological orders with gapped boundaries

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

on the central charge of a factorizable hopf algebra abstract for a semisimple factorizable hopf algebra over a field of characteristic zero we show that the value that an integral takes on the inverse drinfel d element differs from the value that it takes on the drinfel d element itself by at most a fourth root of unity this can be reformulated by saying that the central charge of the hopf algebra is an integer if the dimension of the hopf algebra is odd we show that these two values differ by at most a sign which can be reformulated by saying that the central charge is even we give a precise condition on the dimension that determines whether the plus sign or the minus sign occurs to formulate our results we use the language of modular data

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

pseudo unitary non self dual fusion categories of rank 4 abstract a fusion category of rank 4 has either four self dual simple objects or exactly two self dual simple objects we study fusion categories of rank 4 with exactly two self dual simple objects giving nearly a complete classification of those based rings that admit pseudo unitary categorification more precisely we show that if c is such a fusion category then its grothendieck ring k c must be one of seven based rings six of which have known categorifications in doing so we classify all based rings associated with near group categories of the group z 3 z

time distributed order diffusion wave equation ii applications of laplace and fourier transformations a cauchy problem for a time distributed order multi dimensional diffusion wave equation containing a forcing term is reinterpreted in the space of tempered distributions and a distributional diffusion wave equation is obtained the distributional equation is solved in the general case of weight function or distribution solutions are given in terms of solution kernels green s functions which are studied separately for two cases the first case is when the order of the fractional derivative is in the interval 0 1 while in the second case the order of the fractional derivative is in the interval 0 2 solutions of fractional diffusionwave and fractional telegraph equations are obtained as special cases numerical experiments are also performed an analogue of the maximum principle is also presented

on a class of non linear delay distributed order fractional diffusion equations in this paper we consider a numerical scheme for a class of non linear time delay fractional diffusion equations with distributed order in time this study covers the unique solvability convergence and stability of the resulted numerical solution by means of the discrete energy method the derivation of a linearized difference scheme with convergence order o 4 h 4 in l norm is the main purpose of this study numerical experiments are carried out to support the obtained theoretical results

an error bound for the finite element approximation of the cahn hilliard equation with logarithmic free energy an error bound is proved for a fully practical piecewise linear finite element approximation using a backward euler time discretization of the cahn hilliard equation with a logarithmic free energy

modified methods for solving two classes of distributed order linear fractional differential equations abstract this paper introduces two methods for the numerical solution of distributed order linear fractional differential equations the first method focuses on initial value problems ivps and based on the th caputo fractional definition with the shifted chebyshev operational matrix of fractional integration by applying this method the ivps are converted into simple linear differential equations which can be easily handled the other method focuses on boundary value problems bvps based on picard s method frame this method is based on iterative formula contains an auxiliary parameter which provides a simple way to control the convergence region of solution series several numerical examples are used to illustrate the accuracy of the proposed methods compared to the existing methods also the response of mechanical system described by such equations is studied

larson sweedler theorem and the role of grouplike elements in weak hopf algebras abstract we extend the larson sweedler theorem amer j math 91 1969 75 to weak hopf algebras by proving that a finite dimensional weak bialgebra is a weak hopf algebra iff it possesses a non degenerate left integral we show that the category of modules over a weak hopf algebra is autonomous monoidal with semisimple unit and invertible modules we also reveal the connection of invertible modules to left and right grouplike elements in the dual weak hopf algebra defining distinguished left and right grouplike elements we derive the radford formula amer j math 98 1976 333 for the fourth power of the antipode in a weak hopf algebra and prove that the order of the antipode is finite up to an inner automorphism by a grouplike element in the trivial subalgebra a t of the underlying weak hopf algebra a

finite element approximation of the cahn hilliard equation with concentration dependent mobility we consider the cahn hilliard equation with a logarithmic free energy and non degenerate concentration dependent mobility in particular we prove that there exists a unique solution for sufficiently smooth initial data further we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions finally some numerical experiments are presented

actions of taft s algebras on finite dimensional algebras abstract let f be a field containing a primitive m th root of the unit we characterize the actions of a taft s algebra h m of a certain order m on finite dimensional arbitrary algebras we describe the action in terms of gradings and actions by skew derivations moreover we prove the associative algebra u t 2 of 2 2 upper triangular matrices with entries from f does not generate a variety of h m module algebras of almost polynomial growth

finite element approximation of a model for phase separation of a multi component alloy with non smooth free energy a model for the phase separation of a multi component alloy with non smooth free energy is considered an error bound is proved for a fully practical piecewise linear finite element approximation using a backward euler time discretization an iterative scheme for solving the resulting nonlinear algebraic system is analysed finally numerical experiments with three components in one and two space dimensions are presented in the one dimensional case we compare some steady states obtained numerically with the corresponding stationary solutions of the continuous problem which we construct explicitly

an integral equation method for the cahn hilliard equation in the wetting problem abstract we present an integral equation approach to solving the cahn hilliard equation equipped with boundary conditions that model solid surfaces with prescribed young s angles the discretization of the system in time using convex splitting leads to a modified biharmonic equation at each time step to solve it we split the solution into a volume potential computed with free space kernels plus the solution to a second kind integral equation skie the volume potential is evaluated with the help of a box based volume fmm method for non box domains the source density is extended by solving a biharmonic dirichlet problem the near singular boundary integrals are computed using quadrature by expansion qbx with fmm acceleration our method has linear complexity in the number of surface volume degrees of freedom and can achieve high order convergence in space with adaptive refinement to manage error from function extension

dynamic transitions for quasilinear systems and cahn hilliard equation with onsager mobility the main objectives of this article are two fold first we study the effect of the nonlinear onsager mobility on the phase transition and on the well posedness of the cahn hilliard equation modeling a binary system it is shown in particular that the dynamic transition is essentially independent of the nonlinearity of the onsager mobility however the nonlinearity of the mobility does cause substantial technical difficulty for the well posedness and for carrying out the dynamic transition analysis for this reason as a second objective we introduce a systematic approach to deal with phase transition problems modeled by quasilinear partial differential equations following the ideas of the dynamic transition theory developed in ma and wang phase transition dynamics in nonlinear sciences springer to be published bifurcation theory and applications world scientific series on nonlinear science series a monographs and treatises vol 53 world scientific hackensack nj 2005

quantum subgroups of the haagerup fusion categories we answer three related questions concerning the haagerup subfactor and its even parts the haagerup fusion categories namely we find all simple module categories over each of the haagerup fusion categories in other words we find the quantum subgroups in the sense of ocneanu we find all irreducible subfactors whose principal even part is one of the haagerup fusion categories and we compute the brauer picard groupoid of morita equivalences of the haagerup fusion categories in addition to the two even parts of the haagerup subfactor there is exactly one more fusion category which is morita equivalent to each of them this third fusion category has six simple objects and the same fusion rules as one of the even parts of the haagerup subfactor but has not previously appeared in the literature we also find the full lattice of intermediate subfactors for every irreducible subfactor whose even part is one of these three fusion categories and we discuss how our results generalize to izumi subfactors

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

compact difference scheme for distributed order time fractional diffusion wave equation on bounded domains in this paper we derive and analyse a compact difference scheme for a distributed order time fractional diffusion wave equation this equation is approximated by a multi term fractional diffusion wave equation which is then solved by a compact difference scheme the unique solvability of the difference solution is discussed using the discrete energy method we prove the compact difference scheme is unconditionally stable and convergent finally numerical results are presented to support our theoretical analysis

an improved error bound for a finite element approximation of a model for phase separation of a multi component alloy with a concentration dependent mobility matrix using a slightly different discretization scheme in time and adapting the approach in nochetto et al 1998 for analysing the time discretization error in the backward euler method we improve on the error bounds derived in i barrett and blowley 1998 and ii barrett and blowey 1999c for a fully practical piecewise linear finite element approximation of a model for phase separation of a multi component alloy with a concentration dependent mobility matrix and i a logarithmic free energy and ii a non smooth free energy the deep quench limit respectively moreover the improved error bound in the deep quench limit is optimal numerical experiments with three components illustrating the above error bounds are also presented

modular data for the extended haagerup subfactor we compute the modular data that is the s and t matrices for the centre of the extended haagerup subfactor bmps12 the full structure i e the associativity data also known as 6 j symbols or f matrices still appears to be inaccessible nevertheless starting with just the number of simple objects and their dimensions obtained by a combinatorial argument in mw14 we find that it is surprisingly easy to leverage knowledge of the representation theory of sl 2 mathbb z into a complete description of the modular data we also investigate the possible character vectors associated with this modular data this is the published version of arxiv 1606 07165

near group fusion categories and their doubles a near group fusion category is a fusion category c where all but 1 simple objects are invertible examples of these include the tambara yamagami categories and the even sectors of the e6 and affine d5 subfactors though there are infinitely many others we classify the near group fusion categories and compute their doubles and the modular data relevant to conformal field theory among other things we explicitly construct over 40 new finite depth subfactors with jones index ranging from around 6 85 to around 14 93 we expect all of these doubles to be realised by rational conformal field theories

monoidal 2 structure of bimodule categories abstract we define a notion of tensor product of bimodule categories and prove that with this product the 2 category of c bimodule categories for fixed tensor c is a monoidal 2 category in the sense of kapranov and voevodsky 1991 1 we then provide a monoidal structure preserving 2 equivalence between the 2 category of c bimodule categories and z c module categories module categories over the center of c for a finite group g we show that de equivariantization is equivalent to the tensor product over rep g we derive rep g module fusion rules and show that the group of invertible rep g module categories is isomorphic to h 2 g k extending results in etingof et al 2

numerical solution of distributed order fractional differential equations by hybrid functions in this paper a new numerical method for solving the distributed fractional differential equations is presented the method is based upon hybrid functions approximation the properties of hybrid functions consisting of block pulse functions and bernoulli polynomials are presented the riemann liouville fractional integral operator for hybrid functions is introduced this operator is then utilized to reduce the solution of the distributed fractional differential equations to a system of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

identification of a time dependent source term in a distributed order time fractional equation from a nonlocal integral observation abstract the ultra slow diffusion process from internal source is governed by the inhomogeneous distributed order time fractional diffusion equation we consider an inverse problem of recovering the time dependent internal source from a nonlocal integral observation which can be regarded as the spatial average of a pointwise observation the uniqueness of the source reconstruction using observation data in a finite time interval is proven based on the energy estimate then the reconstruction problem is reformulated as a minimization problem involving a tikhonov regularizing term by deriving the explicit representation of the gradient of the cost functional in terms of an adjoint problem the conjugate gradient method is applied to obtain the numerical solution numerical results are presented to show the validity of the proposed algorithm

two alternating direction implicit difference schemes with the extrapolation method for the two dimensional distributed order differential equations the grunwald formula is used to solve the two dimensional distributed order differential equations two alternating direction implicit adi difference schemes are derived it is proved that the schemes are unconditionally stable and convergent with the convergence orders o h 1 2 h 2 2 2 and o h 1 4 h 2 4 4 in h 1 norm respectively where h 1 h 2 and are the step sizes in time space in x and y direction and distributed order the extrapolation method is applied to improve the approximate accuracy to the orders o 2 ln 2 h 1 2 h 2 2 2 and o 2 ln 2 h 1 4 h 2 4 4 respectively several numerical examples are given to confirm the theoretical results

numerical analysis for distributed order differential equations in this paper we present and analyse a numerical method for the solution of a distributed order differential equation of the general form 0 ma r d ru t dr f t where m is a positive real number and where the derivative d r is taken to be a fractional derivative of caputo type of order r we give a convergence theory for our method and conclude with some numerical examples

doubles of quasi quantum groups in dr1 drinfeld showed that any finite dimensional hopf algebra extends to a quasitriangular hopf algebra the quantum double of based on the construction of a so called diagonal crossed product developed by the authors in hn we generalize this result to the case of quasi hopf algebras as for ordinary hopf algebras as a vector space the quasi quantum double is isomorphic to where denotes the dual of we give explicit formulas for the product the coproduct the r matrix and the antipode on and prove that they fulfill drinfeld s axioms of a quasitriangular quasi hopf algebra in particular becomes an associative algebra containing as a quasi hopf subalgebra on the other hand is not a subalgebra of unless the coproduct on is strictly coassociative it is shown that the category of finite dimensional representations of coincides with what has been called the double category of modules by s majid m2 thus our construction gives a concrete realization of majid s abstract definition of quasi quantum doubles in terms of a tannaka krein like reconstruction procedure the whole construction is shown to generalize to weak quasi hopf algebras with now being linearly isomorphic to a subspace of

an improved error bound for a finite element approximation of a reaction diffusion system of λ ω type in this paper we study a nonlinear in time finite element discretization of a reaction diffusion system of type using the approach in nochetto et al 2000 for analyzing the time discretization error in the backward euler method we improve on the error bound derived in garvie and blowey 2005 we also present some numerical experiments to illustrate the improved error bound

some high order difference schemes for the distributed order differential equations two difference schemes are derived for both one dimensional and two dimensional distributed order differential equations it is proved that the schemes are unconditionally stable and convergent in an l 1 l norm with the convergence orders o 2 h 2 2 and o 2 h 4 4 respectively where h and are the step sizes in time space and distributed order variables several numerical examples are given to confirm the theoretical results

finite dimensional cosemisimple hopf algebras in characteristic 0 are semisimple finite dimensional cosemisimple hopf algebras are a natural generalization of group algebras much of the representation theory of finite groups can be extended to such hopf algebras however there are still many unanswered basic questions for such hopf algebras one such question is a conjecture mentioned by kaplansky i the square of the antipode of a finite dimensional cosemisimple hopf algebra is the identity in this paper we prove that the fourth power of the antipode is the identity in a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 in the course of proving this we investigate two elements of the hopf algebra which are roughly analogous to the sum of the group elements in a group algebra we show that kaplansky s conjecture is equivalent to these two elements being equal for a finite dimensional hopf algebra a with antipode s over a field let l p q pq for p q e a denote the left module action of a on itself in this paper we study two elements of a defined by the trace function the element i defined by p a tr l p s for all pe a and the element x defined by p x tr l p for all p e a we show that 2 is a left integral and use it to prove theorem 3 3 the main result of this paper a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 is also semisimple hence its antipode has order 1 2 or 4 we extend the first corollary to 3 proposition 91 in theorems 4 3 and 4 4 by finding necessary and sufficient conditions for s to be a non zero left integral in which case x a and also study how x relates to the structure of a in particular when dim a 1 0 the right ideal xa may be thought of as a measure of the extent to which s i the elements ji and k are two examples of elements of a associated with an endomorphism of a in general for f end a we define a e a by p a tr l p cf for all pea thus i as2 and x a in section 2 267 0021 8693 88 33 00

logarithmic conformal field theory beyond an introduction this article aims to review a selection of central topics and examples in logarithmic conformal field theory it begins with the remarkable observation of cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory cardy s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators for this a short introduction to the fusion algorithm of nahm gaberdiel and kausch is provided while the percolation logarithmic conformal field theory is still not completely understood there are several examples for which the formalism familiar from rational conformal field theory including bulk partition functions correlation functions modular transformations fusion rules and the verlinde formula has been successfully generalized this is illustrated for three examples the singlet model related to the triplet model symplectic fermions and the fermionic bc ghost system the fractional level wess zumino witten model based on at related to the bosonic ghost system and the wess zumino witten model for the lie supergroup related to at and 1 the bershadsky polyakov algebra and the feigin semikhatov algebras these examples have been chosen because they represent the most accessible and most useful members of the three best understood families of logarithmic conformal field theories the logarithmic minimal models the fractional level wess zumino witten models and the wess zumino witten models on lie supergroups excluding in this review the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum which turns out to be continuous as well as a selection of natural reducible but indecomposable modules this is followed by a detailed description of how to obtain character formulae for each irreducible a derivation of the action of the modular group on the characters and an application of the verlinde formula to compute the grothendieck fusion rules in each case the genuine fusion rules are known so comparisons can be made and favourable conclusions drawn in addition each example admits an infinite set of simple currents hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed the spectrum of such an extended theory is typically discrete and this is how the triplet model arises for example moreover simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory finally each example is concluded by a brief description of the computation of some bulk correlators a discussion of the structure of the bulk state space and remarks concerning more advanced developments and generalizations the final part gives a very short account of the theory of staggered modules the simplest class of representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins these modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven then the important quantities known as logarithmic couplings are reviewed for virasoro staggered modules and their role as fundamentally important parameters akin to the three point constants of rational conformal field theory is discussed an appendix is also provided in order to introduce some of the necessary but perhaps unfamiliar language of homological algebra

a quantitative comparison between c0 and c1 elements for solving the cahn hilliard equation the cahn hilliard ch equation is a time dependent fourth order partial differential equation pde when solving the ch equation via the finite element method fem the domain is discretized by c 1 continuous basis functions or the equation is split into a pair of second order pdes and discretized via c 0 continuous basis functions in the current work a quantitative comparison between c 1 hermite and c 0 lagrange elements is carried out using a continuous galerkin fem formulation the different discretizations are evaluated using the method of manufactured solutions solved with newton s method and jacobian free newton krylov it is found that the use of linear lagrange elements provides the fastest computation time for a given number of elements while the use of cubic hermite elements provides the lowest error the results offer a set of benchmarks to consider when choosing basis functions to solve the ch equation in addition an example of microstructure evolution demonstrates the different types of elements for a traditional phase field model

a novel legendre operational matrix for distributed order fractional differential equations abstract in this paper for the first time the shifted legendre operational matrix of distributed order fractional derivative has been derived also this new operational matrix is used together with tau method for approximation of solutions of linear distributed order fractional differential equations and diffusion equations with distributed order in time moreover eight numerical examples are implemented in order to show the validity and reliability of the suggested methods

on the antipode of a quasitriangular hopf algebra let a r be a quasitriangular hopf algebra with antipode s in the category of vector spaces over a field k in this paper we show that s2 is inner which means s is bijective and when a is finite dimensional there is a grouplike element h e a such that s4 a h r for all a e a suppose that a is any hopf algebra with antipode s over the field k proposition 1 of section 1 gives a sufficient condition for 3 to be inner which we apply to quasitriangular hopf algebras in section 2 its statement and proof are based on calculations found in 2 pp 66671 generally s2 need not be inner suppose that a is finite dimensional if s2 were inner then all ideals of a would be invariant under s2 and hence all subcoalgebras of the dual hopf algebra a would be invariant under s where s s is the antipode of a there are examples 5 over algebraically closed lields in which the latter is not the case section 2 begins with a discussion of properties of quasit angular hopf algebras a r over the held k used in this paper write r cr c r ea a and set u cs r r the main result of section 2 is that u is invertible and s2 a uau for all a e a this result was established in z exercise 7 3 61 under the hypothesis that s is bijective if a is any finite dimensional hopf algebra with antipode s over the field k there are distinguished grouplike elements g e a and tl e a which relate left and right integrals and which together describe s4 in section 3 we consider s4 when a r is a unite dimensional quasit angular hopf algebra let h iiu where u c s r r l is defined as above and u s u i by relating h to g and 1 we are able to show that h is a grouplike element and that s4 a hah for all a e a when a is unimodular we prove that h g thus the product vu does not depend on r in the unimodular case we assume that the reader has some familiarity with the elementary aspects of the theory of hopf algebras a recommended reference is 6

the bem for numerical solution of partial fractional differential equations a numerical method is presented for the solution of partial fractional differential equations fdes arising in engineering applications and in general in mathematical physics the solution procedure applies to both linear and nonlinear problems described by evolution type equations involving fractional time derivatives in bounded domains of arbitrary shape the method is based on the concept of the analog equation which in conjunction with the boundary element method bem enables the spatial discretization and converts a partial fde into a system of coupled ordinary multi term fdes then this system is solved using the numerical method for the solution of such equations developed recently by katsikadelis the method is illustrated by solving second order partial fdes and its efficiency and accuracy is validated

a conservative numerical method for the cahn hilliard equation in complex domains we propose an efficient finite difference scheme for solving the cahn hilliard equation with a variable mobility in complex domains our method employs a type of unconditionally gradient stable splitting discretization we also extend the scheme to compute the cahn hilliard equation in arbitrarily shaped domains we prove the mass conservation property of the proposed discrete scheme for complex domains the resulting discretized equations are solved using a multigrid method numerical simulations are presented to demonstrate that the proposed scheme can deal with complex geometries robustly furthermore the multigrid efficiency is retained even if the embedded domain is present