stability of time stepping methods for abstract time dependent parabolic problems we consider an abstract nonautonomous parabolic problem u t a t u t quad u t 0 u 0 where a t d t subset x to x t 0 le t le t 1 is a family of sectorial operators in a banach space x this problem is discretized in time by means of either an a theta stable runge kutta or an a theta stable linear multistep method we prove that the resulting discretization is stable under some natural assumptions on the relative total variation of a t with respect to t for strongly a theta stable runge kutta methods stability holds even for variable time step sizes our results are applicable to the analysis of time dependent parabolic problems in the lp norms

a general procedure for the adaptation of multistep algorithms to the integration of oscillatory problems this paper introduces a general technique for the construction of multistep methods capable of integrating without local truncation error homogeneous linear odes with constant coefficients including those in particular that result in oscillatory solutions moreover these methods can be further adapted through coefficient modification for the exact integration of forced oscillations in one or more frequencies even confluent ones that occur from nonhomogeneous terms in the differential equation our procedure allows the derivation of many of the existing codes with similar properties as well as the improvement of others that in their original design were only able to integrate oscillations in a single frequency the properties of the methods are studied within a general framework and numerical examples are presented these demonstrate the way in which the new algorithms perform distinctly better than the general purpose codes particularly when integrating the class of equations with perturbed oscillatory solutions the methods developed are mainly applicable to the accurate and efficient integration of problems for which the oscillation frequencies are known as occurs in satellite orbit propagation the underlying ideas have already been applied to the improvement of some chebyshev methods that are not multistep

subnormality for arbitrary powers of 2 variable weighted shifts whose restrictions to a large invariant subspace are tensor products the lifting problem for commuting subnormals lpcs asks for necessary and sufficient conditions for a pair of subnormal operators onhilbert space to admit commuting normal extensions we study lpcs within the class of commuting 2 variable weighted shifts t t1 t2 with subnormal components t1 and t2 acting on the hilbert space 2 z 2 with canonical orthonormal basis fe k1 k2 gk1 k2 0 the core of a commuting 2 variable weighted shift t c t is the restriction of t to the invariant subspace generated by all vectors e k1 k2 with k1 k2 1 we say that c t is of tensor form if it is unitarily equivalent to

zero stability properties of the three ordinate variable stepsize variable formula methods consider the systemy f x y y a x a b y r n wheref is continuous and lipschitzian with respect to the second argument very often linear multistep variable stepsize variable formula methods lm vsvfm s are used to computey k y xk on the points of the grid a x 0 x1 x2 xn b the general lm vsvfm is based on formulae of the following type y k sum limits i 1 s k alpha i bar h k sk y k i sum limits i 0 s k h k i beta i bar h k s k f x k i y k i whereh k xk xk 1 bar h k h k h k 1 ldots h k s k s k k k 1 1 n the coefficients i and i depend on the lasts k 1 stepsizes and on the formula used at stepk only the zero stability properties of some special classes of lm vsvfm s as for example those based on adams formulae were investigated in the literature a class of three ordinate lm vsvfm s is defined in this paper some results concerning the zero stability properties of these methods are proved it is shown that some well known results are simple corollaries of the results found for the three ordinate lm vsvfm s it is easily seen that similar results hold for the corresponding one leg vsvfm s finally the use of the theoretical results in the practical implementations is briefly discussed

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

on classification of modular tensor categories we classify all unitary modular tensor categories umtcs of rank 4 there are a total of 35 umtcs of rank 4 up to ribbon tensor equivalence since the distinction between the modular s matrix s and s has both topological and physical significance so in our convention there are a total of 70 umtcs of rank 4 in particular there are two trivial umtcs with s 1 each such umtc can be obtained from 10 non trivial prime umtcs by direct product and some symmetry operations explicit data of the 10 non trivial prime umtcs are given in sect 5 relevance of umtcs to topological quantum computation and various conjectures are given in sect 6

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

classification of metaplectic modular categories abstract we obtain a classification of metaplectic modular categories every metaplectic modular category is a gauging of the particle hole symmetry of a cyclic modular category our classification suggests a conjecture that every weakly integral modular category can be obtained by gauging a symmetry including the fermion parity of a pointed super modular category

categories premodulaires modularisations et invariants des varietes de dimension 3 an abelian k linear semisimple category having a finite number of simple objects and endowed with a ribbon structure is called premodular it is modular in the sense of turaev if the so called s matrix is invertible a modular category defines invariants of 3 manifolds and a tqft t when is it possible to construct a modularisation of a given premodular category i e a functor to a modular category preserving the structures and dominant in a certain sense it turns out 2 3 that this amounts essentially to making transparent objects trivial we give a full answer to this problem in the case when k is a field of char 0 as well as partial answers in char p under a few obvious hypotheses a premodular category admits a modularisation which is unique th 3 1 and cor 3 5 in char 0 the proof relies on two main ingredients a new and very simple criterion for the s matrix to be invertible 1 1 and deligne s internal characterization of tannakian categories in char 0 d when simple transparent objects are invertible the criterion is simpler 4 2 and the modularisation can be described more explicitly prop 4 4 we conclude with two examples the premodular categories associated with quantum hbox sf sl n and hbox sf pgl n at roots of unity in the first case we obtain modular categories which were built independently by c blanchet b in the second case we obtain modularizations in all the cases where y yokota y found reshetikhin turaev invariants of 3 manifolds thereby improving as well as explaining yokota s results

on the zero stability of variable stepsize multistep methods the spectral radius approach in this paper we illustrate a novel approach for studying the asymptotic behaviour of the solutions of linear difference equations with variable coefficients in particular we deal with the zero stability of the 3 step bdf method on grids with variable stepsize for the numerical solution of ivps for odes our approach is based on the theory of the spectral radius of a family of matrices and yields almost optimal results which give a slight improvement to the best results already known from the literature the success got on the chosen example suggests that our approach has a good potential for more general and harder stability analyses of numerical methods

stability restrictions on time stepsize for numerical integration of first order partial differential equations mathematical models described by partial differential equations appear often when different phenomena related to atmospheric transport for example advection and diffusion processes are studied truncated fourier series can be used in the discretization of the space derivatives resulting in a pseudospectral method in this way the partial differential equation is transformed into a system of ordinary differential equations this system is normally solved by so called step by step integration methods the stability properties of these methods are studied some bounds for the time stepsize are derived several formulae and predictor corrector schemes with large stability intervals on the imaginary axis are constructed numerical experiments with eight time integration algorithms are carried out some recommendations concerning the choice of the time integration algorithms in three different situations are given it is emphasized that in a package for the solution of partial differential equations describing transport processes in the atmospheric environment one should have several time integration algorithms each of which can be chosen optionally finally the possibility of using variable stepsize variable formula time integration algorithms is briefly discussed

implementation of a variable stepsize variable formula method in the time integration part of a code for treatment of long range transport of air pollutants abstract a mathematical model consisting of two partial differential equations is used to study the long range transport of sulphur di oxide and sulphate over europe the discretization of the first order space derivatives the advection terms is carried out by a pseudospectral fourier algorithm a special technique is applied in the discretization of the second order space derivatives the diffusion terms two large systems of ordinary differential equations are solved at each time step it is shown that these systems can efficiently be treated by a variable stepsize variable formula method vsvfm based on the use of predictor corrector schemes the stepsize selection strategy and the formula selection strategy are discussed in detail an attempt to carry out both an accuracy control and a stability control is made at each time step the great efficiency of the vsvfm implemented in our software as well as the reliability of the results are illustrated by numerical experiments in which real meteorological data for 1979 at the grid points of a space domain covering the whole of europe were used the main ideas implemented in the time integration part might be applied in many other situations where the systems of ordinary differential equations arising after the space discretization are only moderately stiff so that the stability requirements are dominant over the accuracy requirements on a large part of the time interval but the use of implicit time integration algorithms that require solving systems of algebraic equations at each time step is not justified as an illustration only it should be mentioned that such an application has been carried out in connection with models describing long range transport of nitrogen pollutants over europe

jointly hyponormal pairs of commuting subnormal operators need not be jointly subnormal we construct three different families of commuting pairs of subnormal operators jointly hyponormal but not admitting commuting normal extensions each such family can be used to answer in the negative a 1988 conjecture of r curto p muhly and j xia we also obtain a sufficient condition under which joint hyponormality does imply joint subnormality

a splitting technique of higher order for the navier stokes equations this article presents a splitting technique for solving the time dependent incompressible navier stokes equations using nested finite element spaces which can be interpreted as a postprocessing step the splitting method is of more than second order accuracy in time the integration of adaptive methods in space and time in the splitting are discussed in this algorithm a gradient recovery technique is used to compute boundary conditions for the pressure and to achieve a higher convergence order for the gradient at different points of the algorithm results on the flow around a cylinder s and the driven cavity s problem are presented

schur product techniques for the subnormality of commuting 2 variable weighted shifts abstract in this paper we study the subnormality of 2 variable weighted shifts using the schur product techniques in matrices and the convolution of two continuous functions as a consequence we find the berger measure of the subnormal weighted shift obtained from the schur product of two subnormal weighted shifts as applications we first give non trivial large classes satisfying the curto muhly xia conjecture see the conjecture given below for 2 variable weighted shifts we next show when the 2 hyponormality of 2 variable weighted shifts becomes subnormality

some behaviors of fsz groups under central products central quotients and regular wreath products abstract we show that any group g with a non f s z m quotient by a central cyclic subgroup also provides a non f s z m group of order m g obtained as a central product of g with a cyclic group we then construct for every prime p 3 and j n an f s z p j group f such that there is a central cyclic subgroup a with f a not f s z p j we apply these results to regular wreath products to construct an fsz p group which is not f s z for any prime p 3 these give the first known examples of fsz groups that are not f s z we are also able to prove a few partial results concerning the fsz properties for the sylow subgroups of symmetric groups in the appendix we enumerate all non fsz groups of order 57

on the frobenius schur indicators for quasi hopf algebras abstract mason and ng have given a generalization to semisimple quasi hopf algebras of linchenko and montgomery s generalization to semisimple hopf algebras of the classical frobenius schur theorem for group representations we give a simplified proof in particular a somewhat conceptual derivation of the appropriate form of the frobenius schur indicator that indicates if and in which of two possible fashions a given simple module is self dual

on the stability of interpolatory variable stepsize adams methods in nordsieck form this paper considers the stability of adams methods that vary the stepsize using the interpolation technique of nordsieck sufficient conditions sometimes necessary on the stepsize ratios h n 1 h n that imply the stability of adams methods with different strategies of stepsize variation are obtained this study extends previous results on this subject by gear and tu siamj numer anal 11 1974 pp 1025 1043 and skeel and jackson siam j numer anal 20 1983 pp 840 853

implementation of a type multistep methods for stiff differential equations the type multistep methods have the form of yn k nyn k 1 n 1 yn k 2 hn k i 0k i nfn 1 where 0 n 2 they are stable for all order and step size changing sequences when implemented in a variable coefficient form this paper will discuss some of the results of when these methods are applied to stiff differential equations

two high order multistep methods with offstep points applied with a variable stepsize abstract in this paper we derive new generalized multistep methods with variable stepsize for initial value problems in ordinary differential equations some numerical examples are given

hp adaptive time integration based on the bdf for viscous flows this paper presents a procedure based on the backward differentiation formulas of order 1 to 5 to obtain efficient time integration of the incompressible navier stokes equations the adaptive algorithm performs both stepsize and order selections to control respectively the solution accuracy and the computational efficiency of the time integration process the stepsize selection h adaptivity is based on a local error estimate and an error controller to guarantee that the numerical solution accuracy is within a user prescribed tolerance the order selection p adaptivity relies on the idea that low accuracy solutions can be computed efficiently by low order time integrators while accurate solutions require high order time integrators to keep computational time low the selection is based on a stability test that detects growing numerical noise and deems a method of order p stable if there is no method of lower order that delivers the same solution accuracy for a larger stepsize hence it guarantees both that 1 the used method of integration operates inside of its stability region and 2 the time integration procedure is computationally efficient the proposed time integration procedure also features a time step rejection and quarantine mechanisms a modified newton method with a predictor and dense output techniques to compute solution at off step points

s4 symmetry of 6j symbols and frobenius schur indicators in rigid monoidal c categories we show that a left rigid monoidal c category with irreducible monoidal unit is also a sovereign and spherical category defining a frobenius schur type indicator we obtain selection rules for the fusion coefficients of irreducible objects as a main result we prove s4 invariance of 6j symbols in such a category

time integration for ale simulations of fluid structure interaction problems stepsize and order selection based on the bdf abstract we present an adaptive algorithm for time integration of fluid structure integration problems the method relies on a fully coupled procedure to solve fsi problems in which a naturally gcl compliant ale formulation for the finite element spatial discretization is used the main originality of the proposed solution procedure is that time integration is performed using automatic order and stepsize selections hp adaptivity based on the backward differentiation formulas bdf the stepsize selection is based on a local error estimate an error controller and a step rejection mechanism it guarantees that the solution precision is within the user targeted tolerance the order selection is based on a stability test and a quarantine mechanism the selection is performed to ensure that no other methods within the family of 0 stable bdf methods would produce a solution of the targeted precision for a larger stepsize and thus a lower computational time to improve efficiency the time integration procedure also relies on a modified newton method and a predictor the time adaptive algorithm behaviors and performances are assessed on the vortex induced translational and rotational vibrations of a square cylinder and on the wake induced vibrations of 3 cylinders in an in line arrangement the algorithm yields substantial cpu time savings compared to constant stepsize and order integration while delivering solutions of prescribed accuracies

on the stability of variable stepsize nordsieck bdf methods in this paper the stability of backward differentiation methods that vary the stepsize using the interpolation technique of nordsieck is considered conditions for the stability of the methods are obtained the study is an extension of previous results given by gear and tu 3 and skeet and jackson 14

stability of multistep methods on variable grids this paper is concerned with the stability of multistep methods for ordinary initial value problems on grids with variable mesh sizes a necessary and sufficient condition for stability is given from which generalizations of recent results by gear et al and by zlatev can be obtained as special cases as an application the stability of the variable bdf formulas is treated

classification of rank 5 premodular categories we survey a number of classification tools developed in recent years and employ them to classify pseudo unitary rank 5 premodular categories up to grothendieck equivalence

mathematical model for studying the sulphur pollution over europe the long range transport of sulphur pollutants in the atmosphere is an important environmental problem a mathematical model consisting of a system of partial differential equations has been developed at the danish agency of environmental protection for studying this phenomenon the model is handled numerically a pseudospectral fourier algorithm is used in the space discretization phase a variable stepsize variable formula method vsvfm based on predictor corrector schemes may be specified during the time discretization part the time integration formulae are especially constructed so that a the predictor corrector schemes have good absolute stability properties on the imaginary axis and b the fundamental properties consistency zero stability and convergence of the global vsvfm are not affected by the changes of the stepsize and or the predictor corrector scheme both the accuracy requirements and the stability requirements are taken into account in the stepsize selection strategy as well as in the formula selection strategy this double check leads to a great efficiency in the numerical treatment of problems arising in atmospheric environments this is demonstrated by many numerical results the question of the reliability of the results obtained in runs of real physical problems is discussed some results obtained in the comparison of calculated results with measurements are reported an illustration where the model is applied in studying the air pollution in the area around denmark due to danish sources is presented

a family of multistep methods to integrate orbits on spheres a trajectory problem is an initial value problem where the interest lies in obtaining the curve traced by the solution rather than in finding the actual correspondence between the values of the parameter and the points on that curve this paper introduces a family of multi stage multi step numerical methods to integrate trajectory problems whose solution is on a spherical surface it has been shown that this kind of algorithms has good numerical properties consistency stability convergence and others that are not standard the latest ones make them a better choice for certain problems

on some properties of quantum doubles of finite groups abstract we prove two results about quantum doubles of finite groups over the complex field the first result is the integrality theorem for higher frobenius schur indicators for wreath product groups s n a n where a is a finite abelian group a proof of this result for a 1 appears in a paper by iovanov montgomery and mason the second result is a lower bound for the largest possible number of irreducible representations of the quantum double of a finite group with at most n conjugacy classes this answers a question asked by eric rowell

k hyponormality of multivariable weighted shifts we characterize joint k hyponormality for 2 variable weighted shifts using this characterization we construct a family of examples which establishes and illustrates the gap between k hyponormality and k 1 hyponormality for each k 1 as a consequence we obtain an abstract solution to the lifting problem for commuting subnormals

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

which 2 hyponormal 2 variable weighted shifts are subnormal it is well known that a 2 hyponormal unilateral weighted shift with two equal weights must be flat and therefore subnormal by contrast a 2 hyponormal 2 variable weighted shift which is both horizontally flat and vertically flat need not be subnormal in this paper we identify a large class s of flat 2 variable weighted shifts for which 2 hyponormality is equivalent to subnormality one measure of the size of s is given by the fact that within s there are hyponormal shifts which are not subnormal

numerical treatment of large scale air pollution models abstract studying air pollution phenomena over a large space domain by a fairly general mathematical model is discussed the space discretization of such a model leads to huge system of odes the integration algorithm used in the solution of these systems must be efficient if a 3 d model is considered then a vector processor is to be used moreover the most time consuming parts of the code are to be vectorized the calculated results concentrations of different air pollutants must be reliable because they have to be used by other specialists therefore different checks concerningthe reliability of the results are carried out in this paper it is shown that an efficient and reliable code for studying both sulphur and nitrogen pollution has been developed at the danish agency of environmental protection some simulation processes are also described breifly

the kernel of the modular representation and the galois action in rcft it is shown that for the modular representations associated to rational conformal field theories the kernel is a congruence subgroup whose level equals the order of the dehn twist an explicit algebraic characterization of the kernel is given it is also shown that the order of the dehn twist is bounded by a function of the number of primary fields allowing for a systematic enumeration of the modular representations coming from rcfts restrictions on the spectrum of the dehn twist and arithmetic properties of modular matrix elements are presented

central invariants and frobenius schur indicators for semisimple quasi hopf algebras abstract in this paper we obtain a canonical central element h for each semi simple quasi hopf algebra h over any field k and prove that h is invariant under gauge transformations we show that if k is algebraically closed of characteristic zero then for any irreducible representation of h which affords the character h takes only the values 0 1 or 1 moreover if h is a hopf algebra or a twisted quantum double of a finite group then h is the corresponding frobenius schur indicator we also prove an analog of a theorem of larson radford for split semi simple quasi hopf algebras over any field k using this result we establish the relationship between the antipode s the values of h and certain associated bilinear forms when the underlying field k is algebraically closed of characteristic zero

module tensor product of subnormal modules need not be subnormal let d d c d d c be a diagonal positive definite kernel and let h h denote the associated reproducing kernel hilbert space of holomorphic functions on the open unit disc dd assume that zf hzf h whenever f hf h then hh is a hilbert module over the polynomial ring c z c z with module action p f pfp f pf we say that h h is a subnormal hilbert module if the operator mzmz of multiplication by the coordinate function z on h h is subnormal n salinas 1988 11 asked whether the module tensor product h 1 c z h 2h 1 c z h 2 of subnormal hilbert modules h 1h 1 and h 2h 2 is again subnormal in this regard we describe all subnormal module tensor products la2 d ws1 c z la2 d ws2 where la2 d ws denotes the weighted bergman hilbert module with radial weight ws z 1s z 2 1 s s z d s 0 in particular the module tensor product la2 d ws c z la2 d ws is never subnormal for any s 6s 6 thus the answer to this question is no

highly continuous interpolants for one step ode solvers and their application to runge kutta methods we suggest a general method for the construction of highly continuous interpolants for one step methods applied to the numerical solution of initial value problems of odes of arbitrary order for the construction of these interpolants one uses along with the numerical data of the discrete solution of a problem provided by a typical one step method at endstep points high order derivative approximations of this solution this approach has two main advantages it allows an easy way of construction of high order runge kutta and nystrom interpolants with reduced cost in additional function evaluations that also preserve the one step nature of the underlying discrete ode solver moreover for problems which are known to possess a solution of high smoothness the approximating interpolant resembles this characteristic a property that on occasion might be desirable an analysis of the stability behavior of such interpolatory processes is carried out in the general case a new numerical technique concerning the accurate determination of the stability behavior of numerical schemes involving higher order derivatives and or approximations of the solution from previous grid points over nonequidistant meshes is presented this technique actually turns out to be of a wider interest as it allows us to infer in certain cases more accurate results concerning the stability of for example the bdf formulas over variable stepsize grids moreover it may be used as a framework for analyzing more complex and supposedly more promising types of methods as they are the general linear methods for first and second order differential equations many particular variants of the new method for first order differential equations that have good prospects of finding a practical implementation are fully analyzed with respect to their stability characteristics a detailed application concerning the construction of c 2 and c 3 continuous extensions for some fifth and sixth order runge kutta pairs supplemented by a detailed study of the local truncation error characteristics of a class of interpolants of this type is also provided various numerical examples show in these cases several advantages of the newly proposed technique with respect to function evaluation cost and global error behavior in comparison with others currently in use

construction of variable stepsize multistep formulas a systematic way of extending a general fixed stepsize multistep formula to a minimum storage variable stepsize formula has been discovered that encompasses fixed coefficient interpolatory variable coefficient variable step and fixed leading coefficient as special cases in particular it is shown that the interpolatory stepsize changing technique of nordsieck leads to a truly variable stepsize multistep formula which has implications for local error estimation and formula changing and it is shown that the variable step stepsize changing technique applicable to the adams and backward differentiation formulas has a reasonable generalization to the general multistep formula in fact it is shown how to construct a variable order family of variable coefficient formulas finally it is observed that the first dahlquist barrier does not apply to adaptable multistep methods if storage rather than stepnumber is the key consideration

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

schur product techniques for commuting multivariable weighted shifts abstract in this paper we study the hyponormality and subnormality of 2 variable weighted shifts using the schur product techniques in matrices as applications we generalize the result in r curto j yoon jointly hyponormal pairs of subnormal operators need not be jointly subnormal trans amer math soc 358 2006 5135 5159 theorem 5 2 and give a non trivial large class satisfying the curto muhly xia conjecture r curto p muhly j xia hyponormal pairs of commuting operators oper theory adv appl 35 1988 1 22 for 2 variable weighted shifts further we give a complete characterization of hyponormality and subnormality in the class of flat contractive 2 variable weighted shifts t t 1 t 2 with the condition that the norm of the 0th horizontal 1 variable weighted shift of t is a given constant

nearly subnormal operators and moment problems abstract we use separation of cones techniques and ideas from multivariable operator theory to show that polynomial hyponormality does not imply subnormality for hilbert space operators as an application we obtain a new result in the theory of power moments in two dimensions

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

a new characterization of subnormality for a class of 2 variable weighted shifts with 1 atomic core abstract given a pair t t 1 t 2 of commuting subnormal hilbert space operators the lifting problem for commuting subnormals lpcs calls for necessary and sufficient conditions for the existence of a commuting pair n n 1 n 2 of normal extensions of t 1 and t 2 this is an old problem in operator theory the aim of this paper is to study lpcs there are three well known subnormal characterizations for operators the berger theorem the bram halmos characterization and franks result in our paper we study a new subnormal characterization which is related to these three well known ones for a class of 2 variable weighted shifts thus we can provide a large nontrivial class of 2 variable weighted shifts in which k hyponormal some k 1 and subnormal are equal and the class is invariant under the action h l t h l t 1 h t 2 l h l 1

existence of non subnormal polynomially hyponormal operators in 1950 p r halmos motivated in part by the successful development of the theory of normal operators introduced the notions of subnormality and hyponormality for bounded hilbert space operators an operator t is subnormal if it is the restriction of a normal operator to an invariant subspace t is hyponormal if t t tt it is a simple matrix calculation to verify that subnormality implies hyponormality but the converse is false one reason is that subnormality is invariant under polynomial calculus indeed analytic functional calculus while hyponormality is not if one then defines t to be polynomially hyponormal when p t is hyponormal for every polynomial p e c z the following question arises naturally

hyponormality and subnormality for powers of commuting pairs of subnormal operators let h0 respectively h denote the class of commuting pairs of subnormal operators on hilbert space respectively subnormal pairs and for an integer k 1 let hk denote the class of k hyponormal pairs in h0 we study the hyponormality and subnormality of powers of pairs in hk we first show that if t1 t2 h1 the pair t12 t2 may fail to be in h1 conversely we find a pair t1 t2 h0 such that t12 t2 h1 but t1 t2 h1 next we show that there exists a pair t1 t2 h1 such that t1mt2n is subnormal for all m n 1 but t1 t2 is not in h this further stretches the gap between the classes h1 and h finally we prove that there exists a large class of 2 variable weighted shifts t1 t2 namely those pairs in h0 whose cores are of tensor form cf definition 3 4 for which the subnormality of t12 t2 and t1 t22 does imply the subnormality of t1 t2

the number of solutions of decomposable form equations has only finitely many solutions the diophantine approximation techniques of thue and mahler and improvements by siegel dyson roth and bombieri made it possible to derive good explicit upper bounds for the number of solutions of 1 1 the best such upper bound to date due to bombieri 1 is 2 12r bombieri assumed that f is irreducible and r 6 which was not essential in his proof for t 0 i e f x y 1 in x y z bombieri and schmidt 2 derived the upper bound constant r which is best possible in terms of r

on classification of extremal non holomorphic conformal field theories rational chiral conformal field theories are organized according to their genus which consists of a modular tensor category and a central charge c a long term goal is to classify unitary rational conformal field theories based on a classification of unitary modular tensor categories we conjecture that for any unitary modular tensor category there exists a unitary chiral conformal field theory so that its modular tensor category is in this paper we initiate a mathematical program in and around this conjecture we define a class of extremal vertex operator algebras with minimal conformal dimensions as large as possible for their central charge and non trivial representation theory we show that there are finitely many different characters of extremal vertex operator algebras possessing at most three different irreducible modules moreover we list all of the possible characters for such vertex operator algebras with

stability of explicit time discretizations for solving initial value problems stability regions of explicit linear time discretization methods for solving initial value problems are treated if an integration method needsm function evaluations per time step then we scale the stability region by dividing bym we show that the scaled stability region of a method satisfying some reasonable conditions cannot be properly contained in the scaled stability region of another method bounds for the size of the stability regions for three different purposes are then given for general nonlinear ordinary differential systems for systems obtained from parabolic problems and for systems obtained from hyperbolic problems we also show how these bounds can be approached by high order methods

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

multistep numerical methods based on the scheifele g functions with application to satellite dynamics in 1971 scheifele designed a method for the integration of perturbed oscillators by refining the classical method of power series this method consists of defining a certain sequence of entire functions and finding the sought solution in the form of a linear combination of them using coefficients obtained through recurrence formulas in spite of the good behavior of the scheifele method it is only practical for use in a few particular cases in which the force function is very simple given the complexity of the preliminary calculations needed to obtain the recurrence formulas this problem is resolved in this article by converting the g functions method into a multistep scheme the key property is that those algorithms are able to integrate without truncation error harmonic oscillators involving various frequencies and that for perturbed problems the local error contains the small perturbation parameter as a factor therefore the methods are useful for finding highly accurate solutions to problems whose solutions are close to quasi periodic ones an application is performed in investigating the behavior of the proposed schemes for the integration of the orbital motion of earth satellites

a 0 stability of variable stepsize bdf methods abstract in this paper the a0 stability of interpolatory variable coefficient and fixed leading coefficient bdf formulas is considered a comparison of the stability properties of the three methods based on their spectral radius is presented in addition we study the effect of some strategies of stepsize variation employed in usual codes on the stability of these methods

multiderivative variable stepsize variable formula methods abstract in this paper we present a study of consistency stability and convergence properties of linear multiderivative multistep variable stepsize variable formula methods

variable step implementation of etd methods for semilinear problems abstract spatial discretization of many evolutionary partial differential equations leads to stiff semilinear systems of ordinary differential equations exponential integrators solve exactly the linear part which is assumed to be responsible of the stiffness of the problem while the nonlinear terms are approximated numerically in this paper a variable step implementation of the exponential time differencing methods of multistep type is considered a local error estimate of exponential type whose computation is for free is introduced and some implementation issues for diagonalizable matrices are addressed numerical results showing the behaviour of the proposed variable step implementation are provided too

frobenius schur indicators and exponents of spherical categories we obtain two formulae for the higher frobenius schur indicators one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist the first one is a categorical generalization of an analogous result by kashina sommerhauser and zhu for hopf algebras and the second one extends bantay s 2nd indicator formula for a conformal field theory to higher degrees these formulae imply the sequence of higher indicators of an object in these categories is periodic we define the notion of frobenius schur fs exponent of a pivotal category to be the global period of all these sequences of higher indicators and we prove that the fs exponent of a spherical fusion category is equal to the order of the twist of its center consequently the fs exponent of a spherical fusion category is a multiple of its exponent in the sense of etingof by a factor not greater than 2 as applications of these results we prove that the exponent and the dimension of a semisimple quasi hopf algebra h have the same prime divisors which answers two questions of etingof and gelaki affirmatively for quasi hopf algebras moreover we prove that the fs exponent of h divides dim h 4 in addition if h is a group theoretic quasi hopf algebra the fs exponent of h divides dim h 2 and this upper bound is shown to be tight

computable explicit bounds for the discretization error of variable stepsize multistep methods abstract this paper deals with the achievement of explicit computable bounds for the global discretization error of variable stepsize multistep methods which are perturbation of strongly stable fixed stepsize methods the approach is based on the study of the growth of solutions of certain variable coefficient difference equations satisfied by the global discretization error

advance in the theory of variable stepsize variable formula methods for ordinary differential equations the theory of the constant stepsize constant formula methods cscfm s for solving initial value problems for ordinary differential equations ode s is well developed however the integration methods are often used as variable stepsize variable formula methods vsvfm s in the codes for automatic solution of ode s this fact explains why it is necessary to develop a rigorous theory for the vsvfm s also results obtained until now in the efforts to develop a theory for the vsvfm s are presented in a systematic way and compared with the well known results for cscfm s the comparison is used to outline the directions in which further efforts in the attempt to improve the theoretical results and to obtain new results are most needed the integration methods can be separated into two large groups implicit methods and explicit methods including here the predictor corrector schemes it is discussed what kind of results are needed for each of these two groups assume that a set f of cscfm s is given from a practical point of view it is highly desirable that the cscfm s belonging to set f have the following property if every cscfm from set f is consistent zero stable and convergent then any vsvfm induced by the cscfm s of set f is also consistent zero stable and convergent for a sufficiently large class of grids a theorem concerning a class of cscfm s with this property is presented the application of this theorem in the selection of methods for solving large systems of ode s arising after the space discretization of air pollution models is discussed it is explained why the vsvfm s based on this class provide a very efficient computational process in the treatment of air pollution models

disintegration of measure techniques for commuting multivariable weighted shifts we employ techniques from the theory of disintegration of measures to study the lifting problem for commuting n tuples of subnormal weighted shifts we obtain a new necessary condition for the existence of a lifting and generate new pathology associated with bringing together the berger measures associated to each individual weighted shift for subnormal 2 variable weighted shifts we then find the precise relation between the berger measure of the pair and the berger measures of the shifts associated to horizontal rows and vertical columns of weights

central invariants and higher indicators for semisimple quasi hopf algebras in this paper we define the higher frobenius schur fs indicators for finite dimensional modules of a semisimple quasi hopf algebra h via the categorical counterpart developed in a 2005 preprint when h is an ordinary hopf algebra we show that our definition coincides with that introduced by kashina sommerhauser and zhu we find a sequence of gauge invariant central elements of h such that the higher fs indicators of a module v are obtained by applying its character to these elements as an application we show that fs indicators are sufficient to distinguish the four gauge equivalence classes of semisimple quasi hopf algebras of dimension eight corresponding to the four fusion categories with certain fusion rules classified by tambara and yamagami three of these categories correspond to well known hopf algebras and we explicitly construct a quasi hopf algebra corresponding to the fourth one using the kac algebra we also derive explicit formulae for fs indicators for some quasi hopf algebras associated to group cocycles