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

inverse shape and surface heat transfer coefficient identification in this paper an inverse geometric problem for the modified helmholtz equation arising in heat conduction in a fin is considered this problem which consists of determining an unknown inner boundary of an annular domain and possibly its surface heat transfer coefficient from one or two pairs of boundary cauchy data boundary temperature and heat flux is solved numerically using the meshless method of fundamental solutions mfs a nonlinear unconstrained minimisation of the objective function is regularised when noise is added to the input boundary data the stability of the numerical results is investigated for several test examples with respect to noise in the input data and various values of the regularisation parameters

bounds for blow up time in nonlinear parabolic problems abstract a first order differential inequality technique is used on suitably defined auxiliary functions to determine lower bounds for blow up time in initial boundary value problems for parabolic equations of the form u t div u grad u f u if blow up occurs in addition conditions which ensure that blow up occurs or does not occur are presented

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

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

crossed s matrices and fourier matrices for coxeter groups with automorphism abstract we study crossed s matrices for braided g crossed categories and reduce their computation to a submatrix of the de equivariantization we study the more general case of a category containing the symmetric category rep a z with a a finite cyclic group and z a such that z 2 1 we give two examples of such categories which enable us to recover the fourier matrix associated with the big family of unipotent characters of the dihedral groups with automorphism as well as the fourier matrix of the big family of unipotent characters of the ree group of type f 4 2

virasoro frames and their stabilizers for the e8 lattice type vertex operator algebra the concept of a framed vertex operator algebra fvoa is new cf dgh this article contributes to this theory with a full analysis of all virasoro frame stabilizers in v the important example of the e8 level 1 affine kac moody voa which is isomorphic to the lattice voa for the root lattice of e8 c we analyze the frame stabilizers both as abstract groups and as subgroups of aut v e8 c each frame stabilizer is a finite group contained in the normalizer of a 2b pure elementary abelian 2 group in aut v but is not usually a maximal finite subgroup of this normalizer in particular we prove that there are exactly five orbits for the action of aut v on the set of virasoro frames thus settling an open question about v in section 5 of dgh the results about the group structure of the frame stabilizers can be stated purely in terms of modular braided tensor categories so this article contributes also to this theory there are two main viewpoints in our analysis the first is the theory of codes lattices markings and the resulting groups of automorphisms the second is the theory of finite subgroups of lie groups we expect our methods to be applicable to the study of other fvoas and their frame stabilizers appendices present aspects of the theory of automorphism groups of voas in particular there is a general result of independent interest on embedding lattices into unimodular lattices so as to respect automorphism groups and definiteness

sl 2 z action for ribbon quasi hopf algebras abstract we study the universal hopf algebra l of majid and lyubashenko in the case that the underlying ribbon category is the category of representations of a finite dimensional ribbon quasi hopf algebra a we show that l a with coadjoint action and compute the hopf algebra structure morphisms of l in terms of the defining data of a we give explicitly the condition on a which makes rep a factorisable and compute lyubashenko s projective s l 2 z action on the centre of a in this case the point of this exercise is to provide the groundwork for the applications to ribbon categories arising in logarithmic conformal field theories in particular symplectic fermions and w p models and to test a conjectural non semisimple verlinde formula

bounds for blow up time for the heat equation under nonlinear boundary conditions a differential inequality technique is used to determine a lower bound on the blow up time for solutions to the heat equation subject to a nonlinear boundary condition when blow up of the solution does occur in addition a sufficient condition which implies that blow up does occur is determined

a meshless method for solving the cauchy problem in three dimensional elastostatics the application of the method of fundamental solutions to the cauchy problem in three dimensional isotropic linear elasticity is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularized by employing the first order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for both under and equally determined cauchy problems in a piece wise smooth geometry the convergence accuracy and stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

invariant rings through categories abstract we formulate a notion of geometric reductivity in an abstract categorical setting which we refer to as adequacy the main theorem states that the adequacy condition implies that the ring of invariants is finitely generated this result applies to the category of modules over a bialgebra the category of comodules over a bialgebra and the category of quasi coherent sheaves on an algebraic stack of finite type over an affine base

global existence and blow up phenomena for nonlinear divergence form parabolic equations with inhomogeneous neumann boundary conditions abstract in this paper we consider nonlinear divergence form parabolic equations with inhomogeneous neumann boundary conditions we establish respectively the conditions on nonlinearities to guarantee that u x t exists globally or blows up at some finite time if blow up occurs we obtain upper and lower bounds of the blow up time

blow up solutions for reaction diffusion equations with nonlocal boundary conditions abstract we devote to investigating the following nonlinear reaction diffusion equations with nonlocal boundary conditions u t u u k 1 t f u in d 0 t u k 2 t d g u d x on d 0 t u x 0 u 0 x 0 in d where d is a bounded convex region in r n n 2 and the boundary d is smooth by constructing some auxiliary functions and using differential inequality technique we derive that the solution blows up at some finite time moreover upper and lower bounds of the blow up time are obtained

on the classification of the grothendieck rings of non self dual modular categories abstract we develop a symbolic computational approach to classifying low rank modular fusion categories up to finite ambiguity by a generalized form of ocneanu rigidity due to etingof ostrik and nikshych it is enough to classify modular fusion algebras of a given rank that is to determine the possible grothendieck rings with modular realizations we use this technique to classify modular categories of rank at most 5 that are non self dual i e those for which some object is not isomorphic to its dual object

regularized method of fundamental solutions for boundary identification in two dimensional isotropic linear elasticity we investigate the stable numerical reconstruction of an unknown portion of the boundary of a two dimensional domain occupied by an isotropic linear elastic material from a prescribed boundary condition on this part of the boundary and additional displacement and traction measurements i e cauchy data on the remaining known portion of the boundary this inverse geometric problem is approached by combining the method of fundamental solutions mfs and the tikhonov regularization method whilst the optimal value of the regularization parameter is chosen according to the discrepancy principle various geometries are considered i e convex and non convex domains with a smooth or piecewise smooth boundary in order to show the numerical stability convergence consistency and computational efficiency of the proposed method

a spin decomposition of the verlinde formulas for type a modular categories a modular category is a braided category with some additional algebraic features the interest of this concept is that it provides a topological quantum field theory in dimension 3 the verlinde formulas associated with a modular category are the dimensions of the tqft modules we discuss reductions and refinements of these formulas for modular categories related with su n our main result is a splitting of the verlinde formula corresponding to a brick decomposition of the tqft modules whose summands are indexed by spin structures modulo an even integer we introduce here the notion of a spin modular category and give the proof of the decomposition theorem in this general context

blow up phenomena for some nonlinear parabolic problems abstract we consider the blow up of solutions of equations of the form u t div u 2 grad u f u by means of a differential inequality technique a lower bound for blow up time is determined if blow up does occur as well as a criterion for blow up and conditions which ensure that blow up cannot occur

global and blow up solutions for nonlinear parabolic equations with robin boundary conditions abstract in this paper we discuss the blow up for classical solutions to the following class of parabolic equations with robin boundary condition b u t g u u f u in 0 t u n u 0 on 0 t u x 0 h x 0 in where is a bounded domain of r n n 2 with smooth boundary by constructing some appropriate auxiliary functions and using a first order differential inequality technique we derive conditions on the data which guarantee the blow up or the global existence of the solution for the blow up solution a lower bound on blow up time is also obtained moreover some examples are presented to illustrate the applications

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

an adaptive greedy technique for inverse boundary determination problem in this paper the method of fundamental solutions mfs is employed for determining an unknown portion of the boundary from the cauchy data specified on parts of the boundary we propose a new numerical method with adaptive placement of source points in the mfs to solve the inverse boundary determination problem since the mfs source points placement here is not trivial due to the unknown boundary we employ an adaptive technique to choose a sub optimal arrangement of source points on various fictitious boundaries afterwards the standard tikhonov regularization method is used to solve ill conditional matrix equation while the regularization parameter is chosen by the l curve criterion the numerical studies of both open and closed fictitious boundaries are considered it is shown that the proposed method is effective and stable even for data with relatively high noise levels

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

z 2z extensions of hopf algebra module categories by their base categories abstract starting with a self dual hopf algebra h in a braided monoidal category s we construct a z 2 z graded monoidal category c c 0 c 1 the degree zero component is the category rep s h of representations of h and the degree one component is the category s the extra structure on h needed to define the associativity isomorphisms is a choice of self duality map and cointegral subject to certain conditions we also describe rigid braided and ribbon structures on c in hopf algebraic terms our construction permits a uniform treatment of tambara yamagami categories and categories related to symplectic fermions in conformal field theory

bem first order regularisation method in linear elasticity for boundary identification in this paper we study the identification of an unknown portion of the boundary of a two dimensional domain occupied by an isotropic linear elastic material from additional cauchy data on the remaining known portion of the boundary this inverse problem is approached using the boundary element method bem in conjunction with the tikhonov first order regularisation procedure the choice of the regularisation parameter is based on the l curve criterion the numerical results obtained show that the proposed method produces a convergent and stable solution

the method of fundamental solutions for the cauchy problem in two dimensional linear elasticity in this paper the application of the method of fundamental solutions to the cauchy problem in two dimensional isotropic linear elasticity is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularised by employing the first order tikhonov functional while the choice of the regularisation parameter is based on the l curve method numerical results are presented for both smooth and piecewise smooth geometries as well as for constant and linear stress states the convergence and the stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

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

meshless methods for the inverse problem related to the determination of elastoplastic properties from the torsional experiment abstract the problem of determining the elastoplastic properties of a prismatic bar from the given experimental relation between the torsional moment m and the angle of twist per unit length of the rod s length is investigated as an inverse problem the proposed method to solve the inverse problem is based on the solution of some sequences of the direct problem by applying the levenberg marquardt iteration method in the direct problem these properties are known and the torsional moment is calculated as a function of the angle of twist from the solution of a non linear boundary value problem this non linear problem results from the saint venant displacement assumption the ramberg osgood constitutive equation and the deformation theory of plasticity for the stress strain relation to solve the direct problem in each iteration step the kansa method is used for the circular cross section of the rod or the method of fundamental solutions mfs and the method of particular solutions mps are used for the prismatic cross section of the rod the non linear torsion problem in the plastic region is solved using the picard iteration

blow up analysis in quasilinear reaction diffusion problems with weighted nonlocal source abstract in this paper we consider the blow up of solutions to a class of quasilinear reaction diffusion problems g u t u 2 u a x f u in 0 t u u 0 on 0 t u x 0 u 0 x in where is a bounded convex domain in r n n 2 weighted nonlocal source satisfies a x f u x t a 1 a 2 u x t p u x t l d x m and a 1 a 2 p l and m are positive constants by utilizing a differential inequality technique and maximum principles we establish conditions to guarantee that the solution remains global or blows up in a finite time moreover an upper and a lower bound for blow up time are derived furthermore two examples are given to illustrate the applications of obtained results

method of fundamental solutions with optimal regularization techniques for the cauchy problem of the laplace equation with singular points the purpose of this study is to propose a high accuracy and fast numerical method for the cauchy problem of the laplace equation our problem is directly discretized by the method of fundamental solutions mfs the tikhonov regularization method stabilizes a numerical solution of the problem for given cauchy data with high noises the accuracy of the numerical solution depends on a regularization parameter of the tikhonov regularization technique and some parameters of the mfs the l curve determines a suitable regularization parameter for obtaining an accurate solution numerical experiments show that such a suitable regularization parameter coincides with the optimal one moreover a better choice of the parameters of the mfs is numerically observed it is noteworthy that a problem whose solution has singular points can successfully be solved it is concluded that the numerical method proposed in this paper is effective for a problem with an irregular domain singular points and the cauchy data with high noises

non degeneracy conditions for braided finite tensor categories abstract for a braided finite tensor category c with unit object 1 c lyubashenko considered a certain hopf algebra f c endowed with a hopf pairing f f 1 to define the notion of a non semisimple modular tensor category we say that c is non degenerate if the hopf pairing is non degenerate in this paper we show that c is non degenerate if and only if it is factorizable in the sense of etingof nikshych and ostrik if and only if its muger center is trivial if and only if the linear map hom c 1 f hom c f 1 induced by the pairing is invertible as an application we prove that the category of yetter drinfeld modules over a hopf algebra in c is non degenerate if and only if c is

jones witten invariants for nonsimply connected lie groups and the geometry of the weyl alcove abstract the quotient process of muger and bruguieres is used to construct modular categories and tqfts out of closed subsets of the weyl alcove of a simple lie algebra in particular it is determined at which levels closed subsets associated to nonsimply connected groups lead to tqfts many of these tqfts are shown to decompose into a tensor product of tqfts coming from smaller subsets the prime subsets among these are classified and apart from some giving tqfts depending on homology as described by murakami ohtsuki and okada they are shown to be in one to one correspondence with the tqfts predicted by dijkgraaf and witten to be associated to chern simons theory with a nonsimply connected lie group thus in particular a rigorous construction of the dijkgraaf witten tqfts is given as a byproduct a purely quantum groups proof of the modularity of the full weyl alcove for arbitrary quantum groups at arbitrary levels is given

correspondences of ribbon categories much of algebra and representation theory can be formulated in the general framework of tensor categories the aim of this paper is to further develop this theory for braided tensor categories several results are established that do not have a substantial counterpart for symmetric tensor categories in particular we exhibit various equivalences involving categories of modules over algebras in ribbon categories finally we establish a correspondence of ribbon categories that can be applied to and is in fact motivated by the coset construction in conformal quantum field theory

the small quantum group as a quantum double we prove that the quantum double of the quasi hopf algebra aq g of dimension ndimg attached in p etingof s gelaki on radically graded finite dimensional quasi hopf algebras mosc math j 5 2 2005 371 378 to a simple complex lie algebra g and a primitive root of unity q of order n2 is equivalent to lusztig s small quantum group uq g under some conditions on n we also give a conceptual construction of aq g using the notion of de equivariantization of tensor categories

dichromatic state sum models for four manifolds from pivotal functors a family of invariants of smooth oriented four dimensional manifolds is defined via handle decompositions and the kirby calculus of framed link diagrams the invariants are parametrised by a pivotal functor from a spherical fusion category into a ribbon fusion category a state sum formula for the invariant is constructed via the chain mail procedure so a large class of topological state sum models can be expressed as link invariants most prominently the crane yetter state sum over an arbitrary ribbon fusion category is recovered including the nonmodular case it is shown that the crane yetter invariant for nonmodular categories is stronger than signature and euler invariant a special case is the four dimensional untwisted dijkgraaf witten model derivations of state space dimensions of tqfts arising from the state sum model agree with recent calculations of ground state degeneracies in walker wang models relations to different approaches to quantum gravity such as cartan geometry and teleparallel gravity are also discussed

blow up phenomena in parabolic problems with time dependent coefficients under neumann boundary conditions this paper deals with the blow up of solutions to a class of parabolic problems with time dependent coefficients under homogeneous neumann boundary conditions for one set of problems in this class we show that no global solution can exist for another we derive lower bounds for the time of blow up when blow up occurs

blow up of solutions for nonlinear parabolic equation with nonlocal source and nonlocal boundary condition this paper is devoted to a nonlinear parabolic equation subject to nonlocal boundary condition and with nonlocal reaction source some criteria on this problem which determine whether the solutions blow up in a finite time or exist globally are given moreover the blow up rate estimates are also obtained