on group theoretical hopf algebras and exact factorizations of finite groups abstract we show that a semisimple hopf algebra a is group theoretical if and only if its drinfeld double is a twisting of the dijkgraaf pasquier roche quasi hopf algebra d for some finite group and some z 3 k we show that semisimple hopf algebras obtained as bicrossed products from an exact factorization of a finite group are group theoretical we also describe their drinfeld double as a twisting of d for an appropriate 3 cocycle coming from the kac exact sequence

certified reduced basis method for electromagnetic scattering and radar cross section estimation we study nontrivial applications of the reduced basis method rbm for electromagnetic applications with an emphasis on scattering and the estimation of radar cross section rcs the method and several extensions are explained with two examples with different characteristics parameters that are allowed to vary within the model include frequency incident angle and measurement angle as well as the geometry of the scatterers with appropriate applications of the empirical interpolation method eim transformation of the domain configuration of perfectly matched layer exponential convergence of the reduced basis solution over the entire parameter domain is achieved moreover we demonstrate that this approach allows for the effective capture of the critical behavior in this case through shapes that minimize scattering this further highlights the robustness and quality of the greedy approximation and the reduced basis method approach

natural norm a posteriori error estimators for reduced basis approximations we present a technique for the rapid and reliable prediction of linear functional outputs of coercive and noncoercive linear elliptic partial differential equations with affine parameter dependence the essential components are i rapidly convergent global reduced basis approximations galerkin projection onto a space wn spanned by solutions of the governing partial differential equation at n judiciously selected points in parameter space ii a posteriori error estimation relaxations of the error residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest and iii offiine online computational procedures methods which decouple the generation and projection stages of the approximation process the operation count for the online stage in which given a new parameter value we calculate the output of interest and associated error bound depends only on n typically very small and the parametric complexity of the problem in this paper we propose a new natural norm formulation for our reduced basis error estimation framework that a greatly simplifies and improves our inf sup lower bound construction offline and evaluation online a critical ingredient of our a posteriori error estimators and b much better controls significantly sharpens our output error bounds in particular through deflation for parameter values corresponding to nearly singular solution behavior we apply the method to two illustrative problems a coercive laplacian heat conduction problem which becomes singular as the heat transfer coefficient tends to zero and a non coercive helmholtz acoustics problem which becomes singular as we approach resonance in both cases we observe very economical and sharp construction of the requisite natural norm inf sup lower bound rapid convergence of the reduced basis approximation reasonable effectivities even for near singular behavior for our deflated output error estimators and significant several order of magnitude online computational savings relative to standard finite element procedures

on semisimple hopf algebras of dimension 2q3 abstract let q be a prime number k an algebraically closed field of characteristic 0 and h a semisimple hopf algebra of dimension 2 q 3 this paper proves that h is always semisolvable that is such hopf algebras can be obtained by a number of extensions from group algebras or duals of group algebras

a jordan holder theorem for weakly group theoretical fusion categories we prove a version of the jordan holder theorem in the context of weakly group theoretical fusion categories this allows us to introduce the composition factors and the length of such a fusion category mathcal c which are in fact morita invariants of mathcal c

lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups we classify lagrangian subcategories of the representation category of a twisted quantum double d g where g is a finite group and is a 3 cocycle on it in view of results of dgno this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups we also establish a canonical bijection between lagrangian subcategories of rep d g and module categories over the category rm vec g omega of twisted g graded vector spaces such that the dual tensor category is pointed this can be viewed as a quantum version of v drinfeld s characterization of homogeneous spaces of a poisson lie group in terms of lagrangian subalgebras of the double of its lie bialgebra d as a consequence we obtain that two group theoretical fusion categories are weakly morita equivalent if and only if their centers are equivalent as braided tensor categories

estimation of the error in the reduced basis method solution of nonlinear equations the reduced basis method is a projection technique for approximating the solution curve of a finite system of nonlinear algebraic equations by the solution curve of a related system that is typically of much lower dimension in this paper the reduced basis error is shown to be dominated by an approximation error this in turn leads to error estimates for projection onto specific subspaces for example subspaces related to taylor lagrange and discrete least squares approximation

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

hopf bimodules coquasibialgebras and an exact sequence of kac abstract based on the ideas of tannaka krein reconstruction we present a categorical construction that assigns to any cleft hopf algebra inclusion k h a coquasibialgebra having k as a hopf subalgebra as a special case the construction gives an intrinsic connection between the bismash product k q and the double cross product q k constructed from the same combinatorial data a cocommutative coquasibialgebra is the same as a cocommutative bialgebra equipped with a sweedler three cocycle thus our construction assigns to every bicrossproduct or hopf algebra extension of a commutative and a cocommutative factor a corresponding cocommutative double crossproduct equipped with a sweedler three cocycle based on this observation we use the construction to prove generalizations of kac s exact sequence for the group of hopf algebra extensions of a group algebra by a dual group algebra

on a symmetry of muger s centralizer for the drinfeld double of a semisimple hopf algebra in this paper we prove a formula that relates muger s centralizer in the category of representations of a factorizable hopf algebra to the notion of hopf kernel of a representation of the dual hopf algebra using this relation we obtain a complete description for muger s centralizer of some fusion subcategories of the fusion category of finite dimensional representations of a drinfeld double of a semisimple hopf algebra

certified reduced basis methods and output bounds for the harmonic maxwell s equations we propose certified reduced basis methods for the efficient and reliable evaluation of a general output that is implicitly connected to a given parameterized input through the harmonic maxwell s equations the truth approximation and the development of the reduced basis through a greedy approach is based on a discontinuous galerkin approximation of the linear partial differential equation the formulation allows the use of different approximation spaces for solving the primal and the dual truth approximation problems to respect the characteristics of both problem types leading to an overall reduction in the off line computational effort the main features of the method are the following i rapid convergence on the entire representative set of parameters ii rigorous a posteriori error estimators for the output and iii a parameter independent off line phase and a computationally very efficient on line phase to enable the rapid solution of many query problems arising in control optimization and design the versatility and performance of this approach is shown through a numerical experiment illustrating the modeling of material variations and problems with resonant behavior

finite group factorizations and braiding we compute the quantum double braiding and other canonical hopf algebra constructions for the bicrossproduct hopf algebrahassociated to the factorization of a finite groupxinto two subgroups the representations of the quantum double are described by a notion of bicrossed bimodules generalising the cross modules of whitehead we also show that basis preserving self duality structures for the bicrossproduct hopf algebras are in one one correspondence with factor reversing group isomorphisms the example z6z6is given in detail we show further that the quantum doubled h is the twisting ofd x by a nontrivial quantum cocycle

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

an explicit fusion algebra isomorphism for twisted quantum doubles of finite groups we exhibit an isomorphism between the fusion algebra of the quantum double of an extraspecial p group where p is an odd prime and the fusion algebra of a twisted quantum double of an elementary abelian group of the same order

probabilistic analysis of regularization in order to use interpolated data wisely it is important to have reliability and confidence measures associated with it a method for computing the reliability at each point of any linear functional of a surface reconstructed using regularization is presented the proposed method is to define a probability structure on the class of possible objects and compute the variance of the corresponding random variable this variance is a natural measure for uncertainty and experiments have shown it to correlate well with reality the probability distribution used is based on the boltzmann distribution the theoretical part of the work utilizes tools from classical analysis functional analysis and measure theory on function spaces the theory was tested and applied to real depth images it was also applied to formalize a paradigm of optimal sampling which was successfully tested on real depth images

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

on the gauge equivalence of twisted quantum doubles of elementary abelian and extra special 2 groups abstract we establish braided tensor equivalences among module categories over the twisted quantum double of a finite group defined by an extension of a group g by an abelian group with 3 cocycle inflated from a 3 cocycle on g we also prove that the canonical ribbon structure of the module category of any twisted quantum double of a finite group is preserved by braided tensor equivalences we give two main applications first if g is an extra special 2 group of width at least 2 we show that the quantum double of g twisted by a 3 cocycle is gauge equivalent to a twisted quantum double of an elementary abelian 2 group if and only if 2 is trivial second we discuss the gauge equivalence classes of twisted quantum doubles of groups of order 8 and classify the braided tensor equivalence classes of these quasi triangular quasi bialgebras it turns out that there are exactly 20 such equivalence classes

reduced basis decomposition dimension reduction is often needed in the area of data mining the goal of these methods is to map the given high dimensional data into a low dimensional space preserving certain properties of the initial data there are two kinds of techniques for this purpose the first projective methods builds an explicit linear projection from the high dimensional space to the low dimensional one on the other hand the nonlinear methods utilizes nonlinear and implicit mapping between the two spaces in both cases the methods considered in literature have usually relied on computationally very intensive matrix factorizations frequently the singular value decomposition svd the computational burden of svd quickly renders these dimension reduction methods infeasible thanks to the ever increasing sizes of the practical datasets in this paper we present a new decomposition strategy reduced basis decomposition rbd which is inspired by the reduced basis method rbm given x the high dimensional data the method approximates it by y t x with y being the low dimensional surrogate and t the transformation matrix y is obtained through a greedy algorithm thus extremely efficient in fact it is significantly faster than svd with comparable accuracy t can be computed on the fly moreover unlike many compression algorithms it easily finds the mapping for an arbitrary out of sample vector and it comes with an error indicator certifying the accuracy of the compression numerical results are shown validating these claims

exact factorizations and extensions of fusion categories abstract we introduce and study the new notion of an exact factorization b a c of a fusion category b into a product of two fusion subcategories a c b of b this is a categorical generalization of the well known notion of an exact factorization of a finite group into a product of two subgroups we then relate exact factorizations of fusion categories to exact sequences of fusion categories with respect to an indecomposable module category which was introduced and studied in 7 we also apply our results to study extensions of a group theoretical fusion category by another one provide some examples and propose a few natural questions

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

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

frobenius schur indicators in tambara yamagami categories abstract we introduce formulae of frobenius schur indicators of simple objects of tambara yamagami categories by using techniques of the fourier transform on finite abelian groups we study some arithmetic properties of indicators we also give some applications to hopf algebras

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h