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

lmi based robust control of fractional order uncertain linear systems this paper concerns with the method of observer based control and static output feedback control for fractional order uncertain systems with the fractional commensurate order a 0 a 1 and a email protected email protected 2 via linear matrix inequality lmi approach respectively first the sufficient conditions for robust asymptotical stability of the closed loop control systems are presented next by using matrix s singular value decomposition svd and lmi technics the existence condition and method of designing a robust stabilizing controller for such fractional order control systems are derived unlike previous methods the results are obtained in terms of lmi which can be easily obtained by matlab s lmi toolbox finally two numerical examples demonstrate the validity of this approach

topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

mappin class group actions on quantum doubles we study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non semisimple hopf algebra that arise in the construction of universal extended topological field theories we discuss how for doubles the degeneracy problem of tqft s is circumvented we find compact formulae for thes 1 matrices using the canonical non degenerate forms of hopf algebras and the bicrossed structure of doubles rather than monodromy matrices a rigorous proof of the modular relations and the computation of the projective phases is supplied using radford s relations between the canonical forms and the moduli of integrals we analyze the projectivesl 2 z action on the center ofuq sl2 forq anl 2m 1st root of unity it appears that the 3m 1 dimensional representation decomposes into anm 1 dimensional finite representation and a2m dimensional irreducible representation the latter is the tensor product of the two dimensional standard representation ofsl 2 z and the finite m dimensional representation obtained from the truncated tqft of the semisimplified representation category ofuq sl2

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

on nonsymmetric saddle point matrices that allow conjugate gradient iterations linear systems in saddle point form are usually highly indefinite which often slows down iterative solvers such as krylov subspace methods it has been noted by several authors that negating the second block row of a symmetric indefinite saddle point matrix leads to a nonsymmetric matrix mathcal a whose spectrum is entirely contained in the right half plane in this paper we study conditions so that mathcal a is diagonalizable with a real and positive spectrum these conditions are based on necessary and sufficient conditions for positive definiteness of a certain bilinear form with respect to which mathcal a is symmetric in case the latter conditions are satisfied there exists a well defined conjugate gradient cg method for solving linear systems with mathcal a we give an efficient implementation of this method discuss practical issues such as error bounds and present numerical experiments

technical communique fractional order proportional derivative controller for a class of fractional order systems recently fractional order systems fos have attracted more and more attention in various fields but the control design techniques available for the fos suffer from the lack of direct systematic approaches in this paper we focus on a given type of simple model of fos a fractional order proportional derivative fo pd controller is proposed for this class of fos and a practical and systematic tuning procedure has been developed for the proposed fo pd controller synthesis the fairness issue in comparing with other controllers such as the traditional integer order pid io pid controller and the fractional order proportional derivative fo pd controller has been addressed under the same number of design parameters and the same specifications fair comparisons of the three controllers i e io pid fo pd and fo pd via the simulation tests illustrate that the io pid controller designed may not always be stabilizing to achieve flat phase specification while both fo pd and fo pd controllers designed are always stabilizing furthermore the proposed fo pd controller outperforms fo pd controller for the class of fractional order systems

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

nodal auxiliary space preconditioning in h curl and h div spaces in this paper we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of h curl and h div elliptic variational problems the preconditioners exclusively rely on solvers for discrete poisson problems we prove mesh independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning the main tools are discrete analogues of so called regular decomposition results in the function spaces h curl and h div our preconditioner for h curl is similar to an algorithm proposed in r beck algebraic multigrid by component splitting for edge elements on simplicial triangulations tech rep sc 99 40 zib berlin germany 1999

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

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

bridged links and tangle presentations of cobordism categories abstract we develop a calculus of surgery data called bridged links which involves besides links also pairs of balls that describe one handle attachments as opposed to the usual link calculi of kirby and others this description uses only elementary local moves namely modifications and isolated cancellations and it is valid also on non simply connected and disconnected manifolds in particular it allows us to give a presentation of a 3 manifold by doing surgery on any other 3 manifold with the same boundary bridged link presentations on unions of handlebodies are used to give a cerf theoretical derivation of presentations of 2 1 dimensional cobordisms categories in terms of planar ribbon tangles and their composition rules as an application we give a different more natural proof of the matveev polyak presentations of the mapping class groups and furthermore find systematically surgery presentations of general mapping tori we discuss a natural extension of the reshetikhin turaev invariant to the calculus of bridged links invariance follows now similar as for knot invariants from simple identifications of the elementary moves with elementary categorical relations for invariances or cointegrals respectively hence we avoid the lengthy computations and the unnatural fenn rourke reduction of the original proofs moreover we are able to start from a much weaker modularity condition which implies the one of turaev generalizations of the presentation to cobordisms of surfaces with boundaries are outlined

mixed constraint preconditioners for the iterative solution of fe coupled consolidation equations the finite element fe integration of the coupled consolidation equations requires the solution of linear symmetric systems with an indefinite saddle point coefficient matrix because of ill conditioning the repeated solution in time of the fe equations may be a major computational issue requiring ad hoc preconditioning strategies to guarantee the efficient convergence of krylov subspace methods in the present paper a mixed constraint preconditioner mcp is developed combining implicit and explicit approximations of the inverse of the structural sub matrix with the performance investigated in some representative examples an upper bound of the eigenvalue distance from unity is theoretically provided in order to give practical indications on how to improve the preconditioner the mcp is efficiently implemented into a krylov subspace method with the performance obtained in 2d and 3d examples compared to that of inexact constraint preconditioners and least square logarithm scaled ilut preconditioners two variants of mcp t mcp and d mcp developed with the aim at reducing the cost of the preconditioner application are also tested the results show that the mcp variants constitute a reliable and robust approach for the efficient solution of realistic coupled consolidation fe models and especially so in severely ill conditioned problems

iterative learning control with initial state learning for fractional order nonlinear systems this paper presents a p type iterative learning control ilc scheme with initial state learning for a class of a 0 a 1 fractional order nonlinear systems by introducing the l norm and using a generalized gronwall inequality the sufficient condition for the robust convergence of the tracking errors with respect to initial positioning errors under p type ilc is obtained based on this convergence condition the learning gain of the initial learning and input learning updating law can be determined unlike the existing methods the ilc scheme will not fix the initial value on the expected condition at the beginning of each iteration finally the validity of the methods are verified by a numerical example

galois theory for braided tensor categories and the modular closure abstract given a braided tensor category c with conjugate dual objects and irreducible unit together with a full symmetric subcategory s we define a crossed product c s this construction yields a tensor category with conjugates and an irreducible unit a category is a category enriched over vect c with positive operation a galois correspondence is established between intermediate categories sitting between c and c s and closed subgroups of the galois group gal c s c aut c c s of c the latter being isomorphic to the compact group associated with s by the duality theorem of doplicher and roberts denoting by d c the full subcategory of degenerate objects i e objects which have trivial monodromy with all objects of c the braiding of c extends to a braiding of c s iff s d under this condition c s has no non trivial degenerate objects iff s d if the original category c is rational i e has only finitely many isomorphism classes of irreducible objects then the same holds for the new one the category c c d is called the modular closure of c since in the rational case it is modular i e gives rise to a unitary representation of the modular group sl 2 z if all simple objects of s have dimension one the structure of the category c s can be clarified quite explicitly in terms of group cohomology

fermionic modular categories and the 16 fold way we study spin and super modular categories systematically as inspired by fermionic topological phases of matter which are always fermion parity enriched and modelled by spin topological quantum field theories at low energy we formulate a 16 fold way conjecture for the minimal modular extensions of super modular categories to spin modular categories which is a categorical formulation of gauging the fermion parity we investigate general properties of super modular categories such as fermions in twisted drinfeld doubles verlinde formulas for naive quotients and explicit extensions of psu 2 4m 2 with an eye towards a classification of the low rank cases

conjugate gradient for nonsingular saddle point systems with a maximally rank deficient leading block abstract we consider iterative solvers for large sparse symmetric linear systems with a saddle point structure since such systems are indefinite the conjugate gradient cg method is not naturally designed for solving them however in the case of a maximally rank deficient leading block we prove that there are two sufficient conditions that allow for cg to be used we show that the conditions are satisfied for a model time harmonic maxwell problem to support our analysis we present several numerical experiments for three dimensional problems on complicated computational domains with constant and variable coefficients

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

fractional order nonlinear systems with delay in iterative learning control in this paper we discuss a p type iterative learning control ilc scheme for a class of fractional order nonlinear systems with delay by introducing the norm and using gronwall inequality the sufficient condition for the robust convergence of the tracking errors is obtained based on this convergence the p type ilc updating laws can be determined

on the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths abstract in this paper a new class of stochastic impulsive differential equations involving bernoulli distribution is introduced for tracking the random discontinuous trajectory a modified tracking error associated with a piecewise continuous variable by zero order holder is defined in the sequel a new random ilc scheme by adopting global and local iteration average operators is designed too sufficient conditions to guarantee the convergence of modified tracking error are obtained by using the tools of mathematical analysis via an impulsive gronwall inequality finally two illustrative examples are presented to demonstrate the performance and the effectiveness of the averaging ilc scheme to track the random discontinuous trajectory

a mixed finite element method with exactly divergence free velocities for incompressible magnetohydrodynamics abstract we introduce and analyze a mixed finite element method for the numerical discretization of a stationary incompressible magnetohydrodynamics problem in two and three dimensions the velocity field is discretized using divergence conforming brezzi douglas marini bdm elements and the magnetic field is approximated by curl conforming nedelec elements the h 1 continuity of the velocity field is enforced by a dg approach a central feature of the method is that it produces exactly divergence free velocity approximations and captures the strongest magnetic singularities we prove that the energy norm error is convergent in the mesh size in general lipschitz polyhedra under minimal regularity assumptions and derive nearly optimal a priori error estimates for the two dimensional case we present a comprehensive set of numerical experiments which indicate optimal convergence of the proposed method for two dimensional as well as three dimensional problems