wavy taylor vortex flows via multigrid continuation methods abstract we present a multigrid continuation method for computing the transition from two dimensional taylor vortex to three dimensional wavy vortex flow in the taylor experiment the method is also used to continue the path of travelling wave solutions as the reynolds number is varied the steady navier stokes equations in a rotating frame of reference and an integral constraint are approximated by second order accurate finite differences for the convective terms in the conservation equations an upwind discretization controlled by the eigenvalues of their jacobians is used an iterative method for finding the speed of the travelling wave involves switching between the unsteady equations and the steady equations in a rotating frame this is used only once to get the first wavy vortex solution a comparison of the results with experimental and numerical data shows good accuracy for the method and the validity of the travelling wave formulation

twisted dual group algebras equivariant deformations of c0 g abstract we consider a family of twisted fourier algebras a g of a locally compact group g which in the case of a abelian group g are the fourier transforms of the usual twisted group algebras of over cap g o the corresponding c algebras c g o are deformations of c 0 g which are equivariant in the sense that g still acts by left translation the main examples come from cocycles on the dual of an abelian subgroup h of g we prove that for such cocycles the twisted dual group algebras c g o are induced from the twisted group algebras c h o and we give detailed formulas for the multiplication on a g which extend to larger dense subalgebras of c 0 g and c b g we anticipate that these larger subalgebras will be useful for constructing deformations of homogeneous spaces c 0 g

representations of the quantum group suq 2 and the little q jacobi polynomials abstract in this paper we study the finite dimensional unitary representations of the quantum group suq 2 then we obtain the peter weyl theorem for suq 2 and the matrix elements of these unitary representations are explicitly expressed in terms of the little q jacobi polynomials which are known as q analogues of orthogonal polynomials using these expressions the orthogonality relations of these polynomials are obtained in terms of the haar measure on the quantum group suq 2

deformations of compact quantum groups via rieffel s quantization it is shown that compact quantum groups containing torus subgroups can be deformed into new compact quantum groups under rieffel s quantization this is applied to showing that the two classes of compact quantum groupskqu andkq studied by levendorkii and soibelman are strict deformation quantization of each other and that the quantum groupsau m have many deformations

equivariant deformations of homogeneous spaces abstract c algebraic deformations of homogeneous spaces g are constructed by completing dense subspaces of c 0 g in a different multiplication and c norm these deformations are equivariant in the sense that they still carry a natural action of g by left translation the motivating examples are the noncommutative heisenberg manifolds of rieffel but the construction given here is based on the authors twisted dual group algebras which are equivariant deformations of c 0 g the procedure is general enough to give other recently studied families of noncommutative spaces and some interesting new examples of simple c algebras

the role of anomalous modes in taylor couette flow one of the classic examples of a fluid mechanical bifurcation is the appearance of taylor vortex flow in a viscous fluid contained between concentric rotating cylinders a natural consequence of the mathematical model due to d g schaeffer which has been most successful in describing this phenomenon in a finite system is the presence of anomalous modes in the steady solution set these are cellular flows that exist in contradiction to the commonly held belief that the radial flow close to the ends is necessarily directed towards the inner cylinder because of the presence of an ekman boundary layer to date anomalous modes have only been studied in a restricted parameter range but here we report the results of numerical and experimental studies that cover a wide range of parameter space it is shown that the lower stability limits for anomalous modes are removed to high reynolds numbers when the gap between the cylinders is reduced this perhaps explains why this class of flow remained undiscovered in over fifty years of study of the taylor couette problem until the work of t b benjamin we also consider the implications of these results for the organization of dynamical behaviour in the taylor couette system

k amenability of hnn extensions of amenable discrete quantum groups abstract we construct hnn extensions of discrete quantum groups we study their representation theory and we show that an hnn extension of amenable discrete quantum groups is k amenable

free products of compact quantum groups we construct and study compact quantum groups from free products ofc algebras in this connection we discover two mysterious classes of natural compact quantum groups a u m anda o m thea u m s respectivelya o m s are non isomorphic to each other for differentm s and are not obtainable by the ordinary quantization method we also clarify some basic concepts in the theory of compact quantum groups

a note on the galois correspondence for commutative rings introductionthe basic correspondence of galois theory for commutative rings de scribed in chapter v of the book by magid 8 is incorrect it relies on awxproperty of the category of profinite spaces lemma v 9 which unfortu nately does not hold in this note we show with an example that thisproperty is crucial for the correctness of the correspondence and weestablish the correct one corollary 5 freely using the work on general or abstract galois theory performed in the meantime by the second named author 2 5 whose first purpose was to generalize the results of chapterwxiv of 8 wxthe standard reference we use for general categorical concepts is thebook by mac lane 7 and for internal category theory the book bywxjohnstone 6 wx

approximation of a nondifferentiable nonlinear problem related to mhd equilibria we present a simple method based on a variant of the implicit function theorem which leads to the existence of a part of a nontrivial solution branch of the nonlinear eigenvalue problem u u in u 1 on where is a two dimensional domain with boundary the advantage of this method is that we can apply it for analysing the approximation of the above problem by a finite element method the error analysis of the discrete problem appears immediately we give also an iteration scheme which allows to solve the approximate problem

tannaka krein duality for compact matrix pseudogroups twisted su n groups the notion of concrete monoidalw category is introduced and investigated a generalization of the tannaka krein duality theorem is proved it leads to new examples of compact matrix pseudogroups among them we have twistedsu n groups denoted bys u n it is shown that the representation theory fors u n is similar to that ofsu n irreducible representations are labeled by young diagrams and formulae for dimensions and multiplicity are the same as in the classical case

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

deformation quantization for actions of qpd abstract the main objective of this article is to develop the theory of deformation of c algebras endowed with a group action from the perspective of non formal equivariant quantization this program initiated in 4 aims to extend rieffel s deformation theory 27 for more general groups than r d in 4 we have constructed such a theory for a class of non abelian lie groups in the present article we study the somehow opposite situation of abelian but non lie groups more specifically we construct here a deformation theory of c algebras endowed with an action of a finite dimensional vector space over a non archimedean local field of characteristic different from 2 at the root of our construction stands the p adic version of the weyl quantization introduced by haran 12 and further extended by bechata 1 and unterberger 34

the free wreath product of a compact quantum group by a quantum automorphism group in this thesis we define and study the free wreath product of a compact quantum group by a quantum automorphism group and in this way we generalize the previous notion of free wreath product by the quantum symmetric group introduced by bichon our investigation is divided into two part in the first we define the free wreath product of a discrete group by a quantum automorphism group we show how to describe its intertwiners by making use of decorated noncrossing partitions and from this thanks to a result of lemeux we deduce the irreducible representations and the fusion rules then we prove some properties of the operator algebras associated to this compact quantum group such as the simplicity of the reduced c algebra and the haagerup property of the von neumann algebra the second part is a generalization of the first one we start by defining the notion of free wreath product of a compact quantum group by a quantum automorphism group we generalize the description of the spaces of the intertwiners obtained in the discrete case and by adapting a monoidal equivalence result of lemeux and tarrago we find the irreducible representations and the fusion rules then we prove some stability properties of the free wreath product operation in particular we find under which conditions two free wreath products are monoidally equivalent or have isomorphic fusion semirings we also establish some analytic and algebraic properties of the dual quantum group and of the operator algebras associated to a free wreath product as a last result we prove that the free wreath product of two quantum automorphism groups can be seen as the quotient of a suitable quantum automorphism group

2 cocycles and twisting of kac algebras we describe the twisting construction with the help of 2 cocycles on hopf von neumann and george kac algebras we show that twisted kac algebras are again kac algebras using this construction we give a wide class of new quantizations of the heisenberg group and describe several series of non trivial finite dimensional hopf algebras kac algebras of dimensions 4n and as twisting of finite groups

centrifugal instabilities of circumferential flows in finite cylinders the wide gap problem the remarkable results obtained by benjamin 1978 who investigated experimentally the nature of taylor vortex flows in short cylinders have stimulated much theoretical work on the role of end effects in stability theory most of this work has been in connection with the rayleigh benard problem in finite rectangular or circular containers where the side wall conditions are simple enough for perturbation methods to be used the experiments of benjamin were performed in cylinders of variable length having end walls held fixed to the stationary outer cylinder in this case the basic state set up when the inner cylinder rotates is never close to the purely circumferential flow of the corresponding infinite problem thus perturbation methods cannot be used directly to investigate benjamin s problem but schaeffer 1980 has proposed a model problem having simpler end wall conditions the conditions used by schaeffer in fact correspond to porous end walls some predictions about benjamin s problem can be made by perturbing the end wall conditions of the model problem towards those of benjamin s apparatus schaeffer s analysis is not valid for most of the available experimental results which correspond to cylinders so short that only two or four taylor cells occur however the bifurcation picture in the length reynolds number plane obtained by benjamin for this configuration is probably similar to those appropriate to longer cylinders using qualitative methods and assuming certain numerical constants have the required behaviour schaeffer argues that benjamin s results are plausible in this paper we investigate quantitatively the model problem proposed by schaeffer using perturbation methods and determine explicit values for the numerical constants which are so crucial in determining the possible equilibrium configurations moreover the present formulation allows us to calculate the stability properties of these equilibrium configurations results are obtained for a wide range of possible ratios of cylinder diameters we also investigate the possibility of equilibrium configurations having an odd number of cells the results obtained are compared with the observations of benjamin

perturbed homoclinic solutions in reversible 1 1 resonance vector fields we consider a smooth reversible vector field in r 4 such that the origin is a fixed point the differential at the origin has two double pure imaginary eigenvalues iq for the critical value 0 of the parameter we show by a normal form analysis that the vector field can be approximated by an integrable field in r 4 for which we know all solutions specially interesting ones are the homoclinics to 0 and homoclinics to periodic solutions depending on the sign of a leading nonlinear coefficient we prove in particular the persistence of these homoclinics for the full vector field

onset of wavy vortices in taylor couette flow with imperfect reflection symmetry we reveal experimentally a mechanism that alters the critical behavior of a hopf bifurcation substantially due to the presence of imperfections of the reflection symmetry in a hydrodynamic system the onset of rotating waves in taylor vortex flow which is widely considered as a classical example for a hopf bifurcation in hydrodynamics is investigated primarily by transient response experiments while wavy vortex flow is not influenced by such unavoidable experimental imperfections the critical behavior of the axially subharmonic rotating wave with wavy outflow boundaries also called the small jet mode is qualitatively altered experimental evidence is provided that the modified critical behavior at the hopf bifurcation is associated with imperfections of the reflection symmetry of the taylor couette setup the experimental results on hopf bifurcation are discussed in the context of a cusp hopf bifurcation model recently proposed by harlim and langford int j bifurcation chaos appl sci eng 17 2547 2007 and compared to experimental results on imperfect pitchfork bifurcation in small aspect ratio taylor couette flow

a tannakian context for galois theory abstract strong similarities have been long observed between the galois and tannaka theories of the representation of groups in this paper we construct an explicit neutral tannakian context for the galois theory of atomic topoi and prove equivalence for the fundamental theorem since the theorem is known for the galois context this yields a proof of the fundamental recognition theorem for a new tannakian context this example is different from the additive cases or their generalization for which the theorem is known to hold and for which the unit of the tensor product is always an object of finite presentation which is not the case in our context

a numerical investigation of the schaeffer homotopy in the problem of taylor couette flows numerical methods are used to study the 4 6 cell exchange process in the taylor vortex problem with particular reference to the homotopy devised by schaeffer the homotopy describes a transformation between two models one incorporating periodic boundary conditions and so referring to flows in an infinite annulus the other with realistic boundary conditions our calculations indicate that the former model is more complicated than previously suspected and lead to a better understanding of the consequences of schaeffer s device

chaos from hopf bifurcation in a fluid flow experiment results of an experimental study of a hopf bifurcation with broken translation symmetry that organizes chaotic homoclinic dynamics from a t2 torus in a fluid flow as a direct consequence of physical boundaries are presented it is shown that the central features of the theory of hopf bifurcation in o 2 symmetric systems where the translation symmetry is broken are robust and are appropriate to describe the appearance of modulated waves homoclinic bifurcation takens bogdanov point and chaotic dynamics in a fluid flow experiment

pure galois theory in categories abstract galois theory j pure appl algebra 19 1980 21 42 2 m barr abstract galois theory ii j pure appl algebra 25 1982 227 247 3 m barr and r diaconescu on locally simply connected toposes and their fundamen tal groups cahiers topologie gt om diffprentielle catigoriques 22 i98 i 301 3 14 4 s u chase d k harrison and a rosenberg galois theory and cohomology of commutative rings mem amer mafh sot 52 1965 15 33 5 s u chase and m e sweedler hopf algebras and galois theory in lecture notes in math vol 97 springer verlag new york berlin 1969 6 i diers categories of boolean sheaves of simple algebras in lecture notes in math vol 1187 springer verlag new york berlin 1986 7 a grothendieck revetements itales et groupe fondamental sga 1 in lecture notes in math vol 269 springer verlag new york berlin 1972 8 j f kennison galois theory and theaters of action in a topos j pure appl algebra 18 1980 149 164 9

finite aspect ratio taylor couette flow shil nikov dynamics of 2 tori abstract the nonlinear dynamics of the flow in a short annular container driven by the rotation of the inner cylinder is studied using direct numerical simulations of the three dimensional navier stokes equations the basic state is s o 2 z 2 symmetric for aspect ratios between 3 6 and 4 4 we have located three codimension two bifurcations a cusp a double hopf and a fold hopf bifurcation all these local bifurcations are z 2 invariant the breaking of z 2 symmetry involves very complex shil nikov type dynamics not directly connected to any of the three codimension two bifurcations but associated with five unstable limit cycles and a wealth of heteroclinic connections between them period adding cascades both direct and reverse of 2 tori have been found narrow regions of chaotic dynamics are interspersed within these quasiperiodic solutions

hopf von neumann algebra bicrossproducts kac algebra bicrossproducts and the classical yang baxter equations two groups g1 g2 are said to be a matched pair if each acts on the space of the other and these actions say obey a certain compatibility condition for every matched pair of locally compact groups g1 g2 we construct an associated coinvolutive hopf von neumann algebra m g1 l g2 by simultaneous cross product and cross coproduct for non trivial these bicrossproduct hopf von neumann algebras are non commutative and non cocommutative if the modules for the actions are also matched then these bicrossproducts are kac algebras in this case we show that the dual kac algebra is of the same form with the roles of g1 g2 and of interchanged examples exist with g1 a simply connected lie group and choices of g2 determined by suitable solutions of the classical yang baxter equations on the complexification of the lie algebra of g1

boundary effects and the onset of taylor vortices it is well established that the onset of spatially periodic vortex states in the taylor couette flow between rotating cylinders occurs at the value of reynolds number predicted by local bifurcation theory however the symmetry breaking induced by the top and bottom plates means that the true situation should be a disconnected pitchfork indeed experiments have shown that the fold on the disconnected branch can occur at more than double the reynolds number of onset this leads to an apparent contradiction why should taylor vortices set in so sharply at the reynolds number predicted by the symmetric theory given such large symmetry breaking effects caused by the boundary conditions this paper offers a generic explanation the details are worked out using a swift hohenberg pattern formation model that shares the same qualitative features as the taylor couette flow onset occurs via a wall mode whose exponential tail penetrates further into the bulk of the domain as the driving parameter increases in a large domain of length l we show that the wall mode creates significant amplitude in the centre at parameter values that are o l 2 away from the value of onset in the problem with ideal boundary conditions we explain this as being due to a hamiltonian hopf bifurcation in space which occurs at the same parameter value as the pitchfork bifurcation of the temporal dynamics the disconnected anomalous branch remains o 1 away from the onset parameter since it does not arise as a bifurcation from the wall mode c 2003 elsevier b v all rights reserved

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

solutions for nonlinear convection in the presence of a lateral boundary abstract a method is described for calculating steady state patterns in large scale nonlinear systems taking into account the effect of a lateral boundary and without the need for extensive numerical calculations the key feature is the determination of the phase shift of the nonlinear periodic form distant from the boundary as a function of wavelength this is found by analyzing the solution close to the boundary where floquet theory is used to describe the departure of the solution from its periodic form for a restricted band of wavelengths lying within the eckhaus boundary dual solutions for the phase shift are found one of which corresponds to an unstable state results are presented here for the one dimensional swift hohenberg equation in a semi infinite domain but in principle the method can be applied to more general pattern forming systems the results are compared with the predictions of weakly nonlinear theory and with nonlinear computations on a large but finite domain

discretization errors at free boundaries of the grad schluter shafranov equation the numerical error of standard finite difference schemes is analyzed at free boundaries of the grad schluter shafranov equation of plasma physics a simple correction strategy is devised to eliminate to leading order the errors which arise as the free boundary crosses the rectangular grid at irregular locations the resulting scheme can be solved by gauss newton or inverse iterations or by multigrid iterations extrapolation from 2nd to 3rd order of accuracy is possible for the new scheme

metric aspects of noncommutative homogeneous spaces for a closed cocompact subgroup of a locally compact group g given a compact abelian subgroup k of g and a homomorphism k g satisfying certain conditions landstad and raeburn constructed equivariant noncommutative deformations c g of the homogeneous space g generalizing rieffel s construction of quantum heisenberg manifolds we show that when g is a lie group and g is connected given any norm on the lie algebra of g the seminorm on c g induced by the derivation map of the canonical g action defines a compact quantum metric furthermore it is shown that this compact quantum metric space depends on continuously with respect to quantum gromov hausdorff distances

traces on noncommutative homogeneous spaces abstract we study properties of c algebraic deformations of homogeneous spaces g which are equivariant in the sense that they preserve the natural action of g by left translation the center is shown to be isomorphic to c g g 0 for a certain subgroup g 0 of g and there is a 1 1 correspondence between normalised traces and probability measures on g g 0 this makes it possible to represent the deformed algebra as operators over l 2 g applications to k theory are also mentioned

quantum group of orientation preserving riemannian isometries we formulate a quantum group analogue of the group of orientation preserving riemannian isometries of a compact riemannian spin manifold more generally of a possibly r twisted and of compact type spectral triple the main advantage of this formulation which is directly in terms of the dirac operator is that it does not need the existence of any good laplacian as in our previous works on quantum isometry groups several interesting examples including those coming from rieffel type deformation as well as the equivariant spectral triples on su 2 and s c2 are discussed

galois theory in symmetric monoidal categories abstract the galois theory presented here is at a level of generality essentially between that of g janelidze d schumacher and r street 1993 appl categ structures 1 103 110 and g janelidze 1991 lecture notes in mathematics vol 1488 pp 157 173 however our purpose is to concentrate on symmetric monoidal categories and so provide a new approach to tannaka duality

noncommutative geometry and reality we introduce the notion of real structure in our spectral geometry this notion is motivated by atiyah s kr theory and by tomita s involution j it allows us to remove two unpleasant features of the connes lott description of the standard model namely the use of bivector potentials and the asymmetry in the poincare duality and in the unimodularity condition

an experimental observation of a new type of intermittency abstract a novel hysteretic form of regular intermittency has been observed experimentally in a variant of the taylor couette problem where the fluid flow is confined between a stationary outer cylinder and a rotating inner cylinder and end plates the results cannot all be explained within the standard pomeau manneville classification of intermittency and an extension to this theory is proposed to account for the observed phenomena a satisfactory agreement is found between experiment and theory both for the reinjection burst frequency scaling law and the influence of noise on the regularity of the bursts

deformation of a kac algebra by an abelian subgroup some examples of quantum groups in literature arise as deformations of a locally compact group by a dual 2 cocycle we make this construction in the framework of kac algebras we show that these deformations are still kac algebras using this construction we give new quantizations of the heisenberg group from this point of view we analyse the dimension 8 non trivial example of kac and paljutkin and give a new example of non trivial dimension 12 semi simple hopf algebras a dimension 12 kac algebra

krein duality for compact quantum groups we define noncommutative krein algebras and extend the classical krein duality for compact groups to compact quantum groups

rieffel type discrete deformation of finite quantum groups we introduce a discrete deformation of rieffel type for finite quantum groups using this we give an example of a finite quantum group a of order 18 such that neither a nor its dual can be expressed as a crossed product of the form c g1 g2 with g1 and g2 ordinary finite groups we also give a deformation of finite groups of lie type by using their maximal abelian subgroups

free products of c algebras small spatial free products of c algebras are constructed under certain conditions they have properties similar to those proved by paschke and salinas for the algebras c gi g2 where g1 g2 are discrete groups the freeproduct analogs of noncommutative bernoulli shifts are discussed 0 introduction let k be a field consider the category of unital algebras over k it is well known that this category admits coproducts free products of algebras 2 heuristically the free product of algebras is the algebra generated by them with no relations except for the identification of unit elements if k c the complex numbers and we consider unital algebras we can easily define a operation on the free products let a b be unital c algebras and a b their free product which is a unital algebra the question arises in what ways may one define a pre c norm on a b that extends the norms on a and b guided by analogy with tensor products we expect to have a choice among many pre c norms giving rise to many c free products of a and b one natural norm is 1c i i sup 1it c ih st representation of a b the representations of a b are in 1 1 correspondence with pairs of representations of a and b which act on the same hilbert space let a b be the completion of a b in this norm it is easy to see that this construction defines a coproduct in the category of c algebras and that a b is the biggest free product of a and b analogous to the biggest tensor product a 0 b if g 1 g2 are discrete groups we obtain c g1 c g2 c gi g2 where g1 g2 is the free product group and this is analogous to the relation c g1 0 c g2 c g1 x g2 this paper is motivated by the question is there a smallest c product a b in analogy to the smallest tensor product a b satisfying a relation cr gi g2 cr gi cr g2 received by the editors january 28 1981 and in revised form may 19 1981 1980 mathematics subject classification primary 46l05 secondary 46l55 1 982 american mathematical society 0002 9947 82 00001022 04 25

anomalous modes in the taylor experiment an experimental investigation and a theoretical appraisal are presented which enlarge upon previous observations of multiple steady flows in a taylor apparatus where the fluid filled annulus is comparatively short the aim of the experiments was a systematic exploration of all anomalous modes in this apparatus namely those stable flows that exist only at sufficiently high reynolds number r and are always distinct from the primary flow developed by gradually increasing r from small values having been established at high r an anomalous mode will always collapse eventually if r is then gradually reduced the theoretical material bearing on the interpretation of the experiments is reviewed in s 2 where mathematical details are largely deferred but several new conclusions are demonstrated in particular some curious facts are revealed about the limits of stability for anomalous modes comprising even numbers of taylor cells and about the properties of flows comprising a single cell only following a description of the apparatus in s 3 which includes an account of several improvements that have been introduced since the earlier experiments the new experimental results are reported in s 4 they include stability curves critical r against aspect ratio of annulus for anomalous modes consisting of from two to seven taylor cells s 4 1 observations on collapse routes for all these modes s 4 3 and measurements of various phenomena exhibited by the single cell mode s 4 5 these and other phenomena recorded here for the first time are seen to accord with the theoretical prospectus

the radial masa in free orthogonal quantum groups abstract we prove that the radial subalgebra in free orthogonal quantum group factors is maximal abelian and mixing and we compute the associated bimodule the proof relies on new properties of the jones wenzl projections and on an estimate of certain scalar products of coefficients of irreducible representations

nonlinear magnetohydrodynamics in axisymmetric heterogeneous domains using a fourier finite element technique and an interior penalty method the maxwell equations in the mhd limit in heterogeneous axisymmetric domains composed of conducting and non conducting regions are solved by using a mixed fourier lagrange finite element technique finite elements are used in the meridian plane and fourier modes are used in the azimuthal direction parallelization is made with respect to the fourier modes continuity conditions across interfaces are enforced using an interior penalty technique the performance of the method is illustrated on kinematic and full dynamo configurations

computation of anomalous modes in the taylor experiment abstract multigrid continuation methods are used to solve the steady axisymmetric incompressible navier stokes equations for taylor couette flows between cylinders of finite or infinite length using schaeffer s homotopy we compute for finite cylinders anomalous modes those seeming to have an odd number of vortices or seeming to have the wrong direction of rotation at an endplate the results show that these modes possess extra vortices not observed in the experiments of benjamin and mullin proc roy soc london ser a 377 1981 221 j fluid mech 121 1982 219 our computations verify conjectures of schaeffer and of benjamin and mullin on the unfolding of the bifurcation diagram as periodic infinite cylinder flows are continuously transformed to flows with rigid ends we obtain five distinct solutions with the same reynolds number aspect ratio and radius ratio and give a systematic procedure for obtaining them the numerical results show qualitative agreement with the experiments of benjamin and mullin they also show quantitative agreement with the more recent experiments of cliffe and mullin j fluid mech 153 1985 243

quantum groups acting on 4 points we classify the compact quantum groups acting on 4 points these are the quantum subgroups of the quantum permutation group q4 our main tool is a new presentation for the algebra c q4 corresponding to an isomorphism of type q4 so 1 3 the quantum subgroups of q4 are subject to a mckay type correspondence that we describe at the level of algebraic invariants

localic galois theory abstract in proposition i of memoire sur les conditions de resolubilite des equations par radicaux galois considers the splitting field a of a polynomial with rational coefficients given any intermediate extension x he proves that the action of the galois group gal a q on the set of morphisms x a is transitive and that x is the fixed field of its galois group gal a x in this article we first state and prove a dual categorical formulation of these statements which turns out to be a theorem about atomic sites with a representable point these abstract developments correspond exactly to classical galois theory we then consider the same situation but with the point no longer representable it determines a proobject and it becomes tautologically prorepresentable mutatis mutandis we state and prove the prorepresentable version of the categorical formulation of galois theorems in this case the classical group of automorphisms has to be replaced by the localic group of automorphisms these developments form the content of a theory that we call localic galois theory from our results it immediately follows the theorem a topos with a point is connected atomic if and only if it is the classifying topos of a localic group and this group can be taken to be the locale of automorphisms of the point the hard implication in this equivalence was first proved in print in joyal and tierney mem amer math soc 151 1984 theorem 1 section 3 chapter viii and it follows from a characterization of atomic topoi in terms of open maps and from a theory of descent for morphisms of topoi and locales our proof of this theorem is completely independent of descent theory and of any other result in joyal and tierney 1984 the theorem follows as a straightforward consequence of our direct generalization of the fundamental results of galois we explain and give the necessary definitions to understand this paper without previous knowledge of the theory of locales other than this our principal results theorems 2 2 and 2 3 require only elementary notions of category theory

on cocycle twisting of compact quantum groups abstract we provide a class of examples of compact quantum groups and unitary 2 cocycles on them such that the twisted quantum groups are non compact but still locally compact quantum groups in the sense of kustermans and vaes this also gives examples of cocycle twists where the underlying c algebra of the quantum group changes

the fusion rules of some free wreath product quantum groups and applications abstract in this article we find the fusion rules of the free wreath product quantum groups s n for any discrete group to do this we describe the spaces of intertwiners between basic corepresentations which allows us to identify the irreducible corepresentations we then apply the knowledge of the fusion rules to prove in most cases several operator algebraic properties of the associated reduced c algebras such as simplicity and uniqueness of the trace we also prove that the associated von neumann algebra is a full type ii 1 factor and that the dual of s n has the haagerup approximation property for all finite groups

a spectral numerical method for the navier stokes equations with applications to taylor couette flow a new spectral method for solving the incompressible navier stokes equations in a plane channel and between concentric cylinders is presented the method uses spectral expansions which inherently satisfy the boundary conditions and the continuity equation and yield banded matrices which are efficiently solved at each time step in addition the number of dependent variables is reduced resulting in a reduction in computer memory requirements several test problems have been computed for the channel flow and for flow between concentric cylinders including taylor couette flow with axisymmetric taylor vortices and wavy vortices in all cases agreement with available experimental and theoretical results is very good

a periodically forced flow displaying symmetry breaking via a three tori tori gluing birurcation and two tori resonances the dynamics due to a periodic forcing harmonic axial oscillations in a taylor couette apparatus of finite length is examined numerically in an axisymmetric subspace the forcing delays the onset of centrifugal instability and introduces a z2 symmetry that involves both space and time this paper examines the influence of this symmetry on the subsequent bifurcations and route to chaos in a one dimensional path through parameter space as the centrifugal instability is enhanced we have observed a well known route to chaos via frequency locking and torus break up on a two tori branch once the z2 symmetry has been broken however this branch is not connected in a simple manner to the z2 invariant primary branch an intermediate branch of three tori solutions exhibiting heteroclinic and homoclinic bifurcations provides the connection on this three tori branch a new gluing bifurcation of three tori is seen to play a central role in the symmetry breaking process 2001 elsevier science b v all rights reserved

the baum connes conjecture for free orthogonal quantum groups we prove an analogue of the baum connes conjecture for free orthogonal quantum groups more precisely we show that these quantum groups have a element and that 1 it follows that free orthogonal quantum groups are k amenable we compute explicitly their k theory and deduce in the unimodular case that the corresponding reduced c algebras do not contain nontrivial idempotents our approach is based on the reformulation of the baum connes conjecture by meyer and nest using the language of triangulated categories an important ingredient is the theory of monoidal equivalence of compact quantum groups developed by bichon de rijdt and vaes this allows us to study the problem in terms of the quantum group suq 2 the crucial part of the argument is a detailed analysis of the equivariant kasparov theory of the standard podle

non compact quantum groups associated with abelian subgroups letg be a lie group for any abelian subalgebra mathfrak h of the lie algebra g ofg and any r in mathfrak h lambda mathfrak h the difference of the left and right translates ofr gives a compatible poisson bracket ong we show how to construct the corresponding quantum group in thec algebra setting the main tool used is the general deformation quantization construction developed earlier by the author for actions of vector groups onc algebras

deformation of c algebras by cocycles on locally compact quantum groups given a c algebra a with a left action of a locally compact quantum group g on it and a unitary 2 cocycle on g we define a deformation a of a the construction behaves well under certain additional technical assumptions on the most important of which is regularity meaning that c0 g g is isomorphic to the algebra of compact operators on some hilbert space in particular then a is stably isomorphic to the iterated twisted crossed product g op g a also in good situations the c algebra a carries a left action of the deformed quantum group g and we have an isomorphism g a g a when g is a genuine locally compact group we show that the action of g on c0 g cr g is always integrable stronger assumptions of properness and saturation of the action imply regularity as an example we make a preliminary analysis of the cocycles on the duals of some solvable lie groups recently constructed by bieliavsky et al and discuss the relation of our construction to that of bieliavsky and gayral

a finite difference procedure for a class of free boundary problems finite difference schemes loose accuracy when free boundaries cross over rectangular grids for a class of second order equations the leading error term at such a boundary can be eliminated by a simple correction strategy this procedure works in any number of space dimensions and offers an alternative to more costly and complicated adaptive grid techniques

wavenumber selection and eckhaus instability in couette taylor flow abstract we review a number of experimental results which pertain to pattern selection in couette taylor flow they include measurements of the harmonic contents of taylor vortices of the location of the eckhaus boundary of wavenumber selection by a spatial ramp of the reynolds number and of the axial variation of the wavenumber in the finite system we show that many of the results can be understood quantitatively in terms of the ginzburg landau amplitude equation

deformation quantization of heisenberg manifolds form a smooth manifold equipped with a poisson bracket we formulate ac algebra framework for deformation quantization including the possibility of invariance under a lie group of diffeomorphisms preserving the poisson bracket we then show that the much studied non commutative tori give examples of such deformation quantizations invariant under the usual action of ordinary tori going beyond this the main results of the paper provide a construction of invariant deformation quantizations for those poisson brackets on heisenberg manifolds which are invariant under the action of the heisenberg lie group and for various generalizations suggested by this class of examples interesting examples are obtained of simplec algebras on which the heisenberg group acts ergodically

k groups of c algebras deformed by actions of rd abstract we show that the topological k groups of a c algebra deformed by an action of r d are isomorphic to those of the original c a1gebra we do this by exhibiting the deformed algebra as a suitable generalized fixed point algebra this also shows that the deformed algebra is nuclear if the original one is

steady state bifurcation with euclidean symmetry we consider systems of partial differential equations equivariant under the euclidean group e n and undergoing steady state bifurcation with nonzero critical wavenumber from a fully symmetric equilibrium a rigorous reduction procedure is presented that leads locally to an optimally small system of equations in particular when n 1 and n 2 and for reaction diffusion equations with general n reduction leads to a single equation our results are valid generically with perturbations consisting of relatively bounded partial differential operators in analogy with equivariant bifurcation theory for compact groups we give a classification of the different types of reduced systems in terms of the absolutely irreducible unitary representations of e n the representation theory of e n is driven by the irreducible representations of o n 1 for n 1 this constitutes a mathematical statement of the universality of the ginzburg landau equation on the line in recent work we addressed the validity of this equation using related techniques when n 2 there are precisely two significantly different types of reduced equation scalar and pseudoscalar corresponding to the trivial and nontrivial one dimensional representations of o 1 there are infinitely many possibilities for each n equal to or greater than 3

highly entangled non random subspaces of tensor products from quantum groups in this paper we describe a class of highly entangled subspaces of a tensor product of finite dimensional hilbert spaces arising from the representation theory of free orthogonal quantum groups we determine their largest singular values and obtain lower bounds for the minimum output entropy of the corresponding quantum channels an application to the construction of d positive maps on matrix algebras is also presented

a finite difference method for free boundary problems fornberg and meyer spasche proposed some time ago a simple strategy to correct finite difference schemes in the presence of a free boundary that cuts across a cartesian grid we show here how this procedure can be combined with a minimax based optimization procedure to rapidly solve a wide range of elliptic type free boundary value problems