numerical computation of heteroclinic orbits we give a numerical method for the computation of heteroclinic orbits connecting two saddle points in 2 these can be computed to very high period due to an integral phase condition and an adaptive discretization we can also compute entire branches one dimensional continua of such orbits the method can be extended to compute an invariant manifold that connects two fixed points in n as an example we compute branches of traveling wave front solutions to the huxley equation using weighted sobolev spaces and the general theory of approximation of nonlinear problems we show that the errors in the approximate wave speed and in the approximate wave front decay exponentially with the period

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

splitting comodules over hopf algebras and application to representation theory of quantum groups of type a0 0 abstract in this work we study some properties of comodules over hopf algebras possessing integrals co frobenius hopf algebras in particular we give a necessary and sufficient condition for a simple comodule to be injective we apply the result obtained to the classification of representations of quantum groups of type a 0 0

a comparison of one dimensional traveling waves in inverse and normal fluidized beds abstract the state of uniform fluidization is usually unstable to small disturbances and this can lead to the formation of vertically traveling voidage waves in inverse fluidization when particle density is less than fluid density s f particles fluidize in the direction of gravity when the drag force exerted by the fluid overcomes buoyancy inverse fluidization thus provides a unique parameter space which augments the study of instability behavior in normal fluidization when f s using continuum equations of continuity and motion we compared the linear stability of normal and inverse bed modes to examine the effect of the froude number fr and fluid to solid density ratio f s making use of numerical bifurcation analysis and continuation periodic solutions in the form of one dimensional traveling waves 1d tws were computed based on wave growth rates and bifurcation structure we identified the fr as an important parameter for predicting instability strength however affects instability onset or the point at which the base state is rendered unstable in the case studies we examined traveling waves were shown to propagate in the direction of fluidization and asymmetrical high amplitude 1d tw profiles suggest fully developed bubble like structures are oriented in the direction of fluidization

bifurcation of plane voidage waves in fluidized beds abstract a two dimensional fluidized bed model is investigated from a bifurcation point of view it is shown that bifurcation to periodic travelling waves occurs when the basic state of uniform fluidization becomes unstable the nature of these waves may be one or two dimensional here the main focus is on the bifurcation of oblique travelling waves i e plane voidage waves propagating into various directions in space these waves can be described by a two dimensional dynamical system which can be analyzed extensively using phase plane methods several stationary and periodic solutions as well as homoclinic and heteroclinic orbits are shown to exist in the model studied

infinite dimensional serial algebras and their representations abstract we study arbitrary algebras all of whose indecomposable finite dimensional representations are uniserial or more generally comodules over serial coalgebras we show that the infinite representation theory for such algebras parallels and non trivially extends the theory of primary infinite abelian groups we obtain a kulikov and a prufer decomposability criterion and determine indecomposables in the category of locally finite modules over such an algebra as applications we obtain consequences on the infinite representation theory of the extended affine dynkin quiver a n and the infinite line and half line quivers and obtain jordan form results for operators on infinite dimensional spaces we also find a class of counterexamples to an infinite version of the pure semisimplicity conjecture there exist algebras all of whose locally finite left modules are direct sums of indecomposables but not all of whose right locally finite modules decompose as such on the way we also provide classifications of one sided serial co algebras and introduce and classify a class of quantum groups whose indecomposable representations are uniserial namely coserial pointed hopf algebras these are one parameter deformations of group algebras and are obtained as certain ore extensions of these

semilattice graded weak hopf algebra and its related quantum g double the major purpose of this paper is to construct a weak hopf algebra with grading from a family of hopf algebras and then to gain a related quantum g double with regular r matrix first over a field k we introduce a so called semilattice graded weak hopf algebra h yh then the quantum g double d h of h is obtained in case that h is commutative moreover it is shown that d h is semisimple respectively von neumann regular if and only if h is a semisimple respectively von neumann regular hopf algebra at last a nontrivial example of semilattice graded weak hopf algebras is obtained

on the structure of semi invariant polynomials in ore extensions let d be an x outer s derivation of a prime ring r where s is an automorphism of r the following is proved among other things the degree of the minimal semi invariant polynomial of the ore extension r x s d is if charr 0 and is pk for some k 0 if charr p 2 where is the least integer 1 such that s ds d is x inner a similar result holds for cv polynomials these are done by introducing the new notion of k basic polynomials for each integer k 0 which enable us to analyze semi invariant polynomials inductively

the hopf bifurcation with symmetry for the navier stokes equations in l p ω n with application to plane poiseuille flow the existence of periodic solutions of the navier stokes equations in function spaces based upon lp nis proved the paper has three parts a a proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of lp n b the evolution equation is restricted to a space of time periodic functions and a fredholm integral equation on this space is formed the lyapunov schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry c the theory is applied to the bifurcation of periodic solutions from planar poiseuille flow in the presence of symmetry so 2 x o 2 x s1 yielding new results for this classic problem the o 2 invariance is in the spanwise direction with the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction there are however points of degeneracy on the neutral surface a numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi periodic solutions

quantum game players can have advantage without discord the last two decades have witnessed a rapid development of quantum information processing a new paradigm which studies the power and limit of quantum advantages in various information processing tasks problems such as when quantum advantage exists and if existing how much it could be are at a central position of these studies in a broad class of scenarios there are implicitly or explicitly at least two parties involved who share a state and the correlation in this shared state is the key factor to the efficiency under concern in these scenarios the shared entanglement or discord is usually what accounts for quantum advantage in this paper we examine a fundamental problem of this nature from the perspective of game theory a branch of applied mathematics studying selfish behaviors of two or more players we exhibit a natural zero sum game in which the chance for any player to win the game depends only on the ending correlation we show that in a certain classical equilibrium a situation in which no player can further increase her payoff by any local classical operation whoever first uses a quantum computer has a big advantage over its classical opponent the equilibrium is fair to both players and as a shared correlation it does not contain any discord yet a quantum advantage still exists this indicates that at least in game theory the previous notion of discord as a measure of non classical correlation needs to be reexamined when there are two players with different objectives

finiteness conditions co frobenius hopf algebras and quantum groups abstract some finiteness conditions for infinite dimensional coalgebras particularly right or left semiperfect coalgebras or co frobenius hopf algebras are studied as well examples of co frobenius hopf algebras are constructed via a hopf algebra structure on an ore extension of a group algebra and it is shown that bicrossproducts of co frobenius hopf algebras are themselves co frobenius

semilinear evolution equations in banach spaces with application to parabolic partial differential equations a theory for a class of semilinear evolution equations in banach spaces is developed which when applied to certain parabolic partial differential equations with nonlinear terms in divergence form gives strong solutions even for nondifferentiable data

primitive cohomology of hopf algebras abstract primitive cohomology of a hopf algebra is defined by using a modification of the cobar construction of the underlying coalgebra among many of its applications two classifications are presented firstly we classify all non locally pi pointed hopf algebra domains of gelfand kirillov dimension two and secondly we classify all pointed hopf algebras of rank one the first classification extends some results of brown goodearl and others in an ongoing project to understand all hopf algebras of low gelfand kirillov dimension the second generalizes results of krop radford and wang you chen which classified hopf algebras of rank one under extra hypothesis properties and algebraic structures of the primitive cohomology are discussed

peristaltic transport of a heat conducting viscous fluid as an application of abstract differential equations and semigroup of operators abstract a recent abstract result of rankin trans amer math soc to appear is applied to the peristaltic transport of a heat conducting fluid in a flexible tube the oberbeck boussinesq equations are used as the governing equations a pressure drop over a wavelength of the tube is prescribed to ensure uniqueness and newton s cooling law for the temperature is imposed on the boundary with the notion of a weak solution introduced local existence uniqueness regularity and continuation of solutions are considered in particular w r 2 regularity of the solutions is established

upper bound on the characters of the symmetric groups let c be a conjugacy class in the symmetric group s n and be a partition of n let f be the degree of the irreducible representation s c the character of s at c and r c the normalized character c f we prove that there exist constants b 0 and 1 q 0 such that for n 4 for every conjugacy class c in s n and every irreducible representation s of s n r c max q 1 n 1 n b supp c where supp c is the number of non fixed digits under the action of a permutation in c 1 is the size of the largest part in and 1 is the number of parts in the proof is obtained by enumeration of rim hook tableaux the hook formula and probabilistic arguments combinatorial algebraic and statistical applications follow this result in particular we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes

nonlinear waves in two fluid flows abstract typically the interaction of two miscible fluids leads to the formation of segregation patterns in the form of plane concentration waves and two or three dimensional voids or clusters the development of small amplitude plane travelling waves and the subsequent onset of higher dimensional instabilities can be elucidated analytically in addition numerical methods allow the computation of higher amplitude solutions as both direct approaches become costly although at different amplitude levels one is tempted to look for a simpler model problem therefore a nonlinear wave equation has been derived for two dimensional dispersed two phase flow in fluidized beds near the stability boundary using the froude number as small parameter this equation is a two dimensional perturbation of the korteweg de vries equation containing all low order terms it allows to identify the transition point between voids and clusters and to investigate these two regions for solitary and periodic waves a ginzburg landau equation is derived to describe transverse perturbations of one dimensional wavetrains bifurcating at the stability boundary

strengths and weaknesses of quantum computing recently a great deal of attention has been focused on quantum computation following a sequence of results bernstein and vazirani in proc 25th annual acm symposium theory comput 1993 pp 11 20 siam j comput 26 1997 pp 1277 1339 simon in proc 35th annual ieee symposium foundations comput sci 1994 pp 116 123 siam j comput 26 1997 pp 1340 1349 shor in proc 35th annual ieee symposium foundations comput sci 1994 pp 124 134 suggesting that quantum computers are more powerful than classical probabilistic computers following shor s result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time it is natural to ask whether all of np can be efficiently solved in quantum polynomial time in this paper we address this question by proving that relative to an oracle chosen uniformly at random with probability 1 the class np cannot be solved on a quantum turing machine qtm in time o 2 n 2 we also show that relative to a permutation oracle chosen uniformly at random with probability 1 the class np cap conp cannot be solved on a qtm in time o 2 n 3 the former bound is tight since recent work of grover in it proc 28 th annual acm symposium theory comput 1996 shows how to accept the class np relative to any oracle on a quantum computer in time o 2 n 2

numerical computation and continuation of invariant manifolds connecting fixed points a numerical method for the computation of an invariant manifold that connects two fixed points of a vector field in mathbb r n is given extending the results of an earlier paper comput appl math 26 1989 pp 159 170 by the authors basically a boundary value problem on the real line is truncated to a finite interval the method applies in particular to the computation of heteroclinic orbits the emphasis is on the systematic computation of such orbits by continuation using the fact that the linearized operator of our problem is fredholm in appropriate banach spaces the general theory of approximation of nonlinear problems is employed to show that the errors in the approximate solution decay exponentially with the length of the approximating interval several applications are considered including the computation of traveling wave solutions to reaction diffusion problems computations were done using the software package auto

on the brauer picard group of a finite symmetric tensor category let cn denote the representation category of finite supergroup kn z 2z we compute the brauer picard group brpic cn of cn this is done by identifying brpic cn with the group of braided tensor autoequivalences of the drinfeld center of cn and studying the action of the latter group on the categorical lagrangian grassmannian of cn we show that this action corresponds to the action of a projective symplectic group on a classical lagrangian grassmannian

new lattice based cryptographic constructions we introduce the use of fourier analysis on lattices as an integral part of a lattice based construction the tools we develop provide an elegant description of certain gaussian distributions around lattice points our results include two cryptographic constructions that are based on the worst case hardness of the unique shortest vector problem the main result is a new public key cryptosystem whose security guarantee is considerably stronger than previous results o n1 5 instead of o n7 this provides the first alternative to ajtai and dwork s original 1996 cryptosystem our second result is a family of collision resistant hash functions with an improved security guarantee in terms of the unique shortest vector problem surprisingly both results are derived from one theorem that presents two indistinguishable distributions on the segment 0 1 it seems that this theorem can have further applications as an example we use it to solve an open problem in quantum computation related to the dihedral hidden subgroup problem

bifurcations of poiseuille flow between parallel plates three dimensional solutions with large spanwise wavelength the spatial dynamics approach is applied to the analysis of bifurcations of the three dimensional poiseuille flow between parallel plates in contrast to the classical studies we impose time periodicity as well as spatial periodicity with period 2 in the streamwise direction however we make no assumptions on the behavior in the spanwise direction except the uniform closeness of the bifurcating solution to the basic flow in an abstract setting it is shown how the dimension of the critical eigenspace of the spatial dynamics analysis can be uniquely determined from the classical linear stability problem for the three dimensional poiseuille problem we are able to find all relevant coefficients from the analysis of the purely two dimensional problem moreover we are able to analyze precisely the influence of a spanwise pressure gradient and the associated spanwise mass flux the study of the reduced problem shows that there are two different kinds of solutions spirals and ribbons which are 2 p periodic in the spanwise direction as in the couette taylor problem and both of them bifurcate in the same direction

hamiltonian hopf bifurcation with symmetry in this paper we study the appearance of branches of relative periodic orbits in hamiltonian hopf bifurcation processes in the presence of compact symmetry groups that do not generically exist in the dissipative framework the theoretical study is illustrated with several examples

lifting of quantum linear spaces and pointed hopf algebras of orderp3 abstract we propose the following principle to study pointed hopf algebras or more generally hopf algebras whose coradical is a hopf subalgebra given such a hopf algebraa consider its coradical filtration and the associated graded coalgebra gr a then gr ais a graded hopf algebra since the coradicala0ofais a hopf subalgebra in addition there is a projection gr a a0 letrbe the algebra of coinvariants of then by a result of radford and majid ris a braided hopf algebra and gr ais the bosonization or biproduct ofranda0 gr a r a0 the principle we propose to studyais first to studyr then to transfer the information to gr avia bosonization and finally to lift toa in this article we apply this principle to the situation whenris the simplest braided hopf algebra a quantum linear space as consequences of our technique we obtain the classification of pointed hopf algebras of orderp3 pan odd prime over an algebraically closed field of characteristic zero with the same hypothesis the characterization of the pointed hopf algebras whose coradical is abelian and has indexporp2 and an infinite family of pointed nonisomorphic hopf algebras of the same dimension this last result gives a negative answer to a conjecture of i kaplansky

cross product bialgebras i abstract the subject of this article is bialgebra factorizations or cross product bialgebras without cocycles we establish a theory characterizing cross product bialgebras universally in terms of projections and injections especially all known types of biproduct double cross product and bicross product bialgebras can be described by this theory furthermore the theory provides new families of cocycle free cross product bialgebras besides the universal characterization we find an equivalent co modular description of certain types of cross product bialgebras in terms of so called hopf data with the help of hopf data construction we recover again all known cross product bialgebras as well as new and more general types of cross product bialgebras we are working in the general setting of braided monoidal categories which allows us to apply our results in particular to the braided category of hopf bimodules over a hopf algebra majid s double biproduct is seen to be a twisting of a certain tensor product bialgebra in this category this resembles the case of the drinfel d double which can be constructed as a twist of a specific cross product

infinite wavelength analysis for two phase flow a three parameter computer assisted study of global bifurcations abstract a computer assisted analysis of the long wave solutions of a two phase flow model in packed beds is presented based on the numerical study of global bifurcations a time dependent pulsing flow arises spontaneously for relatively high flow rates of gas and liquid through a packed bed a recently developed nonlinear model for this phenomenon invokes a travelling wave assumption and associates the observed wavelike pulses with the existence of long period periodic solutions in the travelling wave frame in parameter space such periodic solutions are located in the neighborhood of certain infinite period global bifurcations such as homoclinic and double heteroclinic loops in order to completely map out the travelling wave flow behavior for a given set of fluids and packing it is necessary to perform a three parameter analysis of these global bifurcations using techniques for the continuation of homoclinic and heteroclinic connections we have examined the relation between homoclinic heteroclinic both codimension 1 and double heteroclinic codimension 2 connections in the model in parameter space the families of these bifurcations interact and undergo further qualitative changes these changes include a crossing of homoclinic connection branches and the birth of a pair of double heteroclinic connections we discuss briefly how double heteroclinic loops apparently arise from a local codimension three singularity where a double zero eigenvalue occurs simultaneously with the vanishing of a higher second order term we also discuss the relation between the nature of the predicted solutions and experimental observations this computational study of global bifurcations provides a concise description of the operating region in which pulsing flow can be observed in packed beds

a subexponential time quantum algorithm for the dihedral hidden subgroup problem we present a quantum algorithm for the dihedral hidden subgroup problem dhsp with time and query complexity 2 o sqrt log n in this problem an oracle computes a function f on the dihedral group d n which is invariant under a hidden reflection in d n by contrast the classical query complexity of dhsp is o sqrt n the algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group the algorithm begins as usual with a quantum character transform which in the case of d n is essentially the abelian quantum fourier transform this yields the name of a group representation of d n which is not by itself useful and a state in the representation which is a valuable but indecipherable qubit the algorithm proceeds by repeatedly pairing two unfavorable qubits to make a new qubit in a more favorable representation of d n once the algorithm obtains certain target representations direct measurements reveal the hidden subgroup

weak solutions for a class of parabolic volterra integrodifferential equations abstract we define weak solutions for a class of volterra integrodifferential equations of the form u t au t 0t a t s g s u s ds f t u t t 0 u 0 0 the operator a is the negative infinitesimal generator of an analytic semigroup in a banach space x the operator g t u is related to a by a special form g t u a 1 2 q t u where q t u is an appropriate lower order operator we show the existence and uniqueness of weak solutions and their continuability to infinity under suitable conditions using our results we study the asymptotic behavior as time goes to infinity of strong solutions of a second order initial boundary value problem

hochschild cohomology and the coradical filtration of pointed coalgebras applications abstract in this paper we show that there is a close connection between the coradical filtration of a pointed coalgebra and the hochschild cohomology of that coalgebra with coefficients in some one dimensional bicomodules as an application for a given prime numberpand an algebraically closed fieldkof characteristic 0 we classify all pointed hopf algebras of dimensionp3overk

nonlinear periodic waves in a two phase flow model a partial differential equation is presented as a model for a two phase flow a nonlinear analysis is done using perturbation theory formally yielding periodic traveling wave solutions in light of previous work these solutions under certain restrictions are interpreted as linearly unstable waves which have equilibrated

representations of pointed hopf algebras and their drinfel d quantum doubles abstract we study representations of nontrivial liftings of nilpotent type of quantum linear spaces and their drinfel d quantum doubles we construct a family of verma type modules in both cases and prove a parametrization theorem for simple modules we compute the loewy and socle series of verma modules under a mild restriction on the datum of a lifting we find bases and dimensions of simple modules

periodic solutions of nonlinear parabolic equations abstract in this paper we study the periodic neumann boundary value problem for semilinear parabolic equations we present some existence results and study the structure of the set of solutions giving sufficient conditions under which that solution set is compact connected or acyclic

co frobenius hopf algebras integrals doi koppinen modules and injective objects abstract we investigate hopf algebras with non zero integral from a coalgebraic point of view categories of doi koppinen modules are studied in the special case where the defining coalgebra is left and right semiperfect and several pairs of adjoint functors are constructed as applications we give a very short proof for the uniqueness of the integrals and provide information about injective objects in the category of doi koppinen modules

the power of strong fourier sampling quantum algorithms for affine groups and hidden shifts many quantum algorithms including shor s celebrated factoring and discrete log algorithms proceed by reduction to a hidden subgroup problem in which an unknown subgroup h of a group g must be determined from a quantum state psi over g that is uniformly supported on a left coset of h these hidden subgroup problems are typically solved by fourier sampling the quantum fourier transform of psi is computed and measured when the underlying group is nonabelian two important variants of the fourier sampling paradigm have been identified the weak standard method where only representation names are measured and the strong standard method where full measurement i e the row and column of the representation in a suitably chosen basis as well as its name occurs it has remained open whether the strong standard method is indeed stronger that is whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak or any other known method in this article we settle this question in the affirmative we show that hidden subgroups h of the q hedral groups i e semidirect products mathbb z q ltimes mathbb z p where q mid p 1 and in particular the affine groups a p can be information theoretically reconstructed using the strong standard method moreover if h p rm polylog p these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation we compare our algorithms to two weaker methods that have been discussed in the literature the forgetful abelian method and measurement in a random basis and show that both of these are weaker than the strong standard method thus at least for some families of groups it is crucial to use the full power of representation theory and nonabelian fourier analysis namely to measure the high dimensional representations in an adapted basis that respects the group s subgroup structure we apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems generalizing the work of van dam hallgren and ip on shifts of multiplicative characters finally we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently

duals of pointed hopf algebras in this paper we study the duals of some finite dimensional pointed hopf algebras working over an algebraically closed field k of characteristic 0 in particular we study pointed hopf algebras with coradical k for a finite abelian group and with associated graded hopf algebra of the form b v k where b v is the nichols algebra of v iv igi k k yd as a corollary to a general theorem on duals of coradically graded hopf algebras we have that the dual of b v k is b w k where w iwgi i k k yd this description of the dual is used to explicitly describe the drinfel d double of b v k we also show that the dual of a nontrivial lifting a of b v k which is not itself a radford biproduct is never pointed for v a quantum linear space of dimension 1 or 2 we describe the duals of some liftings of b v k we conclude with some examples where we determine all the irreducible finite dimensional representations of a lifting of b v k by computing the matrix coalgebras in the coradical of the dual

on the number of types of finite dimensional hopf algebras for an odd prime p we construct an infinite class of non isomorphic hopf algebras of dimension p 4 over an infinite field containing primitive p th roots of unity answering in the negative a long standing conjecture of kaplansky

theory and application of a nongeneric heteroclinic loop bifurcation homoclinic and heteroclinic bifurcations from a heteroclinic loop are considered the system under consideration has three parameters two of which are not suitable for generic unfoldings analytical criteria in terms of derivatives to melnikov s functions are given for nongeneric parameters four qualitatively distinct bifurcation diagrams are obtained the result is used to give an explanation of a numerical finding on the generation of traveling pulsing waves in a two phase flow problem

cocycle deformations for hopf algebras with a coalgebra projection abstract let h be a hopf algebra over a field k of characteristic 0 and let a be a bialgebra or hopf algebra such that h is isomorphic to a sub hopf algebra of a and there is an h bilinear coalgebra projection from a to h which splits the inclusion then a r h where r is the pre bialgebra of coinvariants in this paper we study the deformations of a by an h bilinear cocycle if is a cocycle for a then can be restricted to a cocycle r for r and a r r h as examples we consider liftings of b v k where is a finite abelian group v is a quantum plane and b v is its nichols algebra and explicitly construct the cocycle which twists the radford biproduct into the lifting

pointed hopf algebras of dimensionp3 abstract we give a structure theorem for pointed hopf algebras of dimension p 3 having coradical kc p where k is an algebraically closed field of characteristic zero combining this with previous results we obtain the complete classification of all pointed hopf algebras of dimension p 3

limitations of quantum coset states for graph isomorphism it has been known for some time that graph isomorphism reduces to the hidden subgroup problem hsp what is more most exponential speedups in quantum computation are obtained by solving instances of the hsp a common feature of the resulting algorithms is the use of quantum coset states which encode the hidden subgroup an open question has been how hard it is to use these states to solve graph isomorphism it was recently shown by moore et al 2005 that only an exponentially small amount of information is available from one or a pair of coset states a potential source of power to exploit are entangled quantum measurements that act jointly on many states at once we show that entangled quantum measurements on at least n log n coset states are necessary to get useful information for the case of graph isomorphism matching an information theoretic upper bound this may be viewed as a negative result because in general it seems hard to implement a given highly entangled measurement our main theorem is very general and also rules out using joint measurements on few coset states for some other groups such as gl n fpm and gn where g is finite and satisfies a suitable property

quantum lines over non cocommutative cosemisimple hopf algebras the recognition of the importance of braided tensor categories came by the remarkable recent developments of quantum group theory an especially important example of such a category is the braided tensor category hyd of yetter drinfeld modules over a hopf algebra h with bijective antipode it is interesting and delightful to see that old results on ordinary hopf algebras such as those by nichols 18 1978 and by radford 19 1985 are reproduced in terms of the new notion of braided hopf algebra in hyd given a braided hopf algebra r in hyd one can construct an ordinary hopf algebra r h of biproduct this naturally forms a triple r h called a hopf algebra triple over h such that h r h and r h h are hopf algebra maps with id if we define h 1 h x h e x h it was essentially proved by radford 19 and then reproduced by majid 13 that r r h gives a category equivalence called bosonisation from the category of braided hopf algebras in hhyd to the category of hopf algebra triples over h see proposition 1 1

a class of drinfeld doubles that are ribbon algebras andruskiewitsch and schneider classify a large class of pointed hopf algebras with abelian coradical the drinfeld double of each such hopf algebra is investigated the drinfeld doubles of a family of hopf algebras from the above classification are ribbon hopf algebras

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

the hidden subgroup problem and quantum computation using group representations the hidden subgroup problem is the foundation of many quantum algorithms an efficient solution is known for the problem over abelian groups employed by both simon s algorithm and shor s factoring and discrete log algorithms the nonabelian case however remains open an efficient solution would give rise to an efficient quantum algorithm for graph isomorphism we fully analyze a natural generalization of the algorithm for the abelian case to the nonabelian case and show that the algorithm determines the normal core of a hidden subgroup in particular normal subgroups can be determined we show however that this immediate generalization of the abelian algorithm does not efficiently solve graph isomorphism

constructing pointed hopf algebras by ore extensions abstract we present a general construction producing pointed co frobenius hopf algebras and give some classification results for the examples obtained

co frobenius hopf algebras and the coradical filtration we prove that a hopf algebra with a finite coradical filtration is co frobenius we also characterize co frobenius hopf algebras with coradical a hopf subalgebra let h be a hopf algebra whose coradical is a hopf algebra let gr h be the associated graded coalgebra and let r be the diagram of h c f 2 then the following are equivalent 1 h is co frobenius 2 gr h is co frobenius 3 r is finite dimensional 4 the coradical filtration of h is finite this theorem allows us to give systematically examples of co frobenius hopf algebras and opens the way to the classification of ample classes of such hopf algebras

an algebraic framework for group duality abstract a hopf algebra is a pair a whereais an associative algebra with identity and a homomorphism formatoa asatisfying certain conditions if we drop the assumption thatahas an identity and if we allow to have values in the so called multiplier algebram a a we get a natural extension of the notion of a hopf algebra we call this a multiplier hopf algebra the motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group also for these multiplier hopf algebras there is a natural notion of left and right invariance for linear functionals called integrals in hopf algebra theory we show that if such invariant functionals exist they are unique up to a scalar and faithful for a regular multiplier hopf algebra a i e with invertible antipode with invariant functionals we construct in a canonical way the dual it is again a regular multiplier hopf algebra with invariant functionals it is also shown that the dual of is canonically isomorphic with the original multiplier hopf algebra a it is possible to generalize many aspects of abstract harmonic analysis here one can define the fourier transform one can prove plancherel s formula because any finite dimensional hopf algebra is a regular multiplier hopf algebra and has invariant functionals our duality theorem applies to all finite dimensional hopf algebras then it coincides with the usual duality for such hopf algebras but our category of multiplier hopf algebras also includes in a certain way the discrete quantum groups and the compact quantum groups our duality includes the duality between discrete quantum groups and compact quantum groups in particular it includes the duality between compact abelian groups and discrete abelian groups one of the nice features of our theory is that we have an extension of this duality to the non abelian case but within one category this is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type we prove that also these are dual to each other we treat an example that is sufficiently general to illustrate most of the different features of our theory it is also possible to construct the quantum double of drinfel d within this category this provides a still wider class of examples so we obtain many more than just the compact and discrete quantum within this setting

quantum computation and lattice problems we present the first explicit connection between quantum computation and lattice problems namely our main result is a solution to the unique shortest vector problem svp under the assumption that there exists an algorithm that solves the hidden subgroup problem on the dihedral group by coset sampling additionally we present an approach to solving the hidden sub group problem on the dihedral group by using an average case subset sum routine

generalized hopf ore extensions abstract we investigate some hopf algebra structures over an ore extension of a hopf algebra necessary and sufficient conditions for an ore extension of a hopf algebra to have a hopf algebra structure of a certain type are given this construction called a generalized hopf ore extension generalizes hopf ore extensions we describe the generalized hopf ore extensions of the enveloping algebras of lie algebras for some lie algebras g the generalized hopf ore extensions of u g are classified

traveling wave profiles for nonlinear equations abstract exact traveling wave profiles are here obtained for a general reaction diffusion convection equation a two phase flow equation a generalized harry dym equation a generalized korteweg de vries kdv equation a modified mkdv equation a branching network model equation a nagumo equation and a third order equation of fluid mechanics

linearly implicit methods for nonlinear evolution equations we construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear evolution equations and extend thus recent results concerning the discretization of nonlinear parabolic equations the resulting schemes are linearly implicit and include as particular cases implicit explicit multistep schemes as well as the combination of implicit runge kutta schemes and extrapolation we establish optimal order error estimates the abstract results are applied to a third order evolution equation arising in the modelling of flow in a fluidized bed we discretize this equation in space by a petrov galerkin method the resulting fully discrete schemes require solving some linear systems to advance in time with coefficient matrices the same for all time levels