singular perturbations of degenerate diffusions ventsel and freidlin studied small random perturbations dx varepsilon t b x varepsilon t dt sqrt varepsilon sigma x varepsilon t d beta t of the dynamical system dx t b x t dt they proved a large deviations theorem for the family x varepsilon t and used the large deviations estimates to prove among other things a result about exit from a bounded domain containing an equilibrium of the unperturbed system fleming rederived this exit theorem using techniques from stochastic control theory in this paper we study analogous questions for singular perturbations of degenerate diffusions we consider the family of processes dx varepsilon t b x varepsilon t dt tau x varepsilon t circ dz t sqrt varepsilon sigma x varepsilon t d beta t for varepsilon geq 0 where the process x 0 t is degenerate we show that a large deviations principle need not hold but we derive bounds sufficient for obtaining some estimates on probabilities we also adapt fleming s approach to prove an exit theorem

quantum automorphism groups of homogeneous graphs associated to a finite graph x is its quantum automorphism group g the main problem is to compute the poincare series of g meaning the series f z 1 c1z c2z2 whose coefficients are multiplicities of 1 into tensor powers of the fundamental representation in this paper we find a duality between certain quantum groups and planar algebras which leads to a planar algebra formulation of the problem together with some other results this gives f for all homogeneous graphs having 8 vertices or less

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 automorphism group of the lexicographic product of finite regular graphs we study the quantum automorphism group of the lexicographic product of two finite regular graphs providing a quantum generalization of sabidussi s structure theorem on the automorphism group of such a graph

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

simple compact quantum groups i the notion of simple compact quantum group is introduced as non trivial noncommutative and noncocommutative examples the following families of compact quantum groups are shown to be simple a the universal quantum groups bu q for q gl n c satisfying qq in n 2 b the quantum automorphism groups aaut b of finite dimensional c algebras b endowed with the canonical trace when dim b 4 including the quantum permutation groups aaut xn on n points n 4 c the standard deformations kq of simple compact lie groups k and their twists kqu as well as rieffel s deformation kj

quantum symmetry groups of finite spaces we determine the quantum automorphism groups of finite spaces these are compact matrix quantum groups in the sense of woronowicz

two parameter families of quantum symmetry groups we introduce and study natural two parameter families of quantum groups motivated on one hand by the liberations of classical orthogonal groups and on the other by quantum isometry groups of the duals of the free groups specifically for each pair p q of non negative integers we define and investigate quantum groups o p q b p q s p q and h p q corresponding to respectively orthogonal groups bistochastic groups symmetric groups and hyperoctahedral groups in the first three cases the new quantum groups turn out to be related to the dual free products of free quantum groups studied earlier for h p q the situation is different and we show that h p 0 qiso fp where the latter can be viewed as a liberation of the classical isometry group of the p dimensional torus

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

integration over quantum permutation groups abstract we find a combinatorial formula for the haar measure of quantum permutation groups this leads to a dynamic formula for laws of diagonal coefficients explaining the poisson free poisson convergence result for characters

a compositional approach to quantum functions we introduce a notion of quantum function and develop a compositional framework for finite quantum set theory based on a 2 category of quantum sets and quantum functions we use this framework to formulate a 2 categorical theory of quantum graphs which captures the quantum graphs and quantum graph homomorphisms recently discovered in the study of nonlocal games and zero error communication and relates them to quantum automorphism groups of graphs considered in the setting of compact quantum groups we show that the 2 categories of quantum sets and quantum graphs are semisimple we analyze dualisable and invertible 1 morphisms in these 2 categories and show that they correspond precisely to the existing notions of quantum isomorphism and classical isomorphism between sets and graphs

quantum isometries of the finite noncommutative geometry of the standard model we compute the quantum isometry group of the finite noncommutative geometry f describing the internal degrees of freedom in the standard model of particle physics we show that this provides genuine quantum symmetries of the spectral triple corresponding to m f where m is a compact spin manifold we also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries

nouveaux resultats concernant les petites perturbations de systemes dynamiques resume nous etudions les proprietes de grandes deviations associees au comportement quand e 0 du processus de diffusion x e de generateur l e e 2 l l ou l 1 2 i 1 r i 2 l 1 2 j 1 l j 2 b 1 r 1 l et b etant des champs de vecteurs reguliers sur r n les estimations obtenues permettent de resoudre le probleme des goulots de sortie hors d un ouvert d borne lorsque les champs de vecteurs gs 1 gs l sont a support compact k regulier contenu dans d nous formulons enfin et demontrons sous certaines conditions un principe de grandes deviations qui rend compte de la vitesse de convergence de x e vers x 0 e 0 sur un espace approprie valable meme quand x 0 est non degenere

sliding mode control of underactuated multibody systems and its application to shape change control in this article we introduce an approach based on sliding mode control to design full state feedback controllers for stabilisation of underactuated non linear multibody systems we define first order sliding surfaces as a linear combination of actuated and unactuated coordinate tracking errors lyapunov stability analysis guarantees that all system trajectories reach and remain on the sliding surfaces however stability of the sliding surfaces depends on the equilibrium manifold if the system has isolated equilibrium points it is linearly controllable and asymptotic stability can be guaranteed under certain conditions otherwise the control system fails brockett s necessary condition for existence of a smooth stabilising feedback in the latter case if the total momentum is conserved the closed loop control system will be marginally stable consequently a procedure is proposed to achieve an asymptotically stable discontinuous control law through sliding surface redefinition and shape changes it i

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

nonuniform sampling of bandlimited functions in this correspondence we show that a function in the bernstein class bpi p 1 les p les infin can be reconstructed in the lagrange interpolation form with sampling points tn satisfying tn n les d and 4d additional points where d is any nonnegative real number that is not necessarily less than 1 4 here x denotes the integer part of x

an operator space characterization of dual operator algebras this paper is devoted to dual algebras that is w closed algebras of bounded operators on hilbert space we give abstract characterizations of unital dual algebras resp dual algebras up to w homeomorphic unital complete isometry resp complete isomorphism the results we obtain are natural w versions of analogous characterizations due to blecher ruan sinclair and blecher respectively in the category of operator algebras

hopf coactions on odd spheres abstract we prove that the q deformed unitary group i e u q n is the universal compact quantum group in the category of compact quantum groups which coact on the q deformed odd sphere s q 2 n 1 leaving the space spanned by the natural set of generators invariant and preserving the unique s u q n invariant functional on s q 2 n 1 using this we identify u q n as the quantum group of orientation preserving isometries in the sense of bhowmick and goswami 5 for a natural spectral triple associated with s q 2 n 1 constructed by chakraborty and pal 9

ergodic coactions with large multiplicity and monoidal equivalence of quantum groups we construct new examples of ergodic coactions of compact quantum groups in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter these examples are obtained using a bijective correspondence between certain ergodic coactions on c algebras and unitary fiber functors on the representation category of a compact quantum group we classify these unitary fiber functors on the universal orthogonal and unitary quantum groups the associated c algebras and von neumann algebras can be defined by generators and relations but are not yet well understood

nonlocal games and quantum permutation groups we present a strong connection between quantum information and quantum permutation groups specifically we define a notion of quantum isomorphisms of graphs based on quantum automorphisms from the theory of quantum groups and then show that this is equivalent to the previously defined notion of quantum isomorphism corresponding to perfect quantum strategies to the isomorphism game moreover we show that two connected graphs x and y are quantum isomorphic if and only if there exists x v x and y v y that are in the same orbit of the quantum automorphism group of the disjoint union of x and y this connection links quantum groups to the more concrete notion of nonlocal games and physically observable quantum behaviours we exploit this link by using ideas and results from quantum information in order to prove new results about quantum automorphism groups and about quantum permutation groups more generally in particular we show that asymptotically almost surely all graphs have trivial quantum automorphism group furthermore we use examples of quantum isomorphic graphs from previous work to construct an infinite family of graphs which are quantum vertex transitive but fail to be vertex transitive answering a question from the quantum group literature our main tool for proving these results is the introduction of orbits and orbitals orbits on ordered pairs of quantum permutation groups we show that the orbitals of a quantum permutation group form a coherent configuration algebra a notion from the field of algebraic graph theory we then prove that the elements of this quantum orbital algebra are exactly the matrices that commute with the magic unitary defining the quantum group we furthermore show that quantum isomorphic graphs admit an isomorphism of their quantum orbital algebras which maps the adjacency matrix of one graph to that of the other

quantum group of isometries in classical and noncommutative geometry we formulate a quantum generalization of the notion of the group of riemannian isometries for a compact riemannian manifold by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples and then proving the existence of a universal object called the quantum isometry group in the category of compact quantum groups acting smoothly and isometrically on a given possibly noncommutative manifold satisfying certain regularity assumptions the idea of quantum families due to woronowicz and soltan are relevant to our construction a number of explicit examples are given and possible applications of our results to the problem of constructing quantum group equivariant spectral triples are discussed

the morita theory of quantum graph isomorphisms we classify instances of quantum pseudo telepathy in the graph isomorphism game exploiting the recently discovered connection between quantum information and the theory of quantum automorphism groups specifically we show that graphs quantum isomorphic to a given graph are in bijective correspondence with morita equivalence classes of certain frobenius algebras in the category of finite dimensional representations of the quantum automorphism algebra of that graph we show that such a frobenius algebra may be constructed from a central type subgroup of the classical automorphism group whose action on the graph has coisotropic vertex stabilisers in particular if the original graph has no quantum symmetries quantum isomorphic graphs are classified by such subgroups we show that all quantum isomorphic graph pairs corresponding to a well known family of binary constraint systems arise from this group theoretical construction we use our classification to show that of the small order vertex transitive graphs with no quantum symmetry none is quantum isomorphic to a non isomorphic graph we show that this is in fact asymptotically almost surely true of all graphs

quantum isometry groups of the podles spheres abstract for 0 1 c 0 we identify the quantum group so 3 as the universal object in the category of compact quantum groups acting by orientation and volume preserving isometries in the sense of bhowmick and goswami 2009 4 on the natural spectral triple on the podles sphere s c 2 constructed by dabrowski d andrea landi and wagner 2007 in 9