symmetry of the flow around a circular cylinder abstract the symmetry of the two dimensional viscous and incompressible flow past a circular cylinder is studied by disturbing impulsively the steady flow at reynolds number 10 20 40 and 100 the propagation of this disturmbance is studied by dividing the stream and vorticity functions into steady and perturbed stream and vorticity functions the steady stream and vorticity functions are expanded in the finite fourier sine series the steady semianalytical solutions are obtained for the symmetrical flow as a limit of the time dependent equations the perturbed stream and vorticity functions are expanded in the finite fourier sine and cosine series and then along with the steady stream and vorticity functions expansions substituted in the navier stokes equations this leads to a system of coupled parabolic partial differential equations in the coefficient functions of fourier series which is solved numerically this system is solved with the initial condition which corresponds to the applied impulsive velocity to the surface of the cylinder in the perpendicular direction of the flow asymmetric vortices are observed for the reynolds number re 40 and a slight oscillation of the trail of almost symmetrical wakes is seen for the reynolds numbers 20 up to t 100 the symmetrical standing vortices in the strictest sense are not observed for re 10 even up to t 100 vortex shedding is observed for re 100 for reynolds number 10 the disturbance is applied continuously up to t 1 and asymmetric vortices exist up to t 100 the symmetry of the flow depends on the disturbance level in the flow even at reynolds number 10

an affine weyl group approach to the eight parameter discrete painleve equation we present a geometrical construction of the eight parameter discrete painleve equations our starting point is the e 1 8 affine weyl group we assume that the multi dimensional function lives on the vertices of the weight lattice of this group we derive the bilinear equations related to the discrete painleve equation in the form of nonautonomous hirota miwa equations and the elementary miura transformations the compatibility condition of the various miura s that can be written leads to three types of equations difference multiplicative q and another type where the parameters and the independent variable enter through the arguments of elliptic functions we write explicitly the discrete equations in the first two cases and produce their degeneration through coalescence of parameters

universal character and q difference painleve equations the universal character is a polynomial attached to a pair of partitions and is a generalization of the schur polynomial in this paper we introduce an integrable system of q difference lattice equations satisfied by the universal character and call it the lattice q uc hierarchy we regard it as generalizing both q kp and q uc hierarchies suitable similarity and periodic reductions of the hierarchy yield the q difference painleve equations of types a 2g 1 a 1 1 g geq 1 d 5 1 and e 6 1 as its consequence a class of algebraic solutions of the q painleve equations is rapidly obtained by means of the universal character in particular we demonstrate explicitly the reduction procedure for the case of type e 6 1 via the framework of based on the geometry of certain rational surfaces

on the decomposition of tensor products of the representations of the classical groups by means of the universal characters on definit les caracteres universels d une representation rationnelle de gl n c et on donne les formules de decomposition des produits tensoriels entre eux en termes de coefficients de littlewood richardson on demontre les formules de decomposition du produit tensoriel de deux representations irreductibles de sp 2n c et de so n c

10e9 solution to the elliptic painleve equation a function formalism for sakai s elliptic painleve equation is presented this establishes the equivalence between the two formulations by sakai and by ohta ramani grammaticos we also give a simple geometric description of the elliptic painleve equation as a non autonomous deformation of the addition formula on elliptic curves by using these formulations we construct a particular solution of the elliptic painleve equation expressed in terms of the elliptic hypergeometric function 10e9

universal characters and q painleve systems we propose an integrable system of q difference equations satisfied by the universal characters and regard it as a q analogue of the uc hierarchy see 10 via a similarity reduction of this integrable system rational solutions of the q painleve systems are constructed in terms of the universal characters

tropical representation of weyl groups associated with certain rational varieties abstract starting from certain rational varieties blown up from p 1 n we construct a tropical i e subtraction free birational representation of weyl groups as a group of pseudo isomorphisms of the varieties we develop an algebro geometric framework of functions as defining functions of exceptional divisors on the varieties in the case where the corresponding root system is of affine type our construction yields a class of higher order q difference painleve equations and its algebraic degree grows quadratically

q painleve vi equation arising from q uc hierarchy we study the q difference analogue of the sixth painleve equation q pvi by means of tau functions associated with the affine weyl group of type d5 we prove that a solution of q pvi coincides with a self similar solution of the q uc hierarchy as a consequence we obtain in particular algebraic solutions of q pvi in terms of the universal character which is a generalization of the schur polynomial attached to a pair of partitions

semigroup crossed products and the toeplitz algebras of nonabelian groups abstract we consider the quasi lattice ordered groups g p recently introduced by nica we realise their universal toeplitz algebra as a crossed product b p p by a semigroup of endomorphisms and show that the toeplitz representation is faithful precisely when an amenability condition is satisfied we then show directly that many interesting free products are amenable the final result contains theorems of cuntz and dinh on the uniqueness of toeplitz cuntz algebras as special cases

on an integrable system of q difference equations satisfied by the universal characters its lax formalism and an application to q painleve equations the universal character is a generalization of the schur function attached to a pair of partitions we study an integrable system of q difference equations satisfied by the universal characters which is an extension of the q kp hierarchy and is called the lattice q uc hierarchy we describe the lattice q uc hierarchy as a compatibility condition of its associated linear system lax formalism and explore an application to the q painleve equations via similarity reduction in particular a higher order analogue of the q painleve vi equation is presented

algebraic entropy and the space of initial values for discrete dynamical systems a method to calculate the algebraic entropy of a mapping which can be lifted to an isomorphism of a suitable rational surface the space of initial values is presented it is shown that the degree of the nth iterate of such a mapping is given by its action on the picard group of the space of initial values it is also shown by construction that the degree of the nth iterate of every painleve equation in sakai s list is o n2 and therefore its algebraic entropy is zero

universal characters and an extension of the kp hierarchy the universal character is a polynomial attached to a pair of partitions and is a generalization of the schur polynomial in this paper we define vertex operators which play roles of raising operators for the universal character by means of the vertex operators we obtain a series of non linear partial differential equations of infinite order called the uc hierarchy we regard it as an extension of the kp hierarchy we investigate also solutions of the uc hierarchy the totality of the space of solutions forms a direct product of two infinite dimensional grassmann manifolds and its infinitesimal transformations are described in terms of the lie algebra

invariant ideals of twisted crossed products we will prove a result concerning the inclusion of non trivial invariant ideals inside non trivial ideals of a twisted crossed product we will also give results concerning the primeness and simplicity of crossed products of twisted actions of locally compact groups on c algebras

affine weyl groups discrete dynamical systems and painleve equations a new class of representations of affine weyl groups on rational functions are constructed in order to formulate discrete dynamical systems associated with affine root systems as an application some examples of difference and differential systems of painleve type are discussed

enveloping actions and takai duality for partial actions we show that any partial action on a topological space x is the restriction of a suitable global action called enveloping action that is essentially unique in the case of c algebras we prove that any partial action has a unique enveloping action up to morita equivalence and that the corresponding reduced crossed products are morita equivalent the study of the enveloping action up to morita equivalence reveals the form that takai duality takes for partial actions by applying our constructions we prove that the reduced crossed product of the reduced cross sectional algebra of a fell bundle by the dual coaction is liminal postliminal or nuclear if and only if so is the unit fiber of the bundle we also give a non commutative generalization of the well known fact that the integral curves of a vector field on a compact manifold are defined on all of r

lattice equations arising from discrete painleve systems i a2 a1 1 and a1 a1 1 cases we introduce the concept of lattice constructed from functions of painleve systems on which quad equations of abs adler bobenko suris type appear in particular we consider the a5 1 and a6 1 surface q painleve systems corresponding affine weyl group symmetries are of a2 a1 1 and a1 a1 1 types respectively

induced c algebras and landstad duality for twisted coactions suppose n is a closed normal subgroup of a locally compact group g a coaction e a m a c n of n on a c algebra a can be inflated to a coaction s of g on a and the crossed product a x g is then isomorphic to the induced c algebra ind a xe n we prove this and a natural generalization in which a xf n is replaced by a twisted crossed product a x g n g in case g is abelian we recover a theorem of olesen and pedersen we then use this to extend the landstad duality of the first author to twisted crossed products and give several applications in particular we prove that if l n d g n 1 is topologically trivial but not necessarily split as a group extension then every twisted crossed product a x g n g is isomorphic to a crossed product of the form a x n

duality of restriction and induction for c coactions consider a coaction of a locally compact group g on a c algebra a and a closed normal subgroup n of g we prove following results of echterhoff for abelian g that mansfield s imprimitivity between a g n and a g r n implements equivalences between mansfield induction of representations from a g n to a g and restriction of representations froma g rn to a g and between restriction of representations from a g to a g n and green induction of representations from a g to a g r n this allows us to deduce properties of mansfield induction from the known theory of ordinary crossed products

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure