a combinatorial perturbation method and arnold s whiskered tori in vortex dynamics abstract a combinatorial perturbation method for some n body problems is presented this method is used to reformulate a class of n body problems in the form of near integrable hamiltonian systems with many degrees of freedom the kam theory is worked out for n body problems defined on subgraphs g of the complete graph kn the main result in this paper is the existence of arnold s whiskered tori and diffusion for vortex lattice dynamics throughout the paper combinatorial and graph theoretic concepts play important roles in particular the kam nondegeneracy and melnikov transversability conditions are given in terms of binary trees t n one of the main tools here is a combinatorial algorithm for symplectic transformations to jacobi relative coordinates which is based on binary trees t n

trees wreath products and finite gelfand pairs abstract we present a new construction of finite gelfand pairs by looking at the action of the full automorphism group of a finite spherically homogeneous rooted tree of type r on the variety v r s of all spherically homogeneous subtrees of type s this generalizes well known examples as the finite ultrametric space the hamming scheme and the johnson scheme we also present further generalizations of these classical examples the first two are based on harary s notions of composition and exponentiation of group actions finally the generalized johnson scheme provides the inductive step for the harmonic analysis of our main construction

on the numerical solution of the regularized birkhoff equations the birkhoff equations for the evolution of vortex sheets are regularized in a way proposed by krasny the convergence of numerical approximations to a fixed regularization is studied theoretically and numerically the numerical test problem is a two dimensional inviscid jet

general balance and treatment permutations abstract many equi replicate designs whether ordinary block designs or multi stratum designs have a structure which is left unchanged by a group g of permutations of the treatments this group g is often used in the construction of the design a survey of the properties of such designs is given and useful necessary conditions on g are found for such a design to be generally balanced if g is abelian then the necessary conditions are satisfied in this case explicit formulas for the efficiency factors are given

chaotic motion of the n vortex problem on a sphere ii saddle centers in three degree of freedom hamiltonians we study the motion of n point vortices with n on a sphere in the presence of fixed pole vortices which are governed by a hamiltonian dynamical system with n degrees of freedom special attention is paid to the evolution of their polygonal ring configuration called the n ring in which they are equally spaced along a line of latitude of the sphere when the number of the point vortices is n 5n or 6n with n the system is reduced to a two degree of freedom hamiltonian with some saddle center equilibria one of which corresponds to the unstable n ring using a melnikov type method applicable to two degree of freedom hamiltonian systems with saddle center equilibria and a numerical method to compute stable and unstable manifolds we show numerically that there exist transverse homoclinic orbits to unstable periodic orbits in the neighborhood of the saddle centers and hence chaotic motions occur especially the evolution of the unstable n ring is shown to be chaotic

integrable and chaotic motions of four vortices i the case of identical vortices it is shown that the three vortex problem in two dimensional hydrodynamics is integrable whereas the motion of four identical vortices is not a sequence of canonical transformations is obtained that reduces the n degree of freedom hamiltonian which describes the interaction of n identical vortices to one with n 2 degrees of freedom for n 3 a reduction to a single degree of freedom is obtained and this problem can be solved in terms of elliptic functions for n 4 the reduction procedure leads to an effective hamiltonian with two degrees of freedom of the form found in problems with coupled nonlinear oscillators resonant interaction terms in this hamiltonian suggest non integrable behaviour and this is verified by numerical experiments explicit construction of a solution that corresponds to a heteroclinic orbit in phase space is possible the relevance of the results obtained to fundamental problems in hydrodynamics such as the question of integrability of euler s equation in two dimensions is discussed the paper also contains a general exposition of the hamiltonian and poisson bracket formalism for point vortices

a completeness relation for the q analogue coherent states by q integration q integration is defined for the q oscillator realization of quantum groups this is used to prove a completeness relation for the q analogue of the usual coherent states these states are overcomplete

reparametrisation of interest in non uniform factorial designs abstract a definition of factorial effects relying on the treatment structure defined by the hierarchies is proposed it applies to a non uniform situation where the number of levels of a nested factor within the classes defined by each set of levels of its nesting factors may vary a reparametrisation whose parameters belongs to these factorial effects is obtained the development is based on the notion of reference treatment design a conceptual design that can be used as a basis of comparison to assess the properties of any factorial design

wreath product of a complete graph with a cyclic graph topological indices and spectrum abstract in this manuscript we continue the investigations related to the wreath product of graphs by considering the compound graph of a clique with a circuit this product shows nice combinatorial and algebraic properties which permit with reasonable effort to compute some topological indices and the adjacency spectrum

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

quantum deformation of kdv hierarchies and their exact solutions q deformed solitons a complete procedure is given by which one can construct q deformed kdv hierarchies and obtain their exact solutions with arbitrarily many q deformed solitons as an elementary example we obtain the exact solution of the q deformed single soliton of the simplest 3 2 system i e the q deformed kdv equations and investigate this solution in the case of small deformation it is interesting that this method can supply many exact solutions for systems of new conventional nonlinear partial differential equations

quantum deformation of kdv hierarchies and their infinitely many conservation laws a reasonable q deformed differential is defined a set of operation rules are constructed for the q deformed formal pseudo differential operators the complete procedure of constructing the q deformed kdv hierarchies and their infinite conservation laws is given as an important example he obtains a detailed structure of the simplest 3 2 system i e the q deformed kdv equations

conservation laws of discrete lax equations based on the pseudodifference operators ring and lax pair we present a residue formula to calculate the conservation laws of discrete lax equations the details are explained with two illustrative examples furthermore some numerical results are given to verify the method

a wreath product group approach to signal and image processing i multiresolution analysis we propose the use of spectral analysis on certain noncommutative finite groups in digital signal processing and in particular image processing we pay significant attention to groups constructed as wreath products of cyclic groups within this large class of groups our approach recovers the discrete fourier transform dft the haar wavelet transform various multichannel pyramid filter banks and other aspects of multiresolution analysis as special cases of a more general phenomenon in addition the group structure provides a rich algebraic structure that can be exploited for the analysis and manipulation of signals our approach relies on a synthesis of ideas found in the early work of holmes 1987 1990 karpovsky and trachtenberg 1985 and others on noncommutative filtering as well as diaconis s 1989 spectral analysis approach to understanding data

the hamiltonian structure of the maxwell vlasov equations morrison 25 has observed that the maxwell vlasov and poisson vlasov equations for a collisionless plasma can be written in hamiltonian form relative to a certain poisson bracket we derive another poisson structure for these equations by using general methods of symplectic geometry the main ingredients in our construction are the symplectic structure on the co adjoint orbits for the group of canonical transformations and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory our poisson bracket satisfies the jacobi identity whereas morrison s does not 37 our construction also shows where canonical variables can be found and can be applied to the yang mills vlasov equations and to electromagnetic fluid dynamics

complex linear models and cyclic designs abstract many orthogonal factorial designs can be defined by abelian group morphisms by juxtaposition of such designs useful nonorthogonal designs can also be obtained including the classical generalized cyclic designs as well as a new kind of one replicate factorial block designs their efficiencies are easily computed by means of a complex reparametrization based on the irreducible characters of the groups involved the theory extends to the group generated designs defined by bailey and rowley in which the group is not necessarily abelian in some cases we give explicit formulas for the efficiencies of these latter designs

symmetry transformations of an ideal steady fluid flow determined by a potential function first we consider a transformation of 3d trajectories fluid particle paths of inviscid steady flows using the dual stream function approach for the local representation of velocity fields u x y z this enables to derive the equation governing the deformation of trajectories by the gradient field along the surface x y z 0 in fact is a symmetry transformation and it looks formally like the filament motion which preserves the curvature then we investigate in detail a fine structure of a lie algebra associated with an extension of the transformation which creates a visual appearance of sliding stream surfaces x y z 0 along itself the minimal set of generating differential invariants is found this set consists of a single invariant which coincides with a hamiltonian function

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

the vanishing twist in the restricted three body problem abstract this paper demonstrates the existence of twistless tori and the associated reconnection bifurcations and meandering curves in the planar circular restricted three body problem near the lagrangian equilibrium l 4 a twistless torus is created near the tripling bifurcation of the short period family decreasing the mass ratio leads to twistless bifurcations which are particularly prominent for rotation numbers 3 10 and 2 7 this scenario is studied by numerically integrating the regularised hamiltonian flow and finding rotation numbers of invariant curves in a two dimensional poincare map to corroborate the numerical results the birkhoff normal form at l 4 is calculated to eighth order truncating at this order gives an integrable system and the rotation numbers obtained from the birkhoff normal form agree well with the numerical results a global overview for the mass ratio 4 3 is presented by showing lines of constant energy and constant rotation number in action space

the q deformation of akns d hierarchy in this paper we have considered the q deformation of the akns d hierarchy proved the bilinear identity and obtained the function of the q deformed akns d hierarchy

the stability of the lagrange triangular point and a theorem of arnold deprit and deprit bartholome 5 computed the normal form for the hamiltonian of the planar restricted three body problem at the lagrange equilateral triangle equilibrium point up to terms of order 4 this normalization was carried out for all mass ratios p smaller than routh s critical mass ratio p1 i 1 j 9 except for two values p2 and pi at p2 and u the ratio of the linearized frequencies is 1 2 and 1 3 respectively and so low order resonance terms appear this normalization was carried out in order to apply the kam theory to prove the stability of this equilibrium point from the coefficients of the normal form they computed a quantity d defined below which when non zero establishes the stability of the equilibrium point they found that d 0 for 0 cp p u p2 p and p the value u does not correspond to resonance but is simply a point where the quantity d4 changes sign in this paper we shall establish the full stability of the lagrange equilibrium point in the planar restricted three body problem even in the case when u p we do this by computing by machine the normal form up to terms of order 6 and then applying a theorem of arnold 3 which establishes the stability of the equilibrium even in degenerate cases russman lo announced a theorem which implies that the equilibrium is isoenergetic stable when p p by isoenergetic stable he means that the system is stabie only on the energy surface defined by the energy at the equilibrium itself unfortunately there is not even an indication of the proof arnold announced his theorem in a doklady note with only a sketch of a proof the sketch indicates that his proof requires completely redoing all

versal normal form at the lagrange equilibrium l4 a new normal form called versal for the linearized hamiltonian vector field of the planar restricted three body problem at the lagrange equilibrium point l4 depending smoothly on the mass ratio for all values close to the critical routh s ratio is described then a canonical transformation also depending smoothly on the mass ratio which brings the linear hamiltonian vector field into this versal normal form is explicitly calculated

symbolic computation of normal forms for nonlinear hamiltonian systems abstract in this paper a method of computing normal forms for time independent nonlinear hamiltonian systems is given a program using the symbolic manipulator macsyma is used to do an example

nonlinear stability of the equilibria in a double bar rotating system we study the nonlinear stability of the equilibria corresponding to the motion of a particle orbiting around a two finite orthogonal straight segment the potential is a logarithmic function and may be considered as an approximation to the one generated by irregular celestial bodies using arnold s theorem for non definite quadratic forms we determine the nonlinear stability of the equilibria for all values of the parameter of the problem moreover the resonant cases are determined and the stability is investigated

general deformed kdv equations and their exact solutions we introduce the general deformed differential operator using it the general deformed kdv hierarchy is constructed as an elementary example we give the general deformed kdv equations in the form of the general deformation parameter expansion the general deformed kdv equations generates the systems of the non homogeneous linearized kdv equations which are exactly solvable in order to solve the general deformed kdv hierarchy we introduce the general deformed exponential function by using the dressed operator method these exact solutions can be constructed as an elementary example we give some exact solutions of the non homogeneous linearized kdv equation system derived from the general deformed kdv equations

a nonlinear hamiltonian structure for the euler equations abstract the euler equations for inviscid incompressible fluid flow have a hamiltonian structure in eulerian coordinates the hamiltonian operator though depending on the vorticity conservation laws arise from two sources one parameter symmetry groups which are completely classified yield the invariance of energy and linear and angular momenta degeneracies of the hamiltonian operator lead in three dimensions to the total helicity invariant and in two dimensions to the area integrals reflecting the point wise conservation of vorticity it is conjectured that no further conservation laws exist indicating that the euler equations are not completely integrable in particular do not have soliton like solutions

singular manifolds and quasi periodic solutions of hamiltonians for vortex lattices abstract a general method for establishing the existence of quasi periodic solutions of hamiltonian systems for vortex lattices is illustrated in a simple example involving two degrees of freedom the geometry of intersecting singular manifolds of the hamiltonians introduces suitable canonical transformations which put the hamiltonian into the form of singular weakly coupled oscillators as by products of this procedure additional integrals of motion are found for the leading term in the transformed hamiltonian these extra integrals are approximate invariants for the full hamiltonians

harmonic analysis on a finite homogeneous space in this paper we study harmonic analysis on finite homogeneous spaces whose associated permutation representation decomposes with multiplicity after a careful look at frobenius reciprocity and transitivity of induction and the introduction of three types of spherical functions we develop a theory of gelfand tsetlin bases for permutation representations then we study several concrete examples on the symmetric groups generalizing the gelfand pair of the johnson scheme we also consider statistical and probabilistic applications after that we consider the composition of two permutation representations giving a non commutative generalization of the gelfand pair associated to the ultrametric space actually we study the more general notion of crested product finally we consider the exponentiation action generalizing the decomposition of the gelfand pair of the hamming scheme actually we study a more general construction that we call wreath product of permutation representations suggested by the study of finite lamplighter random walks we give several examples of concrete decompositions of permutation representations and several explicit rules of decomposition 1

hyperkahler structures and infinite dimensional grassmannians abstract in this paper we describe an example of a hyperkahler quotient of a banach manifold by a banach lie group although the initial manifold is not diffeomorphic to a hilbert manifold not even to a manifold modelled on a reflexive banach space the quotient space obtained is a hilbert manifold which can be furthermore identified either with the cotangent space of a connected component gr res j j z of the restricted grassmannian or with a natural complexification of this connected component thus proving that these two manifolds are isomorphic hyperkahler manifolds moreover kahler potentials associated with the natural complex structure of the cotangent space of gr res j and with the natural complex structure of the complexification of gr res j are computed using kostant souriau s theory of prequantization

experimental randomization and the validity of normal theory inference abstract this article examines how randomization helps to validate model based probability statements the asymptotic validity of normal theory tests for means in multivariate linear models is established under general conditions numerical results are presented showing how randomization provides a limited degree of robustness in small experiments the conditional relevance of randomization is also discussed consider the following additive model for a comparative experiment y gu where y is a vector of responses is a vector of treatment effects u is a vector of unit effects and g is a permutation representing the assignment of treatments to units the vector gu is obtained by using g to permute the coordinates of u it is assumed that is fixed and g and u are random and independent the experimenter observes y and g but not or u and chooses the distribution of g that is the randomization strategy in this setup three probabilities may be considered conditional given g conditional g

on the q oscillator and the quantum algebra suq 1 1 the authors discuss some problems of the q deformed harmonic oscillator different q bosonisations of the quantum suq 1 1 algebra are given and the corresponding infinite dimensional representations of discrete series are analysed

nearly orthogonal randomized designs a new method of experimental randomization is presented that is suitable underadditive general linear models for treatment and unit effects the method is designed for a standard normal theory analysis of the data the method is motivated by thinking of the randomization strategy which may be represented as a distribution on the group of n n permutation matrices as a discrete approximation to a fictitious ideal randomization strategy corresponding to a continuous distribution on the group of n n orthogonal matrices designs are generated by using a simulated annealing algorithm

on averaging reduction and symmetry in hamiltonian systems abstract the existence of periodic orbits for hamiltonian systems at low positive energies can be deduced from the existence of nondegenerate critical points of an averaged hamiltonian on an associated reduced space alternatively in classical kinetic plus potential energy hamiltonians the existence of such orbits can often be established by elementary geometrical arguments the present paper unifies the two approaches by exploiting discrete symmetries including reversing diffeomorphisms that occur in a given system the symmetries are used to locate the periodic orbits in the averaged hamiltonian and thence in the original hamiltonian when the periodic orbits are continued under perturbations admitting the same symmetries in applications to the henon heiles hamiltonian it is illustrated how higher order averaging can sometimes be used to overcome degeneracies encountered at first order

construction of row column factorial designs the arrangement of 2 n factorials in row column designs to estimate main effects and two factor interactions is investigated single replicate constructions are given which enable estimation of all main effects and maximise the number of estimable two factor interactions constructions and guidance are given for multi replicate designs in single arrays and in multiple arrays consideration is given to constructions for 2 n t fractional factorials

a q analogue of hierarchy and q painleve vi a q analogue of the drinfel d sokolov hierarchy is proposed as a reduction of the q kp hierarchy applying a similarity reduction and a q laplace transformation to the hierarchy one can obtain the q painleve vi equation proposed by jimbo and sakai

confounding in relation to duality of finite abelian groups abstract using the duality in the theory of finite abelian groups we give a precise description of confounded effects in fractional factorial design when the fraction is a subgroup or a coset of a subgroup the result works for a confounded block design considered here as a fraction of a bigger design with complete blocks

poisson brackets and clebsch representations for magnetohydrodynamics multifluid plasmas and elasticity abstract poisson brackets are constructed by the same mathematical procedure for three physical theories ideal magnetohydrodynamics multifluid plasmas and elasticity each of these brackets is given a simple lie algebraic interpretation moreover each bracket is induced to physical space by use of a canonical poisson bracket in the space of clebsch potentials which are constructed for each physical theory by the standard procedure of constrained lagrangians

poisson brackets associated to the conformal geometry of curves in this paper we present an invariant moving frame in the group theoretical sense along curves in the mobius sphere this moving frame will describe the relationship between all conforrnal differential invariants for curves that appear in the literature using this frame we first show that the kac moody poisson bracket on lo n 1 1 can be poisson reduced to the space of conformal differential invariants of curves the resulting bracket will be the conformal analogue of the adler gel fand dikii bracket secondly a conformally invariant flow of curves induces naturally an evolution on the differential invariants of the flow we give the conditions on the invariant flow ensuring that the induced evolution is hamiltonian with respect to the reduced poisson bracket because of a certain parallelism with the euclidean case we study what we call frenet and natural cases we comment on the implications for completely integrable systems and describe conformal analogues of the hasimoto transformation

quasi periodic dynamics of desingularized vortex models sufficient conditions for the existence of quasi pe odic solutions of two different desingularized vortex models for 2 dimensional euler flows are derived one of these modds is the vortex blob model for the evolution of a periodic vortex sheet and the other is a second order elliptic moment model demm for the evolution of widely separated vortex regions the method involves the identification of the well known point vortex hamiltonian term in both models a transformation to new canonical variables the jl coordinates and the desnition of special open sets in phase space the cone sets puts the hamiltonians considered into nearly integrable form kam theory is used to prove the desired results for arbitrary degrees of freedom and almost arbitrary circulations in these modets a rigorous validification of the demm assumption is obtained in view of the lack of a rigorous theory for vortex sheet roll up paat the critical time the dynamical system approach presented here provides an alternative method for studying the macroscopic structures formed in the post critical period

binary trees symplectic matrices and the jacobi coordinates of celestial mechanics a graph theoretic algorithm for constructing the jacobi coordinates in celestial mechanics is given to every full binary tree with n leaves there corresponds a 6n 6n symplectic matrix which defines a jacobi transformation this correspondence yields a direct proof of the symplectic property for all the jacobi coordinates hitherto only special examples of these transformations have been shown to be canonical

on singular hamiltonians the existence of quasi periodic solutions and nonlinear stability on considere l existence de solutions quasiperiodiques pour une classe de hamiltoniens singuliers et la stabilite non lineaire des singularites

restricted randomization a practical example abstract a method of restricted randomization which avoids bad patterns in the treatments but retains validity in the analysis is demonstrated on an example that exhibits many features of real agricultural experiments such as untidy treatment structure and uncertainty about blocking the method uses permutation groups modern farming methods force agricultural experimenters to use long thin lines of plots for blocking rather than more compact areas ordinary randomization of a long thin block can produce a plan with say all low levels of nitrogen at one end of the block the first section of the article discusses why simple rerandomization is not a satisfactory method of avoiding poor layouts restricted randomization has been developed as a method of avoiding badly patterned plans while ensuring validity of the usual analysis one method of restricted randomization uses special groups of permutations of the plots catalogs of good restricted randomizations have been given for symmetric factorial de

integrable discrete systems on r and related dispersionless systems a general framework for integrable discrete systems on r in particular containing lattice soliton systems and their q deformed analogs is presented the concept of regular grain structures on r generated by discrete one parameter groups of diffeomorphisms in terms of which one can define algebra of shift operators is introduced two integrable hierarchies of discrete chains together with bi hamiltonian structures and their continuous limits are constructed the inverse problem based on the deformation quantization scheme is considered

symmetry models and hypotheses for structured data layouts on peut utiliser des proprietes de symetrie pour deriver des tests valides d hypotheses nulles et pour construire des modeles statistiques appropries pour les donnees on developpe des idees generales et on les illustre dans le contexte d une classification a deux criteres

melnikov s method and arnold diffusion for perturbations of integrable hamiltonian systems we start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates we use kolmogorov arnold moser kam theory to show that some of the resulting tori persist under small perturbations and use a vector of melnikov integrals to show that under suitable hypotheses their stable and unstable manifolds intersect transversely this transverse intersection is ultimately responsible for arnold diffusion on each energy surface the method is applied to a pendulum oscillator system