the hamiltonian structure for dynamic free boundary problems hamiltonian structures for 2 or 3 dimensional incompressible flows with a free boundary are determined which generalize a previous structure of zakharov for irrotational flow our poisson bracket is determined using the method of arnold namely reduction from canonical variables in the lagrangian material description using this bracket the hamiltonian form for the equations of a liquid drop with a free boundary having surface tension is demonstrated the structure of the bracket in terms of a reduced cotangent bundle of a principal bundle is explained in the case of two dimensional flows the vorticity bracket is determined and the generalized enstrophy is shown to be a casimir function this investigation also clears up some confusion in the literature concerning the vorticity bracket even for fixed boundary flows

geometrical phase in the cyclic evolution of non hermitian systems the authors derive by a biorthonormal state approach the analogy of berry s phase factor for open non conservative systems for both adiabatic and non adiabatic evolution in the latter case a non unitary evolution operator method is exploited an application is given to the optical supermode propagation in the free electron laser

kinematical symmetries of three dimensional incompressible flows abstract the motion of an incompressible fluid in lagrangian coordinates involves infinitely many symmetries generated by the left lie algebra of group of volume preserving diffeomorphisms of the three dimensional domain occupied by the fluid utilizing a 1 3 dimensional hamiltonian setting an explicit realization of this symmetry algebra is constructed recursively a dynamical connection is used to split the symmetries into reparametrization of trajectories and one parameter family of volume preserving diffeomorphisms of the fluid domain algebraic structures of symmetries and hamiltonian structures of their generators are inherited from the same construction a comparison with the properties of two dimensional flows is included

global regularity for a class of 2d generalized tropical climate models abstract this paper establishes the global existence and regularity of solutions to a two dimensional 2d tropical climate model tcm with fractional dissipation the inviscid counterpart of this model was derived by frierson majda and pauluis 8 as a model for tropical geophysical flows this model reflects the interaction and coupling among the barotropic mode u the first baroclinic mode v of the velocity and the temperature the systems with fractional dissipation studied here may arise in the modeling of geophysical circumstances mathematically these systems allow simultaneous examination of a family of systems with various levels of regularization the aim here is the global regularity with the least dissipation we prove two main results first the global regularity of the system with v and for 1 and 3 2 and second the global regularity of the system with v for 3 2 the proofs of these results are not trivial and the requirements on the fractional indices appear to be optimal the key tools employed here include the maximal regularity for general fractional heat operators the littlewood paley decomposition and besov space techniques lower bounds involving fractional laplacian and simultaneous estimates of several coupled quantities

on global regularity of 2d generalized magnetohydrodynamic equations abstract in this article we study the global regularity of 2d generalized magnetohydrodynamic equations 2d gmhd in which the dissipation terms are u and b we show that smooth solutions are global in the following three cases 1 2 1 0 1 2 2 2 2 0 we also show that in the inviscid case 0 if 1 then smooth solutions are global as long as the direction of the magnetic field remains smooth enough

semidirect products and reduction in mechanics this paper shows how to reduce a hamiltonian system on the cotangent bundle of a lie group to a hamiltonian system in the dual of the lie algebra of a semidirect product the procedure simplifies unifies and extends work of greene guillemin holm holmes kupershmidt marsden morrison ratiu sternberg and others the heavy top compressible fluids magnetohydrodynamics elasticity the maxwell vlasov equations and multifluid plasmas are presented as examples starting with lagrangian variables our method explains in a direct way why semidirect products occur so frequently in examples it also provides a framework for the systematic introduction of clebsch or canonical variables

averaged lagrangians and the mean effects of fluctuations in ideal fluid dynamics we begin by placing the generalized lagrangian mean glm equations for a compressible adiabatic fluid into the euler poincare ep variational framework of fluid dynamics for an averaged lagrangian we then state the ep averag ing result that glm equations arise from glm hamilton s principles in the ep framework next we derive a new set of approximate small amplitude glm equations gm equations at second order in the fluctuating displacement of a lagrangian trajectory from its mean position these equations express the linear and nonlinear back reaction effects on the eulerian mean fluid quantities by the fluctuating displacements of the lagrangian trajectories in terms of their eulerian second moments the derivation of the gm equations uses the linearized relations between eulerian and lagrangian fluctuations in the tradition of lagrangian stability analysis for fluids the gm derivation also uses the method of averaged lagrangians in the tradition of wave mean flow interaction wmfi the gm ep motion equations for compressible and incompressible ideal fluids are compared with the euler alpha turbulence closure equations an alpha model is a glm or gm fluid theory with a taylor hypothesis closure thc such closures are based on the linearized fluctuation relations that determine the dynamics of the lagrangian statistical quantities in the euler alpha closure equations we use the ep averaging result to bridge between the glm equations and the euler alpha closure equations hence combining the small amplitude approximation with thc yields in new gm turbulence closure equations for compressible fluids in the ep variational framework 2002 elsevier science b v all rights reserved

berry s phase and the planar three body problem berry s angular 2 form in seen as a correction to the symplectic structure in a separation of variables type scheme where the variables are canonically non separable this view is applied to the planar three body problem where the rotational and vibrational motions are not separable it is shown that the corrected symplectic structure gives the correct quantisation

global regularity for the 2d mhd equations with mixed partial dissipation and magnetic diffusion abstract whether or not classical solutions of the 2d incompressible mhd equations without full dissipation and magnetic diffusion can develop finite time singularities is a difficult issue a major result of this paper establishes the global regularity of classical solutions for the mhd equations with mixed partial dissipation and magnetic diffusion in addition the global existence conditional regularity and uniqueness of a weak solution is obtained for the 2d mhd equations with only magnetic diffusion

on the quantization of the planar n point vortex problem in this paper the velocity field of a planar fluid with point vortices is recognized as induced by a connection on a principal bundle various geometric topological invariants including braiding of the trajectories of the vortices and the berry s phase are discussed in this bundle context these invariants give rise to quantization schemes for the system

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

on the universal cover of certain exotic kahler surfaces of negative curvature we study a class of examples of negatively curved compact kahler surfaces that are not diffeomorphic to any locally symmetric space from the analysis of certain totally geodesic curves on these surfaces we deduce that for infinitely many examples the natural representation of the fundamental group into pu 2 1 is non faithful we also give a new construction of bounded holomorphic functions on the universal cover of our surfaces based on lifting maps to compact riemann surfaces

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

generalized coherent states and berry s phase the quantum dynamics of time dependent hamiltonians of the kind h t u t h 0 u t is studied in the framework of generalized coherent states it is shown that the quantum and the classical dynamics are isomorphic and that the phase of t is simply the classical action in adiabatic approximation it is easy to extract berry s phase from this a discussion of generalized coherent states cs s necessary to deal with excited states is given in particular for su 2 su 1 1 and e 2 their structure is completely characterized showing that the deep connection with geometric quantization extends to these generalized cs s this technique may be employed to estimate higher order correction to the adiabatic approximation

reflection groups in algebraic geometry after a brief exposition of the theory of discrete reflection groups in spherical euclidean and hyperbolic geometry as well as their analogs in complex spaces we present a survey of appearances of these groups in various areas of algebraic geometry

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

a rigorous derivation of the hamiltonian structure for the nonlinear schrodinger equation abstract we consider the cubic nonlinear schrodinger equation nls in any spatial dimension which is a well known example of an infinite dimensional hamiltonian system inspired by the knowledge that the nls is an effective equation for a system of interacting bosons as the particle number tends to infinity we provide a derivation of the hamiltonian structure which is comprised of both a hamiltonian functional and a weak symplectic structure for the nonlinear schrodinger equation from quantum many body systems our geometric constructions are based on a quantized version of the poisson structure introduced by marsden morrison and weinstein 24 for a system describing the evolution of finitely many indistinguishable classical particles

ideal hydrodynamics on lie groups abstract in this paper we consider the reduction to a lie group of geodesic flows of left or right invariant metrics for a fixed value of momentum we describe the reduced vector field on the lie group as well as its right or left invariant symmetry fields the reduced vector field has an important hydrodynamic interpretation it is a stationary flow of an ideal fluid with a constant pressure we use the explicit expressions for the reduced vector field and its symmetry fields to define secondary hydrodynamics i e we study the reduction for the euler equations of an ideal fluid that describe the geodesics of a right invariant metric on a lie group s diff m of the volume preserving diffeomorphisms of a riemannian manifold m to the group s diff m for a typical coadjoint orbit we find all symmetry fields of a reduced flow and as a corollary we get a simple proof for nonexistence of new invariants of coadjoint orbits which are the integrals of local densities over the flow domain

nonlinear stability of circular vortex patches this paper proves that circular vortex patches in the plane are stable for the nonlinear dynamical system generated by the euler equations of incompressible fluids this is achieved by establishing a relative variational principle in terms of either energy or angular momentum thus we exploit and extend arnold s idea in 1965 1969 to a nonsmooth setting as well

on discontinuous action of monodromy groups on the complex n ball in dm 3 10 3 15 there is defined for each n 3 tuple 1u n 3 of nonintegral real numbers with ei2 i integral a subgroup i acting on a complex n 1 dimensional vector space vj which preserves a hermitian form of signature zi p 1 ei 1 ui 1 dm 2 21 where for any real number r r denotes its fractional part i e 0 0 we will be concerned principally with the case that the signature of the hermitian form is 1 positive n negative in which case the image of v in the projective space pn associated to v is the complex n ball bn also called the n disc we deal almost exclusively with the action of f on bn and pn we call an n 3 tuple u satisfying 0 pi 1 for i 1 n 3 and eiz i 2 a disc n 3 tuple in dm theorem 10 19 it is proved let u be a disc n 3 tuple satisfying condition int for all i a j such that ui uj 1

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

fuzzy clustering with viewpoints in this study we introduce a certain knowledge guided scheme of fuzzy clustering in which domain knowledge is represented in the form of so called viewpoints viewpoints capture a way in which the user introduces his her point of view at the data by identifying some representatives which being treated as externally introduced prototypes have to be included in the clustering process more formally the viewpoints views augment the original data based objective function by including the term that expresses distances between data and the viewpoints depending upon the nature of domain knowledge the viewpoints are represented either in a plain numeric format considering that there is a high level of specificity with regard to how one establishes perspective from which the data need to be analyzed or through some information granules which reflect a more relaxed way in which the views at the data are being expressed the detailed optimization schemes are presented and the performance of the method is illustrated through some numeric examples we also elaborate on a way in which the clustering with viewpoints enhances fuzzy models and mechanisms of decision making in the sense that the resulting constructs reflect the preferences and requirement that are present in the modeling environment

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

remarks on global regularity of 2d generalized mhd equations abstract in this paper we investigate the global regularity of 2d generalized mhd equations in which the dissipation term and magnetic diffusion term are u and b respectively let u 0 b 0 h s with s 2 it is showed that the smooth solution u x t b x t is globally regular for the case 0 1 2 3 2

limiting case of the sobolev inequality in bmo with application to the euler equations we shall prove a logarithmic sobolev inequality by means of the bmo norm in the critical exponents as an application we shall establish a blow up criterion of solutions to the euler equations

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

global small solutions to a tropical climate model without thermal diffusion we obtain the global well posedness of classical solutions to a tropical climate model with only the dissipation of the first baroclinic model of the velocity v under small initial data this model is a modified version of the original system derived by frierson majda pauluis in frierson et al commun math sci 2 591 626 2004 the main difficulty is the absence of thermal diffusion to overcome it we exploit the structure of the equations coming from the coupled terms dissipation term and damp term then we find the hidden thermal diffusion in addition based on the littlewood paley theory we establish a generalized commutator estimate which may be applied to other partial differential equations

a gauge theory for a quantum system with isospin a quantum system with external symmetry is made into a principal fibre bundle with a nonabelian lie group as the structure group and is equipped with a connection a gauge theory for a nonrigid molecule on the basis of the observation that the vector bundle associated with the principal fibre bundle provides a setting for quantum mechanics of the internal molecular motion

on the quantization of the modified poschl teller potential it is shown that the connection one form on the prequantization line bundle that generates the correct energy spectrum for the modified poschl teller potential consists of the canonical one form on the corresponding cotangent bundle together with an adjustment term that arises from the berry s connection

on the coupling between an ideal fluid and immersed particles in this paper we present finite dimensional particle based models for fluids which respect a number of geometric properties of the euler equations of motion specifically we use lagrange poincare reduction to understand the coupling between a fluid and a set of lagrangian particles that are supposed to simulate it we substitute the use of principal connections in cendra et al 2001 13 with vector field valued interpolations from particle velocity data the consequence of writing evolution equations in terms of interpolation is two fold first it provides estimates on the error incurred when interpolation is used to derive the evolution of the system second this form of the equations of motion can inspire a family of particle and hybrid particle spectral methods where the error analysis is built in we also discuss the influence of other parameters attached to the particles such as shape orientation or higher order deformations and how they can help us achieve a particle centric version of kelvin s circulation theorem

remarks on the global regularity of the two dimensional magnetohydrodynamics system with zero dissipation abstract we study the two dimensional magnetohydrodynamics system with generalized dissipation and diffusion in terms of fractional laplacians it is known that the classical magnetohydrodynamics system with full laplacians in both dissipation and diffusion terms admits a unique global strong solution pair making use of the special structure of the system in the two dimensional case we show in particular that the solution pair remains smooth when we have zero dissipation but only magnetic diffusion with its power of the fractional laplacian 3 2

higher order fuzzy entropy and hybrid entropy of a set new entropy measures such as higher order fuzzy entropy and hybrid entropy of a set have been introduced various properties along with their proofs have been included applicability of these new measures to various problems has been highlighted

classical non adiabatic angles if a family of tori in phase space is driven by a time dependent hamiltonian flow in such a way as to return after some time to the original family there generally results a shift in the angle variables one realisation of this process is in the cyclic adiabatic change of a classical hamiltonian and the angle change has previously been shown to separate naturally into a dynamical part and a geometrical part here the same geometrical angle change is extracted when the return is achieved non adiabatically and the dynamical remainder calculated two examples are given the precession of a spin and the rotation of phase space ellipses

an examination of indexes for determining the number of clusters in binary data sets the problem of choosing the correct number of clusters is as old as cluster analysis itself a number of authors have suggested various indexes to facilitate this crucial decision one of the most extensive comparative studies of indexes was conducted by milligan and cooper 1985 the present piece of work pursues the same goal under different conditions in contrast to milligan and cooper s work the emphasis here is on high dimensional empirical binary data binary artificial data sets are constructed to reflect features typically encountered in real world data situations in the field of marketing research the simulation includes 162 binary data sets that are clustered by two different algorithms and lead to recommendations on the number of clusters for each index under consideration index results are evaluated and their performance is compared and analyzed

geometrical analysis of free rotation of a rigid body rotational motion of a free rigid body without action of external torque is governed by the euler equation which is reformulated as a geodesic equation by introducing a riemannian metric and connection on the lie group so 3 the aim of this work is to investigate the stability of the motion from the viewpoint of geodesic variation and also to study the relationship between the instability and the riemannian sectional curvatures of variable signs the stability theorem of steady rotations known in the mechanics dynamical property is recovered by solving the jacobi equation for the variational field y a geometrical field existence of a conjugate point for the rigid body of any shape is confirmed and the condition for any jacobi field concerning the steady rotations such that to have a conjugate time is derived the sectional curvatures k s are calculated for the stable steady rotation the curvatures k s take either positive definite values or both positive and negative values in oscillatory manner depending on the inertia tensor however the time averages s are always positive for any y in the linearly stable case while there exist y s which make negative in the case of linear instability

nonprobabilistic entropies and indetermination measures in the setting of fuzzy sets theory abstract our purpose in this paper is first of all to build an axiomatic generalization for the nonprobabilistic entropy of de luca and termini in the setting of fuzzy sets theory we then build from this entropy an indetermination measure which can be used like discriminant function in pattern recognition when patterns are described by means of fuzzy sets

braid groups and hodge theory this paper gives an account of the unitary representations of the braid group that arise via the hodge theory of cyclic branched coverings of mathbb p 1 highlighting their connections with ergodic theory complex reflection groups moduli spaces of 1 forms and open problems in surface topology

the hamiltonian structure of nonlinear elasticity the material and convective representations of solids rods and plates it is our belief that a thorough understanding of the mathematical underpinnings of elasticity is crucial to its analytical and numerical implementation for example in the analysis of rotating structures the coupling of the equations for geometrically inexact models obtained by linearization or other approximations with those for rotating rigid bodies can easily lead to misleading artificial softening effects that can significantly alter numerical results see simo and vuquoc 1986c especially equations 3 and 5 in this paper we consider fully nonlinear geometrically exact models for rods plates and shells which take into account shear and torsion as well as the usual bending effects in traditional rod and plate models these models can be obtained either from the three dimensional theory by a systematic use of projection methods see e g antman 1972 and naghdi 1972 or by a direct approach within the context of cosserat continuum remarkably the two approaches lead to essentially the same form of the governing field equations in the present context we have chosen as a model problem a particular rod model which may be regarded as an extension of the classical kirchhoff love model see love 1944 to include shear deformations as in

global regularity of the two dimensional incompressible generalized magnetohydrodynamics system abstract in this paper we consider the global regularity of the two dimensional generalized incompressible magnetohydrodynamics system with velocity dissipation exponent 1 4 and magnetic diffusion exponent 1 based on the improved integrability in terms of vorticity this result is a further improvement of very recent works jiu and zhao 2014 and yamazaki 2013 which require 1 3 and 1 therefore our result is more closer to the resolution of the well known global regularity issue on the 2d magnetohydrodynamics system 0 1

an examination of procedures for determining the number of clusters in a data set a monte carlo evaluation of 30 procedures for determining the number of clusters was conducted on artificial data sets which contained either 2 3 4 or 5 distinct nonoverlapping clusters to provide a variety of clustering solutions the data sets were analyzed by four hierarchical clustering methods external criterion measures indicated excellent recovery of the true cluster structure by the methods at the correct hierarchy level thus the clustering present in the data was quite strong the simulation results for the stopping rules revealed a wide range in their ability to determine the correct number of clusters in the data several procedures worked fairly well whereas others performed rather poorly thus the latter group of rules would appear to have little validity particularly for data sets containing distinct clusters applied researchers are urged to select one or more of the better criteria however users are cautioned that the performance of some of the criteria may be data dependent

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

twist dynamics and aubry mather sets around a periodically perturbed point vortex abstract we consider the model of a point vortex under a periodic perturbation and give sufficient conditions for the existence of generalized quasi periodic solutions with rotation number the proof relies on aubry mather theory to obtain the existence of a family of minimal orbits of the poincare map associated to the system

finite mode analogs of 2d ideal hydrodynamics coadjoint orbits and local canonical structure abstract the algebraic finite mode hydrodynamic type systems that have o n integrals of motion for o n n modes and are intrinsically connected with two dimensional ideal fluid flows are studied special emphasis is given on the structure of their phase space and the problem of finding proper dynamical variables for its description in connection with appropriate problems for the original hydrodynamic equations various possibilities for embedding in the space of canonical variables are demonstrated on the simplest example of the su 3 model giving the finite dimensional analogs of clebsch variables

a remark on global regularity of 2d generalized magnetohydrodynamic equations abstract in this paper we study the global regularity of the following 2d two dimensional generalized magnetohydrodynamic equations u t u u p b b u b t u b b u b and get global regular solutions when 0 1 2 1 3 2 3 which improves the results in 5 in particular we obtain the global regularity of the 2d generalized mhd when 0 and 3 2

global regularity for a class of 2d tropical climate model abstract in this paper we establish the global regularity of a modified version of the tropical climate model originally derived by frierson majda pauluis 2004 6 which couples the barotropic and baroclinic modes of the velocity and the typical midtropospheric temperature in proving the global regularity to bypass the difficulty caused by the very weak dissipation any small power 0 of the barotropic mode we utilize the weakly nonlinear energy estimate approach and the maximal l t q l x p regularity for the heat kernel

geometrical aspects of ziglin s non integrability theorem for complex hamiltonian systems on place la methode de yoshida dans un cadre geometrique conceptuel et on etend son applicabilite

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

unitary braid group representations a large class of braid group representations are proven unitary which means a hermitian bilinear form left invariant by the representation under consideration exists this generalizes and improves known results on the burau and gassner representations the elementary methods are naturally suggested by the study of monodromy invariant hermitian forms applications to conformal quantum field theory in two dimensions are mentioned

symmetries and conservation laws of two dimensional hydrodynamics abstract we shall briefly review the hamiltonian and lie algebraic structure of 2 d hydrodynamics and its known conservation laws energy enstrophy the moments of vorticity then we utilize the eigenmode expansion of the euler equation to represented it as an infinite ode system and the detailed analysis of the structure coefficients to show that there are no other nontrivial conserved integrals among all linear and quadratic functional of vorticity this result provides a partial justification of the kraichnan energy enstrophy theory of 2 d turbulence

reduction theory and the lagrange routh equations reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of euler jacobi lagrange hamilton routh poincare and others the modern vision of mechanics includes besides the traditional mechanics of particles and rigid bodies field theories such as electromagnetism fluid mechanics plasma physics solid mechanics as well as quantum mechanics and relativistic theories including gravity symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas to subtle symmetries underlying integrable systems reduction theory concerns the removal of symmetries and their associated conservation laws variational principles along with symplectic and poisson geometry provide fundamental tools for this endeavor reduction theory has been extremely useful in a wide variety of areas from a deeper understanding of many physical theories including new variational and poisson structures to stability theory integrable systems as well as geometric phases this paper surveys progress in selected topics in reduction theory especially those of the last few decades as well as presenting new results on non abelian routh reduction we develop the geometry of the associated lagrange routh equations in some detail the paper puts the new results in the general context of reduction theory and discusses some future directions

on the global evolution of vortex filaments blobs and small loops in 3d ideal flows we consider a wide class of approximate models of evolution of singular distributions of vorticity in three dimensional incompressible fluids and we show that they have global smooth solutions the proof exploits the existence of suitable hamiltonian functions the approximate models we analyze essentially discrete and continuous vortex filaments and vortex loops are related to some problem of classical physics concerning turbulence and also to the numerical approximation of flows with very high reynolds number finally we apply our strategy to discrete models for filaments used in numerical methods

on the topological phases in a planar n point vortex system a topological phase is given for the planar n point vortex problem this phase arises from an element in the pure braid group and is independent of the evolution equations it is deduced from a regular representation of the group of diffeomorphisms on the plane this resulting representation of the pure braid group is the monodromy of the hypergeometric functions in n variables the relations between this topological phase and the usual geometrical phase are discussed

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

area preserving diffeomorphisms of the tetrahedron expressions for the curvature and structure constants of the area preserving diffeomorphism group of the tetrahedron are derived finite dimensional algebras are identified that yield the diffeomorphism lie algebra in an infinite limit

energy conservation for the 3d tropical climate model in bounded domains abstract in this paper we consider the energy conservation to the 3d tropical climate model in bounded domain by exploring the special structure of the nonlinear components and using the ideas introduced in 1 we can give some sufficient conditions for the weak solutions to ensure energy conservation

on the global regularity of two dimensional generalized magnetohydrodynamics system abstract we study the two dimensional generalized magnetohydrodynamics system with dissipation and diffusion in terms of fractional laplacians in particular in the case where the diffusion term has the power 1 in contrast to the previous result of 1 2 we show that 1 3 suffices in order for the solution pair of velocity and magnetic fields to remain smooth for all time

monodromy for the hypergeometric function nfn 1 la fonction hypergeometrique le groupe hypergeometrique la forme hermitienne invariante le cas imprimitif theorie de galois differentielle fonctions hypergeometriques algebriques

clebsch lagrange variational principle and geometric constraint analysis of relativistic field theories inspired by the clebsch optimal control problem we introduce a new variational principle that is suitable for capturing the geometry of relativistic field theories with constraints related to a gauge symmetry its special feature is that the lagrange multipliers couple to the configuration variables via the symmetry group action the resulting constraints are formulated as a condition on the momentum map of the gauge group action on the phase space of the system we discuss the hamiltonian picture and the reduction in the gauge symmetry by stages in this geometric setting we show that the yang mills higgs action and the einstein hilbert action fit into this new framework after a 1 3 splitting moreover we recover the gaus constraint of yang mills higgs theory and the diffeomorphism constraint of general relativity as momentum map constraints inspired by the clebsch optimal control problem we introduce a new variational principle that is suitable for capturing the geometry of relativistic field theories with constraints related to a gauge symmetry its special feature is that the lagrange multipliers couple to the configuration variables via the symmetry group action the resulting constraints are formulated as a condition on the momentum map of the gauge group action on the phase space of the system we discuss the hamiltonian picture and the reduction in the gauge symmetry by stages in this geometric setting we show that the yang mills higgs action and the einstein hilbert action fit into this new framework after a 1 3 splitting moreover we recover the gaus constraint of yang mills higgs theory and the diffeomorphism constraint of general relativity as momentum map constraints

invariants of the euler equations for ideal or barotropic hydrodynamics and superconductivity in d dimensions abstract the hamiltonian formalism for the euler equations of an ideal fluid superconductivity and a barotropic fluid on a d dimensional riemannian manifold is proposed we show that each of these equations has an infinite series of integrals if d is even generalized enstrophies and at least one integral if d is odd generalized helicity we prove that the magnetic hydrodynamics integral f v b is equal to the average linking number of vector fields rot v and b in terms of the ergodic theory all the invariants considered are casimir elements i e invariants of coadjoint action of the corresponding infinite dimensional lie algebras

functional clustering and identifying substructures of longitudinal data summary a functional clustering fc method k centres fc for longitudinal data is proposed the k centres fc approach accounts for both the means and the modes of variation differentials between clusters by predicting cluster membership with a reclassification step the cluster membership predictions are based on a non parametric random effect model of the truncated karhunen loeve expansion coupled with a non parametric iterative mean and covariance updating scheme we show that under the identifiability conditions derived the k centres fc method proposed can greatly improve cluster quality as compared with conventional clustering algorithms moreover by exploring the mean and covariance functions of each cluster thek centres fc method provides an additional insight into cluster structures which facilitates functional cluster analysis practical performance of the k centres fc method is demonstrated through simulation studies and data applications including growth curve and gene expression profile data

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

on coadjoint orbits of rotational perfect fluids in this paper the structure of vortex coadjoint orbits pertaining to perfect fluids having smooth vorticities in r3 within the framework set up by j marsden and a weinstein physica d 7 305 323 1983 in terms of an associated hamiltonian kahler manifold the clebsch manifold described in terms of the so called clebsch variables is investigated the topological quantization of mikhailov and kuznetsov is related to geometric quantization natural candidates for the coherent states on the clebsch manifold are also exhibited

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

a negatively curved kahler threefold not covered by the ball we construct examples of three dimensional compact kahler manifolds with negative curvature not covered by the ball our manifolds are obtained as a natural generalization of the two dimensional examples discovered by mostow and siu using their description in terms of monodromy covers of hypergeometric functions each example is obtained from a hypergeometric monodromy group in pu 3 1 that is not discrete but has finite local monodromy we describe a manifold on which the group acts discretely and check that it has a compact quotient with the above features the examples are locally described as branched covers of the ball with totally geodesic branch locus

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

on the use of fuzzy constraints in semisupervised clustering this paper introduces fuzzy hss a semisupervised hierarchical clustering approach that uses fuzzy instance level constraints these constraints are external information on the shape of fuzzy must link and fuzzy cannot link restrictions they allow uncertainty when indicating whether two instances of a dataset belong to the same group fuzzy must link constraints give a degree of belief of two instances belonging to the same group analogously fuzzy cannot link constraints indicate the degree of belief of two instances not belonging to the same group these constraints have been introduced in a hierarchical clustering process allowing us to obtain the optimal number of groups in a dendrogram when the number of clusters is not known the optimal amount of constraints needed in the process is determined by means of fuzzy entropy an extensive experimental study is provided by comparing this fuzzy semisupervised approach with classic unsupervised methods as well as a crisp semisupervised alternative

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