relative equilibria of point vortices on the hyperbolic sphere relative equilibria of point vortices on a hyperbolic sphere with constant negative curvature is formulated and considered known symmetries for the vortex motion on the hyperbolic sphere are expounded by setting up a geometrical formulation of dynamics in particular the configuration of rings of vortices is investigated in detail

essential self adjointness of liouville operator for 2d euler point vortices abstract we analyse the 2 dimensional euler point vortices dynamics in the koopman von neumann approach classical results provide well posedness of this dynamics involving singular interactions for a finite number of vortices on a full measure set with respect to the volume measure d x n on the phase space which is preserved by the measurable flow thanks to the hamiltonian nature of the system we identify a core for the generator of the one parameter group of koopman von neumann unitaries on l 2 d x n associated to said flow the core being made of observables smooth outside a suitable set on which singularities can occur

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

a continuum of relative equilibria in the five body problem abstract it is generally believed that the set of relative equilibria equivalence classes in the newtonian n body problem for a given set of positive masses is finite however the result has only been proven for n 3 and remains a difficult open question for n 4 a wintner princeton math ser 5 215 princeton university press princeton nj 1941 s smale mathematical intelligencer 20 1998 7 15 we demonstrate that the condition for the masses being positive is a necessary one by finding a continuum of relative equilibria in the five body problem which unfortunately includes one negative mass this family persists in similar potential functions including the logarithmic potential used to describe the motion of point vortices in a plane of fluid

topological bifurcation of relative equilibria for the kirchhoff 8 vortex problem abstract in this article we prove necessary and sufficient conditions for the existence of local and global bifurcations of relative equilibria of the 8 vortex problem applying the equivariant bifurcation theory we prove that from the families of regular concentric squares forming relative equilibria bifurcate relative equilibria whose shape is more complicated

the motion of a vortex on a closed surface of constant negative curvature the purpose of this work is to present an algorithm to determine the motion of a single hydrodynamic vortex on a closed surface of constant curvature and of genus greater than one the algorithm is based on a relation between the laplace beltrami green function and the heat kernel the algorithm is used to compute the motion of a vortex on the bolza surface this is the first determination of the orbits of a vortex on a closed surface of genus greater than one the numerical results show that all the 46 vortex equilibria can be explicitly computed using the symmetries of the bolza surface some of these equilibria allow for the construction of the first two examples of infinite vortex crystals on the hyperbolic disc the following theorem is proved a weierstrass point of a hyperellitic surface of constant curvature is always a vortex equilibrium

instrumental variable estimation in functional linear models in an increasing number of empirical studies the dimensionality measured e g as the size of the parameter space of interest can be very large two instances of large dimensional models are the linear regression with a large number of covariates and the estimation of a regression function with many instrumental variables an appropriate setting to analyze high dimensional problems is provided by a functional linear model in which the covariates belong to hilbert spaces this paper considers the case where covariates are endogenous and assumes the existence of instrumental variables that are functional as well the paper shows that estimating the regression function is a linear ill posed inverse problem with a known but data dependent operator the first contribution is to analyze the rate of convergence of the penalized least squares estimator based on the result we discuss the notion of instrument strength in the high dimensional setting we also discuss a generalized version of the estimator when the problem is premultiplied by an instrument dependent operator this extends the technology of generalized method of moments to high dimensional functional data a central limit theorem is also established on the inner product of the estimator the studied estimators are easy and fast to implement and the finite sample performance is discussed through simulations and an application to the impact of age specific fertility rate curves on yearly economic growth in the united kingdom

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

quasi coherent states for unitary groups the author extends a recent construction of isotropic spin coherent states to unitary groups expectation values in such a quasi coherent state can be obtained from a generating functional which is given explicitly for the fundamental and the adjoint representations of the unitary groups su n and u n the close connection between the corresponding generating functional and the external field problem in qcd is pointed out free quantum fields in such a condensed colour singlet coherent state are formally related to two dimensional lattice gauge theories with in general a mixed action in the large n limit the author exhibits a phase transition for a free quantum field in a singlet quasi coherent state a path integral representation of the transition amplitude in terms of quasi coherent states is also given the corresponding effective action describes dynamics on a complex kahler manifold

on polygonal relative equilibria in the n vortex problem helmholtz s equations provide the motion of a system of n vortices which describes a planar incompressible fluid with zero viscosity a relative equilibrium is a particular solution of these equations for which the distances between the vortices are invariant during the motion in this article we first show that a relative equilibrium formed of a regular polygon and a possible vortex at the center with more than three vertices on the polygon two if there is a vortex at the center requires equal vorticities on the polygon we also provide an 8 vortex configuration formed of two concentric squares making an angle of 45 with uniform vorticity on each square which is in relative equilibrium for any value of the vorticities

synchronization of harmonic oscillators under restorative coupling with applications in electrical networks the role of restorative coupling on synchronization of coupled identical harmonic oscillators is studied necessary and sufficient conditions under which the individual systems solutions converge to a common trajectory are presented through simple physical examples the meaning and limitations of the theorems are expounded also to demonstrate their versatility the results are extended to cover lti passive electrical networks one of the extensions generalizes the well known link between the asymptotic stability of the synchronization subspace and the second smallest eigenvalue of the laplacian matrix

hamiltonian systems near relative equilibria abstract we give explicit differential equations for the dynamics of hamiltonian systems near relative equilibria these split the dynamics into motion along the group orbit and motion inside a slice transversal to the group orbit the form of the differential equations that is inherited from the symplectic structure and symmetry properties of the hamiltonian system is analysed and the effects of time reversing symmetries are included the results will be applicable to the stability and bifurcation theories of relative equilibria of hamiltonian systems

finiteness of stationary configurations of the four vortex problem we show that the number of relative equilibria equilibria and rigidly translating configurations in the problem ot tour point vortices is hnite the proof is based on symbolic and exact integer computations which are carried out by computer we also provide upper bounds for these classes of stationary configurations

thomson s heptagon a case of bifurcation at infinity abstract vortex modeling has a long history descartes 1644 used it as a model for the solar systems j j thomsom 1883 used it as a model for the atom we consider point vortex systems which can be regarded as discrete solutions of the euler equation their dynamics is described by a hamiltonian system of equations we are interested in polygonal configurations and how their stability depends upon various dynamical variables in the plane a polygon with seven vortices has been shown to be a special boundary case polygons with n 7 vortices are linearly and nonlinearly stable while polygons with n 7 vortices are unstable why should n 7 be special celestial mechanics helped us to simplify a problem that has been studied for over a century and to show that the case of thomson s heptagon is actually a case of bifurcation at infinity this becomes particularly clear when considering the corresponding problem of a ring on a sphere with two polar vortices of variable intensities n and s at the north and south pole respectively

point vortices on the hyperbolic plane we investigate the dynamical system of point vortices on the hyperboloid this system has non compact symmetry sl 2 r and a coadjoint equivariant momentum map the relative equilibrium conditions are found and the trajectories of relative equilibria with non zero momentum value are described we also provide the classification of relative equilibria and the stability criteria for a number of cases focusing on 2 and 3 vortices unlike the system on the sphere this system has relative equilibria with non compact momentum isotropy subgroup and these are used to illustrate the different stability types of relative equilibria

hannay berry phase and the restricted three vortex problem abstract asymptotic expansions for the hannay berry phase are computed for a class of hamiltonian systems in perturbed action angle form with slowly varying parameters where the parameters recur on a slow time scale a multi scale perturbation method is used which in principle can be carried to any desired order of accuracy as a specific application we use the method to examine the restricted three vortex problem a completely integrable configuration of two co rotating point vortices a distance d apart and a tracer particle rotating around one of the vortices with distance r in the asymptotic limit r d e we compute the particle trajectory through o e and find that the hannay berry phase is given by 2 3 r 0 d o 1 d 2 where r 0 is the conserved action

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

the n vortex problem on a rotating sphere ii heterogeneous platonic solid equilibria we describe a new method of constructing point vortex equilibria on a sphere made up of n vortices with different strengths such equilibria called heterogeneous equilibria are obtained for the five platonic solid configurations hence for graphic 1 the method is based on calculating a basis set for the nullspace of a matrix obtained by enforcing the necessary and sufficient condition that the mutual distances between each pair of vortices remain constant by symmetries inherent in the platonic solid configurations this matrix is reduced for each case and we call the dimension of the nullspace the degree of heterogeneity of the structure for the tetrahedron n 4 and octahedron n 6 the degree of heterogeneity is 4 and 6 respectively hence we are free to choose each of the vortex strengths independently for the cube n 8 the degree of heterogeneity is 5 for the icosahedron n 12 it is 7 while for the dodecahedron n 20 it is 4 thus the entire set of equilibria based on the platonic solid configurations is obtained including substructures associated with each configuration constructed by taking different linear combinations of the basis elements 1 embed inline graphic 1 gif

high dimensional covariance matrix estimation using a factor model abstract high dimensionality comparable to sample size is common in many statistical problems we examine covariance matrix estimation in the asymptotic framework that the dimensionality p tends to as the sample size n increases motivated by the arbitrage pricing theory in finance a multi factor model is employed to reduce dimensionality and to estimate the covariance matrix the factors are observable and the number of factors k is allowed to grow with p we investigate the impact of p and k on the performance of the model based covariance matrix estimator under mild assumptions we have established convergence rates and asymptotic normality of the model based estimator its performance is compared with that of the sample covariance matrix we identify situations under which the factor approach increases performance substantially or marginally the impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied the asymptotic results are supported by a thorough simulation study

angle variable holonomy in adiabatic excursion of an integrable hamiltonian if an integrable classical hamiltonian h describing bound motion depends on parameters which are changed very slowly then the adiabatic theorem states that the action variables i of the motion are conserved here the fate of the angle variables is analysed because of the unavoidable arbitrariness in their definition angle variables belonging to distinct initial and final hamiltonians cannot generally be compared however they can be compared if the hamiltonian is taken on a closed excursion in parameter space so that initial and final hamiltonians are the same the result shows that the angle variable change arising from such an excursion is not merely the time integral of the instantaneous frequency omega dh di but differs from it by a definite extra angle which depends only on the circuit in parameter space not on the duration of the process the 2 form which describes this angle variable holonomy is calculated

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

methodology and convergence rates for functional linear regression in functional linear regression the slope parameter is a function therefore in a nonparametric context it is determined by an infinite number of unknowns its estimation involves solving an ill posed problem and has points of contact with a range of methodologies including statistical smoothing and deconvolution the standard approach to estimating the slope function is based explicitly on functional principal components analysis and consequently on spectral decomposition in terms of eigenvalues and eigen functions we discuss this approach in detail and show that in certain circumstances optimal convergence rates are achieved by the pca technique an alternative approach based on quadratic regularisation is suggested and shown to have advantages from some points of view

quantum phase corrections from adiabatic iteration the phase change gamma acquired by a quantum state psi t rangle driven by a hamiltonian h 0 t which is taken slowly and smoothly round a cycle is given by a sequence of approximants gamma k obtained by a sequence of unitary transformations the phase sequence is not a perturbation series in the adiabatic parameter epsilon because each gamma k except gamma 0 contains epsilon to infinite order for spin frac 1 2 systems the iteration can be described in terms of the geometry of parallel transport round loops c k on the hamiltonian sphere non adiabatic effects transitions must cause the sequence of gamma k to diverge for spin systems with analytic h 0 t this happens in a universal way the loops c k are sinusoidal spirals which shrink as epsilon k until k sim epsilon 1 and then grow as k the smallest loop has a size exp 1 epsilon comparable with the non adiabaticity

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

functional linear regression with functional response abstract in this paper we develop new estimation results for functional regressions where both the regressor z t and the response y t are functions of hilbert spaces indexed by the time or a spatial location the model can be thought as a generalization of the multivariate regression where the regression coefficient is now an unknown operator we propose to estimate the operator by tikhonov regularization which amounts to apply a penalty on the l 2 norm of we derive the rate of convergence of the mean square error the asymptotic distribution of the estimator and develop tests on as trajectories are often not fully observed we consider the scenario where the data become more and more frequent infill asymptotics we also address the case where z is endogenous and instrumental variables are used to estimate an application to the electricity consumption completes the paper

instrumental variable approach to estimating the scalar on function regression model with measurement error with application to energy expenditure assessment in childhood obesity wearable device technology allows continuous monitoring of biological markers and thereby enables study of time dependent relationships for example in this paper we are interested in the impact of daily energy expenditure over a period of time on subsequent progression toward obesity among children data from these devices appear as either sparsely or densely observed functional data and methods of functional regression are often used for their statistical analyses we study the scalar on function regression model with imprecisely measured values of the predictor function in this setting we have a scalar valued response and a function valued covariate that are both collected at a single time period we propose a generalized method of moments based approach for estimation while an instrumental variable belonging in the same time space as the imprecisely measured covariate is used for model identification additionally no distributional assumptions regarding the measurement errors are assumed while complex covariance structures are allowed for the measurement errors in the implementation of our proposed methods we demonstrate that our proposed estimator is l2 consistent and enjoys the optimal rate of convergence for univariate nonparametric functions in a simulation study we illustrate that ignoring measurement error leads to biased estimations of the functional coefficient the simulation studies also confirm our ability to consistently estimate the function valued coefficient when compared to approaches that ignore potential measurement errors our proposed methods are applied to our motivating example to assess the impact of baseline levels of energy expenditure on body mass index among elementary school aged children

a remark on the kam theorem applied to a four vortex system we consider a planar four vortex system with unit intensities and apply the kam theorem for two dimensional tori with fixed frequency we obtain a rigorous lower bound for the stochasticity threshold of the torus with rotation number 5 1 2 and compare our result with numerical experiments

the correlation coefficient for vortices in the plane abstract for point vortices in the plane we consider the correlation coefficient of ovchinnikov and sigal generalising a result by esposito we show that it vanishes for all vortex equilibria

point vortex equilibria on the sphere via brownian ratchets we describe a brownian ratchet scheme that we use to calculate relative equilibrium configurations of n point vortices of mixed strength on the surface of a unit sphere we formulate it as a proble

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

centroidal frames in dynamical systems i point vortices the dynamics of point vortices is studied in part i of the paper it is well known that the translational centre of mass frame decomposes the motion of a mechanical system into simpler components it is less known however that special rotational frames have also been suggested for the same purpose in contrast to the centre of mass frame the angular velocities of these rotational frames are not given explicitly that limits their application to small perturbations of rigid body rotations a new class of centroidal frames cf related to different groups such as translation rotation dilation etc is introduced in this paper the cfs decompose the motion of point vortices into a group and a relative components without restriction to small perturbations of pure group motions the definition of the cfs is based on an averaging of motion or on minimization of energy of the relative motion where an appropriate energy function is expressed through generators of the group action as a result the linear and angular velocities as well as other characteristics of the cfs can be obtained explicitly part i of the paper presents application of the cfs to a hamiltonian system of point vortices examples of integrable and chaotic motions in the cfs visualize dynamical patterns that are completely hidden in the conventional fixed frame ff motions which look like chaotic in the ff reveal a variety of stable and unstable structures in the cfs quasi periodic and chaotic motions coexist for all energies and the cfs permit to clearly distinguish between them a new phenomenon of asymptotic symmetries in rotational cfs of some chaotic motions is discovered this is related to a permutation symmetry of the hamiltonian

green s function for the laplace beltrami operator on a toroidal surface green s function for the laplacebeltrami operator on the surface of a three dimensional ring torus is constructed an integral ingredient of our approach is the stereographic projection of the toru

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

the n vortex problem on a rotating sphere i multi frequency configurations the problem of n point vortices moving on a rotating unit sphere is considered through a sequence of linear coordinate transformations which takes into account the orientation of the centre of vorticity vector with respect to the axis of rotation we show how to reduce the problem to that on a non rotating sphere where the centre of vorticity vector is aligned with the z axis as a consequence we prove that integrability on the rotating sphere is the same as on the non rotating sphere namely the three vortex problem on the rotating sphere is integrable for all vortex strengths while the four vortex problem is integrable in the special case where the centre of vorticity is zero rigid multi frequency configurations that retain their shape while rotating about two independent axes with two independent frequencies are obtained and necessary conditions for one and two frequency motions are derived examples including dipoles which exhibit global wobbling and tumbling dynamics rings and platonic solid configurations are shown to undergo either periodic or quasi periodic evolution on the rotating sphere depending on the ratio of the solid body rotational frequency to the rotational frequency associated with the rigid structure

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

n body dynamics on closed surfaces the axioms of mechanics a major challenge for our understanding of the mathematical basis of particle dynamics is the formulation of n body and n vortex dynamics on riemann surfaces in this paper we show how the two problems are in fact closely related when considering the role played by the intrinsic geometry of the surface this enables a straightforward deduction of the dynamics of point masses using recently derived results for point vortices on general closed differentiable surfaces m endowed with a metric g we find generally that kepler s laws do not hold what is more even newton s first law the law of inertia fails on closed surfaces with variable curvature e g the ellipsoid

nonlinear stability of a latitudinal ring of point vortices on a nonrotating sphere we study the nonlinear stability of relative equilibria of configurations of identical point vortices on the surface of a sphere in particular we study how the stability changes as a function of the colatitude theta and of the number of vortices n by using the integrals of motion we view the system in asuitable corotating frame where the polygonal vortex configuration is at rest then after a sufficient criterion due to dirichlet the stability ranges are the theta intervals for which the hessian of the hamiltonian evaluated at the equilibrium configuration is positive or negative definite we find that the stability intervals coincide with those for linear stability determined by polvani and dritschel j fluid mech 255 1993 pp 35 64 for n 3 we recover the result previously established by pekarsky and marsden j math phys 39 1998 pp 5894 5907

stationary configurations of point vortices the motion of point vortices in a plane of fluid is an old problem of fluid mechanics which was given a hamiltonian formulation by kirchhoff stationary configurations are those which remain self similar throughout the motion results of two types are presented configurations which are in equilibrium or which translate uniformly are counted using methods of algebraic geometry which establish necessary and sufficient conditions for existence relative equilibria rigidly rotating configurations which lie on a line are studied using a topological construction applicable to other power law systems upper and lower bounds for such configurations are found for vortices with mixed circulations arrangements of three vortices which collide in finite time are well known one dimensional families of such configurations are shown to exist for more than three vortices stationary configurations of four vortices are examined in detail the motion of point vortices in a plane of fluid is an old problem of fluid mechanics which was given a hamiltonian formulation by kirchhoff stationary configurations are those which remain self similar throughout the motion and are of considerable physical interest it is known that a configuration of vortices in equilibrium must have total vortex angular momentum 0 a converse is proved namely that for almost every choice of circulations with zero vortex angular momentum there are exactly n 2 equilibrium configurations a similar statement is proved for rigidly translating configurations with total circulation zero the proofs involve ideas from algebraic geometry relative equilibria rigidly rotating configurations were studied by palmore in the case of positive circulations upper and lower bounds for the number of collinear relative equilibria for arbitrary circulations are obtained by means of a topological construction which is applicable to other power law systems arrangements of three vortices which collide in finite time are well known one dimensional families of such configurations are shown to exist for n vortices n 3 stationary configurations of four vortices are examined in detail chapter 0 introduction the motion of point vortices in the plane is an old problem in fluid mechanics it was first given a hamiltonian formulation by kirchhoff 1876 who proved that n vortices with distinct positions xi yi i 1 n in the x y plane and received by the editors july 31 1985 and in revised form june 13 1986 1980 mathematics subject classification 1985 revision primary 76c05 secondary 70h05 t1987 american mathematical society 0002 9947 87 1 00 25 per page

numerical description of the motion of a point vortex pair on ovaloids this paper presents a numerical study on the dynamics of point vortex pairs moving on convex closed surfaces ovaloids approximated by triangular icosahedral meshes firstly both a discretized conformal mapping between the ovaloid and the unit sphere and a least squares fitting for this approximated conformal map in terms of a spherical harmonic expansion are obtained then an approximation of the conformal spherical metric of the ovaloid is derived by obtaining a spherical harmonic expansion of the corresponding conformal factor and its gradient on the equations of motion for a point vortex pair on with this quasi conformal metric are integrated by using the classical gaussian collocation method for some particular models

minimal polynomial systems for point vortex equilibria point vortices in a two dimensional fluid may be arranged so that all resulting vortex velocities are identical these are equivalent to force free arrangements of 2d coulomb charges in a homogeneous field for generic choice of vortex circulations the number of stationary configurations is finite but a few infinite solution families of arbitrary positive dimension are known to exist this paper extends the classes of point vortex systems known to have a finite or infinite number of stationary configurations respectively the main tool is a new minimal system of polynomial equations characterizing these configurations the polynomial system is extended to include symmetric vortex configurations such as vortex streets and rings as well as circulation symmetries vortices with identical circulations by expressing the systems as a multilinear differential equation one dimensional solution sets can be constructed for a wide range of systems including vortex streets and rings

morse theory and relative equilibria in the planar n vortex problem morse theoretical ideas are applied to the study of relative equilibria in the planar n vortex problem for the case of positive circulations we prove that the morse index of a critical point of the hamiltonian restricted to a level surface of the angular impulse is equal to the number of pairs of real eigenvalues of the corresponding relative equilibrium periodic solution the morse inequalities are then used to prove the instability of some families of relative equilibria in the four vortex problem with two pairs of equal vorticities we also show that for positive circulations relative equilibria cannot accumulate on the collision set

symmetry of planar four body convex central configurations we study the relationship between the masses and the geometric properties of central configurations we prove that in the planar four body problem a convex central configuration is symmetric with respect to one diagonal if and only if the masses of the two particles on the other diagonal are equal if these two masses are unequal then the less massive one is closer to the former diagonal finally we extend these results to the case of non planar central configurations of five particles

three ways to solve the poisson equation on a sphere with gaussian forcing motivated by the needs of vortex methods we describe three different exact or approximate solutions to the poisson equation on the surface of a sphere when the forcing is a gaussian of the three dimensional distance 2 j exp 2 e 2 1 cos q c g a u s s e more precisely the forcing is a gaussian minus the gauss constraint constant c g a u s s this subtraction is necessary because j is bounded for any type of forcing only if the integral of the forcing over the sphere is zero y kimura h okamoto vortex on a sphere j phys soc jpn 56 1987 4203 4206 d g dritschel contour dynamics surgery on the sphere j comput phys 79 1988 477 483 the legendre polynomial series is simple and yields the exact value of the gauss constraint constant but converges slowly for large e the analytic solution involves nothing more exotic than the exponential integral but all four terms are singular at one or the other pole cancelling in pairs so that j is everywhere nice the method of matched asymptotic expansions yields simpler uniformly valid approximations as series of inverse even powers of e that converge very rapidly for the large values of e e 40 appropriate for geophysical vortex computations the series converges to a nonzero o exp 4 e 2 error everywhere except at the south pole where it diverges linearly with order instead of the usual factorial order

motion of three point vortices on a sphere abstract we solve the equations governing the relative motion of three point vortices of arbitrary strength moving on the surface of a sphere of radius r the system is more general than the corresponding one in the plane 1 3 5 19 28 33 34 37 38 and reduces to it in the limit r as long as the three vortices remain sufficiently close to each other during the course of their motion instead of using spherical coordinates we use cartesian coordinates in which the vector i points from the center of the sphere to the vortex with strength i an important conserved quantity is the center of vorticity vector c i i i which with no loss of generality we align with the z axis based on the size of this vector relative to the radius of the sphere we classify the motion into one of five types super radial radial sub radial degenerate or a limiting super radial case this categorization allows us to draw several conclusions about the qualitative motion of the vortices we then fully characterize all fixed and relative equilibria on the sphere for fixed equilibria the vortices must lie on great circles geodesics if the strengths are equal they form an equilateral triangle otherwise the triangle shape is specified once the strength of the vortices is given the relative equilibria are classified as either degenerate c 0 or non degenerate c 0 for each type the shape of the vortex triangle is described and the frequency of rotation is computed as in the planar problem it is possible to introduce trilinear coordinates and study the motion in a phase plane which allows us to locate all the equilibria as well as to characterize more complex relative dynamics we then describe self similar vortex collapse on the sphere stating necessary and sufficient conditions for collapse to occur computing the collapse times and vortex trajectories on the route towards collapse comparisons are made with the collapse formulas in the plane derived in 1 30

time optimal control of linear time invariant systems between two arbitrary states abstract new results are presented for the problem of minimum time control of a general linear time invariant normal system evolving from an arbitrary initial state to an arbitrary final state no rest to no rest problem subjected to bound controls in particular it is demonstrated that the above problem is equivalent to an associated minimum time control problem of transferring the same system from a particular initial state to the state space origin no rest to rest problem where that particular initial state is related to the boundary states of the original problem through a transformation of the state space onto itself if the optimal control history that transfers the system from an arbitrary initial state to the origin is known either analytically or numerically then the new results provide a method to solve the problem of minimum time control between two arbitrary states and moreover to find all of the extremal controlled trajectories a criterion of existence is also given for the solution of the minimum time control between two arbitrary states finally the analytic solution is presented for the minimum time control of the double integrator between arbitrary states that solution provides a significant example of applying the new results

diffusion on a sphere with localized traps mean first passage time eigenvalue asymptotics and fekete points a common scenario in cellular signal transduction is that a diffusing surface bound molecule must arrive at a localized signaling region on the cell membrane before the signaling cascade can be completed the question then arises of how quickly such signaling molecules can arrive at newly formed signaling regions here we attack this problem by calculating asymptotic results for the mean first passage time for a diffusing particle confined to the surface of a sphere in the presence of n partially absorbing traps of small radii the rate at which the small diffusing molecule becomes captured by one of the traps is determined by asymptotically calculating the principal eigenvalue for the laplace operator on the sphere with small localized traps the asymptotic analysis relies on the method of matched asymptotic expansions together with detailed properties of the green s function for the laplacian and the helmholtz operators on the surface of the unit sphere the asymptotic results compare favorably with

on geometric phases for soliton equations this paper develops a new complex hamiltonian structure forn soliton solutions for a class of integrable equations such as the nonlinear schrodinger sine gordon and korteweg de vries hierarchies of equations that yields amongst other things geometric phases in the sense of hannay and berry for example one of the possible soliton geometric phases is manifested by the well known phase shift that occurs for interacting solitons the main new tools are complex angle representations that linearize the corresponding hamiltonian flows on associated noncompact jacobi varieties this new structure is obtained by taking appropriate limits of the differential equations describing the class of quasi periodic solutions a method of asymptotic reduction of the angle representations is introduced for investigating soliton geometric phases that are related to the presence of monodromy at singularities in the space of parameters in particular the phase shift of interacting solitons can be expressed as an integral over a cycle on an associated riemann surface in this setting soliton geometric asymptotics are constructed for studying geometric phases in the quantum case the general approach is worked out in detail for the three specific hierarchies of equations mentioned some links with functions the braid group and geometric quantization are pointed out as well

the topology of reduces phase spaces of the motion of vortices on a sphere abstract the motion of point vortices in a perfect incompressible fluid on a two dimensional sphere can be represented by a hamiltonian flow which is invariant under the action of the special orthogonal group so 3 following the reduction method of marsden and weinstein this motion can be described by hamiltonian flows on reduced phase spaces some topological invariants of these reduced phase spaces are calculated for suitable vorticities the results obtained can be used to give lower bounds on numbers of relative equilibria of the original system

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

the continuation method for rational and deficient systems deficient polynomial systems those with fewer than the full number of solutions cannot be solved efficiently with the usual continuation methods this paper describes a continuation technique which allows some deficient and rational systems of equations to be solved with little wasted computation several examples from the literature are discussed

kam theory analysis of the dynamics of three coaxial vortex rings abstract the equations of motion of three coaxial vortex rings in euclidean 3 space are formulated as a hamiltonian system it is shown that the hamiltonian function for this system can be written as the sum of a completely integrable part h0 related to the motion of three point vortices in the plane and a non integrable perturbation h1 then it is proved that when the vortex strengths all have the same sign and the ratio of the mean distances among the rings is very small in comparison to the mean radius of the rings h1 h0 1 moreover it is shown that h1 h0 is very small for all time for certain initial positions of the rings under the same assumptions it is proved that the decomposition of the hamiltonian and the estimates carry over to a reduced form of the system in coordinates moving with the center of vorticity and having one less degree of freedom then kam theory is applied to prove the existence of invariant two dimensional tori containing quasiperiodic motions the existence of periodic solutions is also demonstrated several examples are solved numerically to show transitions from quasiperiodic and periodic to chaotic regimes in accordance with the theoretical results

the connection whose holonomy is the classical adiabatic angles of hannay and berry and its generalization to the non integrable case we show how averaging defines an ehresmann connection whose holonomy is the classical adiabatic angles which hannay defined for families of completely integrable systems the averaging formula we obtain for the connection only requires that the family of hamiltonians has a continuous symmetry group this allows us to extend the notion of the hannay angles to families of non integrable systems with symmetry we state three geometric axioms satisfied by the connection these axioms uniquely determine the connection thus enabling us to find new formulas for the connection and its curvature two examples are given

uniqueness results for co circular central configurations for power law potentials for a class of potential functions including those used for the planar n body and n vortex problems we investigate co circular central configurations whose center of mass coincides with the center of the circle containing the bodies useful equations are derived that completely describe the problem using a topological approach it is shown that for any choice of positive masses or circulations if such a central configuration exists then it is unique it quickly follows that if the masses are all equal then the only solution is the regular n gon for the planar n vortex problem and any choice of the vorticities we show that the only possible co circular central configuration with center of vorticity at the center of the circle is the regular n gon with equal vorticities

the method of images and green s function for spherical domains motivated by problems in electrostatics and vortex dynamics we develop two general methods for constructing green s function for simply connected domains on the surface of the unit sphere we prove a riemann mapping theorem showing that such domains can be conformally mapped to the upper hemisphere we then categorize all domains on the sphere for which green s function can be constructed by an extension of the classical method of images we illustrate our methods by several examples such as the upper hemisphere geodesic triangles and latitudinal rectangles we describe the point vortex motion in these domains which is governed by a hamiltonian determined by the dirichlet green s function

bang bang hybrid stabilization of perturbed double integrators a bang bang hybrid controller is presented that globally practically stabilizes the origin of a double integrator affected by an unknown bounded uncertainty at the input side the proposed controller has two key features it guarantees a uniformly bounded number of switches over any compact time interval and it is robust against bounded measurement errors when disturbances are absent through a proper choice of the control parameters it reduces to the time optimal bang bang controller for the double integrator

explicit parallel poisson integration of point vortices on the sphere point vortex models are frequently encountered in conceptual studies in geophysical fluid dynamics but also in practical applications for instance in aeronautics in spherical geometry the motion of vortex centres is governed by a dynamical system with a known poisson structure we construct poisson integration methods for these dynamics by splitting the hamiltonian into its constituent vortex pair terms from backward error analysis the method is formally known to provide solutions to a modified poisson system with the correct bracket but with a modified hamiltonian function different orderings of the pairwise interactions are considered and also used for the construction of higher order methods the energy and momentum conservation of the splitting schemes is demonstrated for several test cases for particular orderings of the pairwise interactions the schemes allow scalable parallelization this results in a linear as opposed to quadratic scaling of computation time with system size when scaling the number of processors accordingly

vortex motion on surfaces with constant curvature vortex motion on two dimensional riemannian surfaces with constant curvature is formulated by way of the stereographic projection the relation and difference between the vortex motion on a sphere s 2 and on a hyperbolic plane h 2 can be clearly analysed the hamiltonian formalism is presented for the motion of point vortices on s 2 and h 2 the set of first integrals for each hamiltonian shows a corresponding algebraic property in terms of the poisson bracket defined respectively for s 2 and h 2 as an example of analytic solutions the motion of a vortex pair dipole is considered it is shown that a dipole draws a geodesic curve as its trajectory on s 2 and h 2

non self similar partial and robust collapse of four point vortices on a sphere this paper gives numerical examples showing that non self similar collapse can occur in the motion of four point vortices on a sphere it is found when the four vortex problem is integrable in which the moment of vorticity vector is zero the non self similar collapse has significant properties it is partial in the sense that three of the four point vortices collapse to one point in finite time and the other one moves to the antipodal position to the collapse point moreover it is robust with respect to perturbation of the initial configuration as long as the system remains integrable

invariant dynamical systems embedded in the n vortex problem on a sphere with pole vortices abstract we are concerned with the system of n vortex points on a sphere with two fixed vortex points at the poles this article gives a reduction method of the system to invariant dynamical systems when all the vortex points have the same strength it is carried out by considering the invariant property of the system with respect to the shift and pole reversal transformations for which the polygonal ring configuration of the n vortex points at the line of latitude called the n ring remains unchanged we prove that there exists a 2 p dimensional invariant dynamical system reduced by the p shift transformation for an arbitrary factor p of n the p shift invariant system is equivalent to the p vortex points system generated by the averaged hamiltonian with the modified pole vortices it is also shown that the system can be reduced by the pole reversal transformation when the pole vortices are identical since the reduced dynamical systems are defined in the linear space spanned by the eigenvectors given in the linear stability analysis for the n ring we obtain the inclusion relation among the invariant reduced dynamical systems this allows us to decompose the system of a large number of vortex points into a collection of invariant reduced subsystems

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

consistent estimation with a large number of weak instruments this paper analyzes the conditions under which consistent estimation can be achieved in instrumental variables iv regression when the available instruments are weak and the number of instruments k n goes to infinity with the sample size we show that consistent estimation depends importantly on the strength of the instruments as measured by r the rate of growth of the so called concentration parameter and also on k n in particular when k n the concentration parameter can grow even if each individual instrument is only weakly correlated with the endogenous explanatory variables and consistency of certain estimators can be established under weaker conditions than have previously been assumed in the literature hence the use of many weak instruments may actually improve the performance of certain point estimators more specifically we find that the limited information maximum likelihood liml estimator and the bias corrected two stage least squares b2sls estimator are consistent when k n r n 0 while the two stage least squares 2sls estimator is consistent only if k n r n 0 as n these consistency results suggest that liml and b2sls are more robust to instrument weakness than 2sls

long time symplectic integration the example of four vortex motion detailed comparisons are made between long time numerical integration of the motion of four identical point vortices obtained using both a fourth order symplectic integration method of the implicit runge kutta type and a standard fourth order explicit runge kutta scheme we utilize the reduced hamiltonian formulation of the four vortex problem due to aref pomphrey initial conditions which give both fully chaotic and also quasi periodic motions are considered over integration times of order 10 6 10 7 times the characteristic time scale of the evolution the convergence as the integration time step is decreased of the poincare section is investigated when smoothness of the section compared to the converged image and the fractional change in the hamiltonian h are used as diagnostic indicators it is found that the symplectic scheme gives substantially superior performance over the explicit scheme and exhibits only an apparent qualitative degrading in results up to integration time steps of order the minimum timescale of the evolution it is concluded that this performance derives from the symplectic rather than the implicit character of the method

point vortex dynamics a classical mathematics playground the idealization of a two dimensional ideal flow as a collection of point vortices embedded in otherwise irrotational flow yields a surprisingly large number of mathematical insights and connects to a large number of areas of classical mathematics several examples are given including the integrability of the three vortex problem the interplay of relative equilibria of identical vortices and the roots of certain polynomials addition formulas for the cotangent and the weierstras function projective geometry and other topics the hope and intent of the article is to garner further participation in the exploration of this intriguing dynamical system from the mathematical physics community

relative equilibria of point vortices on the sphere we prove the existence of many different symmetry types of relative equilibria for systems of identical point vortices on a non rotating sphere the proofs use the rotational symmetry group so 3 and the resulting conservation laws the time reversing reflectional symmetries in o 3 and the finite symmetry group of permutations of identical vortices results include both global existence theorems and local results on bifurcations from equilibria a more detailed study is made of relative equilibria which consist of two parallel rings with n vortices in each rotating about a common axis the paper ends with discussions of the bifurcation diagrams for systems of 3 6 identical vortices

the motion of point vortices on closed surfaces we develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere when the sum of the vortex circulations is non ze

streamline topologies for integrable vortex motion on a sphere we describe the instantaneous streamline patterns that occur on the surface of a two dimensional sphere in the presence of point vortices of general strength for cases in which the system is completely integrable after stating some general results based on the spherical topology we categorize all possible instantaneous streamline patterns and describe their stagnation point structure for the cases of two and three vortices it is found that for the case of two vortices the only non degenerate topologies that can arise are a figure eight lemniscate or a limacon which are homotopically equivalent for the case of three vortices there are 12 topologically distinct primitives from which an additional 23 patterns can be produced via continuous deformations on the sphere homotopies all possible streamline patterns that arise from three vortex motion can be obtained via linear superposition of the primitive topologies and their homotopic equivalents in this sense the primitives can be viewed as the building blocks for the general flow patterns the equations of motion and streamline patterns in the stereographic plane are obtained using the projected equations of motion in hamiltonian form we describe streamline patterns for three vortex fixed equilibria and relative equilibria as seen in both a fixed and a rotating frame of reference non equilibrium streamline patterns for a collapsing and special periodic solution are studied although to generally understand and classify the many bifurcations associated with non equilibrium patterns would require more extensive computation we conclude with a discussion of the relevance of the three vortex topological classification scheme and the bifurcation of streamline topologies for understanding global atmospheric weather patterns spherical isobars and large scale mixing phenomena 2000 elsevier science b v all rights reserved

diffeomorphism algebras and the nahm and ward equations the question is investigated as to what equations arise from the reduction of the anti self dual yang mills equations by the imposition of three space time translational symmetries and by the choice of the connection coefficients having values in the infinite dimensional lie algebras associated with one and two dimensional diffeomorphism groups on one and two dimensional auxiliary manifolds special cases of this reduction yield the incompressible euler equations for fluid flow in two spatial dimensions the membrane equations in 4 1 dimensions the continuous version of the heisenberg spin chain equations for antiferromagnets as well as the toda lattice and liouville equations