saari s homographic conjecture of the three body problem saari s homographic conjecture which extends a classical statement proposed by donald saari in 1970 claims that solutions of the newtonian n body problem with constant configurational measure are homographic in other words if the mutual distances satisfy a certain relationship the configuration of the particle system may change size and position but not shape we prove this conjecture for large sets of initial conditions in three body problems given by homogeneous potentials including the newtonian one some of our results are true for n 3

automatic ventricular cavity boundary detection from sequential ultrasound images using simulated annealing an automatic algorithm has been developed for high speed detection of cavity boundaries in sequential 2 d echocardiograms using an optimization algorithm called simulated annealing sa the algorithm has three stages 1 a predetermined window of size n m is decimated to size n m after low pass filtering 2 an iterative radial gradient algorithm is employed to determine the center of gravity cg of the cavity 3 64 radii which originate from the cg defined in stage 2 are bounded by the high probability region each bounded radius is defined as a link in a 1 d 64 member cyclic markov random field this algorithm is unique in that it compounds spatial and temporal information along with a physical model in its decision rule whereas most other algorithms base their decisions on spatial data alone this is the first implementation of a relaxation algorithm for edge detection in echocardiograms results attained using this algorithm on real data have been highly encouraging

adaptive noise smoothing filter for images with signal dependent noise in this paper we consider the restoration of images with signal dependent noise the filter is noise smoothing and adapts to local changes in image statistics based on a nonstationary mean nonstationary variance nmnv image model for images degraded by a class of uncorrelated signal dependent noise without blur the adaptive noise smoothing filter becomes a point processor and is similar to lee s local statistics algorithm 16 the filter is able to adapt itself to the nonstationary local image statistics in the presence of different types of signal dependent noise for multiplicative noise the adaptive noise smoothing filter is a systematic derivation of lee s algorithm with some extensions that allow different estimators for the local image variance the advantage of the derivation is its easy extension to deal with various types of signal dependent noise film grain and poisson signal dependent restoration problems are also considered as examples all the nonstationary image statistical parameters needed for the filter can be estimated from the noisy image and no a priori information about the original image is required

polygonal homographic orbits of the curved n body problem in the 2 dimensional n body problem n 3 in spaces of constant curvature 6 0 we study polygonal homographic solutions we first provide necessary and sufficient conditions for the existence of these orbits and then consider the case of regular polygons we further use this criterion to show that for any n 3 the regular ngon is a polygonal homographic orbit if and only if all masses are equal then we prove the existence of relative equilibria of non equal masses on the sphere of curvature 0 for n 3 in the case of scalene triangles such triangular relative equilibria occur only along fixed geodesics and are generated from fixed points of the sphere finally through a classification of the isosceles case we prove that not any three masses can form a triangular relative equilibrium

saari s conjecture for the collinear n body problem in 1970 don saari conjectured that the only solutions of the newtonian n body problem that have constant moment of inertia are the relative equilibria we prove this conjecture in the collinear case for any potential that involves only the mutual distances furthermore in the case of homogeneous potentials we show that the only collinear and non zero angular momentum solutions are homographic motions with central configurations

optimal gain derivation for the lms algorithm using a visual fidelity criterion this paper analytically derives the optimal feedback gain parameter 0 for the least mean squares lms algorithm used in a video compression application and subject to a visual fidelity criteria experimental results are presented which show these analytically derived values of 0 to be in excellent agreement with experimentally derived 0 over a wide range of filter and data parameters

a remarkable periodic solution of the three body problem in the case of equal masses using a variational method we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane the orbit has zero angular momentum and a very rich symmetry pattern its most surprising feature is that the three bodies chase each other around a flxed eight shaped curve setting aside collinear motions the only other known motion along a flxed curve in the inertial plane is the lagrange relative equilibrium in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle our orbit visits in turns every euler conflguration in which one of the bodies sits at the midpoint of the segment deflned by the other two figure 1 numerical computations

central configurations and total collisions for quasihomogeneous n body problems abstract we consider n body problems given by potentials of the form r a r b with a b constants 0 a b to analyze the dynamics of the problem we first prove some properties related to central configurations including a generalization of moulton s theorem then we obtain several qualitative properties for collision and near collision orbits in the manev type case a 1 at the end we point out some new relationships between central configurations relative equilibria and homothetic solutions

on the singularities of the curved n body problem we study singularities of the n body problem in spaces of constant curvature and generalize certain results due to painleve weierstrass and sundman for positive curvature some of our proofs use the correspondence between total collision solutions of the original system and their orthogonal projection a property that offers a new method of approaching the problem in this par ticular case

homographic solutions of the curved 3 body problem abstract in the 2 dimensional curved 3 body problem we prove the existence of lagrangian and eulerian homographic orbits and provide their complete classification in the case of equal masses we also show that the only non homothetic hyperbolic eulerian solutions are the hyperbolic eulerian relative equilibria a result that proves their instability

a variable step vs adaptive filter algorithm in recent work a new version of an lms algorithm has been developed which implements a variable feedback constant for each weight of an adaptive transversal filter this technique has been called the vs variable step algorithm and is an extension of earlier ideas in stochastic approximation for varying the step size in the method of steepest descents the method may be implemented in hardware with only modest increases in complexity approx 15 percent over the lms widrow hoff algorithm it is shown that an upper bound for the convergence time is the classical mean square error time constant and examples are given to demonstrate that for broad signal classes both narrow band and broad band the convergence time is reduced by a factor of up to 50 in noise canceller applications for the proper selection of variable step parameters finally the vs algorithm is applied to an iir filter and simulations are presented for applications of the vs fir and iir adaptive filters

colouring of graphs with ramsey type forbidden subgraphs a colouring of a graph g v e is a mapping c v 1 2 such that c u c v if email protected e if c v

finiteness of polygonal relative equilibria for generalised quasi homogeneous n body problems and n body problems in spaces of constant curvature we prove for generalisations of quasi homogeneous n body problems with centre of mass zero and n body problems in spaces of negative constant gaussian curvature that if the masses and rotation are fixed there exists for every order of the masses at most one equivalence class of relative equilibria for which the point masses lie on a circle as well as that there exists for every order of the masses at most one equivalence class of relative equilibria for which all but one of the point masses lie on a circle and rotate around the remaining point mass the method of proof is a generalised version of a proof by j m cors g r hall and g e roberts on the uniqueness of co circular central configurations for power law potentials

near collision dynamics for particle systems with quasihomogeneous potentials abstract we study collision and ejection orbits of 3 particle systems having the potential w u v where u and v are homogeneous functions of degree a and b respectively with 1 a b we show that for b 2 collision and ejection orbits tend to form asymptotically a central configuration for the case b 2 which corresponds to maneff s gravitational law we find a set of collision and ejection orbits reaching the triple collision manifold without asymptotic phase this set contains an uncountable union of manifolds and has positive measure within the set of all rectilinear solutions

detecting myocardial boundaries of left ventricle from a single frame 2de image abstract a new method for detecting myocardial boundaries of the left ventricle from a short axis viewed two dimensional echocardiographic 2de image is presented the method extracts the outer boundary based on an ideal echo intensity image model and relaxation algorithm while the endocardial border is detected by the principle employed by klingler et al with some modifications the only a priori information needed in this method is to assign the approximated contraction center of the left ventricle neither the searching limits used in the radial search method nor information about rough positions of myocardial borders used in a plan based detection scheme is required practical examples are given and illustrated methods for the generalization of the proposed method are discussed

saari s conjecture is true for generic vector fields the simplest non collision solutions of the n body problem are the relative equilibria in which each body follows a circular orbit around the centre of mass and the shape formed by the n bodies is constant it is easy to see that the moment of inertia of such a solution is constant in 1970 d saari conjectured that the converse is also true for the planar newtonian n body problem relative equilibria are the only constant inertia solutions a computer assisted proof for the 3 body case was recently given by r moeckel trans amer math soc 2005 we present a different kind of answer proofs that several generalisations of saari s conjecture are generically true our main tool is jet transversality including a new version suitable for the study of generic potential functions

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

on the stability of tetrahedral relative equilibria in the positively curved 4 body problem abstract we consider the motion of point masses given by a natural extension of newtonian gravitation to spaces of constant positive curvature in which the gravitational attraction between the bodies acts along geodesics we aim to explore the spectral stability of tetrahedral orbits of the corresponding 4 body problem in the 2 dimensional case a situation that can be reduced to studying the motion of the bodies on the unit sphere we first perform some extensive and highly precise numerical experiments to find the likely regions of stability and instability relative to the values of the masses and to the latitude of the position of the three equal masses then we support the numerical evidence with rigorous analytic proofs in the vicinity of some limit cases in which certain masses are either very large or negligible or the latitude is close to zero

existence of a lower bound for the distance between point masses of relative equilibria for generalised quasi homogeneous n body problems and the curved n body problem we prove that if for relative equilibrium solutions of a generalisation of quasi homogeneous n body problems the masses and rotation are given then the minimum distance between the point masses of such a relative equilibrium has a universal lower bound that is not equal to zero we furthermore prove that the set of such relative equilibria is compact and prove related results for n body problems in spaces of constant gaussian curvature

an intrinsic approach in the curved n body problem the negative curvature case abstract we consider the motion of n point particles of positive masses that interact gravitationally on the 2 dimensional hyperbolic sphere which has negative constant gaussian curvature using the stereographic projection we derive the equations of motion of this curved n body problem in the poincare disk where we study the elliptic relative equilibria then we obtain the equations of motion in the poincare upper half plane in order to analyze the hyperbolic and parabolic relative equilibria using techniques of riemannian geometry we characterize each of the above classes of periodic orbits for n 2 and n 3 we recover some previously known results and find new qualitative results about relative equilibria that were not apparent in an extrinsic setting

on the n body problem on surfaces of revolution abstract we explore the n body problem n 3 on a surface of revolution with a general interaction depending on the pairwise geodesic distance using the geometric methods of classical mechanics we determine a large set of properties in particular we show that saari s conjecture fails on surfaces of revolution admitting a geodesic circle we define homographic motions and using the discrete symmetries prove that when the masses are equal they form an invariant manifold on this manifold the dynamics are reducible to a one degree of freedom system we also find that for attractive interactions regular n gon shaped relative equilibria with trajectories located on geodesic circles typically experience a pitchfork bifurcation some applications are included

biased motion adaptive temporal filtering for speckle reduction in echocardiography describes a new fully motion adaptive spatio temporal filtering technique to reduce the speckle in ultrasound images the advantages of this approach are demonstrated in echocardiographic boundary detection and in comparison with other techniques the first stage of many automated echocardiographic image interpretation schemes is filtering to reduce the amount of speckle noise the authors show how the two dimensional least mean squares tdlms filter can be configured as a motion compensated filter for a time sequence of ultrasound images that eliminates the blurring associated with direct averaging for an image corrupted by multiplicative speckle noise the mode of the intensity distribution approximates the maximum likelihood estimator in consequence the temporal filter s output is biased towards the mode from the mean using information contained within the speckle itself a new adaptive algorithm for controlling the filter s convergence is also included to evaluate performance application to simulated phantom and an in vivo test sequence of the carotid artery are considered in comparison with other techniques the effect of filtering on edges is of great importance as these are used by subsequent image interpretation schemes quantitative measurements demonstrate the effectiveness of the biased tdlms filter for both noise reduction and edge preservation echocardiographic images have a high noise content and suffer from poor contrast despite this challenging environment the biased tdlms filter is shown to produce images that are better inputs for subsequent feature extraction the benefits for echocardiographic images are highlighted by considering the problems of mitral valve analysis and extraction of the left atrium boundary