convex central configurations of the 4 body problem with two pairs of equal adjacent masses we study the convex central configurations of the 4 body problem assuming that they have two pairs of equal masses located at two adjacent vertices of a convex quadrilateral under these assumptions we prove that the isosceles trapezoid is the unique central configuration

finiteness and bifurcations of some symmetrical classes of central configurations we consider here the classical question of finiteness see smale 13 and wintner 15 given n point masses is the corresponding number of central configurations finite we prove finiteness for a particular family of d dimensional symmetrical configurations of d 2 point masses also we study the bifurcations of these configurations and provide the exact number of central configurations when d 2 3 all our results stem from the application of a new method for studying symmetrical classes of central configurations which is presented in this work

dziobek configurations of the restricted n 1 body problem with equal masses in this article we count the number of n 1 dimensional central configurations of the restricted n 1 body problem with equal masses for n 4 5 there are 25 56 central configurations for n 3 or n 6 the number of central configurations is 2n n 1

fast algorithms for low level vision a recursive filtering structure is proposed that drastically reduces the computational effort required for smoothing performing the first and second directional derivatives and carrying out the laplacian of an image these operations are done with a fixed number of multiplications and additions per output point independently of the size of the neighborhood considered the key to the approach is first the use of an exponentially based filter family and second the use of the recursive filtering applications to edge detection problems and multiresolution techniques are considered and an edge detector allowing the extraction of zero crossings of an image with only 14 operations per output element at any resolution is proposed various experimental results are shown

global existence and large time behavior for a two dimensional chemotaxis navier stokes system abstract this paper concerns the coupled chemotaxis navier stokes system in the two dimensional setting such a system was proposed in 19 to describe the collective effects arising in bacterial suspensions in fluid drops under some basic assumptions on the parameter functions k and the potential function which are consistent with those used by the experimentalists but weaker than those appeared in the known mathematical works we establish the global existence of weak solutions and classical solutions for both the cauchy problem and the initial boundary value problem supplemented with some initial data for the initial boundary value problem we also assert that the solution converges in large time to the spatially homogeneous equilibrium n 0 0 0 with n 0 1 n 0 x d x our results also show that the large diffusion of the cell density or the chemical concentration can rule out the finite time blow up even though the navier stokes fluid is included

a continuum of periodic solutions to the planar four body problem with two pairs of equal masses abstract in this paper we apply the variational method with structural prescribed boundary conditions spbc to prove the existence of periodic and quasi periodic solutions for the planar four body problem with two pairs of equal masses m 1 m 3 and m 2 m 4 a path q t on 0 t satisfies the spbc if the boundaries q 0 a and q t b where a and b are two structural configuration spaces in r 2 4 and they depend on a rotation angle 0 2 and the mass ratio m 2 m 1 r we show that there is a region 0 2 r such that there exists at least one local minimizer of the lagrangian action functional on the path space satisfying the spbc q t h 1 0 t r 2 4 q 0 a q t b for any the corresponding minimizing path of the minimizer can be extended to a non homographic periodic solution if is commensurable with or a quasi periodic solution if is not commensurable with in the variational method with the spbc we only impose constraints on the boundary and we do not impose any symmetry constraint on solutions instead we prove that our solutions that are extended from the initial minimizing paths possess certain symmetries the periodic solutions can be further classified as simple choreographic solutions double choreographic solutions and non choreographic solutions among the many stable simple choreographic orbits the most extraordinary one is the stable star pentagon choreographic solution when 4 5 1 remarkably the unequal mass variants of the stable star pentagon are just as stable as the equal mass choreographies

central configurations of the 4 body problem with masses m 1 m 2 m 3 m 4 m 0 and m small in this paper we give a complete description of the families of central configurations of the planar 4 body problem with two pairs of equals masses and two equal masses sufficiently small in particular we give an analytical proof that this particular 4 body problem has exactly 34 different classes of central configurations moreover for this problem we prove the following two conjectures there is a unique convex planar central configuration of the 4 body problem for each ordering of the masses in the boundary of its convex hull which appears in albouy and fu 2007 3 we also prove the conjecture there is a unique convex planar central configuration having two pairs of equal masses located at the adjacent vertices of the configuration and it is an isosceles trapezoid finally the families of central configurations of this 4 body problem are numerically continued to the 4 body problem with four equal masses

convex central configurations for the n body problem abstract we give a simple proof of a classical result of macmillan and bartky trans amer math soc 34 1932 838 which states that for any four positive masses and any assigned order there is a convex planar central configuration moreover we show that the central configurations we find correspond to local minima of the potential function with fixed moment of inertia this allows us to show that there are at least six local minimum central configurations for the planar four body problem we also show that for any assigned order of five masses there is at least one convex spatial central configuration of local minimum type our method also applies to some other cases

the central configurations of four masses x x y y the configuration of a homothetic motion in the n body problem is called a central configuration in this paper we prove that there are exactly three planar non collinear central configurations for masses x x y y with x y a parallelogram and two trapezoids and two planar non collinear central configurations for masses x x x x two diamonds except the case studied here the only known case where the four body central configurations with non vanishing masses can be listed is the case with equal masses a albouy 1995 1996 which requires the use of a symbolic computation program thanks to a lemma used in the proof of our result we also show that a co circular four body central configuration has non vanishing total mass or vanishing multiplier

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

generic finiteness for dziobek configurations the goal of this paper is to show that for almost all choices of n masses mi there are only finitely many central configurations of the newtonian n body problem for which the bodies span a space of dimension n 2 such a central configuration is called a dziobek configuration the result applies in particular to two dimensional configurations of four bodies and threedimensional configurations of five bodies

finiteness of relative equilibria of the four body problem we show that the number of relative equilibria of the newtonian four body problem is finite up to symmetry in fact we show that this number is always between 32 and 8472 the proof is based on symbolic and exact integer computations which are carried out by computer

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

some notes on four body co circular central configurations abstract there are two symmetric families of four body co circular central configurations namely the kite and isosceles trapezoid using mutual distances as coordinates we prove that if the four body central configuration is an isosceles trapezoid then the diagonals of the isosceles trapezoid cannot be perpendicular to each other furthermore we show that for any four body co circular central configuration the diagonals of the quadrilateral cannot be perpendicular except that the configuration is a kite

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

fast computation of unbiased intensity derivatives in images using separable filters in this paper we prove that all high order non biased spatial intensity derivative operators in images can be computed using linear combinations of separable filters the separable filters are the same as those used by haralick 1984 but different linear combinations are taken a comparison of the number of operations necessary to compute the derivatives using separable and non separable filters is made the conclusion of our analysis is that the optimal way to compute the needed derivatives depends on which derivatives we have to compute on the size of the window and on the order of expansion finally we discuss the performance of an edge detector using these derivatives for unsmoothed and smoothed step edges

four body central configurations with some equal masses we prove firstly that any convex non collinear central configuration of the planar 4 body problem with equal opposite masses 0 such that the diagonal corresponding to the mass is not shorter than that corresponding to the mass must possess a symmetry and therefore must be a kite then by a recent result of bernat llibre and perez chavela this kite is actually a rhombus secondly we prove that a convex non collinear planar 4 body central configuration with three equal masses must be a kite too we also prove that the concave central configuration with three equal masses forming a triangle and the fourth one with any given mass in the interior must be either an equilateral triangle with the fourth mass at its geometric center or an isosceles triangle with the fourth mass on the symmetry axis

the golden ratio and super central configurations of the n body problem abstract in this paper we consider the problem of central configurations of the n body problem with the general homogeneous potential 1 r a configuration q q 1 q 2 q n is called a super central configuration if there exists a positive mass vector m m 1 m n such that q is a central configuration for m with m i attached to q i and q is also a central configuration for m where m m and m is a permutation of m the main discovery in this paper is that super central configurations of the n body problem have surprising connections with the golden ratio let r be the ratio q 3 q 2 q 2 q 1 of the collinear three body problem with the ordered positions q 1 q 2 q 3 on a line q is a super central configuration if and only if 1 r 1 r r 1 and r 1 where r 1 1 is a continuous function such that lim 0 r 1 the golden ratio the existence and classification of super central configurations are established in the collinear three body problem with general homogeneous potential 1 r super central configurations play an important role in counting the number of central configurations for a given mass vector which may decrease the number of central configurations under geometric equivalence

the symmetric central configurations of the 4 body problem with masses m1 m2 m3 m4 we characterize the planar central configurations of the 4 body problem with masses m 1 m 2 m 3 m 4 which have an axis of symmetry it is known that this problem has exactly two classes of convex central configurations one with the shape of a rhombus and the other with the shape of an isosceles trapezoid we show that this 4 body problem also has exactly two classes of concave central configurations with the shape of a kite this proof is assisted by computer

stability in a rhomboidal 5 body problem with generalized central forces we consider a system of five mass points r1 r2 r3 and r4 with masses m1 m2 m and m3 m4 m moving about a single massive body r0 with mass m0 at its center which is assumed to be the origin of the coordinates system we assume that the central body r0 makes a generalized force on the four mass points and that such a force is generated by a manev s type potential i e characterized by a potential of the form 1r er2 on the other hand is assumed that the attraction between the bodies r1 r2 r3 and r4 is of the newtonian type this model represents several cases for instance when the central body is a spheroid or a radiating source first we prove the existence of three different relative rhomboidal solutions and its central configuration is as follows 1 the rhombus is a square and all primaries have equal masses 2 the rhombus is not a square but all masses are equal and 3 the rhombus is not a square and the pairs of primaries have different masses the first two cases present two pa

on optimal infinite impulse response edge detection filters the authors outline the design of an optimal computationally efficient infinite impulse response edge detection filter the optimal filter is computed based on canny s high signal to noise ratio good localization criteria and a criterion on the spurious response of the filter to noise an expression for the width of the filter which is appropriate for infinite length filters is incorporated directly in the expression for spurious responses the three criteria are maximized using the variational method and nonlinear constrained optimization the optimal filter parameters are tabulated for various values of the filter performance criteria a complete methodology for implementing the optimal filter using approximating recursive digital filtering is presented the approximating recursive digital filter is separable into two linear filters operating in two orthogonal directions the implementation is very simple and computationally efficient has a constant time of execution for different sizes of the operator and is readily amenable to real time hardware implementation

classification of four body central configurations with three equal masses it is known that a central configuration of the planar four body problem consisting of three particles of equal mass possesses a symmetry if the configuration is convex or is concave with the unequal mass in the interior we use analytic methods to show that besides the family of equilateral triangle configurations there are exactly one family of concave and one family of convex central configurations which completely classifies such central configurations

stacked central configurations for newtonian n 2p body problems abstract we show the existence of spatial central configurations for the n 2 p body problems in such a configuration n bodies are at the vertices of a regular n gon t 2 p bodies are symmetric with respect to the center of t and located on the straight line which is perpendicular to the regular n gon t and passes through the center of t the masses located on the vertices of the regular n gon are assumed to be equal the masses located on the same line and symmetric with respect to the center of t are equal