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

existence of a class of rotopulsators abstract we prove the existence of a class of rotopulsators for the n body problem in spaces of constant gaussian curvature of dimension k 2

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

rotopulsators of the curved n body problem we consider the n body problem in spaces of constant curvature and study its rotopulsators i e solutions for which the configuration of the bodies rotates and changes size during the motion rotopulsators fall naturally into five groups positive elliptic positive elliptic elliptic negative elliptic negative hyperbolic and negative elliptic hyperbolic depending on the nature and number of their rotations and on whether they occur in spaces of positive or negative curvature after obtaining existence criteria for each type of rotopulsator we derive their conservation laws we further deal with the existence and uniqueness of some classes of rotopulsators in the 2 and 3 body case and prove two general results about the qualitative behaviour of rotopulsators more precisely for positive curvature we show that there is no foliation of the 3 sphere with clifford tori such that the motion of each body is confined to some clifford torus for negative curvature a similar result is proved relative to foliations of the hyperbolic 3 sphere with hyperbolic cylinders

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

super central configurations of the three body problem under the inverse integer power law in this paper we consider the problem of central configurations of the n body problem with the general homogenous potential 1 r where is a positive integer a configuration q q1 q2 qn is called a super central configuration if there exists a positive mass vector m m1 mn such that q is a central configuration for m with mi attached to qi and q is also a central configuration for m where m mandm is a permutation of m the main result in this paper is the existence and classifications of super central configurations in the rectilinear three body problem with general homogenous potential our results extend the previous work xie z j math phys 51 042902 2010 10 1063 1 3345125 from the case in which 2 to the case in which is a positive integer descartes rule of sign is extensively used in the proof of the main theorem

phase portraits of the two body problem with manev potential the manev systems are two body problems defined by a potential of the form a r b r2 where r is the distance between the two particles and a and b are arbitrary constants the hamiltonian h pr2 p 2 r2 2 a r b r2 and the angular momentum p r2 associated with manev systems are two first integrals which are independent and in involution let ih respectively ic be the set of points of the phase space on which h respectively p takes the value h respectively c since h and p are first integrals the sets ih ic and ihc ihic are invariant under the flow of the manev systems we characterize the global flow of these systems when a and b vary thus we describe the foliation of the phase space by the invariant sets ih and the foliation of ih by the invariant sets ihc

phase space structure and regularization of manev type problems phase space structure and regularization of manev type problems florin diacu a vasile mioc b cristina stoica c a department of mathematics and statistics university of victoria victoria canada v8w 3p4 b astronomical institute of the romanian academy astronomical observatory cluj napoca str cire silor 19 3400 cluj napoca romania c institute for gravitation and space sciences laboratory for gravitation str mendeleev 21 25 bucharest romania

existence of a lower bound for the distance between point masses of relative equilibria in spaces of constant curvature abstract we prove that if for the curved n body problem the masses are given the minimum distance between the point masses of a specific type of relative equilibrium solution to that problem has a universal lower bound that is not equal to zero we furthermore prove that the set of all such relative equilibria is compact this class of relative equilibria includes all relative equilibria of the curved n body problem in h 2 and a significant subset of the relative equilibria for s 2 s 3 and h 3

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

star pentagon and many stable choreographic solutions of the newtonian 4 body problem abstract in this paper we give a rigorous proof of the existence of infinitely many simple choreographic solutions in the classical newtonian 4 body problem these orbits are discovered by a variational method with structural prescribed boundary conditions spbc this method provides an initial path that is obtained by minimizing the lagrangian action functional over the spbc we prove that the initial path can be extended to a periodic or quasi periodic solution with computer assistance a family of choreographic orbits of this type is shown to be linearly stable among the many linearly stable simple choreographic orbits the most extraordinary one is the stable star pentagon choreographic solution we also prove the existence of infinitely many double choreographic periodic solutions infinitely many non choreographic periodic solutions and uncountably many quasi periodic solutions each type of periodic solutions has many stable solutions and possibly infinitely many stable solutions our results with spbc largely complement the current results by minimizing the action on a loop space

the flow of the n body problem near a simultaneous binary collision singularity and integrals of motion on the collision manifold we study the flow for the n body problem near a simultaneous binarycollision singularity we blow up the singularity to an invariant collision manifold we show the existence of integrals of motion on the collision manifold we integrate the flow on the collision manifold explicitly in terms of elementary functions we give a complete description of the flow near the singularity

on the c8 3 regularisation of simultaneous binary collisions in the collinear 4 body problem abstract a conjecture on the nature of the simultaneous binary collision is explored in the collinear 4 body problem it is known that any attempt to remove the singularity via block regularisation will result in a regularised flow that is no more than c 8 3 differentiable with respect to initial conditions through a blow up of the singularity this loss of differentiability is investigated and a new proof of the c 8 3 regularity is provided in the process it is revealed that the collision manifold consists of two manifolds of normally hyperbolic saddle singularities which are connected by a manifold of heteroclinics by utilising recent work on transitions near such objects and their normal forms an asymptotic series of the transition past the singularity is explicitly computed it becomes remarkably apparent that the finite differentiability at 8 3 is due to the inability to construct a set of integrals local to the simultaneous binary collision the finite differentiability is shown to be independent of the value of the masses

linear instability of relative equilibria for n body problems in the plane abstract following smale we study simple symmetric mechanical systems of n point particles in the plane in particular we address the question of the linear and spectral stability properties of relative equilibria which are special solutions of the equations of motion our main result is a sufficient condition to detect spectral hence linear instability namely we prove that if the morse index of an equilibrium point with even nullity is odd then the associated relative equilibrium is spectrally unstable the proof is based on some refined formulae for computing the spectral flow as a notable application of our theorem we examine two important classes of singular potentials the homogeneous one with 0 2 which includes the gravitational case and the logarithmic one we also establish for the homogeneous potential an inequality which is useful to test the spectral instability of the associated relative equilibrium

an index theory for asymptotic motions under singular potentials abstract we develop an index theory for parabolic and collision solutions to the classical n body problem and we prove sufficient conditions for the finiteness of the spectral index valid in a large class of trajectories ending with a total collapse or expanding with vanishing limiting velocities both problems suffer from a lack of compactness and can be brought in a similar form of a lagrangian system on the half time line by a regularising change of coordinates which preserve the lagrangian structure we then introduce a maslov type index which is suitable to capture the asymptotic nature of these trajectories as half clinic orbits by taking into account the underlying hamiltonian structure we define the appropriate notion of geometric index for this class of solutions and we develop the relative index theory

symmetric periodic solutions of the anisotropic manev problem we consider the manev potential in an anisotropic space i e such that the force acts differently in each direction using a generalization of the poincare continuation method we study the existence of periodic solutions for weak anisotropy in particular we find that the symmetric periodic orbits of the manev system are perturbed to periodic orbits in the anisotropic problem

resistance and related scaling properties of self avoiding random walks three scaling exponents for random and self avoiding chains are studied by direct renormalisation these describe the end to end resistance of a conducting chain the arc length of the shortest path traceable between the chain ends and the arc length of chain in no loop the scaling behaviours of these quantities are identical to first order in epsilon 4 d d is the space dimension for a long self avoiding chain the authors predict that each one one is proportional to the chain length

infinitely many periodic orbits for the rhomboidal five body problem we prove the existence of infinitely many symmetric periodic orbits for a regularized rhomboidal five body problem with four small masses placed at the vertices of a rhombus centered in the fifth mass the main tool for proving the existence of such periodic orbits is the analytic continuation method of poincare together with the symmetries of the problem

homoclinic orbits and oscillation for the planar three body problem sitnikov proved the existence of oscillation and capture for a special motion of the restricted three body problem m 0 141 alekseev made a systematic study of this situation 11 he also proved the same result for all nonzero masses see section 5 below for a discussion of this case a good exposition of this example following lectures of conley is contained in 9 using the stable manifold result for a degenerately hyperbolic closed orbit of mcgehee 81 easton and mcgehee proposed a planar three body example with negative energy which could possibly exhibit similar oscillation and capture 3 while sitnikov s example has two degrees of freedom this planar example after all the integrals and symmetries are removed has three degrees of freedom they studied a model example which completely decouples when the third body is near infinity and proved oscillation and capture exist for this model example their work left the following three steps undone 1 show the parabolic orbits form a submanifold for the real equations 2 show the a parabolic orbits and w parabolic orbits are transverse in a strong sense see the definition of a hyperbolic homoclinic orbit defined below and 3 show that symbolic dynamics can be used to show oscillation and capture exist using only the fact that the binary asymptotically decouples and not that it completely decouples when the third body is near infinity for negative energy h as the third particle goes to infinity parabolically the asymptotic motion of q r2 rl is that of a two body problem with energy h after regularization the set of all two body motions with energy h 0 is the hopf flow on the three sphere s3 let ivs s3 resp wu s be the set of w parabolic orbits resp w parabolic orbits easton has now shown that ws s3 and wl s3 are lipschitz manifolds 2 this paper proves they are real analytic manifolds and cm at infinity at s

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

triple collision in the quasi homogeneous collinear three body problem abstract we generalize and improve the result of f diacu j differential equations 128 1996 58 77 concerning the dynamics of three points particles moving on the line under the influence of a quasihomogeneous potential w u v where u and v are homogeneous functions of degree a and b respectively for b 2 we found a set of positive measure of collision orbits that do not tend to form asymptotically a central configuration

central configurations and homographic solutions for the quasihomogeneous n body problem for the n body problem given by quasihomogeneous force functions of the form c1 ra c2 rb c1 c2 real and 0 a b we prove that the simultaneous and extraneous central configurations are the only types of central configurations then we focus on attractive attractive force functions c1 and c2 positive and analyze the connection between central configurations and homographic solutions namely we extend the classical lagrange s theorem a solution is homographicif and only if the bodies form equivalent central configurations for all times to attractive attractive quasihomogeneous problems and give sufficient conditions for the construction of homographic solutions

on the anisotropic manev problem we consider the manev potential given by the sum between the inverse and the inverse square of the distance in an anisotropic space i e such that the force acts differently in each direction using mcgehee coordinates we blow up the collision singularity paste a collision manifold to the phase space study the flow on and near the collision manifold and find a positive measure set of collision orbits besides frontal homothetic frontal nonhomothetic and spiraling collisions and ejections we put into the evidence the surprising class of oscillatory collision and ejection orbits using the infinity manifold we further tackle capture and escape solutions in the zero energy case by finding the connection orbits between equilibria and or cycles at impact and at infinity we describe a large class of capture collision and ejection escape solutions

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

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

comet and hill type periodic orbits in restricted n 1 body problems abstract we consider the planar restricted n 1 body problem where the interaction potential between the particle and the primaries is taken to be a finite sum of terms of the form distance with 0 the primaries are assumed to be in a relative equilibrium that is they form a uniformly rotating rigid configuration we show two results first if the infinitesimal particle is far from the primaries and the long range dominant term r of the potential is such that 0 and 2 then there exist two one parameter families of large nearly circular periodic solutions these solutions called comet solutions are elliptic and kam stable for 2 and unstable for 2 second if the infinitesimal particle is close to one of the primaries and the short range dominant term r of the potential near that primary is such that 0 and 2 then there exist two one parameter families of nearly circular periodic solutions called hill solutions that encircle the nearby primary for 2 these orbits are unstable the methodology applied involves appropriate symplectic scalings poincare s continuation method and averaging theory the kam stability of the comet periodic orbits is decided by verifying arnol d s non degeneracy conditions

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

linear stability of relative equilibria in the charged three body problem a relative equilibrium is a periodic orbit of the n body problem that rotates uniformly maintaining the same central configuration for all time in this paper we generalize some results of r moeckel and we apply it to study the linear stability of relative equilibria in the charged three body problem we find necessary conditions to have relative equilibria linearly stable for the collinear charged three body problem for planar relative equilibria we obtain necessary and sufficient conditions for linear stability in terms of the parameters masses and electrostatic charges in the last case we obtain a stability inequality which generalizes the routh condition of celestial mechanics we also proof the existence of spatial relative equilibria and the existence of planar relative equilibria of any triangular shape

structure of clusters generated by random walks the authors study the cluster structure resulting from a nearest neighbour random walk embedded in a d dimensional space each bond visited by the random walks is regarded as belonging to the cluster the diffusion exponent and the fracton dimensional of the fractal cluster in d 3 is found to be dw 3 5 or 0 1 and d 0 57 or 0 02 using a method of exact enumeration of random walks on these fractals

resistance of the edwards walk an analytical and numerical study of the scaling of edwards random walks defined as the zero component limit n to 0 of an n component g0 mod phi 2 mod 2 theory is done in three dimensions the hausdorf dimension 1 nu and the resistance exponent x is calculated using renormalisation group methods in 4 in dimensions the numerical study is done by defining a monte carlo algorithm on the cubic lattice to generate ensembles of edwards walks

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

finiteness of central configurations of five bodies in the plane we prove there are finitely many isometry classes of planar central configurations also called relative equilibria in the newtonian 5 body problem except perhaps if the 5 tuple of positive masses belongs to a given codimension 2 subvariety of the mass space

on the global dynamics of the anisotropic manev problem we study the global flow of the anisotropic manev problem which describes the planar motion of two bodies under the influence of an anisotropic newtonian potential with a relativistic correction term we first find all the heteroclinic orbits between equilibrium solutions then we generalize the poincare melnikov method and use it to prove the existence of infinitely many transversal homoclinic orbits invoking a variational principle and the symmetries of the system we finally detect infinitely many classes of periodic solutions

structural and dynamical properties of random walk clusters the authors study the structural and dynamical properties of the clusters generated by a nearest neighbour random walk embedded in a d dimensional space they have focused on the non trivial case in which the cluster is generated in d 3 the structure of this cluster is characterised by loops for all length scales on the one hand and by the fact that deadends are negligible upon scaling on the other hand the cluster is very dilute and is characterised by fractal dimension df 2 and chemical dimension d1 1 29 or 0 04 from these results it follows that v identical to dldf approximately 2 3 which is consistent with the formula v 2 d 2 or d or 4 obtained using a flory type argument the dynamical diffusion exponents dw and dlw were calculated using the exact enumeration method and found to be dw 3 45 or 0 10 and dlw 2 28 or 0 05 the results suggest that the effect of loops is small but not negligible they also calculated the fracton dimensionality of the cluster and obtained ds 1 14 or 0 02 a scaling function is presented for the end to end mean square displacement of a random walk performed on a random walk cluster this scaling function is supported by their numerical results

random walk on the range of random walk we study the random walk x on the range of a simple random walk on d in dimensions d 4 when d 5 we establish quenched and annealed scaling limits for the process x which show that the intersections of the original simple random walk path are essentially unimportant for d 4 our results are less precise but we are able to show that any scaling limit for x will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions furthermore we demonstrate that when d 4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of x to the origin

central configurations with many small masses abstract by using the method of analytical continuation we find the exact numbers of central configurations for some open sets of n positive masses for any choice of n it turns out that the numbers increase dramatically as n increases e g for some open set of 18 positive masses some 2 08766 1020 classes of distinctive central configurations are found in the mean time we obtained some results about the hausdorff measure for the set of n positive masses where degenerate central configuration arises

polygonal rotopulsators of the curved n body problem we revisit polygonal positive elliptic rotopulsator solutions and polygonal negative elliptic rotopulsator solutions of the n body problem in h3 and s3 and prove the existence of these solutions and prove that the masses of these rotopulsators have to be equal if the rotopulsators are of nonconstant size and show that the number of negative elliptic relative equilibria of this type is finite as is the number of positive elliptic relative equilibria if an upper bound on the size of the relative equilibrium is imposed additionally we prove that a class of negative hyperbolic rotopulsators is in fact a subclass of the class of polygonal negative elliptic rotopulsators

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

lagrange pizzetti theorem for the quasihomogeneous n body problem we consider the n body problem given by quasihomogeneous force functions of the form c1 r a c2 r b c1 c2 a b positive constants and a b and address the fundamentals of homographic solutions generalizing techniques of the classical n body problem we prove that if the homographic solution is not flat then it is homothetic if the homographic solution is flat then it is planar using them we prove an analogue of the lagrange pizzetti theorem for quasihomogeneous potentials

fibonacci and galois representations of feedback with carry shift registers a feedback with carry shift register fcsr with fibonacci architecture is a shift register provided with a small amount of memory which is used in the feedback algorithm like the linear feedback shift register lfsr the fcsr provides a simple and predictable method for the fast generation of pseudorandom sequences with good statistical properties and large periods in this paper we describe and analyze an alternative architecture for the fcsr which is similar to the galois architecture for the lfsr the galois architecture is more efficient than the fibonacci architecture because the feedback computations are performed in parallel we also describe the output sequences generated by the d fcsr a slight modification of the fibonacci fcsr architecture in which the feedback bit is delayed for d clock cycles before being returned to the first cell of the shift register we explain how these devices may be configured so as to generate sequences with large periods we show that the d fcsr also admits a more efficient galois architecture

on the expected value of the linear complexity and the k error linear complexity of periodic sequences rueppel 1986 conjectured that periodic binary sequences have expected linear complexity close to the period length n in this paper we determine the expected value of the linear complexity of n periodic sequences explicitly and confirm rueppel s conjecture for arbitrary finite fields cryptographically strong sequences should not only have a large linear complexity but also the change of a few terms should not cause a significant decrease of the linear complexity this requirement leads to the concept of the k error linear complexity of n periodic sequences we present a method to establish a lower bound on the expected k error linear complexity of n periodic sequences based on the knowledge of the counting function spl nscr sub n sub 0 c i e the number of n periodic sequences with given linear complexity c for some cases we give explicit formulas for that lower bound and we also determine spl nscr sub n 0 c

response to comment on central potentials on spaces of constant curvature the kepler problem on the two dimensional sphere s2 and the hyperbolic plane h2 j math phys 46 052702 2005

the global flow of the manev problem the manev problem a two body problem given by a potential of the form a r b r2 where r is the distance between particles and a b are positive constants comprises several important physical models having its roots in research done by isaac newton we provide its analytic solution then completely describe its global flow using mcgehee coordinates and topological methods and offer the physical interpretation of all solutions we prove that if the energy constant is negative the orbits are generically precessional ellipses except for a zero measure set of initial data for which they are ellipses for zero energy the orbits are precessional parabolas and for positive energy they are precessional hyperbolas in all these cases the set of initial data leading to collisions has positive measure

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

expected pi adic security measures of sequences various measures of security of stream ciphers have been studied that are based on the problem of finding a minimum size generator for the keystream in some special class of generators these include linear and p adic spans as well as adic span which is based on a choice of an element in a finite extension of the integers the corresponding sequence generators are known as linear feedback shift registers feedback with carry shift registers and the more general algebraic feedback shift registers respectively in this paper the average behavior of such security measures when d p 0 or 2 p 0 is studied in these cases if z is the ring of integers in its fraction field and is a ufd it is shown that the average adic span is n o log n for sequences with period n

parabolic orbits in the planar three body problem it is not known whether there exist oscillatory and capture orbits in the planar 3 body problem sitnikov 6 proved such orbits exist for the restricted 3 body problem alekseev i extended this work and related it to the existence of homoclinic orbits mcgehee and easton 3 studied a model problem closely related to the planar 3 body problem and proved the existence of oscillatory and capture orbits near certain orbits biasymptotic to an invariant 3 sphere the goal of this paper is to construct an invariant 3 sphere at infinity in the planar 3 body problem and to prove that the stable and unstable manifolds of this sphere are lipschitz manifolds to complete the study of oscillatory and capture orbits one must investigate the way in which the stable and unstable manifolds of the 3 sphere intersect each other this is a difficult question which is not treated here however using methods of melnikov and using one of the masses as a perturbation parameter such an investigation might be possible the stable manifold theorem which is proven here is not standard since the invariant 3 sphere is not hyperbolic geometric methods of mcgehee 4 are extended and applied to obtain the theorem a special case of the system of equations that we study has the form j x dy lx e x y l f 4 dy l x e x y l 0 1 4

n body spatial parabolic orbits asymptotic to collinear central configurations an orbit for the newtonian n body problem is called parabolic when all the mutual distances between particles approach infinity while the velocities tend to zero as time t goes to infinity the asymptotic growth rate for these mutual distances turns out to be constant multiples of t raised to the 2 3 power or to the power 2 p 1 for the inverse p central force law 1 p 3 111 indeed when the position vectors are scaled by dividing by t213 or by the square root of the moment of inertia then the scaled position vectors tend to the set of central configurations a central configuration is formed whenever for all the particles a fixed constant multiple of the mass times the position vector is equal to the corresponding force vector it turns out that central configurations can be identified with the critical point set of a certain function then the configurations are classified as being either degenerate or nondegenerate to correspond with the classification of the critical point of this function should the central configuration be nondegenerate as is known to be the case for collinear central configurations the equilateral triangle configuration for the three body problem and the equilateral tetrahedron configuration for the four body problem then the limiting configuration of a given motion must be unique for a discussion of central configurations see 15 chap 5 8 121 this uniqueness follows by combining two facts first the nondegeneracy implies that the orbit of the central configuration by the w 3 action must be isolated in the set of all central configurations the so 3 action on configuration space is rotation about the center of mass of the system secondly there can be no rotation of the limiting configuration in the sense of motion along the so 3 action on configuration space see 71 for the planar three body problem and 13 for the general case the lack of

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

specific heats of classical spin systems and inhomogeneous differential approximants derives accurate compact expressions for the high temperature specific heats of classical s infinity 3 vector spin systems on fcc bcc and sc lattices for pure heisenberg xy and ising like couplings respectively the analysis of the appropriate series expansions demonstrates the utility of inhomogeneous differential approximants and supports the estimate alpha h 0 21 or 0 04

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

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

on central configurations this paper concerns an old problem which arose in celestial mechanics and has since taken on a life of its own the probem of central configurations of the n body problem

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

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

algebraic feedback shift registers a general framework for the design of feedback registers based on algebra over complete rings is described these registers generalize linear feedback shift registers and feedback with carry shift registers basic properties of the output sequences are studied relations to the algebra of the underlying ring synthesis of the register from the sequence which has implications for cryptanalysis and basic statistical properties these considerations lead to security measures for stream ciphers analogous to the notion of linear complexity that arises from linear feedback shift registers we also show that when the underlying ring is a polynomial ring over a finite field the new registers can be simulated by linear feedback shift registers with small nonlinear filters

on returns to the starting site in lattice random walks abstract in sufficiently low dimensional systems the conditional mean time to return to the starting site conditional upon return eventually occuring is infinite we examine the conditional mean time n to return in a walk of finite duration n steps for walks of polya type n is found asymptotically proportional to n n log 2 n n and log n in dimensions 1 2 3 and 4 respectively results are also given for walks with long ranged transitions and for a one dimensional walk in a central potential

the existence of oscillatory and superhyperbolic motion in newtonian systems we establish the existence of solutions for the collinear newtonian four body problem where the maximum spacing between particles exceeds any given function of time as t as an example there exist solutions where the maximum spacing separates faster than any constant multiple of time also it is shown that the three and four body problems admit oscillatory solutions

formation of symmetric structures in the dynamics of repelling particles we consider hamiltonian systems corresponding to the motions of a system of n repelling particles evolving in space under the action deriving from a very long range potential energy the asymptotic behavior of the system is analysed for the cases u ln r and only special asymptotic shapes are reached which may present quite interesting symmetries and correspond to the critical points of a gradient system the relationships between the original hamiltonian and the asymptotic gradient system are discussed

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

the global flow of the quasihomogeneous potentials of manev schwarchild type abstract in this paper we study two body problems defined by a potential of the form v r a r b r2 c r3 where r is the distance between the two particles and a b c are arbitrarily chosen constants the hamiltonian h r p r p 1 2 p r 2 p 2 r 2 a r b r 2 c r 3 and the angular momentum p r 2 are two first integrals independent and in involution let ih respectively im be the set of points on the phase space on which h respectively p takes the value h respectively m since h and p are first integrals the sets ih im and ihm ih im are invariant under the systems associated to the hamiltonian h we characterize the global flow of the systems when a b and c vary describing the foliation of the phase space by the invariant sets ih the foliation of ih by the invariant sets ihm and the movement of the flow over ihm

on isometric immersions of the hyperbolic plane into the lorentz minkowski space and the monge ampere equation of a certain type

the existence of noncollision singularities in newtonian systems in this paper we solve a long standing problem in celestial mechanics proposed by painleve and poincare in the last century the problem which concerns the nature of the singularities in the n body problem asks whether there exists a noncollision singularity in the newtonian n body problem here we give an affirmative answer to this problem by proving the existence of noncollision singularities in the 5 body problem we consider n point masses moving in a euclidean space w3 let the mass of the ith particle be mi 0 let its position be qj e r3 and let 4i e 3 be its velocity according to newton s law

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

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

resistance of random walks unrestricted lattice random walks in which a unit conductor is placed along each bond traversed are considered the mean end to end resistance is studied as a function of the number of steps in the walk and the spatial dimension a critical scaling law is found whose exponent is consistently given by four different calculational schemes

bifurcations of the lagrangian orbits from the classical to the curved 3 body problem we consider the 3 body problem of celestial mechanics in euclidean elliptic and hyperbolic spaces and study how the lagrangian equilateral relative equilibria bifurcate when the gaussian curvature varies we thus prove the existence of new classes of orbits in particular we find some families of isosceles triangles which occur in elliptic space

on the manifolds of total collapse orbits and of completely parabolic orbits for the n body problem abstract in this paper we show for most choices of masses in the coplanar n body problem and for certain masses in the three dimensional n body problem that the set of initial conditions leading to complete collapse forms a smooth submanifold in phase space where the dimension depends upon properties of the limiting configuration a similar statement holds for completely parabolic motion by a proper scaling these two motions become dual to each other in the sense that one forms the unstable set while the other forms the stable set of a particular set of points in the new phase space most of the paper is devoted to solving the painleve wintner problem which asserts that these types of orbits cannot enter in an infinite spin

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

classification and stability of relative equilibria for the two body problem in the hyperbolic space of dimension 2 we classify and analyze the stability of all relative equilibria for the two body problem in the hyperbolic space of dimension 2 and we formulate our results in terms of the intrinsic riemannian data of the problem

central configurations with a quasihomogeneous potential function for the classical n body problem if the masses are paced at rest in the configuration that is central then the masses will tend to the origin and result in a simultaneous collision a winter the analytical foundations of celestial mechanics 1st ed princeton university press princeton 1941 however if the potential is quasihomogeneous in nature that is the potential is of the form u v w where w i jamimjrij and v i jbmimjrij then we will dramatically get different results this paper determines the orbits that will result when the potential function is quasihomogeneous it also shows when there existences a relative equilibrium solution for the regular n gon configuration this is one half of perko s theorem for a homogeneous potential finally we show the existence of a central configuration which is not central for either w or v this configuration is the placement of two mutual equal pairs of masses placed at the vertices of a square this shows that the reverse implication of perk