positive periodic solutions of singular systems with a parameter abstract the existence and multiplicity of positive periodic solutions for second order non autonomous singular dynamical systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter our results provide a unified treatment for the problem and significantly improve several results in the literature the proof of our results is based on the krasnoselskii fixed point theorem in a cone

platonic polyhedra topological constraints and periodic solutions of the classical n body problem we prove the existence of a number of smooth periodic motions u of the classical newtonian n body problem which up to a relabeling of the n particles are invariant under the rotation group mathcal r of one of the five platonic polyhedra the number n coincides with the order mathcal r of mathcal r and the particles have all the same mass our approach is variational and u is a minimizer of the lagrangian action mathcal a on a suitable subset mathcal k of the h1t periodic maps u 3n the set mathcal k is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group mathcal r there exist infinitely many such cones mathcal k all with the property that mathcal a mathcal k is coercive for a certain number of them using level estimates and local deformations we show that minimizers are free of collisions and therefore classical solutions of the n body problem with a rich geometric kinematic structure

index and stability of symmetric periodic orbits in hamiltonian systems with application to figure eight orbit in this paper using the maslov index theory in symplectic geometry we build up some stability criteria for symmetric periodic orbits in a hamiltonian system which is motivated by the recent discoveries in the n body problem the key ingredient is a generalized bott type iteration formula for periodic solution in the presence of finite group action on the orbit for second order system we prove under general boundary conditions the close formula for the relationship between the morse index of an orbit in a lagrangian system and the maslov index of the fundamental solution for the corresponding orbit in its hamiltonian system counterpart and the boundary conditions cover the cases which appeared in the n body problem as an application we consider the stability problem of the celebrated figure eight orbit due to chenciner and montgomery in the planar three body problem with equal masses and we clarify the relationship between linear stability and its variational nature on various loop spaces the basic idea is as follows the variational characterization of the figure eight orbit provides information about its morse index based on its relation to the maslov index our stability criteria come into play

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

additive operators preserving idempotent matrices over fields and applications abstract suppose f is a field of characteristic not 2 let m n f be the algebra of all n n matrices over f and let a 2 a 3 a 1 and a be the semigroups of all additive operator on m n f that preserve idempotence preserve tripotence preserve inverses of matrices and preserve group inverses of matrices respectively the main result in this paper is that the semigroup a 2 is generated by transposition similarity the operators x x for fixed arbitrary injective endomorphisms on f and the operators x tr x for fixed arbitrary additive maps from f to m n f with 1 o as applications we determine the structures of a 3 a 1 and a when the characteristic of f is also not 3

some general techniques on linear preserver problems abstract several general techniques on linear preserver problems are described the first one is based on a transfer principle in model theoretic algebra that allows one to extend linear preserver results on complex matrices to matrices over other algebraically closed fields of characteristic 0 the second one concerns the use of some simple geometric technique to reduce linear preserver problems to standard types so that known results can be applied the third one is about solving linear preserver problems on more general operator algebras by reducing the problems to idempotent preservers numerous examples will be given to demonstrate the proposed techniques

non planar minimizers and rotational symmetry in the n body problem abstract the newtonian n body problem admits uniformly rotating relative equilibrium solutions in the plane but not in r 3 when the bodies are allowed to interact in r 3 is it preferable in terms of action to leave the plane and follow a non planar trajectory we use the variational techniques of chenciner and venturelli celestial mech dyn astro 77 2000 139 to show that for an open set of masses there is a class of collision free action minimizing orbits of certain rotational symmetry in the four body problem which are non coplanar i e the planar relative equilibrium is not the least action solution among orbits in r 3 both periodic and quasi periodic solutions are constructed in this way we also discuss constructing collision free action minimizing solutions possessing d rotational symmetry along with various other symmetry constraints

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

highly eccentric hip hop solutions of the 2n body problem abstract we show the existence of families of hip hop solutions in the equal mass 2 n body problem which are close to highly eccentric planar elliptic homographic motions of 2 n bodies plus small perpendicular non harmonic oscillations by introducing a parameter the homographic motion and the small amplitude oscillations can be uncoupled into a purely keplerian homographic motion of fixed period and a vertical oscillation described by a hill type equation small changes in the eccentricity induce large variations in the period of the perpendicular oscillation and give rise via a bolzano argument to resonant periodic solutions of the uncoupled system in a rotating frame for small 0 the topological transversality persists and brouwer s fixed point theorem shows the existence of this kind of solutions in the full system

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

efficient region segmentation on compressed gray images using quadtree and shading representation abstract image segmentation which partitions the image into homogeneous regions is a fundamental operation in image processing suppose the input gray image with size n n has been compressed into the compressed image via quadtree and shading representation assume that the number of blocks in the representation is b commonly b n 2 due to the compression effect this paper first derives some closed forms for computing the mean variance of any block and for calculating the two statistical measures of any merged region in o 1 time it then presents an efficient o b b time algorithm for performing region segmentation on the compressed image directly where b is the inverse of the ackerman s function and is a very slowly growing function with the same time complexity our results extend the pioneering results by dillencourt and samet from the map image to the gray image in addition with four real images experimental results show that our proposed algorithm has about 55 4 execution time improvement ratio when compared to the previous fastest region segmentation algorithm by fiorio and gustedt whose o n 2 time algorithm is run on the original n n gray image

eigenvalue problem of sturm liouville systems with separated boundary conditions let lambda j be the jth eigenvalue of sturm liouville systems with separated boundary conditions we build up the hill type formula which represent prod nolimits j 1 lambda j 1 as a determinant of finite matrix consequently we get the krein type trace formula based on the hill type formula which express sum nolimits j 1 over lambda j m as trace of finite matrices the trace formula can be used to estimate the conjugate point alone a geodesic in riemannian manifold and to get some infinite sum identities

global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for the n body problem abstract given n point masses turning in a plane at a constant speed this paper deals with the global bifurcation of periodic solutions for the masses in that plane and in space as a special case one has a complete study of n identical masses on a regular polygon and a central mass the symmetries of the problem are used in order to find the irreducible representations the linearization and with the help of the orthogonal degree theory all the symmetries of the bifurcating branches

trace formula for linear hamiltonian systems with its applications to elliptic lagrangian solutions in the present paper we build up trace formulas for both the linear hamiltonian systems and sturm liouville systems the formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the hessian of the action functional a natural application is to study the non degeneracy of linear hamiltonian systems precisely by the trace formula we can give an estimation for the upper bound such that the non degeneracy preserves moreover we could estimate the relative morse index by the trace formula consequently a series of new stability criteria for the symmetric periodic orbits is given as a concrete application the trace formula is used to study the linear stability of elliptic lagrangian solutions of the classical planar three body problem which depends on the mass parameter beta in 0 9 and the eccentricity e in 0 1 based on the trace formula we estimate the stable region and hyperbolic region of the elliptic lagrangian solutions

on the existence of infinitely many periodic solutions to some problems of n body type we prove the existence of infinitely many periodic solutions with prescribed period to a class of problems of n body type

action minimizing orbits in the parallelogram four body problem with equal masses in this paper we prove the existence of a new periodic solution for the planar newtonian four body problem with equal masses on this orbit two mass points travel on one star shaped closed curve while the other two travel on another and have the opposite orientation the configuration of the masses changes from square to collinear periodically and remains a parallelogram for all time

simple choreographies of the planar newtonian n body problem in the n body problem a simple choreography is a periodic solution where all masses chase each other on a single loop in this paper we prove that for the planar newtonian n body problem with equal masses n 3 there are at least 2n 3 2 n 3 2 different main simple choreographies this confirms a conjecture given by chenciner et al geometry mechanics and dynamics springer new york pp 287 308 2002 all the simple choreoagraphies we prove belong to the linear chain family

morse index and stability of elliptic lagrangian solutions in the planar three body problem abstract we illustrate a new way to study the stability problem in celestial mechanics in this paper using the variational nature of elliptic lagrangian solutions in the planar three body problem we study the relation between morse index and its stability via maslov type index theory of periodic solutions of hamiltonian system for elliptic lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the morse index which is the key to understand the stability problem as a special case we provide a criterion to spectral stability of relative equilibrium

variational proof of the existence of the super eight orbit in the four body problem using the variational method chenciner and montgomery ann math 152 881 901 2000 proved the existence of an eight shaped periodic solution of the planar three body problem with equal masses just after the discovery gerver numerically found a similar periodic solution called super eight in the planar four body problem with equal mass in this paper we prove the existence of the super eight orbit by using the variational method the difficulty of the proof is to eliminate the possibility of collisions in order to solve it we apply the scaling technique established by tanaka ann inst h poincare anal non lineaire 10 215 238 1993 proc am math soc 122 275 284 1994 and investigate the asymptotic behavior of a binary collision

additive idempotence preservers abstract let a be a local matrix algebra over a field of characteristic different from 2 b an arbitrary algebra and x y banach spaces additive mappings a b which preserve idempotents are characterized the result is then used in classifying additive surjections b x b y which preserve idempotents and do not annihilate finite rank operators

topologically distinct collision free periodic solutions for the n center problem this work concerns the planar n center problem with homogeneous potential of degree alpha alpha in 1 2 the existence of infinitely many topologically distinct non collision periodic solutions with a prescribed energy is proved a notion of admissibility in the space of loops on the punctured plane is introduced so that in any admissible class and for any positive h the existence of a classical periodic solution with energy h for the n center problem with alpha in 1 2 is proven in case alpha 1 a slightly different result is shown it is the case that there is either a non collision periodic solution or a collision reflection solution the results hold for any position of the centres and it is possible to prescribe in advance the shape of the periodic solutions the proof combines the topological properties of the space of loops in the punctured plane with variational and geometrical arguments

collisionless periodic solutions to some three body problems we provide sufficient conditions for the existence of periodic solutions to some three body problems periodic solutions are found as minima of the associated action integral and are shown to be free of double and triple collisions

minimizing periodic orbits with regularizable collisions in the n body problem in the newtonian n body problem there are various subsystems with two degrees of freedom such as the collinear three body problem and the isosceles three body problem after we determine a normal form of the lagrangians of these subsystems we prove the existence of periodic solutions with regularizable collisions for these systems our result includes several examples such as schubart s orbit with or without equal masses among others

characterizations of automorphisms of operator algebras on banach spaces abstract let x be a complex banach space of dimension greater than one and denote by b x the algebra of all the bounded linear operators on x it is shown that if b x b x is a multiplicative map not assumed linear and if is sufficiently close to a linear automorphism of b x in some uniform sense then it is actually an automorphism

new periodic solutions for 2n body problems in r3 for newtonian 2n body problems with equal masses in r3 we prove the existence of new noncollision periodic solution such that 2n bodies move on two different closed curves

symbol recognition by error tolerant subgraph matching between region adjacency graphs we propose an error tolerant subgraph isomorphism algorithm formulated in terms of region adjacency graphs rag a set of edit operations to transform one rag into another one are defined as regions are represented by polylines and string matching techniques are used to measure their similarity the algorithm follows a branch and bound approach driven by the rag edit operations this formulation allows matching computing under distorted inputs and also reaching a solution in a near polynomial time the algorithm has been used for recognizing symbols in hand drawn diagrams

symmetry groups and non planar collisionless action minimizing solutions of the three body problem in three dimensional space periodic and quasi periodic solutions of the n body problem can be found as minimizers of the lagrangian action functional restricted to suitable spaces of symmetric paths the main purpose of this paper is to develop a systematic approach to the equivariant minimization for the three body problem in three dimensional space first we give a finite complete list of symmetry groups fitting to the minimization of the action with the property that any other symmetry group can be reduced to be isomorphic to one of these representatives a second step is to prove that the resulting local and global symmetric action minimizers are always collisionless when they are not already bound to collisions furthermore we prove some results which address the question of whether minimizers are planar or non planar as a consequence of our theory we will give general criteria for a symmetry group to yield planar or homographic minimizers either homographic or not as in the chenciner montgomery eight solution on the other hand we will provide a rigorous proof of the existence of some interesting one parameter families of periodic and quasi periodic non planar orbits these include the choreographic marchal s p12 family with equal masses together with a less symmetric choreographic family which anyway probably coincides with the p12 family

symmetric trajectories for the 2n body problem with equal masses we consider the problem of 2n bodies of equal masses in mathbb r 3 for the newtonian like weak force potential r and we prove the existence of a family of collision free nonplanar and nonhomographic symmetric solutions that are periodic modulo rotations in addition the rotation number with respect to the vertical axis ranges in a suitable interval these solutions have the hip hop symmetry a generalization of that introduced in 19 for the case of many bodies and taking account of a topological constraint the argument exploits the variational structure of the problem and is based on the minimization of lagrangian action on a given class of paths

binary decompositions for planar n body problems and symmetric periodic solutions a binary decomposition for a system of n masses is a way of treating the system as big n atop 2 big binaries with the total action exactly the same as that of the original system by considering binary decompositions we are able to provide effective lower bound estimates for the action of collision paths in several spaces of symmetric loops as applications we use our estimates to prove the existence of some new classes of symmetric periodic solutions for the n body problem

entire minimal parabolic trajectories the planar anisotropic kepler problem we continue the variational approach to parabolic trajectories introduced in our previous paper barutello et al entire parabolic trajectories as minimal phase transitions arxiv 1105 3358v1 2011 which sees parabolic orbits as minimal phase transitions we deepen and complete the analysis in the planar case for homogeneous singular potentials we characterize all parabolic orbits connecting two minimal central configurations as free time morse minimizers in a given homotopy class of paths these may occur for at most one value of the homogeneity exponent in addition we link this threshold of existence of parabolic trajectories with the absence of collisions for all the minimizers of fixed end problems and also with the existence of action minimizing periodic trajectories with nontrivial homotopy type

existence and minimizing properties of retrograde orbits to the three body problem with various choices of masses poincare made the first attempt in 1896 on applying variational calculus to the three body problem and observed that collision orbits do not necessarily have higher values of action than classical solutions little progress had been made on resolving this difficulty until a recent breakthrough by chenciner and montgomery afterward variational methods were successfully applied to the jv body problem to construct new classes of solutions in order to avoid collisions the problem is confined to symmetric path spaces and all new planar solutions were constructed under the assumption that some masses are equal a question for the variational approach on planar problems naturally arises are minimizing methods useful only when some masses are identical this article addresses this question for the three body problem for various choices of masses it is proved that there exist infinitely many solutions with a certain topological type called retrograde orbits that minimize the action functional on certain path spaces cases covered in our work include triple stars in retrograde motions double stars with one outer planet and some double stars with one planet orbiting around one primary mass our results largely complement the classical results by the poincare continuation method and conley s geometric approach

new methods in celestial mechanics and mission design the title of this paper is inspired by the work of poincare 1890 1892 who introduced many key dynamical systems methods during his research on celestial mechanics and especially the three body problem since then many researchers have contributed to his legacy by developing and applying these methods to problems in celestial mechanics and more recently with the design of space missions this paper will give a survey of some of these exciting ideas and we would especially like to acknowledge the work of michael dellnitz frederic gabern katalin grubits oliver junge wang sang koon francois lekien martin lo sina ober blobaum kathrin padberg robert preis and bianca thiere one of the purposes of the ams current events session is to discuss work of others even though we were involved in the research reported on here this short paper is intended to survey many ideas due to our collaborators and others this survey is by no means complete and we apologize for not having time or space to do justice to many important and fundamental works in fact the results reported on here rely on and were inspired by important preceding work of many others in celestial mechanics mission design and in dynamical systems we mention just a few whose work had a positive influence on what is reported here brian barden ed belbruno robert farquhar gerard gomez george haller charles jaffe kathleen howell linda petzold josep masdemont vered rom kedar radu serban carles simo turgay uzer steve wiggins and roby wilson in an upcoming monograph see koon lo marsden and ross 2005 the dynamical systems and computational approach and its application to mission design are discussed in detail one of the key ideas is that the competing gravitational pull between celestial bodies creates a vast array of passageways that wind around the sun planets and moons the boundaries of these passageways are realized geometrically as invariant manifolds attached to equilibrium points and periodic orbits in interlinked three body problems in particular tube like structures form an interplanetary transport network which will facilitate the exploration of mercury the moon the asteroids and the outer solar system including a mission to assess the possibility of life on jupiter s icy moons the use of these methods in problems in molecular dynamics of interest in chemistry is also briefly discussed

detectability of critical points of smooth functionals from their finite dimensional approximations abstract given a critical point of a c 2 functional on a separable hilbert space we obtain sufficient conditions for it to be detectable i e visible from finite dimensional rayleigh ritz galerkin rrg approximations while examples show that even nondegenerate critical points are without any further restriction not visible we single out relevant classes of smooth functionals e g the hamiltonian action on the loop space or the functionals associated with boundary value problems for some semilinear elliptic equations such that their nondegenerate critical points are visible from their rrg approximations

additive drazin inverse preservers let h be a real or complex hilbert space and b h denote the banach algebra of all bounded linear operators on h for t b h if there exists an operator td b h and a positive integer k such that ttd tdt tdttd td tk 1td tk then t is said to be drazin invertible and td is a drazin inverse of t we say a map b h b h preserves drazin inverse if td t d for every drazin invertible operator t b h in this paper we determine the structures of additive maps on b h preserving drazin inverses

zero product preserving additive maps on symmetric operator spaces and self adjoint operator spaces in this note we characterize the additive maps on the space of symmetric operators and the space of self adjoint operators which preserve zero products in both directions and the additive maps on the space of self adjoint operators which preserve jordan zero products in both directions we also give a complete classification of additive maps on the von neumann algebra of all bounded linear operators acting on a hilbert space which preserve square zero in both directions or preserve operators annihilated by a polynomial in both directions

globally minimizing parabolic motions in the newtonian n body problem we consider the n body problem in mathbb r d with the newtonian potential 1 r we prove that for every initial configuration xi and for every minimizing normalized central configuration x0 there exists a collision free parabolic solution starting from xi and asymptotic to x0 this solution is a minimizer in every time interval the proof exploits the variational structure of the problem and it consists in finding a convergent subsequence in a family of minimizing trajectories the hardest part is to show that this solution is parabolic and asymptotic to x0

on the existence of collisionless equivariant minimizers for the classical n body problem we show that the minimization of the lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n body problem provided that some simple conditions on the symmetry group are satisfied more precisely we give a fairly general condition on symmetry groups g of the loop space for the n body problem with potential of homogeneous degree with 0 which ensures that the restriction of the lagrangian action mathcal a to the space g of g equivariant loops is coercive and its minimizers are collisionless without any strong force assumption in proving that local minima of g are free of collisions we develop an averaging technique based on marchal s idea of replacing some of the point masses with suitable shapes see 10 as an application several new orbits can be found with some appropriate choice of g furthermore the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits

variational property of periodic kepler orbits in constant curvature spaces abstract we study the variational property of the periodic kepler orbits on the sphere the plane and the hyperbolic plane we first classify the orbits by the two constants of motion the energy and the angular momentum then we characterize the local variational property of the closed orbits by computing the maslov type indices finally we study the global variational property of the closed orbits we prove that the closed orbits on the hyperbolic plane minimizes the action among all loops which encircle the attracting center

efficient algorithms for coding hilbert curve of arbitrary sized image and application to window query previously several efficient hilbert scan based operations were presented but they all suffer from the constraint that the image size must be 2 rx2 r considering an image with arbitrary size i 1xi 2 this paper first presents an efficient snake scan based algorithm for coding the hilbert curve of the given image the proposed coding algorithm takes o k logu time to code the hilbert order of one pixel where k denotes the number of the quadrants and u min i 1 i 2 next a memory saving hilbert curve representation called hcgl is presented for representing the encoded hilbert curve and it can be constructed in o l 2logl time where l max i 1 i 2 based on the hcgl representation of arbitrary sized image an application to window query is presented and the proposed window query algorithm takes o mlogl p time where m denotes the number of generated maximal quadtree blocks and p denotes the number of output codes under the same psnr peak signal to noise ratio experimental results demonstrate that our proposed hcgl representation outperforms some existing related algorithms in terms of execution time and bpp bit per pixel in addition our proposed window query algorithm has been justified in the gis geographical information system application

symmetry groups of the planar three body problem and action minimizing trajectories we consider periodic and quasi periodic solutions of the three body problem with homogeneous potential from the point of view of equivariant calculus of variations first we show that symmetry groups of the lagrangian action functional can be reduced to groups in a finite explicitly given list after a suitable change of coordinates then we show that local symmetric minimizers are always collisionless without any assumption on the group other than the fact that collisions are not forced by the group itself moreover we describe some properties of the resulting symmetric collisionless minimizers lagrange euler hill type orbits and chenciner montgomery figure eights

additive mappings preserving operators of rank one abstract let b x be the algebra of all bounded linear operators on a nontrivial real or complex banach space and let f x be the subalgebra of all finite rank operators a characterization of additive mappings on f x which preserve operators of rank one or projections of rank one is given in the real case such mappings are automatically linear

image compression by b tree triangular coding this paper describes an algorithm for still image compression called b tree triangular coding bttc the coding scheme is based on the recursive decomposition of the image domain into right angled triangles arranged in a binary tree the method is attractive because of its fast encoding o n log n and decoding spl theta n where n is the number of pixels and because it is easy to implement and to parallelize experimental studies indicate that bttc produces images of satisfactory quality from a subjective and objective point of view one advantage of bttc over jpeg is its shorter execution time

a new branch of mountain pass solutions for the choreographical 3 body problem we prove the existence of a new branch of solutions of mountain pass type for the periodic 3 body problem with choreographical constraint at first we describe the variational structure of the action functional associated to the choreographical three body problem in mathbb r 3 in the second part using a bisection algorithm we provide a numerical non rigorous solution of mountain pass type for this problem in a rotating frame with angular velocity 1 5 the last step consists in the rigorous computer assisted proof of the existence of a full branch of solutions for the problem starting from the mountain pass solution detected numerically

action minimizers under topological constraints in the planar equal mass four body problem abstract it is shown that in the planar equal mass four body problem there exist two sets of action minimizers connecting two planar boundary configurations with fixed symmetry axes and specific order constraints a double isosceles configuration and an isosceles trapezoid configuration while order constraints are introduced on the boundary configurations by applying the level estimate method these minimizers are shown to be collision free and they can be extended to two new sets of periodic or quasi periodic orbits

variational constructions for some satellite orbits in periodic gravitational force fields in a gravitational force field a bounded orbit of an infinitesimal point mass is called a satellite orbit the purpose of this paper is to establish a variational theory for the existence of some relative periodic satellite orbits in periodic gravitational force fields the gravitation field can be generated by a relative equilibrium or a uniformly rotating asteroid our approach is to regard the aggregate of primaries together with the small satellite as a restricted full two body problem and then look for direct and retrograde relative periodic satellite orbits by direct methods of calculus of variations regularity of the action minimizing satellite orbit is obtained by providing satisfactory lower bound estimates for the distance between the satellite and the mass center an upper bound estimate for this distance is also provided as a contrast to classical existence proofs by continuation from infinity

additive mappings that preserve rank one nilpotent operators let b x be the algebra of bounded operators on a complex infinite dimensional banach space x and f x be the subalgebra of all finite rank operators in b x a characterization of additive mappings on f x which preserve rank one nilpotent operators in both directions is given as applications of this result some additive preservers are described

subharmonic pl solutions of first order hamiltonian systems abstract in this paper for any symplectic matrix p the existence of subharmonic p l solutions of the first order non autonomous superquadratic hamiltonian systems is considered under the convex condition the existence of infinitely many geometrically distinct p l solutions is proved

periodic solutions of second order non autonomous singular dynamical systems abstract in this paper we establish two different existence results of positive periodic solutions for second order non autonomous singular dynamical systems the first one is based on a nonlinear alternative principle of leray schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity the second one is based on schauder s fixed point theorem and the result sheds some new light on problems with weak singularities and proves that in some situations weak singularities may help create periodic solutions recent results in the literature are generalized and significantly improved

removing collision singularities from action minimizers for the n body problem with free boundaries for the planar and spatial n body problems it has been proved by marchal and chenciner that solutions for the minimizing problem with fixed ends are free from interior collisions this important result has been extended by ferrario terracini to newtonian type problems and equivariant problems it has also been used to construct many symmetric solutions for the n body problem

on the minimizing triple collision orbits in the planar newtonian three body problem in this paper we study the minimizing triple collision orbits in the planar newtonian three body problem with arbitrary masses we show that for a given non collinear initial configuration the minimizing triple collision orbit is collision free until a simultaneous collision and its limiting configuration is the lagrangian configuration with the same orientation as the initial configuration for the collinear initial configuration under a certain technical assumption there exist two minimizing orbits the limiting configurations of these orbits are the two opposite lagrangian configurations