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

interactions of two dimensionally localized pulses of the regularized long wave equation abstract interactions between positive and or negative cylindrical bell shaped pulses are investigated numerically for a two dimensional regularized long wave equation overtaking collisions of two same signed pulses are found to be fairly elastic head on collisions between positive and negative pulses are inelastic and their overall features are analogous to those in the one dimentional case some properties of collision processes are discussed in terms of the conversed quantities

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

central configurations of the n body problem via equivariant morse theory in this paper we use the equivariant morse theory to give an estimate of the minimal number of central configurations in the n body problem in 3 in the case of equal masses we prove that the planar central configurations are saddle points for the potential energy from this we deduce the presence of non planar central configurations for every n 4

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 study of the apsidal angle and a proof of monotonicity in the logarithmic potential case abstract this paper concerns the behaviour of the apsidal angle for orbits of central force system with homogeneous potential of degree 2 1 and logarithmic potential we derive a formula for the apsidal angle as a fixed end points integral and we study the derivative of the apsidal angle with respect to the angular momentum l the monotonicity of the apsidal angle as function of l is discussed and it is proved in the logarithmic potential case

cylindrical quasi solitons of the zakharov kuznetsov equation abstract evolutions and interactions of two dimensional solitary waves of the zakharov kuznetsov equation are investigated numerically formations of cylindrical bell shaped pulses are observed in the initial value problems a single bell shaped pulse propagates stably without any deformation like a soliton two similar pulses exchange their amplitudes without merging and two dissimilar ones undergo overtaking collision after a collision of two pulses the strong pulse becomes somewhat stronger and the weak one becomes weaker with radiation of ripples so that the collision process is slightly inelastic generated ripples are very small for center to center collision of similar pulses the cylindrically symmetric pulses of the zakharov kuznetov equation are thus found to behave approximately soliton like some properties of collision process are interpreted in terms of the conservation laws

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

algorithms for the certified write all problem in this paper we prove new upper bounds on the complexity of the certified write all problem with respect to an adaptive adversary our strongest result is that for any epsilon 0 there exists an o p 1 epsilon work algorithm for the p processor p memory cell write all we also give a randomized o p 2 log p work algorithm for a p processor p 2 memory cell write all

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

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

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

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

scattering of regularized long wave solitary waves abstract the lagrangian density for the regularized long wave equation also known as the bbm equation is presented using the trial function technique ordinary differential equations that describe the time dependence of the position of the peaks amplitudes and widths for the collision of two solitary waves are obtained these equations are analyzed in the born and equal width approximations and compared with numerical results obtained by direct integration utilizing the split step fast fourier transform method the computations show that collisions are inelastic and that production of solitary waves may occur

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

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

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

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

cylindrical solitary pulses in a two dimensional stabilized kuramoto sivashinsky system abstract by linearly coupling a generalized zakharov kuznetsov equation alias the two dimensional 2d benney equation to an extra linear dissipative equation a 2d extension of a recently proposed stabilized kuramoto sivashinsky system is developed the model applies to the description of surface waves on 2d liquid layers in various physical settings the extra equation provides for the stability of the zero state in the system thus paving a way to the existence of stable 2d localized solitary pulses sps a perturbation theory based on a family of cylindrical solitons existing in the conservative counterpart of the system is developed by treating dissipation and gain in the model as small perturbations it is shown that the system may select two steady state solitons from the continuous family provided by the conservative counterpart of which the one with larger amplitude is expected to be stable numerical simulations support the analytical predictions quite well additionally it is found that a shallow quasi one dimensional 1d trough is attached to the stable sp if the integration domain is not very large and an explanation to this feature is proposed stable double humped bound states of two pulses are found too

efficient gossip and robust distributed computation this paper presents an efficient deterministic gossip algorithm for p synchronous crash prone message passing processors the algorithm has time complexity t o log2p and message complexity m o p1 e for any e 0 this substantially improves the message complexity of the previous best algorithm that has m o p1 77 while maintaining the same time complexity the strength and utility of the new result is demonstrated by constructing a deterministic algorithm for performing n tasks in this distributed setting previous solutions used coordinator or checkpointing approaches immediately incurring a work penalty n f p for f crashes or relied on strong communication primitives such as reliable broadcast or had work too close to the trivial p n bound of oblivious algorithms the new algorithm uses p crash prone processors to perform n similar and idempotent tasks so long as one processor remains active the work of the algorithm is w o n p min f 1 log3p and its message complexity is m o fpepsiv p min f 1 log p for any e 0 this substantially improves the work complexity of previous solutions using simple point to point messaging while meeting or beating the corresponding message complexity bounds the new algorithms use communication graphs and permutations with certain combinatorial properties that are shown to exist the algorithms are correct for any permutations and in particular the same expected bounds can be achieved using random permutations

solving the at most once problem with nearly optimal effectiveness we present and analyze a wait free deterministic algorithm for solving the at most once problem how m shared memory fail prone processes perform asynchronously n jobs at most once our algorithmic strategy provides for the first time nearly optimal effectiveness which is a measure that expresses the total number of jobs completed in the worst case the effectiveness of our algorithm equals n 2m 2 this is up to an additive factor of m close to the known effectiveness upper bound n m 1 over all possible algorithms and improves on the previously best known deterministic solutions that have effectiveness only n logm o n we also present an iterative version of our algorithm that for any m o n logn3 is both effectiveness optimal and work optimal for any constant 0 we then employ this algorithm to provide a new algorithmic solution for the write all problem which is work optimal for any m o n logn3