modelling quasicrystals at positive temperature we consider a two dimensional lattice model of equilibrium statistical mechanics using nearest neighbor interactions based on the matching conditions for an aperiodic set of 16 wang tiles this model has uncountably many ground state configurations all of which are nonperiodic the question addressed in this paper is whether nonperiodicity persists at low but positive temperature we present arguments mostly numerical that this is indeed the case in particular we define an appropriate order parameter prove that it is identically zero at high temperatures and show by monte carlo simulation that it is nonzero at low temperatures

a smoothing method for mathematical programs with equilibrium constraints the mathematical program with equilibrium constraints mpec is an optimization problem with variational inequality constraints mpec problems include bilevel programming problems as a particular case and have a wide range of applications mpec problems with strongly monotone variational inequalities are considered in this paper they are transformed into an equivalent one level nonsmooth optimization problem then a sequence of smooth regular problems that progressively approximate the nonsmooth problem and that can be solved by standard available software for constrained optimization is introduced it is shown that the solutions stationary points of the approximate problems converge to a solution stationary point of the original mpec problem numerical results showing viability of the approach are reported

tacit collusion games in pool based electricity markets under transmission constraints harrington et al math program ser b 104 407 435 2005 introduced a general framework for modeling tacit collusion in which producing firms collectively maximize the nash bargaining objective function subject to incentive compatibility constraints this work extends that collusion model to the setting of a competitive pool based electricity market operated by an independent system operator the extension has two features first the locationally distinct markets in which firms compete are connected by transmission lines capacity limits of the transmission lines together with the laws of physics that guide the flow of electricity may alter firms strategic behavior second in addition to electricity power producers other market participants including system operators and power marketers play important roles in a competitive electricity market the new players are included in the model in order to better represent real world markets and this inclusion will impact power producers strategic behavior as well the resulting model is a mathematical program with equilibrium constraints mpec properties of the specific mpec are discussed and numerical examples illustrating the impacts of transmission congestion in a collusive game are presented

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

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

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

generalized nash games and quasi variational inequalities abstract a generalized nash game is an n person noncooperative game with nondisjoint strategy sets other names for this game form include social equilibria and pseudo nash games this paper explores both the qualitative and quantitative properties of such games through the use of quasi variational inequality theory several interesting relationships between the variational and quasi variational inequality forms of this class of games are described and the practical implementation of generalized nash games are explored at length

partially b regular optimization and equilibrium problems this paper introduces a concept termed partial b regularity for a feasible solution to a bivariate constraint system and shows that this condition leads to the equivalence between the b stationarity of a pair of lifted and unlifted programs in particular for an optimization problem with a univariate pseudoconvex objective function constrained by such a nonconvex bivariate system partial b regularity provides a sufficient condition for a b stationary point to be globally optimal applications of partial b regularity to several classes of optimization and equilibrium problems are presented these include a lexicographic optimization problem a nonconvex mathematical program with equilibrium constraints mpec that arises from a convex implicit value function optimization problem and a nash equilibrium program with equilibrium constraints

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

two stage non cooperative games with risk averse players this paper formally introduces and studies a non cooperative multi agent game under uncertainty the well known nash equilibrium is employed as the solution concept of the game while there are several formulations of a stochastic nash equilibrium problem we focus mainly on a two stage setting of the game wherein each agent is risk averse and solves a rival parameterized stochastic program with quadratic recourse in such a game each agent takes deterministic actions in the first stage and recourse decisions in the second stage after the uncertainty is realized each agent s overall objective consists of a deterministic first stage component plus a second stage mean risk component defined by a coherent risk measure describing the agent s risk aversion we direct our analysis towards a broad class of quantile based risk measures and linear quadratic recourse functions for this class of non cooperative games under uncertainty the agents objective functions can be shown to be convex in their own decision variables provided that the deterministic component of these functions have the same convexity property nevertheless due to the non differentiability of the recourse functions the agents objective functions are at best directionally differentiable such non differentiability creates multiple challenges for the analysis and solution of the game two principal ones being 1 a stochastic multi valued variational inequality is needed to characterize a nash equilibrium provided that the players optimization problems are convex 2 one needs to be careful in the design of algorithms that require differentiability of the objectives moreover the resulting multi valued variational formulation cannot be expected to be of the monotone type in general the main contributions of this paper are as follows a prior to addressing the main problem of the paper we summarize several approaches that have existed in the literature to deal with uncertainty in a non cooperative game b we introduce a unified formulation of the two stage snep with risk averse players and convex quadratic recourse functions and highlight the technical challenges in dealing with this game c to handle the lack of smoothness we propose smoothing schemes and regularization that lead to differentiable approximations d to deal with non monotonicity we impose a generalized diagonal dominance condition on the players smoothed objective functions that facilitates the application and ensures the convergence of an iterative best response scheme e to handle the expectation operator we rely on known methods in stochastic programming that include sampling and approximation f we provide convergence results for various versions of the best response scheme particularly for the case of private recourse functions overall this paper lays the foundation for future research into the class of sneps that provides a constructive paradigm for modeling and solving competitive decision making problems with risk averse players facing uncertainty this paradigm is very much at an infancy stage of research and requires extensive treatment in order to meet its broad applications in many engineering and economics domains

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

convergence properties of a regularization scheme for mathematical programs with complementarity constraints we study the convergence behavior of a sequence of stationary points of a parametric nlp which regularizes a mathematical program with equilibrium constraints mpec in the form of complementarity conditions accumulation points are feasible points of the mpec they are c stationary if the mpec linear independence constraint qualification holds they are m stationary if in addition an approaching subsequence satisfies second order necessary conditions and they are b stationary if in addition an upper level strict complementarity condition holds these results complement recent results of fukushima and pang convergence of a smoothing continuation method for mathematical programs with equilibrium constraints in ill posed variational problems and regularization techniques springer verlag new york 1999 we further show that every local minimizer of the mpec which satisfies the linear independence upper level strict complementarity and a second order optimality condition can be embedded into a locally unique piecewise smooth curve of local minimizers of the parametric nlp

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

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 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

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

an implementable active set algorithm for computing a b stationary point of a mathematical program with linear complementarity constraints erratum in m fukushima and p tseng siam j optim 12 2002 pp 724 739 an active set algorithm was proposed for solving a mathematical program with a smooth objective function and linear inequality complementarity constraints it is asserted therein that under a uniform licq on the feasible set this algorithm generates iterates whose cluster points are b stationary points of the problem however the proof has a gap and shows only that each cluster point is an m stationary point we discuss this gap and show that b stationarity can be achieved if the algorithm is modified and an additional error bound condition holds

three modeling paradigms in mathematical programming celebrating the sixtieth anniversary since the zeroth international symposium on mathematical programming was held in 1949 this paper discusses several promising paradigms in mathematical programming that have gained momentum in recent years but have yet to reach the main stream of the field these are competition dynamics and hierarchy the discussion emphasizes the interplay between these paradigms and their connections with existing subfields including disjunctive equilibrium and nonlinear programming and variational inequalities we will describe the modeling approaches mathematical formulations and recent results of these paradigms and sketch some open mathematical and computational challenges arising from the resulting optimization and equilibrium problems our goal is to elucidate the need for a systematic study of these problems and to inspire new research in the field

modeling and computing two settlement oligopolistic equilibrium in a congested electricity network a model of two settlement electricity markets is introduced which accounts for flow congestion demand uncertainty system contingencies and market power we formulate the subgame perfect nash equilibrium for this model as an equilibrium problem with equilibrium constraints epec in which each firm solves a mathematical program with equilibrium constraints mpec the model assumes linear demand functions quadratic generation cost functions and a lossless dc network resulting in equilibrium constraints as a parametric linear complementarity problem lcp we introduce an iterative procedure for solving this epec through repeated application of an mpec algorithm this mpec algorithm is based on solving quadratic programming subproblems and on parametric lcp pivoting numerical examples demonstrate the effectiveness of the mpec and epec algorithms and the tractability of the model for realistic size power systems