Related papers: Counting Hamiltonian Paths in Transitive Tournamen…
An investigation of classical fields with fractional derivatives is presented using the fractional Hamiltonian formulation. The fractional Hamilton's equations are obtained for two classical field examples. The formulation presented and the…
The set of (ordered) score sheets of a round-robin football tournament played between $n$ teams together with the pointwise addition has the structure of an affine monoid. In this paper we study (using both theoretical and computational…
We present several methods for predicting the dynamics of Hamiltonian systems from discrete observations of their vector field. Each method is either informed or uninformed of the Hamiltonian property. We empirically and comparatively…
In this paper we relate dynamics associated to zero-sum games (Fictitious play) to Hamiltonian dynamics. It turns out that the Hamiltonian dynamics which is induced from fictitious play, has properties which are rather different from those…
For a given differentiable map $(x,y)\to (X(x,y),Y(x,y))$, which has an inverse, we show that there exists a Hamiltonian flow in which x plays the role of the time variable while y is fixed.
For the family of complex rational functions known as "Generalized McMullen maps", F(z) = z^n + a/z^n+b, for complex parameters a and b, with a nonzero, and any integer n at least 3 fixed, we reveal, and provide a combinatorial model for,…
Mean field games models describing the limit of a large class of stochastic differential games, as the number of players goes to $+\infty$, have been introduced by J.-M. Lasry and P.-L. Lions. We use a change of variables to transform the…
A wide variety of problems in combinatorics and discrete optimization depend on counting the set $S$ of integer points in a polytope, or in some more general object constructed via discrete geometry and first-order logic. We take a tour…
In this paper we study the structure of $k$-transitive closures of directed paths and formulate several properties. Concept of $k$-transitive orientation generalize the traditional concept of transitive orientation of a graph.
Fluctuational transitions between two co-existing chaotic attractors, separated by a fractal basin boundary, are studied in a discrete dynamical system. It is shown that the mechanism for such transitions is determined by a hierarchy of…
We study absolutely periodic points and trajectories of Hamiltonian systems. Our main result is a necessary and sufficient for a Hamiltonian system to have the following property: if there exists one absolutely periodic trajectory then all…
We prove that a tournament with $n$ vertices has more than $0.13n^2(1+o(1))$ edge-disjoint transitive triples. We also prove some results on the existence of large packings of $k$-vertex transitive tournaments in an $n$-vertex tournament.…
Statistical mechanics of a small system of cars on a single-lane road is developed. The system is not characterized by a Hamiltonian but by a conditional probability of a velocity of a car for the given velocity and distance of the car…
We give a series of combinatorial results that can be obtained from any two collections (both indexed by $\Z\times \N$) of left and right pointing arrows that satisfy some natural relationship. When applied to certain self-interacting…
In a mathematical context in which one can multiply distributions the "`formal"' nonperturbative canonical Hamiltonian formalism in Quantum Field Theory makes sense mathematically, which can be understood a priori from the fact the so…
The problem of finding the number of ordered commuting tuples of elements in a finite group is equivalent to finding the size of the solution set of the system of equations determined by the commutator relations that impose commutativity…
In any periodic direction on the regular pentagon billiard table, there exists two combinatorially different billiard paths, with one longer than the other. For each periodic direction, McMullen asked if one could determine whether the…
We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of…
We give some formulas of the James-Hopf maps by using combinatorial methods. An application is to give a product decomposition of the spaces $\Omega\Sigma^2(X)$.
Complex trajectories for Hamiltonians of the form H=p^n+V(x) are studied. For n=2 time-reversal symmetry prevents trajectories from crossing. However, for n>2 trajectories may indeed cross, and as a result, the complex trajectories for such…