Topological rigidity of Hamiltonian loops and quantum homology

This paper studies the question of when a loop φ={φ _t }_{0≤ t ≤1} in the group Symp(M,ω) of symplectomorphisms of a symplectic manifold (M,ω) is isotopic to a loop that is generated by a time-dependent Hamiltonian function. (Loops with this property are said to be Hamiltonian.) Our main result is that Hamiltonian loops are rigid in the following sense: if φ is Hamiltonian with respect to ω, and if φ′ is a small perturbation of φ that preserves another symplectic form ω′, then φ′ is Hamiltonian with respect to ω′. This allows us to get some new information on the structure of the flux group, i.e. the image of π₁(Symp(M,ω)) under the flux homomorphism. We give a complete proof of our result for some manifolds, and sketch the proof in general. The argument uses methods developed by Seidel for studying properties of Hamiltonian loops via the quantum homology of M. Oblatum 31-X-1997 & 20-III-1998 / Published online: 14 October 1998     

On a conjecture concerningk-Hamilton-nice sequences

In this paper, we prove that a non-negative rational number sequence (a ₁,a ₂, ...,a _k+1) isk-Hamilton-nice, if (1)a _k+12, and (2) _j ^=1/h (i _j –1)k–1 implies for arbitraryi ₁,i ₂,...i _h {1,2,... ,k}. This result was conjectured by Guantao Chen and R.H. Schelp, and it generalizes several well-known sufficient conditions for graphs to be Hamiltonian.This project is supported by the National Natural Science Foundation of China.     

On the number of hamiltonian cycles in a maximal planar graph

We consider the problem of the minimum number of Hamiltonian cycles that could be present in a Hamiltonian maximal planar graph on p vertices. In particular, we construct a p-vertex maximal planar graph containing exactly four Hamiltonian cycles for every p ≥ 12. We also prove that every 4-connected maximal planar graph on p vertices contains at least p/(log₂ p) Hamiltonian cycles.     

Hamiltonian paths in Cayley graphs

The classical question raised by Lovász asks whether every Cayley graph is Hamiltonian. We present a short survey of various results in that direction and make some additional observations. In particular, we prove that every finite group G has a generating set of size at most log₂|G|, such that the corresponding Cayley graph contains a Hamiltonian cycle. We also present an explicit construction of 3-regular Hamiltonian expanders.     

Diffusion times and stability exponents for nearly integrable analytic systems

For a positive integer n and R>0, we set . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H _j ) of the completely integrable Hamiltonian on , with unstable orbits for which we can estimate the time of drift in the action space. These functions H _j are analytic on a fixed complex neighborhood V of , and setting the time of drift of these orbits is smaller than (C(1/ɛ _j )^1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface, the result is therefore almost optimal since the stability exponent for such orbits is 1/2(n−2). An analogous result for Hamiltonian diffeomorphisms is also proved. Two main ingredients are used in order to deal with the analytic setting: a version of Sternberg's conjugacy theorem in a neighborhood of a normally hyperbolic manifold in a symplectic system, for which we give a complete (and seemingly new) proof; and Easton windowing method that allow us to approximately localize the wandering orbits and estimate their speed of drift.     

On the number of Hamiltonian cycles in triangulations

It is proved that if a planar triangulation different from K₃ and K₄ contains a Hamiltonian cycle, then it contains at least four of them. Together with the result of Hakimi, Schmeichel, and Thomassen [2], this yields that, for n ? 12, the minimum number of Hamiltonian cycles in a Hamiltonian planar triangulation on n vertices is four. We also show that this theorem holds for triangulations of arbitrary surfaces and for 3-connected triangulated graphs.     

Gell-Mann and Low Formula for Degenerate Unperturbed States

The Gell-Mann and Low switching allows to transform eigenstates of an unperturbed Hamiltonian H ₀ into eigenstates of the modified Hamiltonian H ₀ + V. This switching can be performed when the initial eigenstate is not degenerate, under some gap conditions with the remainder of the spectrum. We show here how to extend this approach to the case when the ground state of the unperturbed Hamiltonian is degenerate. More precisely, we prove that the switching procedure can still be performed when the initial states are eigenstates of the finite rank self-adjoint operator P₀VP₀{\mathcal{P}_{0}V\mathcal{P}_{0}} , where P₀{\mathcal{P}_0} is the projection onto a degenerate eigenspace of H ₀.     

Hamiltonian results in K1,3-free graphs

There have been a number of results dealing with Hamiltonian properties in powers of graphs. In this paper we show that the square and the total graph of a K_1,3-free graph are vertex pancyclic. We then discuss some of the relationships between connectivity and Hamiltonian properties in K_1,3-free graphs.     

On Hamiltonian perturbations of hyperbolic systems of conservation laws I: Quasi‐Triviality of bi‐Hamiltonian perturbations

We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one‐dimensional systems of hyperbolic PDEs v _t + [?( v )]_x = 0. Under certain genericity assumptions it is proved that any bi‐Hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tool is in constructing the so‐called quasi‐Miura transformation of jet coordinates, eliminating an arbitrary deformation of a semisimple bi‐Hamiltonian structure of hydrodynamic type (the quasi‐triviality theorem). We also describe, following [35], the invariants of such bi‐Hamiltonian structures with respect to the group of Miura‐type transformations depending polynomially on the derivatives. © 2005 Wiley Periodicals, Inc.     

Hamilton cycles in random graphs and directed graphs

We prove that almost every digraph D_{2–in, 2–out} is Hamiltonian. As a corollary we obtain also that almost every graph G_4–out is Hamiltonian. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16: 369–401, 2000     

Hamiltonian fourfold 1:1 resonance with two rotational symmetries

In this communication we deal with the analysis of Hamiltonian Hopf bifurcations in 4-DOF systems defined by perturbed isotropic oscillators (1-1-1-1 resonance), in the presence of two quadratic symmetries I ₁ and I ₂. As a perturbation we consider a polynomial function with a parameter. After normalization, the truncated normal form gives rise to an integrable system which is analyzed using reduction to a one degree of freedom system. The Hamiltonian Hopf bifurcations are found using the ‘geometric method’ set up by one of the authors.      

Geometric interpretation of Hamiltonian cycles problem via singularly perturbed Markov decision processes

We consider the Hamiltonian cycle problem (HCP) embedded in a singularly perturbed Markov decision process (MDP). More specifically, we consider the HCP as an optimization problem over the space of long-run state-action frequencies induced by the MDP's stationary policies. We also consider two quadratic functionals over the same space. We show that when the perturbation parameter, ? is sufficiently small the Hamiltonian cycles of the given directed graph are precisely the maximizers of one of these quadratic functionals over the frequency space intersected with an appropriate (single) contour of the second quadratic functional. In particular, all these maximizers have a known Euclidean distance of z _m (?) from the origin. Geometrically, this means that Hamiltonian cycles, if any, are the points in the frequency polytope where the circle of radius z _m (?) intersects a certain ellipsoid.     

A Degree Constraint for Uniquely Hamiltonian Graphs

A graph, G, is called uniquely Hamiltonian if it contains exactly one Hamilton cycle. We show that if G=(V, E) is uniquely Hamiltonian then Where #(G)=1 if G has even number of vertices and 2 if G has odd number of vertices. It follows that every n-vertex uniquely Hamiltonian graph contains a vertex whose degree is at most c log₂n+2 where c=(log₂3−1)⁻¹≈1.71 thereby improving a bound given by Bondy and Jackson [3].     

Every connected,locally connected nontrivial graph with no induced claw is hamiltonian

A graph is locally connected if every neighborthood induces a connected subgraph. We show here that every connected, locally connected graph on p ≥ 3 vertices and having no induced K_1,3 is Hamiltonian. Several sufficient conditions for a line graph to be Hamiltonian are obtained as corollaries.     

Aubry‐Mather theory and periodic solutions of the forced Burgers equation

Consider a Hamiltonian system with Hamiltonian of the form H(x, t, p) where H is convex in p and periodic in x, and t and x ∈ ℝ¹. It is well‐known that its smooth invariant curves correspond to smooth Z²‐periodic solutions of the PDE u_t + H(x, t, u)_x = 0. In this paper, we establish a connection between the Aubry‐Mather theory of invariant sets of the Hamiltonian system and Z²‐periodic weak solutions of this PDE by realizing the Aubry‐Mather sets as closed subsets of the graphs of these weak solutions. We show that the complement of the Aubry‐Mather set on the graph can be viewed as a subset of the generalized unstable manifold of the Aubry‐Mather set, defined in (2.24). The graph itself is a backward‐invariant set of the Hamiltonian system. The basic idea is to embed the globally minimizing orbits used in the Aubry‐Mather theory into the characteristic fields of the above PDE. This is done by making use of one‐ and two‐sided minimizers, a notion introduced in [12] and inspired by the work of Morse on geodesics of type A [26]. The asymptotic slope of the minimizers, also known as the rotation number, is given by the derivative of the homogenized Hamiltonian, defined in [21]. As an application, we prove that the Z²‐periodic weak solution of the above PDE with given irrational asymptotic slope is unique. A similar connection also exists in multidimensional problems with the convex Hamiltonian, except that in higher dimensions, two‐sided minimizers with a specified asymptotic slope may not exist. © 1999 John Wiley & Sons, Inc.     

On Hamiltonian-connected regular graphs

In this paper it is shown that any m-regular graph of order 2m (m ≧ 3), not isomorphic to K_m,m, or of order 2m + 1 (m even, m ≧ 4), is Hamiltonian connected, which extends a previous result of Nash-Williams. As a corollary, it is derived that any such graph contains atleast m Hamiltonian cycles for odd m and atleast 1/2m Hamiltonian cycles for even m.     

The dressed nonrelativistic electron in a magnetic field

We consider a nonrelativistic electron interacting with a classical magnetic field pointing along the x₃‐axis and with a quantized electromagnetic field. When the interaction between the electron and photons is turned off, the electronic system is assumed to have a ground state of finite multiplicity. Because of the translation invariance along the x₃‐axis, we consider the reduced Hamiltonian associated with the total momentum along the x₃‐axis and, after introducing an ultraviolet cutoff and an infrared regularization, we prove that the reduced Hamiltonian has a ground state if the coupling constant and the total momentum along the x₃‐axis are sufficiently small. We determine the absolutely continuous spectrum of the reduced Hamiltonian and, when the ground state is simple, we prove that the renormalized mass of the dressed electron is greater than or equal to its bare one. We then deduce that the anomalous magnetic moment of the dressed electron is nonnegative. Copyright © 2006 John Wiley & Sons, Ltd.