Nomadic decompositions of bidirected complete graphs

We use to denote the bidirected complete graph on n vertices. A nomadic Hamiltonian decomposition of is a Hamiltonian decomposition, with the additional property that “nomads” walk along the Hamiltonian cycles (moving one vertex per time step) without colliding. A nomadic near-Hamiltonian decomposition is defined similarly, except that the cycles in the decomposition have length n-1, rather than length n. Bondy asked whether these decompositions of exist for all n. We show that admits a nomadic near-Hamiltonian decomposition when .     

Bifurcations of limit cycles from quintic Hamiltonian systems with a double figure eight loop

This paper deals with Liénard equations of the form , , with P and Q polynomials of degree 5 and 4 respectively. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree six, exhibiting a double figure eight loop. The number of limit cycles and their distributions are given by using the methods of bifurcation theory and qualitative analysis.     

Singularities of the Hamiltonian vectorfield in nonautonomous variational problems

Variational problems with n degrees of freedom give rise (by Pontriaguine maximum principle) to a Hamiltonian vectorfield in , that presents singularities (nonsmoothness points) when the Lagrangian is not convex. For one degree of freedom nonautonomous problems of the calculus of variations where the Hamiltonian vectorfield in depends explicitly on the time, we consider the associated autonomous vectorfield in and classify its singularities up to an equivalence that takes into account the special role played by the time coordinate, i.e., that respects the foliation of into planes of constant time.     

New fermionic formula for unrestricted Kostka polynomials

A new fermionic formula for the unrestricted Kostka polynomials of type is presented. This formula is different from the one given by Hatayama et al. and is valid for all crystal paths based on Kirillov-Reshetikhin modules, not just for the symmetric and antisymmetric case. The fermionic formula can be interpreted in terms of a new set of unrestricted rigged configurations. For the proof a statistics preserving bijection from this new set of unrestricted rigged configurations to the set of unrestricted crystal paths is given which generalizes a bijection of Kirillov and Reshetikhin.     

Asymptotic expansion of the period function

Let P be a not necessarily bounded polycycle of an analytic vector field on an open set of the plane. Suppose that the singularities which appear after desingularization of the vertices of P are formally linearizable. Consider the function T defined by the return time near P. It is shown that the function T and its derivative T′ have asymptotic expansions in and . It is also shown that under some other conditions imposed on the polycycle vertices, the asymptotic expansions of T and T′ converge absolutely and uniformly to these functions, respectively. These results are applied to the polycycles of the analytic vector fields which have a Darboux first integral. In particular, it is obtained that if P is a polycycle of a Hamiltonian vector field with an analytic (polynomial if P is unbounded) Hamiltonian function, T is a nonoscillating function. Another application concerns the nilpotent centers or focus, since the singularities which appear after desingularization of such a singularity have analytic first integrals.     

Comodules and Landweber exact homology theories

We show that, if E is a commutative MU-algebra spectrum such that is Landweber exact over , then the category of -comodules is equivalent to a localization of the category of -comodules. This localization depends only on the heights of E at the integer primes p. It follows, for example, that the category of -comodules is equivalent to the category of -comodules. These equivalences give simple proofs and generalizations of the Miller-Ravenel and Morava change of rings theorems. We also deduce structural results about the category of -comodules. We prove that every -comodule has a primitive, we give a classification of invariant prime ideals in , and we give a version of the Landweber filtration theorem.     

Periodic solutions of superlinear autonomous Hamiltonian systems with prescribed period

In this paper we prove an existence theorem of nonconstant periodic solution of superlinear autonomous Hamiltonian system with prescribed period under an assumption weaker than Ambrosetti-Rabinowitz-type condition:

     

Exact algorithms for the Hamiltonian cycle problem in planar graphs

We construct an exact algorithm for the Hamiltonian cycle problem in planar graphs with worst case time complexity , where c is some fixed constant that does not depend on the instance. Furthermore, we show that under the exponential time hypothesis, the time complexity cannot be improved to .     

Perturbation from an elliptic Hamiltonian of degree four—III global centre

The paper deals with Liénard equations of the form , with P and Q polynomials of degree, respectively, 3 and 2. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree four, exhibiting a global centre. It is proven that the least upper bound of the number of zeros of the related elliptic integral is four, and this is a sharp one.This result permits to prove the existence of Liénard equations of type (3,2) with a quadruple limit cycle, with both a triple and a simple limit cycle, with two semistable limit cycles, with one semistable and two simple limit cycles or with four simple limit cycles.     

On the Korteweg–de Vries Equation: Convergent Birkhoff Normal Form

The Korteweg–de Vries equation (KdV)[formula]is a completely integrable Hamiltonian system of infinite dimension with phase space the Sobolev spaceH^N(S¹; ), (N?1), Hamiltonian (q):=∫_S1((∂_xq(x))²+q(x)³) dx, and Poisson structure ∂/∂x. The functionq≡0 is an elliptic fixed point. We prove that for anyN?1, the Korteweg–de Vries equation (and thus the entire KdV-hierarchy) admits globally defined real analytic action-angle variables. As a consequence it follows that in a neighborhood ofq≡0 inH¹(S¹; ), the KdV-Hamiltonian (and similarly any Hamiltonian in the KdV-hierarchy) admits a convergent Birkhoff normal form; to the best of our knowledge this is the first such example in infinite dimension. Moreover, using the constructed action-angle variables, we analyze the regularity properties of the Hamiltonian vectorfield of KdV.     

On Hamiltonian elliptic systems with periodic or non-periodic potentials

This paper is concerned with the following Hamiltonian elliptic system

     

Fiber Hamiltonians in non-relativistic quantum electrodynamics

A translation invariant Hamiltonian H of non-relativistic quantum electrodynamics is studied. This Hamiltonian is decomposed with respect to the total momentum PT:

     

Analytic-non-integrability of an integrable analytic Hamiltonian system

We introduce the polynomial Hamiltonian and we prove that the associated Hamiltonian system is Liouville-C^∞-integrable, but fails to be real-analytically integrable in any neighbourhood of an equilibrium point. The proof only uses power series expansions, and is elementary.