Starting from a reaction path Hamiltonian, a suitably reduced harmonic bath averaged Hamiltonian is derived by averaging over all the normal mode coordinates. Generalization of the harmonic bath averaged Hamiltonian to any dimensions are performed and the feasibility to use a linear reaction path/surface are investigated and discussed. By use of a harmonic bath averaged Hamiltonian, the tunneli...

Hamiltonian dynamical systems tend to have infinitely many periodic orbits. For example, for a broad class of symplectic manifolds almost all levels of a proper smooth Hamiltonian carry periodic orbits. The Hamiltonian Seifert conjecture is the existence problem for regular compact energy levels without periodic orbits. Very little is known about how large the set of regular energy values witho...

The present paper is a review of counterexamples to the “Hamiltonian Seifert conjecture” or, more generally, of examples of Hamiltonian systems having no periodic orbits on a compact energy level. We begin with the discussion of the “classical” and volume– preserving Seifert conjectures. Then the constructions of counterexamples to the Hamiltonian Seifert conjecture in dimensions greater than o...

An orthogonal double cover (ODC) of the complete graph Kn by a graph G is a collection G of n spanning subgraphs of Kn, all isomorphic toG, such that any two members ofG share exactly one edge and every edge ofKn is contained in exactly two members of G. In the 1980s Hering posed the problem to decide the existence of an ODC for the case that G is an almost-Hamiltonian cycle, i.e. a cycle of le...

We generalize the following two seminal results. 1. Thomassen’s result [19] in 1983, which says thatevery 4-connected planar graph is hamiltonian-connected (which generalizes the old result of Tutte[20] in 1956, which says that every 4-connectedplanar graph is hamiltonian). 2. Thomas and Yu’s result [16] in 1994, which saysthat every 4-connected projective planar graph is<lb...

1 (Dubrovin–Novikov Hamiltonian operator [1]) is compatible with a nondegenerate local Hamiltonian operator of hydrodynamic type K 2 if and only if the operator K 1 is locally the Lie derivative of the operator K 2 along a vector field in the corresponding domain of local coordinates. This result gives, first of all, a convenient general invariant criterion of the compatibility for the Dubrovin...

Approximate Bayesian computation (ABC) is a powerful and elegant framework for performing inference in simulation-based models. However, due to the difficulty in scaling likelihood estimates, ABC remains useful for relatively lowdimensional problems. We introduce Hamiltonian ABC (HABC), a set of likelihood-free algorithms that apply recent advances in scaling Bayesian learning using Hamiltonian...

Comment on " Berry phase correction to electron density in solids " by Xiao et al. Abstract The main result of Xiao et al. In a recent paper on the semiclassical dynamics of a Bloch electron, Xiao, Shi and Niu [1] claim that, due to a Berry curvature term, Liouville's theorem on the conservation of the phase-space volume element would be violated, a fact which would have been overlooked so far....

In this paper we introduce invariants of semi-free Hamiltonian actions of S 1 on compact symplectic manifolds (which satisfy some technical conditions related to positivity) using the space of solutions to certain gauge theoretical equations. These equations generalize at the same time the vortex equations and the holomor-phicity equation used in Gromov–Witten theory. In the definition of the i...

A Hamilton loop is a loop in which every subloop is normal. This definition is somewhat broader than the analogous definition of group theory where a Hamiltonian group is restricted to be non-Abelian. The structure of a Hamiltonian group is well known [4, pp. 129-131 ]l —it is the direct product of an Abelian group whose elements have odd order, an Abelian group with exponent 2, and a quaternio...