Pairs of edge disjoint Hamiltonian circuits in 5-connected planar graphs

Summary A variety of examples of 4-connected 4-regular graphs with no pair of disjoint Hamiltonian circuits were constructed in response to Nash-Williams conjecture that every 4-connected 4-regular graph is Hamiltonian and also admits a pair of edge-disjoint Hamiltonian circuits. Nash-Williams's problem is especially interesting for planar graphs since 4-connected planar graphs are Hamiltonian. Examples of 4-connected 4-regular planar graphs in which every pair of Hamiltonian circuits have edges in common are included in the above mentioned examples.B. Grünbaum asked whether 5-connected planar graphs always admit a pair of disjoint Hamiltonian circuits. In this paper we introduce a technique that enables us to construct infinitely many examples of 5-connected planar graphs, 5-regular and non regular, in which every pair of Hamiltonian circuits have edges in common.     

Hamiltonian spatial structure for three-dimensional water waves in a moving frame of reference

Summary The governing equations for three-dimensional time-dependent water waves in a moving frame of reference are reformulated in terms of the energy and momentum flux. The novelty of this approach is that time-independent motions of the system—that is, motions that are steady in a moving frame of reference—satisfy a partial differential equation, which is shown to be Hamiltonian. The theory of Hamiltonian evolution equations (canonical variables, Poisson brackets, symplectic form, conservation laws) is applied to the spatial Hamiltonian system derived for pure gravity waves. The addition of surface tension changes the spatial Hamiltonian structure in such a way that the symplectic operator becomes degenerate, and the properties of this generalized Hamiltonian system are also studied. Hamiltonian bifurcation theory is applied to the linear spatial Hamiltonian system for capillary-gravity waves, showing how new waves can be found in this framework.     

Symplectic Transformations and a Reduction Method for Asymptotically Linear Hamiltonian Systems

In many fields of applications, especially in applications from mechanics, many equations of motion can be written as Hamiltonian systems. In this paper, we study a class of asymptotically linear Hamiltonian systems. We construct a symplectic transformation which reduces the linear systems of the Hamiltonian systems. This reduction method can be applied to study the existence of periodic solutions for a class of asymptotically linear Hamiltonian systems under weaker conditions on the linear systems of the Hamiltonian systems.     

The Hamiltonian connectivity of rectangular supergrid graphs

A Hamiltonian path of a graph is a simple path which visits each vertex of the graph exactly once. The Hamiltonian path problem is to determine whether a graph contains a Hamiltonian path. A graph is called Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices. In this paper, we will study the Hamiltonian connectivity of rectangular supergrid graphs. Supergrid graphs were first introduced by us and include grid graphs and triangular grid graphs as subgraphs. The Hamiltonian path problem for grid graphs and triangular grid graphs was known to be NP-complete. Recently, we have proved that the Hamiltonian path problem for supergrid graphs is also NP-complete. The Hamiltonian paths on supergrid graphs can be applied to compute the stitching traces of computer sewing machines. Rectangular supergrid graphs form a popular subclass of supergrid graphs, and they have strong structure. In this paper, we provide a constructive proof to show that rectangular supergrid graphs are Hamiltonian connected except one trivial forbidden condition. Based on the constructive proof, we present a linear-time algorithm to construct a longest path between any two given vertices in a rectangular supergrid graph.     

Derivation of Korteweg‐de Vries flow equations from fourth order nonlinear Schrödinger equation

We perform a multiple scale analysis on the fourth order nonlinear Schrödinger equation in the Hamiltonian form together with the Hamiltonian function. We derive, as amplitude equations, Korteweg‐de Vries flow equations in the bi‐Hamiltonian form with the corresponding Hamiltonian functions. Copyright © 2013 John Wiley & Sons, Ltd.     

Generalized fibonacci cubes are mostly hamiltonian

The Hamiltonian problem is to determine whether a graph contains a spanning (Hamiltonian) path or cycle. Here we study the Hamiltonian problem for the generalized Fibonacci cubes, which are a new family of graphs that have applications in interconnection topologies [J. Liuand W.-J. Hsu, ?Distributed Algorithms for Shortest-Path, Deadlock-Free Routing and Broadcasting in a Class of Interconnection Topologies,”? International Parallel Processing Symposium (1992)]. We show that each member of this family contains a Hamiltonian path. Furthermore, we also characterize the members of this family that contain a Hamiltonian cycle.     

二次算子族及非负无穷维Hamilton算子的谱分布

本文讨论了一类在弦和梁的微小振动中出现的二次算子族L(λ)=λ~2MλK-A的谱分布问题,进而将所得结论与无穷维Hamilton算子联系起来,利用无穷维Hamilton算子的特殊结构,得到了一类非负无穷维Hamilton算子的谱分布,这为无穷维Hamilton算子的半群方法提供了理论保证.     

Implicit contact dynamics and Hamilton-Jacobi theory

In this paper, we introduce implicit Hamiltonian dynamics in the framework of contact geometry in two different ways: first, we introduce classical implicit Hamiltonian dynamics on a contact manifold, followed by evolution Hamiltonian dynamics. In the first case, implicit contact Hamiltonian dynamics is defined as a Legendrian submanifold of a tangent contact space, whilst the implicit evolution dynamic is understood as a Lagrangian submanifold of a certain symplectic space embedded into the tangent contact space. To conclude, we propose a geometric Hamilton-Jacobi theory for both of these formulations.     

Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy

We study the existence of quasi-periodic, invariant tori in a nearly integrable Hamiltonian system of high order proper degeneracy, i.e., the integrable part of the Hamiltonian involves several time scales and at each time scale the corresponding Hamiltonian depends on only part of the action variables. Such a Hamiltonian system arises frequently in problems of celestial mechanics, for instance, in perturbed Kepler problems like the restricted and non-restricted 3-body problems and spatial lunar problems in which several bodies with very small masses are coupled with two massive bodies and the nearly integrable Hamiltonian systems naturally involve different time scales. Using KAM method, we will show under certain higher order non-degenerate conditions of Bruno–Rüssmann type that the majority of quasi-periodic, invariant tori associated with the integrable part will persist after the non-integrable perturbation. This actually concludes the KAM metric stability for such a properly degenerate Hamiltonian system.     

大型不正定陀螺系统本征值问题

陀螺动力系统可以导入哈密顿辛几何体系,在哈密顿陀螺系统的辛子空间迭代法的基础上提出了一种能够有效计算大型不正定哈密顿函数的陀螺系统本征值问题的算法．利用陀螺矩阵既为哈密顿矩阵而本征值又是纯虚数或零的特点,将对应哈密顿函数为负的本征值分离开来,构造出对应哈密顿函数全为正的本征值问题,利用陀螺系统的辛子空间迭代法计算出正定哈密顿矩阵的本征值,从而解决了大型不正定陀螺系统的本征值问题,算例证明,本征解收敛得很快．     

Lyapunov inequality for linear Hamiltonian systems on time scales

The principal aim of this paper is to state and prove some Lyapunov inequalities for linear Hamiltonian system on an arbitrary time scale , so that the well-known case of differential linear Hamiltonian systems and the recently developed case of discrete Hamiltonian systems are unified. Applying these inequalities, a disconjugacy criterion for Hamiltonian systems on time scales is obtained.     

On a class of coupled Hamiltonian operators and their integrable hierarchies with two potentials

We discuss at first in this paper the Gauge equivalence among several u‐linear Hamiltonian operators and present explicitly the associated Gauge transformation of Bäcklund type among them. We then establish the sufficient and necessary conditions for the linear superposition of the discussed u‐linear operators and matrix differential operators with constant coefficients of arbitrary order to be Hamiltonian, which interestingly shows that the resulting Hamiltonian operators survive only up to the third differential order. Finally, we explore a few illustrative examples of integrable hierarchies from Hamiltonian pairs embedded in the resulting Hamiltonian operators.     

Ultraviolet renormalization of the Nelson Hamiltonian through functional integration

Starting from the N-particle Nelson Hamiltonian defined by imposing an ultraviolet cutoff, we perform ultraviolet renormalization by showing that in the ultraviolet cutoff limit a self-adjoint operator exists after a logarithmically divergent term is subtracted from the original Hamiltonian. We obtain this term as the diagonal part of a pair interaction appearing in the density of a Gibbs measure derived from the Feynman–Kac representation of the Hamiltonian. Also, we show existence of a weak coupling limit of the renormalized Hamiltonian and derive an effective Yukawa interaction potential between the particles.     

Hamiltonian differencing of fluid dynamics

By analyzing the Hamiltonian structures of several representations of continuous Lagrangian fluid dynamics, a universal Hamiltonian form is developed which unifies those structures and applies both to the continuous and spatially discrete cases. Then the universal Hamiltonian form is used as a “template” for generating numerical differencing schemes which retain the underlying Hamiltonian structure of the continuous theory. Examples are discussed of these spatial differencing schemes for the Euler equations in one, two, and three dimensions. In one dimension, the nondissipative part of the von Neumann-Richtmeyer scheme is recovered as a special case.     

一类无穷维Hamilton算子的半群生成定理

研究了无穷维H am ilton算子生成C0半群的问题,得到了类无穷维H am ilton算子生成C0半群的一个充分条件.把结果应用在一类双曲型混合问题生成的无穷维H am ilton算子上,证明此类算子生成C0半群,并利用H ille-Y osida定理进一步说明了结果的正确性和有效性.另外,还给出了波动方程相应的无穷维H am ilton算子所生成的C0半群的具体表达式.     

An upper bound on the length of a Hamiltonian walk of a maximal planar graph

A Hamiltonian walk of a connected graph is a shortest closed walk that passes through every vertex at least once, and the length of a Hamiltonian walk is the total number of edges traversed by the walk. We show that every maximal planar graph with p(≥ 3) vertices has a Hamiltonian cycle or a Hamiltonian walk of length ≤ 3(p - 3)/2.     

Hamiltonian and Variational Linear Distributed Systems

We use the formalism of bilinear- and quadratic differential forms in order to study Hamiltonian and variational linear distributed systems. It was shown in [1] that a system described by ordinary linear constant-coefficient differential equations is Hamiltonian if and only if it is variational. In this paper we extend this result to systems described by linear, constant-coefficient partial differential equations. It is shown that any variational system is Hamiltonian, and that any scalar Hamiltonian system is contained (in general, properly) in a particular variational system.     

Geometric Aspects of Spatially Periodic Interfacial Waves

Periodic waves at the interface between two inviscid fluids of differing densities are considered from a geometric point of view. A new Hamiltonian formulation is used in the analysis and restriction of the Hamiltonian structure to space-periodic functions leads to an O -invariant Hamiltonian system. Motivated by the simplest O -invariant Hamiltonian system, the spherical pendulum, we analyze the properties of traveling waves, standing waves, interactions between standing and traveling waves (mixed waves) and time-modulated spatially periodic waves. A singularity in the bifurcation of traveling waves leads to a nonlinear resonance and this is investigated numerically.     

Nonlinearization of the Lax pairs for discrete Ablowitz-Ladik hierarchy

The discrete Ablowitz-Ladik hierarchy with four potentials and the Hamiltonian structures are derived. Under a constraint between the potentials and eigenfunctions, the nonlinearization of the Lax pairs associated with the discrete Ablowitz-Ladik hierarchy leads to a new symplectic map and a class of finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the symplectic map and these finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Each member in the discrete Ablowitz-Ladik hierarchy is decomposed into a Hamiltonian system of ordinary differential equations plus the discrete flow generated by the symplectic map.     

Decomposition of a hierarchy of nonlinear evolution equations

The generalized Hamiltonian structures for a hierarchy of nonlinear evolution equations are established with the aid of the trace identity. Using the nonlinearization approach, the hierarchy of nonlinear evolution equations is decomposed into a class of new finite-dimensional Hamiltonian systems. The generating function of integrals and their generator are presented, based on which the finite-dimensional Hamiltonian systems are proved to be completely integrable in the Liouville sense. As an application, solutions for the hierarchy of nonlinear evolution equations are reduced to solving the compatible Hamiltonian systems of ordinary differential equations.