Spectral Properties of Hamiltonians of Charged Systems in a Homogeneous Magnetic Field: II. The Structure of the Pure Point Spectrum

We study the pure point spectrum of the energy operator H(P ) of a many-particle charged quantum system in a homogeneous magnetic field based on the results in our previous work under fixation of the sum P of the pseudomomentum components of the system. We prove that the discrete spectrum H(P ) of a short-range system is infinite under some conditions (which, for example, hold for a system of two oppositely charged particles) even in the case of a finitely supported potential. For a long-range system of the type of a (+)-ion of an atom (including the ion), the discrete spectrum is infinite.     

On κ—ordered Graphs Involved Degree Sum

A graph G is κ-ordered Hamiltonian 2≤κ≤n,if for every ordered sequence S of κ distinct vertices of G,there exists a Hamiltonian cycle that encounters S in the given order,In this article,we prove that if G is a graph on n vertices with degree sum of nonadjacent vertices at least n 3κ-9/2,then G is κ-ordered Hamiltonian for κ=3,4,…,[n/19].We also show that the degree sum bound can be reduced to n 2[κ/2]-2 if κ(G）≥3κ-1/2 or δ（G）≥5κ-4.Several known results are generalized.     

Sparse pseudo‐random graphs are Hamiltonian

In this article we study Hamilton cycles in sparse pseudo‐random graphs. We prove that if the second largest absolute value λ of an eigenvalue of a d‐regular graph G on n vertices satisfies and n is large enough, then G is Hamiltonian. We also show how our main result can be used to prove that for every c >0 and large enough n a Cayley graph X (G,S), formed by choosing a set S of c log⁵ n random generators in a group G of order n, is almost surely Hamiltonian. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 17–33, 2003     

Two-Step High Order Starting Values for Implicit Runge–Kutta Methods

In this article a simple form of expressing and studying the order conditions to be satisfied by starting algorithms for Runge–Kutta methods, which use information from the two previous steps is presented. In particular, starting algorithms of highest order for Runge–Kutta–Gauss methods up to seven stages are derived. Some numerical experiments with Hamiltonian systems to compare the behaviour of the new starting algorithms with other existing ones are presented.     

On generalized hamiltonian systems

1. PreliminaryIt is well known that{1] a 8ymPlectic form is invariant along the trajectory of a Hamilto-nian system. Based on this fundamental property, certain techniques have been developed.The purpose of this paper is to extend such an approach to a wider class of dynamic systeIns,namely, genera1ized Hamiltonian systems. Our purpose is to investigate a class of dynaInicsystems, which possess a certain "geometric structure".Deflnition 1.1[1'2]. Let M be a tIlallifo1d. w E fl'(M) is call…     

Torus actions on symplectic orbi-spaces

Which -dimensional orbi-spaces have effective symplectic - torus actions? As shown by Lerman and Tolman (1997) and Watson (1997), this question reduces to that of characterizing the finite subgroups of centralizers of tori in the real symplectic group . We resolve this question, and generalize our method to a calculation of the centralizers of all tori in .

     

Invariant Subspace Approach to Linear H∞-Control via Measurement Feedback

This paper analyzes, and thus reveals the structure of the stable invariant subspace of the related Hamiltonian matrix arising from the measurement feedback H -control problem. Using this, it presents another method for the verification of the admissibility of the controller derived by Doyle et al. in 1989. The method not only eliminates unnecessary assumptions on stabilizability and detectability, but also gives deeper insight into the relationship among the stable parts of the associated matrices.     

Geometric Dynamics

A kinematic differential system on a Riemann (or semi-Riemann) manifold induces a Lorentz-Udrite world-force law, i.e., any local group with one parameter (any local flow) on a Riemann (or semi-Riemann) manifold induces the dynamics of the given vector field or of an associated particle, which will be called geometric dynamics.The cases of Riemann-Jacobi or Riemann-Jacobi-Lagrange structures are imposed by the behavior of an external tensor field of type (1,1). The case of the Finsler-Jacobi structure appears if the initial metric is chosen such that the energy of the given vector field is constant (Sec. 1). At the end of Sec. 1 are formulated open problems regarding some extensions of geometric dynamics.Adequate structures on the tangent bundle describe the geometric dynamics in the Hamilton language (Sec. 2).Section 3 proves the existence of a Finsler-Jacobi structure induced by an almost contact metric structure.The theory is applied to electromagnetic dynamical systems (the starting point of our theory), offering new principles of unification of the gravitation and the electromagnetism. Also, here, one enounces open problems regarding the geometric dynamics induced by the electric intensity and magnetizing force (Sec. 4).From the geometrical point of view, we create a wider class of Riemann-Jacobi, Riemann-Jacobi-Lagrange, or Finsler-Jacobi manifolds ensuring that all trajectories of a given vector field are geodesics. Having T₁M²ⁿ⁺¹ in mind, the problem of creating a wider class of Riemannian manifolds, in which there exists a vector field such that (1) all trajectories of the vector field are geodesics; (2) the flow defined by is incompressible; (3) the condition which corresponds to the property that is the associate vector field of the contact structure is satisfied;was studied intensively by S. Sasaki. The results were not satisfactory, but Sasaki discovered (, , )-structures [10].AMS Subject Classification (1991): 70H35, 53C22, 58F25, 83C22     

Hamiltonians for the Quantum Hall Effect on Spaces with Non-Constant Metrics

The problem of studying the quantum Hall effect on manifolds with non constant metric is addressed. The Hamiltonian on a space with hyperbolic metric is determined, and the spectrum and eigenfunctions are calculated in closed form. The hyperbolic disk is also considered and some other applications of this approach are discussed as well. PACS numbers: 73.43.-f, 02.30.Jr.     

Hamiltonian Systems and Darboux Transformation Associated with a 3 × 3 Matrix Spectral Problem

Starting from a 3 × 3 matrix spectral problem, we derive a hierarchy of nonlinear equations. It is shown that the hierarchy possesses bi-Hamiltonian structure. Under the symmetry constraints between the potentials and the eigenfunctions, Lax pair and adjoint Lax pairs including partial part and temporal part are nonlinearied into two finitedimensional Hamiltonian systems (FDHS) in Liouville sense. Moreover, an explicit N-fold Darboux transformation for CDNS equation is constructed with the help of a gauge transformation of the spectral problem.