Constructing the light-front QCD Hamiltonian

We propose the light-front Lagrangian and the corresponding Hamiltonian that produce a theory perturbatively equivalent to the conventional QCD in the Lorentz coordinates after the regularization is removed. The regularization used is nonstandard and breaks the gauge invariance.But after the regularization is removed, this invariance is restored by the introduction of a finite number of counterterms with coefficients dependent on the regularization parameters. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 3, pp. 417–437, September, 1999.     

Construction of a perturbatively correct light-front Hamiltonian for a (2+1)-dimensional gauge theory

We discuss a perturbation theory on the light front regularized by a method analogous to Pauli–Villars regularization for the (2+1)-dimensional SU(N)-symmetric gauge theory. This allows constructing a correct renormalized light-front Hamiltonian.     

Hamiltonian chaos in the interaction of moving two-level atoms with cavity vacuum

We study nonlinear dynamics of the fundamental cavity quantum-electrodynamical system consisting of a point-like collection of identical two-level atoms moving through a lossless single-mode cavity. Taking into account the interatomic and the atom-field quantum correlations of the first order, we go beyond the semiclassical model and derive a dynamical system that is able to describe the vacuum Rabi oscillations with atoms moving in a spatially inhomogeneous cavity field. A simple expression for the equilibrium points of this system provides a class of initial conditions for atoms and a cavity mode under which the atomic population and radiation may be trapped. In the strong-coupling limit and the rotating-wave approximation, the model is shown to be integrable with atoms moving through a resonant cavity with an arbitrary spatial profile of the mode along the propagation axis. The general exact solution is derived in an explicit form in terms of Jacobian elliptic functions. Numerical simulation confirms that perturbations, that are produced by a modulation of the coupling between moving atoms and a cavity mode, provide, out of resonance, a mechanism responsible for Hamiltonian chaos in the interaction of two-level atoms with cavity vacuum. These chaotic vacuum Rabi oscillations may be considered as a new kind of reversible spontaneous emission.     

Infiniteness of zero modes for the Pauli operator with singular magnetic field

We establish that the Pauli operator describing a spin-1/2 two-dimensional quantum system with a singular magnetic field has, under certain conditions, an infinite-dimensional space of zero modes, possibly, both spin-up and spin-down, moreover there is a spectral gap separating the zero eigenvalue from the rest of the spectrum. In particular, infiniteness takes place if the field has infinite flux, which settles this previously unknown case of Aharonov-Casher theorem.     

Construction of infinite unimodular sequences with zero autocorrelation

Unimodular waveforms x are constructed on the integers with the property that the autocorrelation of x is one at the origin and zero elsewhere. There are three different constructions: exponentials of the form e^{2 pi na q},e^{2 pi i n^alpha theta}, sequences taken from roots of unity, and sequences constructed from the elements of real Hadamard matrices. The first is expected and elementary and the second is based on the construction of Wiener. The third is the most intricate and is really one of a family of distinct but structurally similar waveforms. A natural error estimate problem is posed for the last construction. The analytic solution is not as useful as the simulations because of the inherent counting problems in the construction.     

Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero

In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first-order nonperiodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we give some new criteria to guarantee that Hamiltonian systems with asymptotically quadratic terms and spectrum point zero have at least one and a finite number of pairs of homoclinic orbits under some adequate conditions, respectively.     

On local strong solutions to the Cauchy problem of the two-dimensional full compressible magnetohydrodynamic equations with vacuum and zero heat conduction

This paper concerns the Cauchy problem of the two-dimensional full compressible magnetohydrodynamic equations with zero heat-conduction and vacuum as far field density. In particular, the initial density can have compact support. We prove that the Cauchy problem admits a local strong solution provided both the initial density and the initial magnetic field decay not too slow at infinity.     

一种构造可积Hamilton系统的新方法

建议了一种新的构造可积Hamilton系统的方法。对于给定的Poisson流形,本文利用Dirac-Poisson结构构造其上的新Poisson括号[1],进而获得了新的可积Hamilton系统。构造的Poisson括号一般是非线的,并且这种方法也不同于通常的方法[2～4]。本文还给出了两个实例。     

Performance modeling of state dependent system with mixed standbys and two modes of failure

The present investigation deals with a multicomponent repairable system with state dependent rates. For smooth functioning of the system, mixed standbys (warm and cold) are provided so that the failed units are immediately replaced by standbys if available. To prevent congestion in the system due to failure of units, permanent along with additional repairmen are provided to restore the failed units. It is assumed that the units may fail in two modes. The units have exponential life time and repair time distributions. The failed unit may balk in case of heavy load of failed units. The failed units may also wait in the queue and renege on finding the repairmen busy according to a pre-specified rule. The Chapman–Kolmogorov equations, governing the model in the form of matrix are constructed using transition flow rates of different states. The steady state solution of queue size distribution is derived using product formula. A cost function is suggested to determine the optimal number of warm and cold standbys units required for the desired level of quality of service. The numerical illustrations are carried out to explore the effect of different parameters on performance measures.     

Construction of asymptotic solutions for weakly nonlinear Hamiltonian systems

A change of variables is constructed for a class of weakly nonlinear Hamiltonian systems which makes it possible to reduce the order of the nonlinearity from O() to O(²) and to construct asymptotic solutions. The results can be applied to some nonlinear partial differential equations of interest in physics.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 36–40, 1984.The author thanks V. M. Babich, V. F. Lazutkin, and I. A. Molotkov for useful remarks.     

Construction of Hamiltonian Cycles in Layered Cubic Planar Graphs

 In this paper, a class of cubic planar graphs is given that have Hamiltonian cycles that can be constructed in linear time. A member of this class is called a layered cubic planar graph, and consists of a sequence of cycles C ₀ ,C ₁ ,…,C _n such that each pair of successive cycles, C _i , C _i+1 , is joined by a matching. The cycles can be pictured as concentric circles, and the edges of the matchings as radial line segments between successive circles. The subgraph bounded by two successive cycles forms a layer; each face in layer i is incident to a fixed number k _i+1 of edges in the matching in layer i+1. The problem that initially motivated this work is that of identifying classes of convex cubic polyhedra that can be easily edge three-colored. Received: September 21, 1998 Final version received: July 21, 1999     

Construction of many Hamiltonian stationary Lagrangian surfaces in Euclidean four-space

We make a large use of a Weierstrass representation formula to describe a variety of Hamiltonian stationary Lagrangian surfaces. Among the examples we give are the already known tori and cones, but also simply periodic cylinders, singularities of non-conical type and branch points of any order. Received: 11 November 2001 / Accepted: 23 January, 2002 / Published online: 5 September 2002