The author constructs new solutions of the vacuum Einstein equations which satisfy the colliding wave conditions of Ernst et al. (1988), and which may therefore be interpreted as colliding gravitational plane waves. T...
The author constructs new solutions of the vacuum Einstein equations which satisfy the colliding wave conditions of Ernst et al. (1988), and which may therefore be interpreted as colliding gravitational plane waves. These solutions have collinear polarisation, and for certain choices of parameters they reduce to previously known solutions.
Consider a reflector antenna system, consisting of a point light source O, a reflecting surface F, and an object T in space, to be illuminated in this system. Under the assumptions of the geometric optics theory it is...
Consider a reflector antenna system, consisting of a point light source O, a reflecting surface F, and an object T in space, to be illuminated in this system. Under the assumptions of the geometric optics theory it is required to construct the surface F when the position of the light source and the object T are given, and the power distribution is a function prescribed in advance on T. In addition, the aperture of the incidence ray cone is also prescribed. Using differential geometric methods the author studies this inverse problem and obtains certain relations between elements of the system. In the radially symmetric case the author establishes conditions for existence and uniqueness of a solution to the problem.
The author studies an inverse problem consisting in recovering a reflecting surface such that for a given point source of light the directions of reflected rays cover a prescribed region of the far sphere and the dens...
The author studies an inverse problem consisting in recovering a reflecting surface such that for a given point source of light the directions of reflected rays cover a prescribed region of the far sphere and the density of the distribution of reflected rays is a function of the reflected directions prescribed in advance. The power density of the source as well as the aperture of the incident cone are also given and the laws of geometric optics are applied. In this form the problem has been posed by Westcott and Norris (1975). For circular far field and aperture and distribution densities close to radially symmetric ones (in some Holder norm) he shows that the above problem can be solved, provided a natural energy conservation condition is satisfied.
We construct self-similar solutions of various soliton equations obtained with the help of the inverse scattering transform with a variable spectral parameter. We demonstrate that corresponding self-similar systems, w...
We construct self-similar solutions of various soliton equations obtained with the help of the inverse scattering transform with a variable spectral parameter. We demonstrate that corresponding self-similar systems, which represent nonlinear ordinary differential equations (ODEs), may be divided into two classes. The first class contains equations that can be directly solved in the framework of the method of isomonodromic deformations. Some of the equations may be regularly reduced to certain Painleve equations. Equations in the second class include variable coefficients that satisfy additional nonlinear ODES. We prove that at least one such additional ODE for the coefficients is without the Painleve property. As far as we know, there is no regular method that can be used to solve these supplementary equations. The second class contains really new equations that one could solve, in principle, by the method of isomonodromic deformations only after one finds a solution to the additional ODE. The most interesting ODE, from the physical point of view, is a self-similar reduction for the Maxwell-Bloch system with pumping, which has applications to nonlinear optics.
We show how to construct new representations of the various R-matrix algebras starting from known representations. For linear r-matrix algebras we investigate a dynamical r-matrix which depends on the spectral paramet...
We show how to construct new representations of the various R-matrix algebras starting from known representations. For linear r-matrix algebras we investigate a dynamical r-matrix which depends on the spectral parameter and half of the dynamical variables (particle coordinates) only. The Toda lattices and the Henon-Heiles systems illustrate the scheme.
We describe a close connection between the localized induction equation hierarchy of integrable evolution equations on space curves and surfaces of constant negative Gauss curvature.
We describe a close connection between the localized induction equation hierarchy of integrable evolution equations on space curves and surfaces of constant negative Gauss curvature.
Here we develop the Topological Approximation Method (TAM) which gives a new description of the mixing and transport processes in chaotic two-dimensional time-periodic Hamiltonian flows. It is based upon the structure...
Here we develop the Topological Approximation Method (TAM) which gives a new description of the mixing and transport processes in chaotic two-dimensional time-periodic Hamiltonian flows. It is based upon the structure of the homoclinic tangle, and supplies a detailed solution to a transport problem for this class of systems, the characteristics of which are typical to chaotic, yet not ergodic dynamical systems. These characteristics suggest some new criteria for quantifying transport and mixing-hence chaos-in such systems. The results depend on several parameters, which are found by perturbation analysis in the near integrable case, and numerically otherwise. The strength of the method is demonstrated on a simple model. We construct a bifurcation diagram describing the changes in the homoclinic tangle as the physical parameters are varied. From this diagram we find special regions in the parameter space in which we approximate the escape rates from the vicinity of the homoclinic tangle, finding non-trivial self-similar solutions as the forcing magnitude tends to zero. We compare the theoretical predictions with brute force calculations of the escape rates, and obtain satisfactory agreement.
Considers the problem of lattice trails, introduced by Malakis (1975, 1976). The author proves the existence of a finite connective constant, and establishes a result for the growth of n-step trails tnanalogous to the...
Considers the problem of lattice trails, introduced by Malakis (1975, 1976). The author proves the existence of a finite connective constant, and establishes a result for the growth of n-step trails tnanalogous to the best known result for self-avoiding walks, tn= lambdanexpt(O(/n)). For the honeycomb (d=2) and Lave's lattice (d=3) the author establishes a counting theorem, from which one can deduce the exact value of the connective constant lambda for the honeycomb lattice, lambda2=2+ square root 2. Further, it follows that the trail problem is in the same universality class as the self-avoiding walk problem for those lattices. An exact amplitude relation between trails and self-avoiding walks, and between dumb-bells, and trails and self-avoiding walks is established. The non-existence of a counting theorem for arbitrary lattices is established. An inequality for the triangular lattice connective constant is proved. A high-density expansion for lambda for the d-dimensional hybercubic lattice is also obtained. The author argues that, contrary to recent suggestions, the model is in the same universality class as the self-avoiding walk model.
The configuration of self-avoiding polymer molecules terminally attached to a rigid boundary is resolved into loop, train and tail components on the basis of a convolution integral analysis. The expectation lengths (l...
The configuration of self-avoiding polymer molecules terminally attached to a rigid boundary is resolved into loop, train and tail components on the basis of a convolution integral analysis. The expectation lengths (lloop), (ltrain) and (ltail) and the component fractions are determined as functions of chain length and chain-plane interaction. In the case of zero chain-plane attraction thre is good quantitative agreement with Monte Carlo estimates, although the predicted dependence of (lloop) upon chain length appears too strong. Results are also presented for hard sphere sequences as a function of chain-plane attraction when it is found that there is a progressive redistribution amongst loop and tail states, the train exponent remaining essentially unmodified over the range of interactions investigated. Exponent representations are proposed and compared with earlier analyses. The various component structures are discussed in terms of the interplay of entropic and energetic processes at the boundary.
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