Class Of Possible Solutions Research Articles

Multivariate Linear Rational Expectations Models

This paper considers the solution of multivariate linear rational expectations models. It is described how all possible classes of solutions (namely, the unique stable solution, multiple stable solutions, and the case where no stable solution exists) of such models can be characterized using the quadratic determinantal equation (QDE) method of Binder and Pesaran (1995, in M.H. Pesaran & M. Wickens [eds.], Handbook of Applied Econometrics: Macroeconomics, pp. 139–187. Oxford: Basil Blackwell). To this end, some further theoretical results regarding the QDE method expanding on previous work are presented. In addition, numerical techniques are discussed allowing reasonably fast determination of the dimension of the solution set of the model under consideration using the QDE method. The paper also proposes a new, fully recursive solution method for models involving lagged dependent variables and current and future expectations. This new method is entirely straightforward to implement, fast, and applicable also to high-dimensional problems possibly involving coefficient matrices with a high degree of singularity.

Numerical inversion of two-dimensional geoelectric conductivity distributions from magnetotelluric data

In this paper, a new inversion technique, called the minimum first-order entropy (MinEnt-1) method, is proposed for the reconstruction of two-dimensional geoelectric conductivity distributions from magnetotelluric (MT) data. The method combines an iterative search with a regularization technique based on the minimization of the entropy measure of the vector of first-differences of the unknown conductivities. Numerical simulations, using synthetic data corrupted with gaussian noise, show that the MinEnt-1 algorithm converges to excellent conductivity reconstructions, yielding in many cases results that are superior to those obtained by the maximum entropy formalism. Unlike other classical regularization schemes, which maximize smoothness for a given data, the proposed method constrains the class of possible solutions into a restricted set of low entropy models, constituted by locally smooth regions separated by sharp discontinuities. This may be an effective approach for the incorporation of prior information about the local smoothness of the real physical model.

Qualitative analysis of ten-dimensional lovelock cosmological models

The Einstein-like field equations obtained from the variation of a Lagrangian containing linear, quadratic (Gauss-Bonnet) and quartic terms for a ten-dimensional cosmological model cannot be solved analytically. However, we can reduce them to a system of dynamical equations for the “Hubble parameters”. The study of the mathematical properties of the fixed points of this system gives a qualitative picture of the behaviour of a class of possible solutions. The inclusion of a quartic term generates an extremely rich structure for the corresponding dynamical system. Some solutions are shown to exhibit the interesting property of dimensional reduction, which has been proposed as a possible explanation of the three-dimensional character of our universe.

Isodynamical (omnigenous) equilibrium in symmetrically confined plasma configurations

Isodynamical or omnigenous equilibrium has the property that the magnitude of the magnetic field is constant on magnetic surfaces. It is shown that in plasma confinement configurations with one ignorable coordinate there are three possible classes of solutions, characterized by the properties of the curvature of the magnetic axis, the magnitude of the magnetic field on axis, and the closure of magnetic surfaces about the magnetic axis. Solutions belonging to class (i) have a straight magnetic axis, a finite field on axis, and closed magnetic surfaces. Solutions in class (ii) have a curved magnetic axis, closed magnetic surfaces, and a magnetic field that vanishes on axis. Finally, solutions in class (iii) have a curved magnetic axis, a finite magnetic field on axis, and open magnetic surfaces.

Shock processes by aftereffects and multivariate lack of memory

If the survival function satisfies the functional equation where e = (1, …, 1), and if the marginal distributions are exponential, then is the multivariate exponential distribution of Marshall and Olkin. The functional equation has many solutions if the requirement of exponential marginals is not imposed, but the class of possible marginals is somewhat limited (e.g., marginals must be absolutely continuous). The class of possible solutions of the equation is characterized in this paper, and several examples are obtained from models for dependence that may be of practical interest.

Shock processes by aftereffects and multivariate lack of memory

If the survival function satisfies the functional equation where e = (1, …, 1), and if the marginal distributions are exponential, then is the multivariate exponential distribution of Marshall and Olkin. The functional equation has many solutions if the requirement of exponential marginals is not imposed, but the class of possible marginals is somewhat limited (e.g., marginals must be absolutely continuous). The class of possible solutions of the equation is characterized in this paper, and several examples are obtained from models for dependence that may be of practical interest.

Group analysis for a linear hyperbolic equation arising from a quasilinear reducible system

The aim of this paper is to carry out infinitesimal group analysis for a second order linear hyperbolic equation in canonical form. To such a kind of equation one could be led in considering the hodograph transformation for a first order reducible system. In the first part of the paper we characterize the most general expression for the generator of the Lie group, which has an infinite dimensional algebra. Then we calculate the invariant surfaces in some special cases of interest, and point out the corresponding ordinary differential equations whose integration allows us to determine possible classes of solutions for the equation we are considering.

Small-angle scattering: direct structure analysis

A direct method for structural interpretation of small-angle scattering data by monodisperse isotropic systems is presented. The spherical harmonics technique proposed by Stuhrmann [Acta Cryst. (1970), A26, 297-306] is used to choose the class of possible solutions. The method enables one to decompose scattering intensity into partial amplitudes corresponding to different multipole components of scattering density using the space and value density restrictions. If a particle possesses a rotational symmetry, its structure may be restored directly using the method; if not, additional information is required to get a set of possible solutions. The method is stable with respect to experimental errors. The efficiency of the procedure is shown in model examples. The application of the method to interpret the small-angle X-ray scattering curve of bacteriophage T7 produced a map of electron density with a resolution of about 12 Å.

Ordering principle for cluster expansions in the theory of quantum fluids, dense gases, and simple classical liquids

A study is made of a series-expansion procedure which gives the leading terms of the $n$-particle distribution function ${p}^{(n)}(1,2,\dots{},n)$ as explicit functionals in the radial distribution function $g(r)$. The development of the series is based on the cluster-expansion formalism applied to the Abe form for ${p}^{(n)}$ expressed as a product of the generalized Kirkwood superposition approximation ${p}_{K}^{(n)}$ and a correction factor $\mathrm{exp}[{A}^{(n)}(1,2,\dots{},n)]$. An ordering parameter $\ensuremath{\mu}$ is introduced to determine ${A}^{(n)}$ and ${p}^{(n)}$ in the form of infinite power series in $\ensuremath{\mu}$, and the postulate of minimal complexity is employed to eliminate an infinite number of possible classes of solutions in a sequential relation connecting ${A}^{(n\ensuremath{-}1)}$ and ${A}^{(n)}$. Derivation of the series for ${p}^{(n)}$ and many other algebraic manipulations involving a large number of cluster integrals are greatly simplified by the use of a scheme which groups together all cluster terms having, in a certain way, the same source term. In particular, the scheme is useful in demonstrating that the nature of the series structure of ${p}^{(3)}$ is such that its three-point Fourier transform ${S}^{(3)}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{1},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{2},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{3})$ has as a factor the product of the three liquid-structure functions $S({k}_{1})S({k}_{2})S({k}_{3})$. The results obtained to order ${\ensuremath{\mu}}^{4}$ for ${A}^{(3)},{p}^{(3)}$, and ${S}^{(3)}$ agree with those derived earlier in a more straightforward but tedious approach. The result for ${p}^{(4)}$ shows that the convolution approximation ${p}_{c}^{(4)}$, which contains ${\ensuremath{\mu}}^{3}$ terms, must be supplemented by a correction of $O({\ensuremath{\mu}}^{3})$ in order to be accurate through third order. The $\ensuremath{\mu}$-expansion approach is also examined for the cluster expansion of the correlation function in the Bijl-Dingle-Jastrow description of a manyboson system, and then compared with the number-density expansion formula by using the Gaussian model for $g(r)\ensuremath{-}1$ to evaluate cluster integrals. A testing procedure based on the requirement ${p}^{(3)}(1,2,2)=0$ is developed to study accuracy of the $\ensuremath{\mu}$-ordered approximations for ${p}^{(3)}$. Numerical results obtained to orders ${\ensuremath{\mu}}^{2},{\ensuremath{\mu}}^{3}$, and ${\ensuremath{\mu}}^{4}$ with the Gaussian model indicate substantial improvements with each increase in the order of truncation in the power series of ${p}^{(3)}$. A brief discussion is presented concerning the asymptotic behavior of $g(r)$ in the context of equilibrium statistical mechanics.

Water Waves in a Nonhomogeneous Incompressible Fluid

After a brief discussion of some undesirable features of a number of different partial differential equations often employed in the existing literature on water waves, a relatively simple restricted theory is constructed by a direct approach which is particularly suited for applications to problems of fluid sheets. The rest of the paper is concerned with a derivation of a system of nonlinear differential equations (which may include the effects of gravity and surface tension) governing the two-dimensional motion of incompressible in-viscid fluids for propagation of fairly long waves in a nonhomogeneous stream of water of variable initial depth, as well as some new results pertaining to hydraulic jumps. The latter includes an additional class of possible solutions not noted previously.

Exact solutions of the magnetohydrodynamic equations

Exact one-dimensional solutions of the magnetohydrodynamic equations of an incompressible fluid are considered. It is shown that one class of plane wave solutions of the linearized equations is also a possible class of solutions of the general equations including the effect of displacement current. A similar result is also established for the solutions for a horizontally stratified fluid. For the particular case when the viscosity is equal to the magnetic diffusivity an exact solution is obtained for the magnetohydrodynamic Rayleigh problem for a semi-infinite plate. It is shown that this solution may be employed directly to give the solution for liquids of small, but not necessarily equal, viscosity and magnetic diffusivity.