Showing papers on "Singularity published in 1970"

General Lattice Model of Phase Transitions

Abstract: A general lattice-statistical model which includes all soluble two-dimensional model of phase transitions is proposed. Besides the well-known Ising and "ice" models, other soluble cases are also considered. After discussing some general symmetry properties of this model, we consider in detail a particular class of the soluble cases, the "free-fermion" model. The explicit expressions for all thermodynamic functions with the inclusion of an external electric field are obtained. It is shown that both the specific heat and the polarizability of the free-fermion model exhibit in general a logarithmic singularity. An inverse-square-root singularity results, however, if the free-fermion model also satisfies the ice condition. The results are illustrated with a specific example.

Exact Results for the Kondo Problem: One-Body Theory and Extension to Finite Temperature

Abstract: Nozi\`eres and De Dominicis's one-body theory of the x-ray singularity is extended to the Kondo effect, and also to the finite-etmperature case. The Kondo problem is shown to be equivalent to the thermodynamics of charged rods moving on a circle, or to that of an Ising model with inverse-square interaction.

On singularities of folding type

Abstract: In this article we find necessary and sufficient conditions for the existence of a mapping with a singularity of folding type on a given submanifold.

On the convergence of finite-difference approximations to one-dimensional singular boundary-value problems

Abstract: Consider a linear ordinary differential equation of the 2nd order which has a singularity at the origin; according to the nature of this singularity we must consider either the two-point boundary-value problem or the one-point boundary value problem. Finite-difference schemes are studied; results are given concerning error analysis and monotone convergence.

Solution of the Poisson–Boltzmann Equation about a Cylindrical Particle

Abstract: An analytical approximation is developed to the solution of the Poisson–Boltzmann equation about a charged cylindrical particle in an electrolyte solution. A symmetrical electrolyte is treated specifically, but the method applies also to nonsymmetrical ones. The analysis preserves the mathematical character of the exact problem and (as is shown by comparison with machine solutions) yields results with errors which are for most purposes negligibly small. The complete solution involves five separate cases depending on the value of limR→∞[Ψ / K0(R)], where R is the radial coordinate and Ψ the electrostatic potential, both in dimensionless form. In two cases the singularity in Ψ is at R = 0, and the Debye–Huckel solution is a uniformly valid, though quantitatively poor, approximation. In the other three cases the (physically relevant) singularity is away from the origin. Similar singular behavior would account for the limited success of iterative and regular perturbation schemes for the solution about the sphere.

Preuve d'une conjecture de Brieskorn

Abstract: In this paper we prove the following conjecture of Brieskorn: “The complex of (holomorphic) differential forms of an isolated hypersurface singularity of dimension n>1 is exact in degree n−1.”

General Cosmological Solution of the Gravitational Equations with a Singularity in Time

Abstract: Publisher Summary This chapter presents a qualitative description of the evolution of the metric in the general cosmological solution of the gravitational equations with a singularity in time. A general solution is one composed of a sufficient number of arbitrary functions to specify arbitrary initial conditions at a given moment of time. As there exists no systematic method for examining the singularities of the solutions of the Einstein equations, the search for increasingly more general solutions of this kind proceeded essentially by trial and error. A negative result from such a procedure could of course never be completely conclusive in itself; the construction of a new solution with the required generality reverses the conclusion without affecting the results pertaining to the concrete solutions considered previously.

Development of a Rossby Wave Critical Level

Abstract: We discuss the time-dependent behavior of a Rossby wave on a latitudinally varying flow, near the point where the steady-state wave equation is singular. The wave is forced by the switch-on of a steady forcing. Analytic solutions are obtained for the latitudinal propagation of nondivergent Rossby waves in a linear shear flow and for a large longitudinal wavelength. It is shown that the north–south eddy velocity v′ approaches the steady-state solution everywhere when nondimensional time >1, this time being a few days or less for atmospheric planetary waves. The east–west eddy velocity u′ takes much longer to approach a steady state near the singularity. One-half the steady-state amplitude of u′ is approached in a time inversely proportional to the square root of the distance from the singularity. The solution for u′ near the singularity settles down to the steady solution only after a time large compared to the inverse of the distance from the singularity. The steady-state solution for u′ is logar...

Is the singularity at separation removable

Abstract: Many numerical integrations support Goldstein's theory of the structure of the solution of a laminar boundary layer near the point of separation O when the mainstream is prescribed, and in particular confirm that the solution is singular there. The existence of the singularity, however, implies that the hypotheses of the boundary layer break down in the neighbourhood of O, and it has been suggested that the disturbance to the mainstream near O is sufficient to smooth out the singularity and enable the solution to pass over into another conventional boundary layer downstream of O containing a region of reversed flow. The aim of this paper is to explore this possibility in detail using the methods of the triple-deck, developed by the author and others, which have proved successful in somewhat related problems.Granted the hypothesis that the interaction between the boundary layer and the mainstream is significant near separation and manifests itself through a triple deck, it is found that its streamwise extent is O(e2l) where e−8 is a characteristic Reynolds number, e [Lt ] 1, and l a characteristic length of the problem. The upper deck is of width O(e2l), lies entirely outside the boundary layer, and in it the flow is inviscid. The main deck is of width O(e4l) and constitutes the majority of the boundary layer near O, and the perturbations in the velocity are largely inviscid. Finally, the lower deck is of lateral extent and is controlled by a linear equation of boundary-layer type. The whole structure is found to be consistent provided a certain integro-differential equation can be solved, which takes different forms according as the mainstream is supersonic or subsonic. When the mainstream is subsonic it is found that there is no solution to this equation that is sufficiently smooth on the downstream side of the triple deck. When the mainstream is supersonic it is found that the triple deck can at best postpone the breakdown of the assumed structure which still must occur within a distance O(e2l) of O.It is concluded that the singularity is not removable by the methods proposed and it is inferred that the singularity is a real phenomenon terminating the flow which, at high Reynolds number, exists upstream of O.

Numerical methods of high-order accuracy for singular nonlinear boundary value problems

Abstract: is a 2n-th order self-adjoint linear differential operator, and it was shown that the Rayleigh-Ritz-Galerkin method is an efficient scheme, both theoretically and numerically, for solving such problems. Our aim here is to extend the results of [2] and [3] to the case of nonsel[adjoint linear differential operators whose coefficients have a singularity at one or both end points of the interval [0, t ]. For ease of exposition, we shall restrict ourselves here to second order operators, as in the particular case of

On the Singularity of Dynamical Response and Critical Slowing Down

Abstract: Phenomenological arguments on the critical slowing down is presented and "similarity law" is proposed on the indices of the critical slowing down The similarity law is confirm­ ed in linear spin chains near the critical field and in the kinetic Ising model near the critical temperature It is exactly shown in the linear spin chains that the critical index of slowing down is different from that of the static susceptibility and that the dynamical susceptibility has a logarithmic singularity with respect to the frequency at the critical field and at zero temperature Although many investigations have been made on singularities of several kinds of dynamical phenomena near the critical point, there are very few that go essentially beyond a dynamical molecular field theory and afford a unified point of view on anomalies of dynamical responses in the vicinity of the criti­ cal point The purpose of this paper is to discuss the critical slowing down charac­ teristic of dynamical critical phenomena from a unified point of view In § 2, the main results of the Kubo linear response theory 1 ) are summarized for con­ venience to discuss the singularity of dynamical response and critical slowing down in the subsequent sections In § 3, the formulation of relaxation time and its relation with dynamical susceptibility are given for the purpose of phenomeno­ logical arguments on the critical slowing down "Similarity law" on the criti­ cal slowing down is proposed as a working assumption In § 4, as an example, the dynamics of linear spin chains is investigated in detail near the critical field The rigorous analysis yields that the critical index of slowing down is different from that of the corresponding susceptibility and that the "similarity law" holds with respect to the magnetization and partial energy In § 5, we discuss the results obtained by our previous perturbational calculation 2 ) and those obtained by a computer simulation 3 ) from our point of view

Discontinuity formation in solutions of homogeneous non-linear hyperbolic equations possessing smooth initial data

Abstract: Large classes of non-linear equations, at which previous breakdown theories have been aimed, are obtainable by differentiation of first order equations. The general solutions of these first order equations are also solutions of the corresponding second order equations. These are displayed and employed to calculate the critical time for singularity occurrence. Examples are discussed from gas dynamics, shallow water waves, wave propagation in solids, and electrical transmission lines. This method, when applicable, is simple and yields results which agree with those obtainable from the Ludford and Lax-Jeffrey theories.

The hydrogen atom in the presence of the Fermi-contact interaction

Abstract: It is shown that the energy associated with the Fermi-contact part of the hamiltonian of the hydrogen atom cannot be expanded as a perturbation series. A variation calculation shows that the total energy is E 0 when the Fermi-contact operation is added to H 0, and -∞ when it is subtracted. In order to do a priori variational calculations on the ‘second-order’ properties of this operator, such as spin-spin coupling constants, it is probably necessary to remove the singularity caused by the δ function.

Use of Splines and Numerical Integration in Geometrical Acoustics

Abstract: Two contributions are made to ray theoretic techniques for propagation through continuously stratified media. The first makes use of spline polynomials to approximate speed‐of‐sound profiles smoothly. The second eliminates the singularity occuring in the usual range integral. A versatile automatic numerical procedure for solving a wide variety of acoustic propagation problems results.

On the singularities of cosmological solutions of the gravitational equations.

Abstract: Publisher Summary This chapter examines the general properties of the cosmological solutions of the gravitational equations near a time singularity. The customarily used (Friedmann) cosmological solution of Einstein's gravitational equations is based on the assumption that matter is distributed in space homogeneously and isotropically. This assumption is very far-fetched mathematically, apart from the fact that its fulfillment in a real world can at best be only approximate. The solution obtained for the centrally symmetrical problem is actually a particular case of a more general class of solutions. In the absence of matter, there is naturally no centrally-symmetrical solution as the free gravitational field cannot have such symmetry.

Physical-Region Discontinuity Equation

Abstract: A Cutkosky‐type formula for the discontinuity around an arbitrary physical‐region singularity is derived from precisely formulated S‐matrix principles.

Evaluation of Some Fermi-Dirac Integrals

Abstract: The evaluation of Fermi‐Dirac integrals is discussed for cases in which the Sommerfeld method fails. Such cases occur when the integrand has a singularity at the Fermi surface and when the integrand is a rapidly oscillating function. As examples, the first‐order exchange integral for electrons and the free‐energy integral of the noninteracting electron gas in a magnetic field are evaluated. The method uses a contour‐integral representation of the Fermi function (previously mentioned by Dingle), supplemented by Mittag‐Leffler type expansions.

The Planar Motion of a Trojan Asteroid

Abstract: The planar motion of an asteroid in an orbit close to that of Jupiter is considered within the framework of the restricted circular three-body problem. The solution is derived asymptotically to second order using the two-variable expansion procedure for the case of small Jupiter-sun mass ratio. It is shown that the solution derived under the assumption of small Jupiter-sun mass ratio becomes singular for orbits that approach Jupiter. The nature of the singularity is exhibited as a guide for future work valid for this case. The results are given in explicit form for the coordinates as functions of time including both short and long periodic terms. Finally, the present solution is specialized to the family of long periodic Trojan orbits about the sun-Jupiter libration points.

A finite element application of the modified Rayleigh-Ritz method

Abstract: Considerable attention has been devoted in the literature on numerical methods towards securing energy convergence of solutions for, say, linearly elastic plate bending problems. Although energy convergence is necessary it by no means follows that the derived bending moments and shearing forces converge uniformly at a given point and it is this kind of feature which the engineer is really seeking. This question is examined in the context of a problem which is of particular interest to the civil engineering field and concerns the bending of a square plate under uniformly distributed load; the plate has simply supported edges and contains a central square hole with free edges. The solution to this multiply connected and mixed boundary value problem is obtained through a recently developed modification to the Rayleigh–Ritz method which has very general application and renders the solution mathematically valid up to the internal corner points where the bending moments are singular. Use is made of triangular equilibrium finite elements in conjunction with continuous eigenfunctions. Although it is already known that the order (i.e. the eigenvalue) of the singularity at the internal corners is available by inspection, it is an interesting feature of the present solution that a good approximation to the amplitude is also obtained by an inspection of the finite element results.

Asymptotic analysis of oscillatory mode of approach to a singularity in homogeneous cosmological models

Abstract: Publisher Summary This chapter presents the evolution of the metric in the oscillatory mode of approach to a singularity in homogeneous cosmological models. It is possible to carry out analytic and statistical investigation of the evolutions of a model with appreciable fullness in the asymptotic limit of times that are arbitrarily close to the singularity. It is appropriate to recall that although the physical applicability of Einstein's equations in their present form can be clarified under the indicated singular conditions only by a future synthesis of physical theories, the existing gravitational theory itself does not lose its logical cohesion at any density of matter. The process of evolution of the metric on approaching the singular point consists, consequently, of successive periods, during each of which the distance scales oscillate along two spatial axes and decrease monotonically along the third axis. On going from one era to another, the direction along which the monotonic decrease of the distances takes place is transferred from one axis to another.