Showing papers on "Singularity published in 1969"

Ising Model with Second-Neighbor Interaction. I. Some Exact Results and an Approximate Solution

Abstract: The square Ising lattice with unequal first- and second-neighbor interactions has been considered. Among the exact results discussed, we show that the lowest energy state can be ferromagnetic, antiferromagnetic, or superantiferromagnetic, and the transition temperature should vanish in some cases. It is also shown that this problem is a special case of a more general problem arising in the statistical consideration of the hydrogen-bonded crystals. A well-defined approximation procedure is then introduced to solve the latter problem and to derive the (approximate) critical condition and expressions for the thermodynamic functions. The critical temperature thus determined is exact for the regular Ising lattice and for the lattices with ${T}_{c}=0$ while for the equivalent neighbor model the error is less than 2%. The specific heat possesses the usual logarithmic singularity in all cases.

Singularities in the X-Ray Absorption and Emission of Metals. II. Self-Consistent Treatment of Divergences

Abstract: The many-body approach of the preceding paper is generalized to include self-energy and vertex renormalization is self-consistent fashion. The algebra of parquet diagrams is developed in terms of an unknown divergent parameter (related to the deep-hole Green's function $\mathcal{G}$, and of an unknown irreducible interaction $R$. The resulting interaction operator $\ensuremath{\gamma}$ is used to calculate $\mathcal{G}$ and $R$; the corresponding coupled equations are solved self-consistently in the weak coupling limit. The spectral densities of the deep-hole Green's function and of the x-ray response function are calculated explicitly; in the weak coupling limit, the Mahan singularity found in the preceding paper is unaffected by renormalization. The self-consistent formalism presented here describes the reaction of divergent fluctuations on themselves, and should, therefore, be useful in other more complicated problems, such as the Kondo effect.

Absolute Zero of Time

Abstract: A singularity involving infinite densities at a finite proper time in the past is strongly suggested for the beginnings of the Universe by Einstein's general relativity theory, and is consistent with the few relevant observational data. There is no reasonable point at which to anticipate a failure of the theory, especially since a simplified quantum calculation in the accompanying paper predicts that quantum effects do not change the nature of the singularity. Therefore, we suggest that the singularity be treated as an essential element of cosmological theory, and indicate how this can be made more palatable by refining our concepts of time.

Pathological oscillations of a rotating fluid

Abstract: A theoretical study is made of the free periods of oscillation of an incompressible inviscid fluid, bounded by two rigid concentric spheres of radii a, b (a > b), and rotating with angular velocity Ω about a common diameter. An attempt is made to use the Longuet-Higgins solution of the Laplace tidal equation as the first term of an expansion in powers of the parameter e = (a − b)/(a + b), of the solution to the full equations governing oscillations in a spherical shell. This leads to a singularity in the second-order terms at the two critical circles where the characteristic cones of the governing equation touch the shell boundaries.A boundary-layer type of argument is used to examine the apparent non-uniformity in the neighbourhood of these critical circles, and it is found that, in order to remove the singularity in the pressure, an integrable singularity in the velocity components must be introduced on the characteristic cone which touches the inner spherical boundary. Further integrable singularities are introduced by repeated reflexion at the shell boundaries, and so, even outside the critical region the velocity terms contain what may reasonably be described as a pathological term, generally of order e½ compared to that found by Longuet-Higgins, periodic with wavelength O(ea) in the radial and latitudinal directions.Some consequences of this result are discussed.

Distortion of sonic bangs by atmospheric turbulence

Abstract: Recorded pressure signatures of supersonic aircraft often show intense, spiky perturbations superimposed on a basic N -shaped pattern. A first-order scattering theory, incorporating both inertial and thermal interactions, is developed to explain the spikes. Scattering from a weak shock is studied first. The solution of the scattering equation is derived as a sum of three terms: a phase shift corresponding to the singularity found by Lighthill; a small local compression or rarefaction; a surface integral over a paraboloid of dependence, whose focus is the observation point and whose directrix is the shock. The solution is found to degenerate at the shock into the result given by ray acoustics, and the surface integral is identified with the scattered waves that make up the spikes. The solution is generalized for arbitrary wave-forms by means of a superposition integral. Eddies in the Kolmogorov inertial subrange are found to be the main source of spikes, and Kolmogorov's similarity theory is used to show that, for almost all times t after a sonic-bang shock passes an observation point, the mean-square pressure perturbation equals $(\Delta p)^2 (t_c/t)^{\frac{7}{6}}$ , where Δ p is the pressure jump across the shock and t c is a critical time predicted in terms of meteorological conditions. For an idealized model of the atmospheric boundary layer, t c is calculated to be about 1 ms, a figure consistent with the qualitative data currently available. The mean-square pressure perturbation just behind the shock itself is found to be finite but enormous, according to first-order scattering theory. It is conjectured that a second-order theory might explain the shock thickening that actually occurs.

Amelioration of divergence difficulties in the theory of weak interactions

Abstract: A new approach to the problem of the singularity of the weak interactions is presented. Its aim is to provide a theoretical interpretation of the extreme smallness of the violation of selection rules associated with the weak-vector-current operator appearing in the conventional Fermi or intermediate-vector-boson interaction Lagrangian. To illustrate what we have in mind, we note that on account of this singular character, the conventional theories have not yet yielded an understanding of the weakness of strangeness and parity violation in hadronic processes and the weakness of semileptonic neutral decays. We begin with an interaction Lagrangian in which the constituents of the conventional weak current (e.g., strangenesschanging, axial-vector, muonic, etc.) are coupled to possibly distinct local vector operators. This is done in such a way that the effective weak interaction between two currents decomposes into two parts, one having the universality of the weak interaction, the other, called diagonal, acting only between a constituent and itself. It is then possible to transfer the singularity of the weak interaction to the diagonal interaction and to impose any desired degree of symmetry upon the singular part of the diagonal interaction. Two realizations of this approach are presented. Both are intermediate-boson theories involving gradient-coupled spin-0 bosons as well as spin-1 bosons. An important consequence of these theories is that, apart from implying a lower bound, the weak interactions give no indication of the magnitude of the diagonal interactions. Thus while the scattering of μ-neutrinos by electrons should be governed by the conventional universality formula, there is no reason to expect universality to hold for the scattering of e-neutrinos by electrons.

Quadratic Corrections to the Lagrangian Density of the Gravitational Field and the Singularity

Abstract: where the first two terms lead to (1 ), and the remaining terms are quadratic corrections in R, which can become important only for large R (these values of R are, however, not of the order of the inverse square of the Planck length l 02 = c3/Gtl ~ 1066 cm-2, but somewhat smaller: R ~ lii2/137). The general form of the Lagrangian density of the gravitational field has been considered by PolievktovNikoladzeE2l for the case of weak gravitational fields; a correction to the Lagrangian density of the form R2ln (R/R0 ) due to quantum effects has been considered in unpublished work by T. Hill (cf _(31 ), with the aim of obtaining a nonsingular solution for the collapse of a dust-like sphere. In the present paper we consider the possibility of applying (2) near the Friedman singularity in connection with the problem of the nonsingular transition from constriction to expansion in the simplest case of a homogeneous and isotropic space filled with matter with the equation state p = €/3. 1. Let us show first that not all quadratic invariants in (2) are independent. Indeed, multiplying the identity R;kzm + R;m~tz + Rilmh = 0 by Riklm and using Rikzm = -Rkilm, we find

On the real singularities of the N-body problem

Abstract: Theorems on collision singularities in many body problem, and bibliography on equations of motion in celestial mechanics

A singularity-free empty universe.

Abstract: Einstein equations solution for empty space without singularities containing metric, with closed homogeneous space type hypersurfaces expanding anisotropically

Duality and the pomeranchuk singularity.

Abstract: The Pomeranchuk singularity plays an exceptional role within the framework of the duality hypothesis for strong interaction scattering amplitudes. While the socalled 'ordinary' Regge trajectories in a given channel are supposed to be 'built from resonances in another channel, the pomeron is said to be constructed from nonresonating background (Harari I968; Freund I968; Gilman, Harari & Zarmi I968). Many consequences of this idea were explored in the last year or so, and a large number of successful predictions have been derived. The purpose of this report is to review this special status of the Pomeranchuk singularity, to discuss its various implications and to raise some related problems which have remained unsolved.

Irrotational flow about ship forms

Abstract: A method of computing the potential flow about a ship model in terms of a source distribution on the hull surface is described. After a formulation and discussion of the flow with wavemaking, the zero-froude number, double-ship-model case is treated in detail. The Fredholm singular integral equation of the second kind for the source distribution is solved numerically by removing the singularity, replacing the integral by a quadrature formula, and solving the resulting high-order set of linear equations by an iteration formula for which convergence is proved. The corresponding velocity potential on the hull surface, the evaluation of which from the source distribution also requires the calculation of a singular integral, is obtained by first solving the equipotential problem for the hull form (which employs the same kernel as the original integral equation). The solution of this Dirichlet problem is then used to remove the singularity from the velocity-potential integral, and then the latter is computed by means of a quadrature formula. The method is applied to an ellipsoid, results from which are compared with the exact solution. A fortran program is included. (Author)

On the singularity method of subsonic lifting- surface theory.

Abstract: Subsonic lifting surface theory including leading edge, discussing singularities in solution of integral equation for determination of aerodynamic properties

S‐Matrix Singularity Structure in the Physical Region. I. Properties of Multiple Integrals

Abstract: We obtain the real singularities and corresponding discontinuities of a class of multiple integrals over real contours. Our aim is to give a unified treatment, obtaining, by elementary mathematical methods, both previously known results and some new generalizations. In a subsequent paper the results are applied to unitarity integrals.

Macroscopic causality conditions and properties of scattering amplitudes.

Abstract: Two causality conditions that refer only to mass‐shell quantities are formulated and their consequences explored. The first condition, called weak asymptotic causality, expresses the requirement that some interaction between the initial particles must occur before the last interaction from which final particles emerge. This condition is shown to imply that if a two‐body scattering function is analytic except for singularities in the energy variable at normal thresholds, then (a) the physical scattering functions in two adjacent parts of the physical region separated by any normal threshold are parts of a single analytic function; (b) the path of continuation joining these two parts bypasses the singularity in the upper half‐plane of the energy variable; and (c) the integral over the physical function can be represented as an integral over a contour that is distorted into the upper‐half energy plane (hence not, for example, by a principal‐value integral). Singularities possessing finite derivatives of all ...

On a Method to Subtract off a Singularity at a Corner for the Dirichlet or Neumann Problem

Abstract: Let D be a plane domain partly bounded by two line segments which meet at the origin and form there an interior angle 7ra > 0. Let U(x, y) be a solution in D of Poisson's equation such that either U or a U/an (the normal derivative) takes prescribed values on the boundary segments. Let U(x, y) be sufficiently smooth away from the corner and bounded at the corner. Then for each positive integer N there exists a function VN(X, y) which satisfies a related Poisson equation and which satisfies related boundary conditions such that U - VN is N-times con- tinuously differentiable at the corner. If 1/a is an integer VN may be found ex- plicitly in terms of the data of the problem for U. a In solving an elliptic partial differential equation by numerical methods the results proved about convergence of the numerical approximation to the actual solution frequently depend on differentiability properties of the (unknown) solu- tion. In the work of Gerschgorin (2) and other papers written since, it is assumed that the solution of the partial differential equation has derivatives of order four which are continuous up to the boundary. If the boundary and all the data are sufficiently smooth there is, of course, no problem. In many cases, however, the boundary pos- sesses a finite number of singularities, usually (in the two-dimensional case) in the. form of corners; occasionally too, the boundary data may have jumps. Laasonean (3) has proved that convergence of the discrete solution to the actual solution holds for the Dirichlet problem, but that the convergence is slow in a neighborhood of the corner. In this paper we will consider a method to subtract off the singularity. The method is quite old (see Fox (1)), but includes results on the asymptotic behavior of solutions near a corner. In this light see the works of Lewy (4), Lehman (5), Wasow (6), and the author (7). We consider a problem for which the solution is not known too be smooth. We then find, explicitly in terms of the boundary data, a solution to a related problem; then the difference between these two solutions is a solution to a. third problem, and is sufficiently well-behaved to insure convergence of difference schemes. Finally, the sought solution can be found by adding the explicitly given one to the numerically-solved one. Let D be a plane domain partly bounded by two open line segments ri and r2, which share the origin as a common endpoint and form there an interior angle 7ra > 0. We assume that ri is a subset of the positive x-axis and r2 makes an angle 7ra > 0 with the positive x-axis. Let F(x, y) be given in D and ib(x, y) (respectively

The collision singularity in a perturbed two- body problem

Abstract: Collision of all bodies in a perturbed n-body problem is analyzed by an extension of the author's results for a perturbed two-body problem (1969). A procedure is set forth to prove that the absolute value of energy in a perturbed n-body system remains bounded until the moment of collision. It is shown that the characteristics of motion in both perturbed problems are basically the same.

High-temperature critical properties of the Ising model on a triple of related lattices

Abstract: Three regular three-dimensional lattices of coordination numbers, q = 3, 4, and 6 are introduced. Exact relations are derived among the specific-heat singularity amplitudes and among the susceptibility singularity amplitudes. Exact high-temperature series expansions for the partition function and the susceptibility are derived for the q = 3 and q = 6 lattices. Precise values of the critical temperature, susceptibility amplitude, critical energy, and critical entropy are obtained for all three lattices. The variation of Ising critical parameters with coordination number is discussed.

Statistical Mechanics of the Finite Ising Model with Higher Spin

Abstract: The partition functions of the finite Ising models of spin S =1, 3/2, 2, 5/2 and 3 have been calculated by a high speed computer. The periodicity condition at the boundaries has been imposed, and only the nearest neighbor interaction has been assumed. The specific heats of these finite systems have been calculated as a function of temperature. The maxima of the specific heat increase proportionally to long N in the two-dimensional Ising model with spin S =1, N being the number of lattice points, which indicates that the specific heat for the above lattice has a singularity such as C v / k ∼ A log | T - T c |+ B ± : A ∼0.10 and B - - B + =0.24. The difference ( B + - B - ) has been estimated from the asymmetric distribution of the zeros of the partition function in the complex temperature plane.

S-Matrix Singularity Structure in the Physical Region. II. Unitarity Integrals

Abstract: The general techniques developed in an earlier paper [J. Math. Phys. 10, 494 (1969)] are applied to evaluate the singularities and discontinuities of unitarity integrals. The results are conveniently expressed in terms of what we call mechanism (or M) diagrams.

On the singularity method of subsonic lifting- surface theory

Abstract: Subsonic lifting surface theory including leading edge, discussing singularities in solution of integral equation for determination of aerodynamic properties

S‐Matrix Singularity Structure in the Physical Region. III. General Discussion of Simple Landau Singularities

Abstract: Drawing on the understanding of unitarity integrals acquired in previous papers [J. Math. Phys. 10, 494, 545 (1969)], we show that unitarity demands that connected amplitudes be singular on the positive‐α arcs of all ``simple'' Landau curves in their physical regions. Further, we show that the amplitudes are nonsingular on mixed‐α arcs, while on the positive‐α arcs their discontinuities are given by the Cutkosky rule. This confirms arguments from perturbation theory and demonstrates how a weak analyticity assumption can generate in an exact way a singularity scheme relating to causality.

Critical‐Point Singularities of the Perturbation Series for the Ground State of a Many‐Fermion System

Abstract: We show that the many‐fermion ground‐state energy with an attractive potential has a critical singularity. This singularity destroys the validity of ``low‐density'' approximations. We also find that the K‐matrix formalism is, in principle, not applicable to attractive potentials because of the presence of ``Emery singularities.'' We introduce an R‐matrix formalism which is numerically very close to the K matrix and free from manifest ``Emery singularities.'' A model calculation is performed on the lattice gas to try to anticipate what quality of results can be expected from summing an R‐matrix expansion with fixed density.

Model for the Derivation of Kinetic Theory

Abstract: A soluble model for the derivation of kinetic theory is discussed. It is shown that Bogoliubov's adiabatic assumption is not valid even to leading order in the kinetic regime. Furthermore, the adiabatic assumption gives rise to a singular behavior in the correlation function which is sufficiently strong to violate the ordering assumed in the expansion. The exact result for the model correlation is neither adiabatic nor singular throughout the kinetic regime. We first show that the singularity is removed by properly reordering the correlation in the relevant domains; then a method for obtaining uniformly valid asymptotic solutions for a wide class of integrodifferential problems (that includes our model) is presented. We apply our technique to the BBGKY hierarchy and show that for a weakly coupled gas (i) the familiar Landau equation is satisfied in lowest order by the velocity distribution function and (ii) the two‐particle correlation function can be calculated consistently to leading order, yielding a n...

On the behaviour of the wave function for the 23S state of the two‐electron atom in the region where both electron–nuclear distances are small

Abstract: The expansion of the wave function for the 23S state of the two-electron atom in the neighbourhood of the singularity at r1 = r2 = 0 is considered. The restrictions imposed on the variational functions by this expansion are discussed. For the 23S state of He, Li+, N5+ the behaviour of the variational function based on the Fock expansion in the neighbourhood of this singularity is investigated. The agreement of the variational coefficients with the theoretical coefficients is satisfactory. The calculated values of E and 〈δ(r2)〉 for He, Li+, N5+ are given.