Showing papers on "Singularity published in 1968"

Theory of a Two-Dimensional Ising Model with Random Impurities. I. Thermodynamics

Abstract: Recent experiments demonstrate that at the Curie temperature the specific heat may be a smooth function of the temperature. We propose that this effect can be due to random impurities and substantiate our proposal by a study of an Ising model containing such impurities. We modify the usual rectangular lattice by allowing each row of vertical bonds to vary randomly from row to row with a prescribed probability function. In the case that this probability is a particular distribution with a narrow width, we find that the logarithmic singularity of Onsager's lattice is smoothed out into a function which at ${T}_{c}$ is infinitely differentiable but not analytic. This function is expressible in terms of an integral involving Bessel functions and is computed numerically.

On the radio occultation method for studying planetary atmospheres

Abstract: The problem of determining the refractivity profile of a planetary atmosphere from optical or radio occultation data is identical in principle to the problem of determining the variation of seismic velocities in the earth from the observed travel times of seismic body waves. In either case, a complete set of data can be inverted uniquely, the only constraints being those fundamental to geometric optics. Expressions are given for converting observed Doppler shifts to the index of refraction as a function of depth in the atmosphere. The effect of various approximations on the analysis is discussed; it is found that a ‘thin atmosphere’ approximation simplifies the mathematics and preserves the singularity at the critical ray curvature.

Local Characterization of Singularities in General Relativity

Abstract: We formulate a new approach to singularities: their local description. Given any incomplete space‐time M, we define a topological space, the ``g boundary,'' whose points consist of equivalence classes of incomplete geodesics of M. The points of the g boundary may be thought of as the ``singular points'' of M. Local properties of the singularity may now be described in a well‐defined way in terms of local properties of the g boundary. For example, the notions: ``dimensionality of a singularity,'' ``past and future of a singular point,'' ``neighborhood of a singular point,'' ``spacelike or timelike character of a singularity,'' and ``metric structure of a singularity'' may all be expressed as properties of the g boundary. Two applications of the g boundary outside of the realm of singularities are discussed: (1) In the case in which the space‐time M is extendable (for example, Taub space), the g boundary is shown to be that regular 3‐surface across which M may be extended [in this case, the Misner boundary between Taub and Newman‐Unti‐Tamburino (NUT) space]. (2) With a slight modification of the definitions, the g boundary of an asymptotically simple space‐time is shown to be Penrose's surface at ``conformal infinity.'' The application of the g boundary technique to singularities is illustrated with a number of examples. The g‐boundary structure of one particular example leads to our consideration of non‐Hausdorff space‐times.

Scattering of Massless Scalar Waves by a Schwarzschild ``Singularity''

Abstract: This paper investigates the scattering and absorption of scalar waves satisfying the equation φ;μ;μ=0 in the Schwarzschild metric. This problem has been previously considered by Hildreth. We find, for a Schwarzschild mass m, the following cross sections in the zero‐frequency limit for s‐waves: σ(absorption) = 0, dσ/dΩ ≃ [c + ⅓(2m) ln (2mω)]2, where c is a constant of order m. These results disagree with the previous calculation. We exhibit a method of solution for the equation. Its limiting (Newtonian) form, with suitable identification of the coefficients, is the problem of Coulomb scattering in non‐relativistic quantum mechanics. By demanding coordinate conditions which for large l allow the usual Coulomb results in a partial‐wave expansion, we are able to define a partial‐wave cross section. The (summed) differential cross section for small frequencies inherits the logarithmic behavior of the s‐wave part, which is the only contribution explicitly calculated. (The l ≠ 0 contributions and the behavior of...

Condensation of the Ideal Bose Gas as a Cooperative Transition

Abstract: The thermodynamic properties of the noninteracting Bose gas in the neighborhood of its transition are examined in detail. The order parameter is a complex extensive variable, but the thermodynamic properties depend only on its amplitude under simple boundary conditions. As the dimensionality or the single-particle energy spectrum is varied, the critical singularity displays a variety of forms. The equation of state has a simple structure, different from the homogeneous form often discussed for critical systems but asymptotically reducing to the latter except when logarithmic singularities are involved. The correlation function in the critical region is a homogeneous function of the distance and a correlation length. Only for a quadratic energy spectrum is the Ornstein-Zernike theory result valid at the critical temperature. A precise correspondence is noted between the asymptotic properties of the ideal Bose gas transition and those of the spherical model of ferromagnetism.

Singularities of the n-body problem. I

Abstract: Little is known about the nature of the singularities of the n-body problem. While it is plausible to suppose that they are due to collisions, this has never been established, except when n = 2 or n = 3. In the general case the best that can be said at present is the fact, due to PAINLEV~ [5], that a singularity occurs at the time to if and only if the minimum of the mutual distances between pairs of particles approaches zero as the time t approaches to. In the present paper we shall investigate the problem of singularities due to collisions. We define a singularity at time to to be due to collisions i f as t ~ to each particle approaches a definite position in the inertial coordinate f rame. This means, in view of PAINLEV~'S theorem, that at least two particles approach the same point. In 1908 VON ZEIPEL [4] published a statement to the effect that if the system remains bounded as t--+ to, then a singularity at time t o is due to collisions. His proof is erroneous, and the assertion still stands as a conjecture. The purpose of the present paper is to obtain necessary and sufficient conditions for a singularity due to collisions. It will be supposed that the origin of coordinates is fixed at the center of mass, and that the singularity occurs as t ~ 0 +. The following notation will be used. The symbols m k, re, vk denote respectively the mass, position and velocity of the kth particle. We define further

Torsional Vibration of an Elastic Solid Containing a Penny‐Shaped Crack

Abstract: The axisymmetric wave equation is solved for the problem of torsional elastic waves impinging on a penny‐shaped crack the periphery of which is assumed to be infinitely sharp. Using Hankel transforms, the problem is reduced to the solution of two simultaneous integral equations of the Fredholm type. The proposed method of solution permits an examination of the complete scattered‐wave field at points both near to and far from the penny‐shaped plane of discontinuity. In elastodynamics, however, it is the nearfield stress solution that is of chief interest. To this end, the singular nature of the local dynamic stress field is determined in elementary closed form, while the magnitude of this stress field, which can be adequately described by a singularity parameter k3, is calculated numerically. The important results are that (1) the stresses are singular of the order r1−12 as r1 → 0 at the diffracting edge of the crack and (2) k3 is found to be proportional to the material constants, the crack radius, and th...

Numerical solutions of the flow field for conical bodies in a supersonic stream

Abstract: : A numerical procedure for solving the problem of steady supersonic inviscid flow around smooth conical bodies is presented. Results are obtained by solving the elliptic partial differential equations that define the conical flow between the body and the shock. Results are given for circular cones up to moderately high relative incidences, including some cases for incidences beyond a critical value at which the entropy singularity moves from the surface. Also presented are a few results for elliptic cones at zero and non-zero incidence, as well as results for another conical body whose cross section is defined by a fourth order even cosine Fourier series. The applicability of the method can be limited by the entropy singularity moving too far away from the surface by the flow field containing regions of locally conically supersonic flow, or by the shock wave approaching very close to the Mach wave. Comparison of results shows excellent agreement with other theoretical methods and also with experimental results. The method is efficient in computer time.

Pressure-loading functions for oscillating wings with control surfaces.

Abstract: In the linearized formulation of the oscillating-surf ace problem, singularities in the lift distribution occur at subsonic leading edges, at control surface leading edges, and in general wherever the up wash prescribed by the wing deformations is discontinuous. These singularities are examined by use of the method of matched asymptotic expansions. It is shown that for a known discontinuity in upwash (as in the case of the control surface) the form, as well as the strength, of the singularity are determined uniquely. For subsonic leading edges only the form, but not the strength, of the singularity can be determined. A discussion is also given of the proper shape of the loading functions near side edges. Nomenclature b = reference length (root semi chord) Cp = pressure coefficient k = reduced frequency, ub/Um M = Mach number of freestream MN = M cos A, Mach number normal to edge p — pressure amplitude, Cp = peikt t = time Um = freestream velocity w = amplitude of prescribed upwash on the wing X) y} z = Cartesian coordinates with x in the freestream and in the span wise direction xc = location of control-surfa ce leading edge and hinge Xj y, z = stretched variables

Low-Energy-Theorem Approach to Single-Particle Singularities in the Presence of Massless Bosons

Abstract: By the use of low-energy theorems, which follow from gauge invariance and analyticity assumptions, we determine the nature of the single-particle singularity of a meson propagator in the presence of the known massless bosons:Photons and gravitons. In addition to regaining the well-known results for covariant gauges in electrodynamics, we present new results for covariant gauges in gravity theory, and for radiation gauges in both electrodynamics and gravity theory. The gauges in which no infrared singularities are present are found: For covariant electrodynamics it is of course the Yennie gauge; for covariant gravity theory a similar gauge is given. In radiation gauges it is shown that Schwinger's new gauge has this desirable property for photons, and an analogous gauge is constructed for gravitons. It is established that in these gauges the single-particle singularity of the meson propagator becomes a simple pole.

Crossing, hermitian analyticity, and the connection between spin and statistics.

Abstract: The analytic S‐matrix framework is further developed. First some results of earlier works are collected and the physical‐region analyticity properties recently derived from macroscopic causality conditions are described. These entail that scattering functions are analytic at physical points except on positive‐α Landau surfaces, and that there they are ie limits of analytic functions from certain well‐defined directions, except possibly at certain points where four or more positive‐α surfaces intersect. A general ie rule that also covers these exceptional points is then stated. It is then shown that the scattering function defined by analytic continuation is either symmetric or antisymmetric under interchange of variables describing identical particles and that the sign induced by the interchange is independent of the particular scattering function in which the variables appear. The physical‐region analyticity properties of bubble‐diagram functions are then derived from the general ie rule. These functions are products of scattering functions and conjugate scattering functions integrated over physical internal‐particle variables, as in the terms of unitarity equations. They are shown to be analytic in the physical region except on Landau surfaces and, more specifically, except on those Landau surfaces that correspond to Landau diagrams that are supported by the bubble diagram in question, with the further restriction that the Landau α's must be positive or negative for lines lying within positive or negative bubbles, respectively. Also, the basic rule for continuation around these singularities is derived. A new general derivation of the pole‐factorization theorem is given, which is based on slightly weaker assumptions than earlier proofs. Particular attention is paid to the over‐all sign. A general derivation of the crossing and Hermitian analyticity properties of scattering functions is then given. On the basis of the deduced general rules for constructing the paths of continuation that connect the crossed and Hermitian conjugate points, the various related points are found to be boundary values of a single physical sheet. In particular, a certain sequence of continuations is shown to take one back to the original point. From this fact it follows that abnormal statistics are incompatible with simultaneous unitarity in both the direct and crossed channels. The proof given here does not depend on the notion of interchange of variables other than those of identical particles. Earlier proofs depended on the unphysical notion of interchange of variables representing conjugate particles. Finally it is shown that the analytically continued M functions with normal‐ordered variables are precisely the scattering functions: no extra signs are needed or permitted. Aside from the general i rule, the analyticity assumptions are these: (1) The discontinuity around a singularity of a bubble diagram B has no residue at a physical‐particle mass value (in an appropriate variable) unless the singularity corresponds to a diagram that is supported by B and has the single‐particle‐exchange form that corresponds to a pole at that mass value. (2) The residue just described has the pole‐factorization property. (3) Confluences of infinite numbers of singularity surfaces do not invalidate the results established by assuming that this number is locally finite. Assumption (1) entails that all relevant singularities of scattering functions lie on Landau surfaces. That is the basic assumption.

Equivalence of Markov processes

Abstract: continuity of one Markov process with respect to another in terms of multiplicative functional. In this paper we investigate the equivalence of Hunt processes from a local point of view. After introducing the basic notation and definitions in §2 we discuss three special cases in §3 which motivate and illustrate the general theory which is to be developed. In §4 a class of martingales is associated with a Markov process ; this leads to the proof of an extended Markov property in §5. The extended Markov property is central in proving our results on the equivalence of Hunt processes. §§6 and 7 are concerned with certain technical results which lead to the statement and proof of the main results in §§8 and 9. In §8 it is shown that if two Hunt processes arising from Feller semigroups are not equivalent then at least one of four basic types of singularity must occur. The four basic types of singularity are : singularity on the germ field, singularity on the tail field, local singularity at a stopping time and jump singularity at a stopping time. §9 contains a deeper study of local singularities. In particular, if the processes have the additional property which we call local smoothness then the existence of local singularities at a stopping time is equivalent to the existence of singularities on the germ field. Finally, in §10 the property of local smoothness is characterized in potential theoretic terms. The author would like to thank Professer P. A. Meyer and an anonymous

On the asymptotic decay of coupling for infinite phased arrays

Abstract: The asymptotic decay of coupling with distance from a single excited element it an infinite linear array has been analyzed for several special cases and is shown to behave according to exp (-jkr)/r3/2where r is the distance from the excited element. This behavior is also characteristic of propagation over a lossy surface. By geometric extrapolation one would expect an asymptotic decay of coupling for planar arrays to behave according to exp (-jkr)/r2. This extension is carried out for an arbitrary infinite periodic planar current sheet with the expected result. In addition, this behavior is shown to be valid for an infinite parallel plate array immersed in a magnetized cold plasma. In all cases, the asymptotic behavior is shown to depend on the form of the singularity of the derivative of the reflection coefficient with scan angle. This singularity occurs at grazing incidence of the radiated beams.

Dynamical Treatment of the Thermal Diffuse Scattering of X-Rays

Abstract: A dynamical theory of the thermal diffuse scattering of X-rays from thick crystals is developed which consistently takes into account the Bragg scattering of both the incident and diffusely scattered X-rays. Special attention is paid to the analysis of the intensity distribution of the diffusely scattered waves near the Bragg peak. The known singularity of the one-phonon scattering of type 1/q2, where q is the phonon wave vector, is shown to be transformed at small q to the more weak singularity of type1/q| for the case of the Bragg diffraction (here,q| is the projection of the vector q ona plane parallel to the crystal surface). [Russian Text Ignored].

Singular variations near the contact discontinuity in the theory of interplanetary blast waves

Abstract: This report is limited to calling attention to a feature of the ideal interplanetary blastwave according to the Parker type of formal theory which has not been scrutinized in the literature before It is shown analytically that, on account of inertial effects, a gradient singularity of the radial momentum transfer rate arises in association with the local coronal accumulation in front of the rear propelling contact surface postulated in the theory and that such a local accumulation is a general result for power-law explosive waves extruding against an ambient hydrogen solar wind having a strictly inverse-square radial decay in density The usual numerical schemes are rendered ineffective for the determination of the expected singular local-wave behavior To circumvent the difficulty, a combined numerical-integral technique has been developed and in this work it has been applied on six specific model waves, including both the decelerating and the non-decelerating, which have energies increasing with time as t K with K = 28, 4, 1, 1/2, 1/4 and 1/8, respectively Also, a local solution corresponding to the limiting approximation of a constant material velocity gradient has been analytically constructed The importance of the local singular behavior is quantitatively appraised and certain interplanetary implications are given cursory inferences It is considered that the new results should have inescapable relevance with regard to (1) the quantitative determination of the progressive peak transverse magnetic field during an interplanetary storm and (2) the theoretical determination of the macroscopic stability of the postulated contact surface

Interaction of Free‐Surface Waves with Viscous Wakes

Abstract: A method of investigating the interaction of free‐surface waves with viscous wakes is described. The method consists in constructing a viscous‐wake solution to the Oseen equations that satisfies the three linearized free‐surface conditions appropriate to a viscous fluid. The solution is characterized by a singularity which simulates approximately the effect of a body. More general flows of the same type can be formed by superposition. The solution obtained is believed to be the first one to represent explicitly a viscous wake in the presence of a free surface.

The ferromagnetic transition of iron according to a Landau-theory of phase transitions of second order

Abstract: The properties of iron at its ferromagnetic transition point are described by a type of Landau-theory developed for He II. The theory includes spatial fluctuations of the spontaneous magnetisation which lead to a logarithmic singularity of the specific heat nearTc in agreement with measurements byKraftmakher andRomashina.

On Final Value Control

Abstract: Final value control of linear plants is discussed. An expression for the closed loop fundamental matrix in terms of the open loop transition matrix is presented. It is shown that for scalar minimum energy final value control the singularities in the feedback loops are inversely proportional to the order of the derivative being fed back. That is, the position loop has an ηth order singularity, the velocity loop an η—lth order singularity, etc. A partially closed loop control is developed that allows the highest-order feedback singularity to be set between η and zero. Those results are illustrated by examples.

On the logarithmic singularity of the specific heat of He II

Abstract: The well known singular behaviour of the specific heat at the λ-point of liquid helium can be understood on the basis of an extended Ginzburg-Landau type theory in which spatial fluctuations of the order parameter are incorporated.