Showing papers on "Singularity published in 1971"
TL;DR: In this paper, a system of either fermions or bosons interacting in one dimension by a two-body potential with periodic boundary conditions was investigated, and expressions for the one-particle density matrix at zero temperature and particular (nontrivial) values of the coupling constant $g, as a determinant of order $N\ifmmode\times\else\texttimes\fi{}N$ were presented.
Abstract: We investigate exactly a system of either fermions or bosons interacting in one dimension by a two-body potential $V(r)=\frac{g}{{r}^{2}}$ with periodic boundary conditions. In addition to rederiving known results for correlation functions and thermodynamics in the thermodynamic limit, we present expressions for the one-particle density matrix at zero temperature and particular (nontrivial) values of the coupling constant $g$, as a determinant of order $N\ifmmode\times\else\texttimes\fi{}N$. These concise expressions allow a discussion of the momentum distribution in the thermodynamic limit. In particular, for a case of repulsive bosons, the determinant is evaluated explicitly, exhibiting a weak (logarithmic) singularity at zero momentum, and vanishing outside of a "Fermi" surface.
993 citations
TL;DR: The solution of the zero-field "eight-vertex" model is presented in this article, which includes the square lattice Ising, dimer, ice, $F, and KDP models as special cases.
Abstract: The solution of the zero-field "eight-vertex" model is presented. This model includes the square lattice Ising, dimer, ice, $F$, and KDP models as special cases. It is found that in general the free energy has a branch-point singularity at a phase transition, with a continuously variable exponent.
530 citations
11 Dec 1971
TL;DR: In this paper, a new method for the solution of EMP interaction problems is developed, which involves expanding the solution in terms of its singularities in the Laplace transform or complex frequency (or s) plane.
Abstract: : This note develops a new method for the solution of EMP interaction problems. Basically it involves expanding the solution in terms of its singularities in the Laplace transform or complex frequency (or s) plane. In the time domain each term comes from an inverse transform of the corresponding term in the singularity expansion. Finite size objects with well behaved media have only poles in the finite s plane for their delta function response. These factor into terms involving the classical natural frequencies and modes but in addition bring out factors which we call coupling coefficients as well as the possiblity of higher order poles besides simple poles, but still of finite order in the finite s plane. If the incident waveform has singularities in the finite s plane the response can be generally split into an object part (containing object poles) and a waveform part containing the waveform singularities. The object poles directly give amplitudes, frequencies, damping constants, and phases for the damped sinusoidal waveforms seen so commonly in EMP tests using pulsed waveforms. There is some latitude in the calculation of coupling coefficients and some difficulties are discussed. (Author)
284 citations
TL;DR: In this paper, the authors considered the plane problem of two materially dissimilar isotropic, homogeneous, elastic wedges that are bonded together along both of their common faces so that the cross section forms a composite fullplane with a single corner in the otherwise straight interface boundary.
225 citations
TL;DR: In this paper, the behaviour near to and far from an initial singularity in a broad subclass of the models studied in previous papers is examined and the influence of the matter on the evolution at these times is discussed.
Abstract: The behaviour near to and far from an initial singularity in a broad subclass of the models studied in previous papers [1–3] is examined. The influence of the matter on the evolution at these times is discussed. The singularity types for the various models, which are mostly of cigar or oscillatory nature, are found. It is discovered that among these models, only those of the same Bianchi type as a Robertson-Walker model can become “approximately Robertson-Walker” in a sense defined in the paper. Qualitative conclusions concerning black-body isotropy, the Hubble relation, helium abundance and horizon structure are given.
202 citations
TL;DR: In this article, a phase transition from a superfluid state to a non-superfluid state was observed in the rotational frequency of the deformed nucleus of the nucleus 160 Dy.
195 citations
TL;DR: In this paper, the complete solution of the ridged waveguide eigenvalue problem is presented by the formulation of an integral eigen value equation which is subsequently solved numerically by application of the Ritz-Galerkin method.
Abstract: The complete solution of the ridged waveguide eigen-value problem is presented. The solution is obtained by the formulation of an integral eigenvalue equation which is subsequently solved numerically by application of the Ritz-Galerkin method. The significance of the eigenvalue spectrum is discussed and the modes are classified as either hybrid or trough modes. Equations are given for the electric and magnetic fields and a brief discussion of the edge singularity is presented. The theory is illustrated by computing the dominant eigenvalues and characteristic impedances of various unsymmetrical ridged waveguides.
177 citations
TL;DR: In this paper, the dependence of the singularities on the local geometry and on the types of boundary conditions is investigated, and it is found that the type of boundary condition determines the singularity but the geometry largely influences the magnitude of singularity.
160 citations
TL;DR: In this paper, the ground-state energy of a many-fermion system is derived from the many-body Schrodinger equation via perturbation theory, and the analytic singularities of the energy function are analyzed, and procedures designed to avoid or overcome them.
Abstract: Calculational procedures are developed for the ground-state energy of a many-fermion system. The entire theory is developed from the many-body Schr\"odinger equation via perturbation theory. The linked cluster expansion is derived by elementary methods. The analytic singularities of the energy function are analyzed, and procedures designed to avoid or overcome them. Results for the calculation of the energy in the presence of a purely repulsive two-body force are reviewed, and the situation is found to be reasonably satisfactory, though good results are restricted to weak potentials when the system is not dilute. Results for an attractive potential surrounding a strongly repulsive core are reviewed, and procedures discribed for calculations in this case. A sample calculation with an assessment of the probable errors is reviewed.
93 citations
TL;DR: In this paper, the contact problem of an elastic stiffener bonded to an elastic half plane with different mechanical properties is considered and the governing integral equation is reduced to an infinite system of linear algebraic equations.
Abstract: The contact problem of an elastic stiffener bonded to an elastic half plane with different mechanical properties is considered. The governing integral equation is reduced to an infinite system of linear algebraic equations. It is shown that, depending on the value of a parameter which is a function of the elastic constants and the thickness of the stiffener, the system is either regular or quasi-regular. A complete numerical example is given for which the strength of the stress singularity and the contact stresses are tabulated.
55 citations
TL;DR: In this article, the vacuum expectation value of the time-ordered product of two exponentials of free fields is defined as a distribution using minimal singularity as a criterion, and the implication of this definition for an exponentially self-coupled scalar field is studied in second order of a perturbation expansion.
Abstract: We define the vacuum expectation value of the time-ordered product of two exponentials of free fields as a distribution using minimal singularity as a criterion. The implication of this definition for an exponentially self-coupled scalar field is studied in second order of a perturbation expansion.
TL;DR: In this article, a one-dimensional Fredholm integral equation is derived for the ground state solution of the delta function model for two-electron helium-like ions, and the perturbation series is analyzed in terms of the singularity which determines its radius of convergence.
Abstract: A one‐dimensional Fredholm integral equation is derived for the ground state solution of the delta‐function model for two‐electron heliumlike ions. This equation is solved numerically; the perturbation series is developed through E(20) and compared with the solution of the integral equation. The series is further analyzed in terms of the singularity which determines its radius of convergence.
TL;DR: The analytical continuation to complex value of t from Maradudin's result for t > 1, and the value of real and imaginary parts of the integral I(t ± ie) for 0 < t < 1, e → 0, and t ∼ 0, are given in this paper.
Abstract: The lattice Green's function for the body‐centered cubic (bcc) lattice I(t)=1π3∫ ∫ 0π∫dx dy dzt±ie−cosxcosycosz is considered. With the use of the analytic continuation to complex value of t from Maradudin's result for t > 1, the value of the real and imaginary parts of the integral I(t ± ie) for 0 < t < 1, e → 0, is obtained. The expressions valid for t → ∞, t≳1, t≲1, and t ∼ 0 are given. They are useful for analyzing the nature of the singularity and for carrying out numerical calculations in all regions of t.
TL;DR: In this article, the authors considered the linear inviscid reflexion of internal/inertial waves from smooth bumpy surfaces where a characteristic (or ray) is tangent to the surface at some point.
Abstract: This paper considers the linear inviscid reflexion of internal/inertial waves from smooth bumpy surfaces where a characteristic (or ray) is tangent to the surface at some point. There are two principal cases. When a characteristic associated with the incident wave is tangent to the surface we have diffraction; when the tangential characteristic is associated with a reflected wave we have split reflexion, a phenomenon which has no counterpart in classical non-dispersive wave theory. In both these cases the problem of determining the wave field may be reduced to a set of coupled integral equations with two unknown functions. These equations are solved for the simplest topography for each case, and the properties of the wave fields for more general topographies are discussed. For both split reflexion and diffraction, the fluid velocity has an inverse-square-root singularity on the tangential characteristic, and the energy density has a corresponding logarithmic singularity. The diffracted wave field penetrates into the shadow region a distance which is of the order of the incident wavelength. Possibilities for instability and mixing are discussed.
TL;DR: In this paper, a solution of the Einstein equations for a universe whose three-space sections are spaces of Bianchi type IX in the sense of a pictorial or diagrammatic solution was obtained, by means of which the singularity in this case is very close in character to that of universes which have only expansion and shear.
CERN1
TL;DR: In this paper, dual one-loop amplitudes for external pions are constructed from the Born term of Neveu and Schwarz, and non-orientable diagrams are needed, although the model has SU(2) symmetry.
TL;DR: In this article, the authors used the theory of elastic dielectrics with polarization gradient along with the values of material coefficients obtained by Askar et al. to solve two types of problems involving cylindrical and spherical cavities.
TL;DR: In this article, the concepts of absolute continuity and singularity for operator-valued measures are introduced and Radon-Nikodym and Lebesgue decomposition theorems for such measures are established.
TL;DR: In this paper, the steady flow of an incompressible elastic power law fluid in a two-dimensional channel with non-parallel walls was investigated and a method was developed for estimating the pressure drop in the naturally converging flow field before the entrance to a capillary.
Abstract: Similarity solutions are determined for the steady flow of an incompressible elastic power law fluid in a two-dimensional channel with nonparallel walls. The possibility of wall suction or injection is considered. Solutions are found to exist only for power law indices of unity. A method is developed for estimating the pressure drop in the naturally converging flow field before the entrance to a capillary. In diverging flows a singularity is found to arise due to the elasticity of the fluid. The singularity corresponds to a Deborah number of unity. It is postulated that the singularity is, for the constitutive equation used here, a possible source of the flow instability commonly referred to as melt fracture.
01 Jan 1971
TL;DR: In this article, the convergence of the method of ignoring the singularity was studied for two important classes of functions: those having an algebraic singularity of a fixed order and those having a logarithmic singularity in the range of integration.
Abstract: The convergence of the method of ignoring the singularity was first discussed in [1]. This paper deals with the degree of this convergence for two important classes of functions : those having an algebraic singularity of a fixed order and those having a logarithmic singularity in the range of integration. For the latter class the degree of convergence is found to depend explicitly on the quadrature rule chosen for the approximation process, whereas for the former class it depends solely on the order of the singularity.
TL;DR: Convergence theorem for singular integrands numerical quadratures, assuming domination near singularity by monotone integrable function as discussed by the authors, assuming domination of singularity in quadrature.
Abstract: Convergence theorem for singular integrands numerical quadratures, assuming domination near singularity by monotone integrable function
TL;DR: In this article, the effects of general relativity on the Jacobian figures of uniformly rotating homogeneous masses are determined, and it is shown that the post-Newtonian figure is obtained by a deformation of the Jacobi ellipsoid by a suitable Lagrangian displacement cubic in the coordinates.
Abstract: The effects of general relativity, in the post-Newtonian approximation, on the Jacobian figures of equilibrium of uniformly rotating homogeneous masses are determined, It is shown, for example, that the post-Newtonian figure is obtained by a deformation of the Jacobi ellipsoid by a suitable Lagrangian displacement cubic in the coordinates. The solution of the post-Newtonian equations exhibits an indeterminacy at the point of bifurcation M 2 , where the Jacobian sequence branches off from the Maclaurin sequence, and a singularity at a point J 4 , where the axes of the Jacobi ellipsoid are in the ratios 1:02972 :0,2575. The indeterminacy in the solution at M 2 arises from the fact that at this point the Maclaurin spheroid is neutral to an infinitesimal deformation proportional to (X 1 , -X 2 , 0); and the singularity at J 4 arises from the fact that at this point the Jacobi ellipsoid is unstable to the deformation induced by the effects of general relativity.
TL;DR: In this article, it is shown that any inhomogeneity and anisotropy in the distribution and motion of a non-rotating ideal fluid accelerates collapse to a singularity which will be reached in a finite proper time.
Abstract: The problem of singularities is examined from the stand-point of a local observer. A singularity is defined as a state with an infinite proper rest mass density. The approach consists of three steps: (i) The complete system of equations describing a non-symmetric motion of a perfect fluid under assumption of adiabatic thermodynamic processes and of no release of nuclear energy is reduced to six Einstein field equations and their four first integrals for six remaining unknown componentsgik. (ii) A differential relation for the behavior of the rest mass density is deduced. It shows that any inhomogeneity and anisotropy in the distribution and motion of a non-rotating ideal fluid accelerates collapse to a singularity which will be reached in a finite proper time. Collapse is also inevitable in a rotating fluid in the case of extremely high pressure when the relativistic limit of the equation of state must be applied. In the case of a lower or zero pressure the relation does not give an unambiguous answer if the matter is rotating. (iii) The influence of rotation on the motion of an incoherent matter is investigated. Some qualitative arguments are given for a possible existence of a narrow class of singularity-free solutions of Einstein equations. Assuming rotational symmetry the Einstein partial differential equations together with their first integrals are reduced to a system of simultaneous ordinary differential equations suitable for numerical integration. Without integrating this system the existence of the class of singularity-free solutions is confirmed and exactly delimited. These solutions, representing a new general relativistic effect, are, however, of no importance for the application in cosmology or astrophysics. It is proved that in all the other cases interesting from the point of view of application the occurrence of a point singularity in incoherent matter with a rotational symmetry is inevitable even if the rotation is present.
CERN1
TL;DR: In this article, the pomeron-like singularity, present in orientable nonplanar loops, is a cut rather than a pole, and the corresponding counterterms are complex.
Abstract: Off-mass-shell orientable nonplanar loops are used to suggest a specific way of regularizing one-loop amplitudes. The corresponding counterterms are complex because the pomeronlike singularity, present in orientable nonplanar loops, is a cut rather than a pole.