Showing papers on "Singularity published in 1972"
TL;DR: In this article, ESHELBY deduced a surface-integral representation for the force on an elastic singularity or inhomogeneity, which gives rise to a conservation law for regular elastostatic fields appropriate to homogeneous but not necessarily isotropic solids in the presence of infinitesimal deformations.
Abstract: Several years ago ESHELBY [1] (1956), in a paper devoted to the continuum theory of lattice defects, deduced a surface-integral representation for the "force on an elastic singularity or inhomogeneity", which-in the absence of such
defects-gives rise to a conservation law for regular elastostatic fields appropriate to homogeneous but not necessarily isotropic solids in the presence of infinitesimal deformations. Morevoer, ESHIELBY noted that his result, when suitably interpreted, remains strictly valid for finite deformations of elastic solids.
583 citations
TL;DR: In this paper, the integral equation is formulated as a constraint upon the local picture-frame solutions, whence these local solutions are solved directly by a variational method, using finite elements, in a manner such that the problem of the Green's-function singularity is side-stepped.
Abstract: An unbounded region is divided into local picture-frame regions where a partial differential-equation solution is obtained, with the remaining unbounded region represented by an integral equation. (The method permits the use of free-space Green's functions, and thus special problem-dependent Green's functions need not be found.) The integral equation is formulated as a constraint upon the local picture-frame solutions, whence these local solutions are solved directly by a variational method, using finite elements, in a manner such that the problem of the Green's-function singularity is side-stepped. The technique is applicable where sources and media inhomogeneities and anisotropies are local, and can all be placed within one or several picture frames. It is in these cases that the integral-equation approach is at a particular disadvantage, and the use of a partial differential-equation technique is advisable if not necessary. Examples presented include the static and harmonic fields of a parallel-plate capacitor, a microstrip line on a dielectric substratum, and a radiating antenna with dielectric obstacles.
241 citations
TL;DR: In this paper, the authors considered irrotational dust solutions of the Einstein equations and defined velocity dominated singularities of these solutions, which can be considered as a three-dimensional manifold with an invariantly and uniquely defined inner metric tensor, extrinsic curvature tensor and scalar bang time function.
Abstract: We consider irrotational dust solutions of the Einstein equations. We define ``velocity‐dominated'' singularities of these solutions. We show that a velocity‐dominated singularity can be considered as a three‐dimensional manifold with an invariantly and uniquely defined inner metric tensor, extrinsic curvature tensor, and scalar bang time function. We compute this structure for a variety of known exact models. The structure of the singularity uniquely determines the solution in a certain class of spatially inhomogeneous models. We briefly discuss the b boundary (Schmidt boundary). In an appendix we generalize conformal transformations to ``stretch'' transformations and calculate the curvature form of a stretched metric.
183 citations
01 Jan 1972
166 citations
TL;DR: Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K; for the triangular lattice to v16, for the square lattice and for the honeycomb lattice, inclusive.
Abstract: Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K; for the triangular lattice to v16, for the square lattice to v21 and for the honeycomb lattice to v32, inclusive. The asymptotic behaviour of the ferromagnetic and antiferromagnetic susceptibility is studied. It is concluded that the ferromagnetic singularity is not exactly factorizible. The antiferromagnetic susceptibility of the square and honeycomb lattices has a singularity of the same type as the energy at the antiferromagnetic critical temperature.
105 citations
90 citations
TL;DR: In this article, the growth of a pair of parallel infinite vortices is studied and the shape of the wave at this instant is calculated using linear theory, and it is shown that its gross properties are given roughly by linear theory.
Abstract: Numerical methods are used to study the growth of waves of finite amplitude on a pair of parallel infinite vortices. The vortices are treated as lines except in so far as the detailed structure of the core is needed to remove consistently the singularity in the line integrals for the velocities of the vortices. It is shown that the vortices eventually touch and the shape of the wave at this instant is calculated. The wave is quite distorted at this instant, but it is shown that its gross properties are given roughly by linear theory.
83 citations
82 citations
TL;DR: In this article, the authors derived exact equations of motion for a system interacting with a reservoir by means of projection-operator techniques, and the kernel of the master equation obtained in case a) is investigated in the thermodynamic limit of the reservoir.
78 citations
TL;DR: In this article, it is shown that a line singularity distribution should be associated with the surface singularity distributions over the hull of an actual ship hull, that is of a hull neither thin nor flat, nor slender, seems to be one of the main causes of the discrepancy between the measured and calculated values of the wave resistance.
Abstract: The inaccuracy of the usual representations of an actual ship hull, that is of a hull neither thin, nor flat, nor slender, seems to be one of the main causes of the discrepancy between the measured and calculated values of the wave resistance. It is shown in the first part of this paper that a line singularity distribution should be associated with the surface singularity distribution over the hull. In the second part, the order of magnitude of the contribution of the line singularity is evaluated in the particular cases of a vertical, elliptical, infinite cylinder. It is considerably high when normal doublet distribution is used. It becomes of the first order at low Froude numbers in the case of a source distribution. The problem still requires extensive research in several directions outlined at the end of the paper.
76 citations
TL;DR: In this article, a new solution of the Einstein field equations depending on four arbitrary constants was obtained by the method of complex coordinate transformations, which is of type II and possesses a wire-like singularity.
TL;DR: In this article, the authors studied the number of selfavoiding walks and self-avoiding returns to the origin on two and three dimensional lattices and studied numerically by the ratio method.
Abstract: New data for the number of selfavoiding walks and selfavoiding returns to the origin on two and three dimensional lattices are presented and studied numerically by the ratio method. Estimates for the critical attrition and critical indices are given. For a loose-packed lattice the selfavoiding walk generating function appears to have a singularity on the negative real axis. This singularity is at the same distance from the origin as the physical singularity, and is found to be cusp-like with an exponent of 1/2 in two dimensions and 3/4 in three dimensions. This behaviour enables a close analogy to be drawn between the behaviour of the Ising model high temperature susceptibility and the walk generating function.
TL;DR: The simplest covariant generalization of the scalar wave equation leads to significant pion creation and annihilation processes near an isotropic Friedmann-type singularity (such processes are negligible for particles of nonzero spin) as mentioned in this paper.
Abstract: The simplest covariant generalization of the scalar wave equation leads to significant pion creation and annihilation processes near an isotropic Friedmann-type singularity (such processes are negligible for particles of nonzero spin). Estimates for a plausible initial state yield pion creation of the same order of magnitude as obtained by Zeldovich near an anisotropic Kasner-type singularity.
TL;DR: In this paper, the laminar boundary-layer equations were solved for incompressible flow past a parabola at angle of attack, and it was concluded that a singularity is always present at separation independent of the mildness of the pressure gradient at that point.
Abstract: : The laminar boundary-layer equations were solved for incompressible flow past a parabola at angle of attack. Such flow experiences a region of adverse pressure gradient and thus can be employed to study the boundary-layer separation process. The present solutions were obtained numerically using both implicit and Crank Nicholson type difference schemes. It was found that in all cases the point of vanishing shear stress (the separation point) displayed a Goldstein type singularity. Based on this evidence, it is concluded that a singularity is always present at separation independent of the mildness of the pressure gradient at that point. (Author)
TL;DR: In this article, a class of inviscid irrotational flows which satisfy the conditions at a time-dependent free surface exactly were investigated. But the results were limited to the elliptical and hyperbolic cases.
Abstract: Attention is drawn to a class of inviscid irrotational flows which satisfy the conditions at a time-dependent free surface exactly. The flows are related to the ellipsoids of Dirichlet (1860).Depending on a parameter P, the cross-section may take the form of a variable ellipse (P 0) or a pair of parallel lines (P = 0). The elliptical case was investigated both theoretically and experimentally by Taylor (1960). The hyperbolic case (P > 0) is remarkable in that the flow develops a singularity when the angle between the asymptotes approaches a right-angle. It is suggested that this solution represents a possible instability near the crest of a standing gravity wave of large amplitude.In the intermediate case (P = 0) the solution describes an open-channel flow in which the fluid filaments are stretched uniformly in a horizontal direction. The latter flow is demonstrated experimentally.
TL;DR: In this paper, the conformal group is relized on Hilbert spaces of holomoprhic or antiholomorphic functions over a tube domain in complex Minkowski space.
Abstract: Unitary analytic representations of the conformal group are relized on Hilbert spaces of holomoprhic or antiholomorphic functions over a tube domain in complex Minkowski space. The distributional boundary values of these functions are tempered distributions on real Minkowski space. The representations are characterized by an integral scale dimension labeln and two spin labelsj
1 andj
2. The connection between the dimensionn and the degree of singularity of the tempered distribution is investigated. We propose an application to inclusive reactions of elementary particles.
TL;DR: In this paper, the structure functions of deep inelastic electro-and-neutrino-production on nucleons are derived from the quark parton model or equivalently from a hypothesis of Fritzsch and Gell-Mann on the leading light cone singularity of current commutators.
TL;DR: The region near the initial singularity of the most general case of Bianchi-type IX universes is investigated in this paper, and the simplified equations of motion which obtain are studied by two means, an analytical method and a pictorial description which makes use of a point moving in a set of potentials.
TL;DR: In this article, the diffraction matrix for a surface singularity where the curvature but not the slope is discontinuous is rigorously derived, and the results differ significantly from the physical optics estimates and are analogous to those for a wedge-like singularity.
Abstract: To enlarge the scope of the geometrical theory of diffraction, the diffraction matrix for a surface singularity where the curvature but not the slope is discontinuous is rigorously derived. The model that is employed consists of two parabolic cylinders of different latus recta joined together at the front, thereby creating a line discontinuity of the required form. For each of the two principal polarizations, asymptotic developments of the surface fields in the vicinity of the join are calculated, from which the diffraction coefficients are then obtained by integration. The results differ significantly from the physical optics estimates and are analogous to those for a wedge-like singularity. This analogy permits a trivial deduction of the complete diffraction matrix.
TL;DR: In this paper, a method of simultaneously regularizing the three types of binary collisions in the planar problem of three bodies is developed: the coordinates are transformed by means of certain fourth degree polynomials, and a new independent variable is introduced, too.
Abstract: A new method of simultaneously regularizing the three types of binary collisions in the planar problem of three bodies is developed: The coordinates are transformed by means of certain fourth degree polynomials, and a new independent variable is introduced, too. The proposed transformation is in each binary collision locally equivalent to Levi-Civita's transformation, whereas the singularity corresponding to a triple collision is mapped into infinity. The transformed Hamiltonian is a polynomial of degree 12 in the regularized variables.
TL;DR: In this article, the authors proposed a method which uses complex variable techniques to determine the shape of the free boundary near to the separation point, which is also used to improve the accuracy of the finite-difference solution in the neighbourhood of the singularity.
Abstract: The numerical solution of free boundary problems gives rise to many computational difficulties. One such difficulty is due to the singularity at the separation point between the fixed and free boundaries. A method is suggested which uses complex variable techniques to determine the shape of the free boundary near to the separation point. This complex variable solution is also used to improve the accuracy of the finite-difference solution in the neighbourhood of the singularity. The analytical study was incorporated into an algorithm for the numerical solution of a particular free boundary problem concerning the percolation of a fluid through a porous dam. Some numerical results for this problem are presented.
TL;DR: In this article, an analytical scheme is presented which yields an approximate solution of the Poisson Boltzmann equation, which is more precise when the singularity of the solution is further away from the origin and when the dimensionality of the problem is lower.
Abstract: An analytical scheme is presented which yields an approximate solution of the Poisson‐Boltzmann equation. The result is more precise when the singularity of the solution is further away from the origin and when the dimensionality of the problem is lower. The solution is compared with the results of numerical integration in the three‐dimensionless case.
TL;DR: In this article, two different perturbation series (the polarization expansion and a united-atom expansion) of the ground state energy of the delta-function model for one-electron diatoms are studied and the radii of convergence are determined.
Abstract: Two different perturbation series (the polarization expansion and a united-atom expansion) of the ground state energy of the delta-function model for one-electron diatoms are studied and the radii of convergence are determined For both expansions the singularity in the energy which limits the radius of convergence is a branch point with exponent one-half The physical significance of the branch point is that for particular values of the perturbation parameter, two different energy eigenvalues coalesce The positions of the branch points are computed as a function of the internuclear separation R For all values of R, both series converge for all physical values of the perturbation parameters A lower bound to the radius of convergence of the polarization expansion has been computed previously by Claverie It is proved in the present paper that the lower bound calculation is in fact an exact determination of the radius of convergence The results of the model study are applied to real one-electron diatoms to suggest the possible location of a branch point singularity in the energy of the ground state
TL;DR: In this paper, it is shown how to describe the process of alternation of Kasner epochs in the oscillatory regime of approach to a singularity of the general (nonhomogeneous) solution to the Einstein equation.
Abstract: It is shown how to describe the process of alternation of Kasner epochs in the oscillatory regime _of approach to a singularity of the general (nonhomogeneous) solution to the Einstein equation. The umversahty of the previously established law of alternation of Kasner exponents is proved (for homogeneous models) and a general description of th•e rotations of the Kasner axes under the change of epochs is given. This achieves the existence proof of a general solution with a time singularity.
TL;DR: In this article, singular differential operators are considered, and for convenience the finite singularity is taken to be zero, and conditions are given on the functions qi for which the number of linearly independent solutions y of L(y) = λhy (Re λ = 0) satisfying.
Abstract: We consider here singular differential operators, and for convenience the finite singularity is taken to be zero. One operator discussed is the operator L defined by where q 0 > 0 and the coefficients q t are real, locally Lebesgue integrable functions defined on an interval (a, b). For a given positive, continuous weight function h, conditions are given on the functions qi for which the number of linearly independent solutions y of L(y) = λhy (Re λ = 0) satisfying.
TL;DR: In this paper, an account is given of some developments in the asymptotic evaluation of integrals of a single variable, and estimates of the errors in Laplace-type integrals and the method of steepest descents are provided.
Abstract: An account is given of some developments in the asymptotic evaluation of integrals of a single variable. After a discussion of Laplace integrals and quadrature formulas, estimates are provided of the errors in Laplace-type integrals and the method of steepest descents. Then Fourier transforms and the method of stationary phase are considered; integrands which are generalized functions are included and there is also a brief description of integrals of convolution type. Finally, uniformly valid formulas for the coalescence of two saddle points or of a saddle point and singularity are derived.
TL;DR: In this article, the X-ray critical scattering in NiCr 2 O 4 in the vicinity of its Jahn-Teller phase transition has been investigated in order to clarify the nature of the interaction between local distortions.
Abstract: The X-ray critical scattering in NiCr 2 O 4 in the vicinity of its Jahn-Teller phase transition has been investigated in order to clarify the nature of the interaction between local distortions. The results are summarized as follows: (i) The critical scattering due to correlation of local distortions has been observed up to 100 K above the transition temperature. (ii) Relative intensities of the critical scattering around various reciprocal lattice points confirm that the correlation of local distortions propagates through acoustic type displacement field, showing that the indirect pair interaction via strain field is dominant to cause Jahn-Teller phase transition in NiCr 2 O 4 . (iii) The characteristics of the anisotropy of the critical scattering show a singularity at k =0, indicating that the interaction is of long range. The above results are in very good agreement with the theory recently developed by Kataoka and Kanamori.
01 Jan 1972
TL;DR: In this paper, the authors lay down procedures which are valid in all situations and which do not require prior examination of the ranks of V and X. The general Gauss-Markoff model (Y, Xβ, σ 2 V), is considered, where V is possibly singular and X possibly deficient in rank.
Abstract: Some recent work of the author on 'Unified Theory of Linear Estimation' is described. The general Gauss-Markoff (GGM) model (Y, Xβ, σ 2 V), is considered, where V=E[(Y-Xβ) (Y-Xβ)'] is possibly singular and X possibly deficient in rank. Aitken's procedure of least squares is not applicable when V is singular. The object of the paper is to lay down procedures which are valid in all situations and which do not require prior examination of the ranks of V and X. Two unified methods are suggested. One is a numerical approach called the Inverse Partitioned Matrix (IPM) method. Another is an analogue of the least squares theory, called the Unified Least Squares (ULS) method. It has been pointed out that singularity of V imposes some restriction on the parameter β, which have to be taken into account in constructing unbiased estimators.
TL;DR: In this article, the optimal profile of a two-dimensional plate that produces the maximum hydrodynamic lift while planing on a water surface, under the condition of no spray formation and no gravitational effect, was determined, with the latter assumption serving as a good approximation for operations at large Froude numbers.
Abstract: This paper attempts to determine the optimum profile of a two-dimensional plate that produces the maximum hydrodynamic lift while planing on a water surface, under the condition of no spray formation and no gravitational effect, the latter assumption serving as a good approximation for operations at large Froude numbers. The lift of the sprayless planing surface is maximized under the isoperimetric constraints of fixed chord length and fixed wetted arc-length of the plate. Consideration of the extremization yields, as the Euler equation, a pair of coupled nonlinear singular integral equations of the Cauchy type. These equations are subsequently linearized to facilitate further analysis. The analytical solution of the linearized problem has a branch-type singularity, in both pressure and flow angle, at the two ends of plate. In a special limit, this singularity changes its type, emerging into a logarithmic one, which is the weakest type possible. Guided by this analytic solution of the linearized problem, approximate solutions have been calculated for the nonlinear problem using the Rayleigh-Ritz method and the numerical results compared with the linearized theory.