Showing papers on "Gravitational singularity published in 1984"
Book•
25 May 1984
TL;DR: In this paper, the authors give a coherent account of the theory of isolated singularities of complete intersections, and show that the discriminant of the semi-universal deformation of an A-D-E singularity is isomorphic to the associated Coxeter group.
Abstract: This monograph gives a coherent account of the theory of isolated singularities of complete intersections. One encounters such singularities often as the central fibres of analytic map-germs; that is why such map-germs (deformations) receive here a great deal of attention. The work treats both the topological side--including vanishing cycles and monodromy--and the analytic side--including properties of the discriminants of deformations--and explores the connections between them. It ends with a proof that the discriminant of the semi-universal deformation of an A-D-E singularity is isomorphic to the discriminant of the associated Coxeter group.
486 citations
TL;DR: In this paper, a new method to compute normal forms of vector-field singularities is proposed, which are simpler than those known as Arnold-Takens normal forms and are uniquely determined from the original singularity in the category of (jets of) coordinate transformations.
Abstract: A new method to compute normal forms of vector-field singularities is proposed. Normal forms for some degenerate singularities of vector fields are computed. These normal forms are simpler than those known as Arnold-Takens normal form. Parameters in the normal forms are uniquely determined from the original singularity in the category of (jets of) coordinate transformations.
108 citations
TL;DR: A specific singularity of a vector field on R 3 is considered in this article, of codimension 2 in the dissipative case and of codeimension 1 in the conservative case.
Abstract: A specific singularity of a vector field on R 3 is considered, of codimension 2 in the dissipative case and of codimension 1 in the conservative case. In both contexts in generic unfoldings the existence is proved of subordinate Sil'nikov bifurcations, which have codimension 1. Special attention is paid to the C°°-flatness of this subordinate phenomenon.
92 citations
TL;DR: In this article, a comprehensive study of spatially homogeneous and SO(3)-isotropic exact solutions of the 10-parameter Lagrangian of the 'Poincare gauge theory' is presented.
Abstract: This is a comprehensive study of spatially homogeneous and SO(3)-isotropic exact solutions of the 10-parameter Lagrangian of the 'Poincare gauge theory'. Some sets of new exact solutions are presented. In particular, all solutions following from the so-called modified double quality ansatz are obtained, up to integration of some familiar ordinary differential equations. For certain classes of solutions, the occurrence of torsion singularities is discussed in detail. Furthermore, the authors investigate whether some solutions without metric singularity can provide reasonable cosmological models.
92 citations
TL;DR: For n ≥ 3, there exists an embedded minimal hypersurface in Rn+1 which has an isolated singularity but which is not a cone as discussed by the authors, and each example constructed here is asymptotic to a given, completely arbitrary, nonplanar minimal cone.
Abstract: For n≥3, there exists an embedded minimal hypersurface in Rn+1 which has an isolated singularity but which is not a cone. Each example constructed here is asymptotic to a given, completely arbitrary, nonplanar minimal cone and is stable in case the cone satisfies a strict stability inequality.
89 citations
TL;DR: In this article, the authors investigated the general problem of stress singularity near the free edge of two bonded anisotropic materials, at the tip of a crack between two materials and in the vicinity of a broken layer.
85 citations
TL;DR: In this article, the evolution of a class of exact spatially homogeneous cosmological models of Bianchi type VI.............. h677 is discussed, and the existence of this quasi-isotropic stage implies that these models can be compatible in principle with the observed universe.
Abstract: The evolution of a class of exact spatially homogeneous cosmological models of Bianchi type VI
h
is discussed. It is known that solutions of type VI
h
cannot approach isotropy asymptotically at large times. Indeed the present class of solutions become asymptotic to an anisotropic vacuum plane wave solution. Nevertheless, for these solutions the initial anisotropy can decay, leading to a stage of finite duration in which the model is close to isotropy. Depending on the choice of parameters in the solution, this quasi-isotropic stage can commence at the initial singularity, in which case the singularity is of the type known as “isotropic” or “Friedmann-like.” The existence of this quasi-isotropic stage implies that these models can be compatible in principle with the observed universe.
55 citations
TL;DR: In this article, the structure of two-dimensional variational integrals which are invariant under conformal mappings of the parameter domain was analyzed and it was shown that classical solutions of the corresponding Euler equations cannot have isolated singularities if their Dirichlet integral is finite.
Abstract: We analyze the structure of two-dimensional variational integrals which are invariant under conformal mappings of the parameter domain. This allows us to prove that classical solutions of the corresponding Euler equations cannot have isolated singularities if their Dirichlet integral is finite.
46 citations
TL;DR: In this article, isolated point singularities of the coupled Yang-Mills equations in R3 were considered and a removable singularity theorem was proved under appropriate conditions on the curvature and the Higgs field.
Abstract: We consider isolated point singularities of the coupled Yang-Mills equations inR3. Under appropriate conditions on the curvature and the Higgs field, a removable singularity theorem is proved.
26 citations
TL;DR: In this paper, activity series for hard-particle lattice gases and hard particles in continuous space were examined with respect to the singularity on the negative activity axis, and it was shown that the nature of singularity depends only on the dimensionality of space.
Abstract: Activity series for hard-particle lattice gases and hard particles in continuous space are examined with respect to the singularity on the negative activity axis. For approximately spherical particles it is found that the nature of the singularity depends only on the dimensionality of space.
21 citations
TL;DR: A survey of theories of singularities of systems of rays and wave fronts near caustics is given in this paper, where the problem of passing about an obstacle bounded by a smooth surface of general position is studied in detail.
Abstract: A survey is given of theories of singularities of systems of rays and wave fronts, that is, singularities of systems of extremals of variational problems and solutions of the Hamilton-Jacobi equations near caustics The problem of passing about an obstacle bounded by a smooth surface of general position is studied in detail Theorems are proved on the normal forms of Lagrangian manifolds with singularities formed by rays of the system of extremals of a variational problem in the symplectic space of all oriented lines which tear off from the surface of the obstacle as well as theorems on Legendre manifolds with singularities formed by contact elements of a wave front and 1-jets of a solution of the Hamilton-Jacobi equation
TL;DR: In this paper, the authors present a reasonable solution to this problem by making as unknown quantities, not only the unknown amplitude coefficients involved in the linear combination of the solutions, but also the coordinates of the singularities, the scattered wave is matched on the boundary in the sense of the least squares.
Abstract: As a method of solving the electromagnetic scattering by perfectly conducting cylinders, the discrete singularity method is discussed. For this method, a finite number of singular points is distributed within the scatterer and the wave function composed of a linear combination of solutions (each of which is derived from a Helmholtz equation with each single singularity among the many) is matched with the incident wave function on the boundary. the remaining important problem in this method is how to distribute these singularities in order to obtain the accurate solution with faster convergence and with fewer number of singular points for an arbitrary configuration of the scatterer. the purpose of this paper is to present a reasonable solution to this problem. the essential point of the method is that by making as unknown quantities, not only the unknown amplitude coefficients involved in the linear combination of the solutions, but also the coordinates of the singularities, the scattered wave is matched on the boundary in the sense of the least squares. This problem is linear in the amplitude coefficient but nonlinear in the singularity coordinate. Thus we solve this problem by using the algorithm by Marquardt et al. of a nonlinear optimization process. This paper considers the general description and indicates its effectiveness by showing a few examples of the scattering problem of rectangular cylinders.
TL;DR: In this article, the effect of potential energy singularities on the existence of analytic first integrals of mechanical systems with two degrees of freedom is investigated and applications to the restricted many-body problem are presented.
TL;DR: In this article, a hybrid of dimensional and analytic regularization is used to regulate and uncover a Meijer's G -function representation for a class of massless, divergent Feynman integrals in an axial gauge.
01 Aug 1984
TL;DR: In this article, it was shown that the electric field singularity depends on the first even Dirichlet eigenvalue while the first Neumann is related to the magnetic field singularities.
Abstract: The field singularities near the tip of an elliptic cone are presented. We prove that the electric field singularity depends on the first even Dirichlet eigenvalue while the first Neumann is related to the magnetic field singularity. Some data on the eigenvalues are given.
TL;DR: The results of A N Varchenko regarding the zeta-function of the monodromy operator for a singular point of a hypersurface are generalized to the case of a complete intersection singularity.
Abstract: The results of A N Varchenko regarding the zeta-function of the monodromy operator for a singular point of a hypersurface are generalized to the case of a complete intersection singularity
TL;DR: In this paper, all classical solutions of the Liouville theory for strings having finite stable minimum energies are calculated explicitly together with their minimal energies, including the set of natural solitonlike singularities described by Jorjadze, Pogrebkov, and Polivanov.
Abstract: All classical solutions of the Liouville theory for strings having finite stable minimum energies are calculated explicitly together with their minimal energies. Our treatment automatically includes the set of natural solitonlike singularities described by Jorjadze, Pogrebkov, and Polivanov. Since the number of such singularities is preserved in time, a sector of solutions is not only characterized by its boundary conditions but also by its number of singularities. Thus, e.g., the Liouville theory with periodic boundary conditions has three different sectors of solutions with stable minimal energies containing zero, one, and two singularities. (Solutions with more singularities have no stable minimum energy.) It is argued that singular solutions do not make the string singular and therefore may be included in the string quantization.
TL;DR: In this article, the authors considered the case of a delta-function plane wave impinges upon a three-dimensional moving body, where the body can change shape in any smooth manner, as long as it remains in a fixed sphere and moves slower than the wave speed.
TL;DR: In this paper, the thermal stresses at the free edge of finite cylinders made of dissimilar materials were analyzed using the finite element method, and the results of the analysis showed that the radial stress and the hoop stress exhibit a logarithmic singularity.
TL;DR: In this paper, the problem of matching the B-Z simple solitons to flat space-time is analyzed and found to have more than one solution depending on the type of singularity admitted in the Ricci tensor.
Abstract: The simple soliton solutions of Einstein's equations obtained by Belinski and Zakharov using the inverse scattering method have been interpreted as gravitational (solitary) shock waves, partly on the basis of an analysis of certain (coordinate) singularities apparently inherent to the method. A closer study reveals, however, that such singularities can be removed by an appropriate extension of the solutions, which is given explicitly. A classification of inequivalent flat space-time metrics appropriate for the applications of the method is derived. The problem of matching the Belinski-Zakharov (B-Z) simple solitons to flat space-time is analyzed and found to have more than one solution depending on the type of singularity admitted in the Ricci tensor. This is further illustrated by considering a three-parameter solution, inequivalent to that of Belinski and Zakharov. For negative values of one of these parameters the ranges of the coordinates are limited only by curvature singularities. For positive values of the parameter, coordinate singularities, similar to those in the B-Z solution, are also present. In this case, however, a matching to flat space-time leads to a shock front whose intersection with any spacelike hypersurface is bounded, in contrast with the behavior displayed by the B-Z solutions. The limiting case when the parameter is zero is found to have some special properties. A smooth extension is also shown to exist.
TL;DR: In this paper, the authors study weakly rational singularities of complex analytic spaces that are rational if and only if they are Cohen-Macauley singularities, and they study invariance under morphisms and deformations of these singularities.
Abstract: We study particular singularities of complex analytic spaces that we call weakly rational and that contain rational singularities. In fact, a weakly rational singularity is rational if and only if it is Cohen-Macauley. Invariance under morphisms and deformations of weakly rational singularities is also studied.
TL;DR: In this article, the authors discuss the next-to-leading (but dominant for dimension less than six) corrections to the large-N$ behavior of the magnetization at the Yang-Lee edge singularity.
Abstract: We discuss the next-to-leading (but dominant for dimension less than six) corrections to the large-$N$ behavior of the magnetization at the Yang-Lee edge singularity. The $N$ dependence of the corresponding amplitude is valid for all dimensions below six.
Abstract: The closed universe begins with a point singularity at the Big Bang and returns to a point singularity at the Big Crunch. These singularities present serious mathematical difficulties in extrapolating to times before the initial and after the final events. Such situations violate Einstein's singularity-free field theory. A possible solution to the crisis posed by gravitational collapse and the singularity problem is to regard Einstein's field equations for the problem as analogous to a dynamical system that requires stabilization. Attention is presently given to a dynamically stabilized model which views the universe as pulsating, with a collapse that rebounds at some minimum value of the expansion parameter. 10 references.
Journal Article•
TL;DR: In this article, the conditions générales d'utilisation (http://www.compositio.org/conditions) of the agreement with the Foundation Compositio Mathematica are defined.
Abstract: © Foundation Compositio Mathematica, 1984, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
TL;DR: A recent approach to quantum gravity leading to the concept of quantum stationary geometries is reviewed in this paper, where a stationary states equation is presented for homogeneous relativistic cosmologies of the various Bianchi types, and a Friedmann universe filled with a massless scalar field.
Abstract: A recent approach to quantum gravity leading to the concept of quantum stationary geometries is reviewed. A stationary states equation is presented for: (a) homogeneous relativistic cosmologies of the various Bianchi types; (b) a Friedmann universe filled with a massless scalar field. The equation is solved near the singularity to show that stationary states avoid the singularity. The result is discussed and compared with other approaches.
01 Apr 1984
TL;DR: In this article, the authors show that for a 2-dimensional polyhedral surface immersed in 4-space, the normal curvature can be expressed in terms of singularities of projections into oriented 3-spaces, leading to a new curvature quantity for polyhedral immersion f: M2 -v R4.
Abstract: Using the approach of singularities of projections into lower dimensional spaces it is possible to define nonintrinsic local curvature quantities at each vertex of a polyhedral surface immersed in 4-space which add up to the normal Euler number of the immersion. Related uniqueness results for lattice polyhedra have been established by B. Yusin. For a surface immersed in Euclidean n-space the total (or tangential) curvature can be expressed in terms of singularities of projections into lines, and this interpretation makes it possible to give a unified treatment of curvature for smooth and polyhedral embeddings. In this note we show that for a 2-dimensional surface immersed in 4-space we can carry out a similar construction for the normal curvature in terms of singularities of projections into oriented 3-spaces, recapturing the standard definition for smooth immersions and leading to a new curvature quantity for a polyhedral immersion f: M2 -v R4. We show how to assign to each vertex v of u2 a real number ivf (v) such that the normal Euler class v(f) of the immersion is the sum of the normal curvature at the vertices of M2. In contrast to the case of tangential curvatures where the quantities involved are intrinsic, depending only on the metric of the surface in a neighborhood of a vertex, the normal curvature will depend on the immersion f. This is to be expected since the (tangential) Euler characteristic is a topological invariant but the normal Euler class depends on the immersion. The constructions depend on the author's paper [2] defining the normal Euler class in terms of singularities of projections. Working independently, B. Yusin has constructed curvature quantities for vertex stars of lattice polyhedral surfaces in 4-space, with all edges parallel to the four coordinate axes [4]. His values agree with those described in this note and they establish a uniqueness result, showing that the curvature quantities described here are the only ones which can sum to the normal Euler class of an immersed polyhedral surface. Construction of the curvature quantities. For a smooth immersion f: M2 R4 we may define curvature quantities by using moving frames {ei, e2, e3, e4} with el, e2 tangent to f(M2) and e3, e4 normal to the surface. We define the 1forms Jij by dej = Ej=1 wijej. The tangential curvature of an open set U of M2 Received by the editors December 29, 1979 and, in revised form, August 12, 1983. 1980 Mathematics S*ect Classifiation. Primary 57R20, 57Q35, 53C42; Secondary 52A25.
TL;DR: In this paper, the authors considered the load transfer problem of a long cylindrical elastic bar partially embedded in an elastic half space and found that the effects of the stress singularity are not significant, especially for long bars and when the main concern is only on the force-displacement relationship at the top end of the bar.
Abstract: This paper is a reconsideration of elastostatic load transfer problem of a long cylindrical elastic bar partially embedded in an elastic half space. Each problem is considered as consisting of two interacting systems, an extended half space and a one-dimensional fictitious bar. A compatibility condition is imposed near the interface of the interacting systems. In order to incorporate the real phenomenon of stress singularities at the ends of the bar, without carrying our a complicated derivation of the stress singularity factor, the basic unknown force at both ends of the fictitious bar is set to zero. However, the effects of the stress singularity are found to be not significant, especially for long bars and when the main concern is only on the force-displacement relationship at the top end of the bar.
01 Jan 1984
TL;DR: Recently substantial progress has been achieved in the theory of vacuum quantum effects in external gravitational field Here we may mention Hawking's discovery of the black holes quantum evaporation and the complete analysis of vacuum Quantum effects in isotropic Universes It was often pointed out that quantum theory may yield the resolution of the main problems of modern theoretical physics as mentioned in this paper.
Abstract: Recently substantial progress has been achieved in the theory of vacuum quantum effects in external gravitational field Here we may mention Hawking’s discovery of the black holes quantum evaporation and the complete analysis of vacuum quantum effects in isotropic Universes It was often pointed out that quantum theory may yield the resolution of one of the main problems of modern theoretical physics — the problem of gravitational singularities Here we shall make an attempt to eliminate the cosmological singularity in the framework of the theory of quantum fields in curved space-time