Showing papers on "Singularity published in 1977"
Journal Article•
TL;DR: An integral equation for the t-channel partial wave amplitudes in the investigation of the multi-Regge form of the 2..-->..2+n amplitude was derived in this article.
Abstract: An integral equation is derived for the t-channel partial wave amplitudes in the investigation of the multi-Regge form of the 2..-->..2+n amplitude. For a t-channel state with isospin T=1 the solution of this equation is a Regge pole. The analytic properties of the isospin T=0, 2 partial wave amplitudes are investigated near the threshold for the production of two or three particles. It is shown that in the j-plane there are moving poles and cuts. For the T=0 vacuum channel it was found that the partial wave amplitude has a fixed square-root type branch point to the right of j=1.
838 citations
TL;DR: Triangular and prismatic quadratic isoparametric elements, formed by collapsing one side and placing the mid-side node near the crack tip at the quarter point, are shown to embody the (1/√r) singularity of elastic fracture mechanics and the ( 1/r)-singularity of perfect plasticity as discussed by the authors.
Abstract: Triangular and prismatic quadratic isoparametric elements, formed by collapsing one side and placing the mid-side node near the crack tip at the quarter point, are shown to embody the (1/√r) singularity of elastic fracture mechanics and the (1/r) singularity of perfect plasticity. The procedure of performing the fracture analysis for the case of small scale yielding is discussed, and the finite element results are compared with theoretical results.
The proposed elements have wide application in the fracture analysis of structures where ductile fracture is investigated. They permit a determination of the relationship between crack tip field parameters, loading, and geometry. And for a given fracture criterion can be applied to the prediction of fracture in structures such as pressure vessels under in service conditions.
563 citations
TL;DR: In this article, it was shown that the energy density must diverge at least as fast as 1/t along certain null geodesics near a strong curvature singularity in any conformally flat spacetime.
317 citations
TL;DR: In this paper, the field of a thin massive wire can be characterized in general relativity, in terms of the extrinsic curvature of a tube of constant geodesic radius centered on the wire in the limit when the radius shrinks to zero.
Abstract: This paper is a preliminary study of how the field of a thin massive "wire" can be characterized in general relativity. For a class of "simple" line sources, a linear stress-energy-momentum tensor can be defined in terms of the extrinsic curvature of a tube of constant geodesic radius centered on the wire, in the limit when the radius shrinks to zero. A number of examples are considered, including the ring singularity of the Kerr metric. The Kerr ring is composed of dustlike material circulating about it with the speed of light. The mass distribution in a cross section is proportional tc $cos(\frac{\ensuremath{\psi}}{2})$, where $\ensuremath{\psi}$ is the angle of rotation about the ring in a plane normal to it. The "half-pole" structure is compatible with single-valuedness because of the two-sheeted character of the Kerr manifold.
190 citations
TL;DR: In this paper, a discussion of the Kantowski-Sachs cosmological models is given locally as admitting a four-parameter continuous isometry group which acts on spacelike hypersurfaces, and which possesses a threeparameter subgroup whose orbits are 2-surfaces of constant curvature.
Abstract: A discussion is given of the ’’Kantowski–Sachs’’ cosmological models; these are defined locally as admitting a four‐parameter continuous isometry group which acts on spacelike hypersurfaces, and which possesses a three‐parameter subgroup whose orbits are 2‐surfaces of constant curvature (i.e., the models possess spherical symmetry, combined with a translational symmetry, and can thus be regarded as nonempty analogs of part of the extended Schwarzschild manifold). It is shown that all general relativistic models in which the matter content is a perfect fluid satisfying reasonable energy conditions are geodesically incomplete, both to the past and to the future, and that at each resulting singularity the fluid energy density is infinite. In the case where the fluid obeys a barotropic equation of state (which includes all known exact perfect fluid solutions) the field equations are shown to decouple to form a plane autonomous subsystem. This subsystem is examined using qualitative (Poincare–Bendixson) theory, and phase–plane diagrams are drawn depicting the behavior of the fluid’s energy density and shear anisotropy in the course of the models’ evolution. Further diagrams depict the conformal structure of these universes, and a table summarizes the asymptotic properties of all physically relevant variables.
138 citations
TL;DR: In this paper, a finite element formulation is described for problems with solution functions known to have local rλ variation (s), 0<λ<1, and thus singular gradients. But the conditions of continuity, low order solution capability, and accurate numerical integration of the singularity element are discussed with a view towards establishing the general range of applicability.
Abstract: A finite element formulation is described for problems with solution functions known to have local rλ variation (s), 0<λ<1, and thus singular gradients. Special 3-node triangular elements encircle the singularity and focus to share a common node at the singular point. The shape function of each triangle has the appropriate r λ mode and a smooth angular mode expressed in element natural co-ordinates. As with standard elements, the unknowns are the nodal values of the function. Even if the precise angular form of the asymptotic solution is known, the formulation makes no attempt to embed it, but instead piecewise approximates it. This allows assembly of the element coefficient matrix using standard procedures without nodeless variables and bandwidth complications.
The conditions of continuity, low order solution capability, and accurate numerical integration of the singularity element are discussed with a view towards establishing the general range of applicability of the formulation. Numerical applications to the elastic fracture mechanics problems of composite bondline cracking and crack branching are discussed.
109 citations
TL;DR: In this paper, the authors studied the structure of n dimensional rectifiable currents in R n+l which minimize the integrals of parametric elliptic integrands and gave a geometric construction from which regularity estimates can be obtained for minimizing hypersurfaces.
Abstract: Introduction In this paper we study the structure of n dimensional rectifiable currents in R n+l which minimize the integrals of parametric elliptic integrands. The existence of such minimizing surfaces is well known [7, 5.1.6] as is their regularity almost everywhere [7, 5.3.19]. In Par t I of the present paper we give a new geometric construction from which regularity estimates can be obtained for minimizing hypersurfaces. In this construction we replace the parametric problem for n dimensional surfaces in R ~§ by a nonparametric problem for which the minimizing hypersurfaee is a graph in R n§ with horizontal slices closely approximating in a certain sense the hypersuffaee(s) minimizing the original problem. Analysis of the associated Euler-Lagrange partial differential equation carried out in w 2 of Part I yields an upper bound for the integral of the square of the second fundamental form over the approximating graphs, hence over the regular parts of the original surface. Since a neighbourhood of a singular point must contribute substantially to this integral (see Theorem 1.3 and the remark following it), we can thus conclude by an argument similar to that given by Miranda [13] tha t the Hausdorff ( n 2)-dimensional measure of the interior singular set is locally finite (Theorem 3.1). In Par t I I of this work we show that the singular sets in question must have Hausdorff
85 citations
TL;DR: Gabrielov and Varchenko as discussed by the authors proposed a conjecture on the signature of the quadratic form of a quasihomogeneous singularity, which is a conjecture based on the inertia index of the singularity.
Abstract: ContentsIntroductionSome definitions § 1. Picard-Lefschetz theory § 2. The monodromy group and variation operator of a singularity § 3. The intersection matrices of specific singularities § 4. Intersection matrices of singularities of functions of two variablesAppendix 1. The connection between the modality of a singularity and the inertia index of its quadratic formAppendix 2. V. I. Arnol'd, A conjecture on the signature of the quadratic form of a quasihomogeneous singularityAppendix 3. A. M. Gabrielov, Monodromy groups and bordering of singularitiesAppendix 4. A. N. Varchenko, The characteristic polynomial of the monodromy operator and the Newton diagram of a singularityReferences
83 citations
81 citations
TL;DR: In this paper, the authors considered an infinite system of Newton's equation of motion for one-dimensional particles interacting by a finite-range hard-core potential of singularity like an inverse power of distance between the hard cores.
Abstract: An infinite system of Newton's equation of motion is considered for one-dimensional particles interacting by a finite-range hard-core potential of singularity like an inverse power of distance between the hard cores. Existence of limiting solutions is proved for initial configurations of finite specific energy and the semigroup of motion is constructed if energy fluctuations near infinity increase only as a small power of distance from the origin. In this case uniqueness of solutions is also proved and the solution is a weakly continuous function of initial data. The allowed set of initial configurations carries a wide class of probability measures including Gibbsian fields with different potentials. In the absence of hard cores limiting solutions are constructed for initial configurations with a logarithmic order of energy and density fluctuations.
74 citations
TL;DR: In this article, the infinite system of Newton's equations of motion is considered for two-dimensional classical particles interacting by conservative two-body forces of finite range, and the existence and uniqueness of solutions are proved for initial configurations with a logarithmic order of energy fluctuation at infinity.
Abstract: The infinite system of Newton's equations of motion is considered for two-dimensional classical particles interacting by conservative two-body forces of finite range. Existence and uniqueness of solutions is proved for initial configurations with a logarithmic order of energy fluctuation at infinity. The semigroup of motion is also constructed and its continuity properties are discussed. The repulsive nature of interparticle forces is essentially exploited; the main condition on the interaction potential is that it is either positive or has a singularity at zero interparticle distance, which is as strong as that of an inverse fourth power.
01 Jan 1977
TL;DR: Two examples of rational singularities of schemes over an algebraically closed field of characteristic zero are given: Singularities occurring as the quotient of a regular scheme by a finite group and singularities in the type u2 v2-g(t. tN).
Abstract: Two examples of rational singularities of schemes over an algebraically closed field of characteristic zero are given: Singularities occurring as the quotient of a regular scheme by a finite group and singularities of the type u2 v2-g(t. tN). Unless otherwise explicitly mentioned, we assume that all schemes are irreducible, reduced and of finite type over an algebraically closed field k of characteristic zero and that all points are k-rational. Let U be a scheme, X a regular scheme and g: X -U a proper, surjective, birational morphism. We call g: X -U a resolution of U. DEFINITION 1. Let Y be a scheme and y E Y. Then Y has a rational singularity at y if there exists a neighbourhood U of y in Y, such that for every resolution g: X-> U we have g*9x-t ?u and Rig*?9x = 0 for i =# 0 (henceforth we write Rg Cx? u) In this paper we want to discuss two examples of rational singularities needed in [5, ?5]. REMARKS. (i) Using "flat base change" [1] it is easy to see that the question whether y E Y is a rational singularity depends only on ?9 r ("y " denotes the completion with respect to the maximal ideal). (ii) If the base field has positive characteristic, one needs additional conditions to define rational singularities [3]. (iii) Every regular point of a scheme is a rational singularity [2]. (iv) In Definition 1 it is sufficient to consider one resolution of U. This last statement follows from (iii) and the Leray spectral sequence. Using the same kind of argument one gets LEMMA 1. Let h: Y'-Y be a proper, surjective, birational morphism of schemes. Assume that Y' has only rational singularities; then Rh*c9y _ ? if and only if Y has only rational singularities. We first consider quotient singularities: DEFINITION 2. Let Y be a scheme and y E Y. Then Y has a quotient singularity at y if there exist a regular scheme Y' and a finite group G acting Received by the editors June 17, 1976. AMS (MOS) subject classifications (1970). Primary 14E15, 14B05; Secondary 14E20.
TL;DR: An axially symmetric, stationary exact solution of Einstein's equations for dust is studied in this article, which is asymptotically flat and represents a rotating dust cloud extending tenuously to infinity, containing a singularity at the centre.
Abstract: An axially symmetric, stationary exact solution of Einstein's equations for dust is studied. It is asymptotically flat, and represents a rotating dust cloud extending tenuously to infinity, containing a singularity at the centre. An explanation is given as to why there exists no corresponding solution in Newtonian theory.
TL;DR: In this paper, it was shown that there is an upper bound to the rate of growth of the Ricci curvature near a singularity, and that the curvature can be maintained near the singularity.
Abstract: It is shown that there is an upper bound to the rate of growth of the Ricci curvature near a singularity.
TL;DR: In this paper, it was shown that if the dipolar energy is ignored, superfluid He-A exhibits one additional line singularity besides the one obtained by Toulouse and Kleman.
Abstract: It is shown that if the dipolar energy is ignored, superfluid He-A exhibits one additional line singularity besides the one obtained by Toulouse and Kleman. In addition, point singularities in the spin vector are now possible.
CERN1
TL;DR: In this article, the quantum-mechanical problem of reggeon field theory in zero transverse dimensions is re-examined in order to set up a precise mathematical framework for the case μ = α(0) − 1 > 0.
TL;DR: In this article, the authors generalize the well-known criterion of Kakutani to measures corresponding to arbitrary random sequences and give a description of the set of convergence of a submartingale with bounded increments.
Abstract: The basic result in this paper (Theorem 1) generalizes the well-known criterion of Kakutani to measures corresponding to arbitrary random sequences. The proof is based on Theorem 6, which gives a description of the set of convergence of a submartingale with bounded increments. The question of absolute continuity and singularity of measures corresponding to solutions of stochastic difference equations is studied. The dichotomy for Gaussian measures is obtained as a corollary.Bibliography: 13 titles.
TL;DR: A brief but precise and unified account of the results that have been rigorously established at the time of writing concerning the existence and nature of singularities in classical general relativity is given in this paper.
Abstract: A brief, but precise and unified account is given of the results that have been rigorously established at the time of writing concerning the existence and nature of singularities in classical general relativity.
TL;DR: In this article, a relation between the internal Van Hove singularities in a band and the asymptotic behaviour of the coefficients of the continued fraction which represents the density of states was derived.
Abstract: A relation is derived between the internal Van Hove singularities in a band and the asymptotic behaviour of the coefficients of the continued fraction which represents the density of states. Thus one may show that these coefficients oscillate about their infinite limit values with a frequency related to the position of the singularity within the band, a result first obtained empirically by Gaspard and Cyrot-Lackmann. The present method enables one to derive in addition the amplitude, phase and decay law of these oscillations.
TL;DR: In this paper, the behavior of linearized acoustic theory at the throat of a converging-diverging duct with a quasi-one-dimensional steady flow with a high subsonic throat Mach number is studied.
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 described.
Abstract: © Foundation Compositio Mathematica, 1978, 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: In this article, the Hamiltonian is written down in the strict Coulomb gauge having div A = 0 everywhere, and it is then transformed canonically to a more popular gauge where div A has a delta -function singularity on the interface, and where the scalar potential is zero in absence of the particle.
Abstract: The interaction between a quantized model of a semi-infinite plasma and a charged non-relativistic particle outside it is considered allowing fully for relativistic retardation in the electromagnetic field. The choice of gauge, which in the past has occasioned difficulty and imprecision, is elucidated. The Hamiltonian is written down in the strict Coulomb gauge having div A=0 everywhere. It is then transformed canonically to a more popular gauge where div A has a delta -function singularity on the interface, and where the scalar potential is zero in absence of the particle. The effective interaction ('dynamic image potential') is evaluated to second order in nu / omega pZ but exactly in omega pZ/c; (v is the particle velocity, Z is distance to interface, omega p is the plasma frequency) and the results discussed.
TL;DR: In this paper, the authors give a formula which determines directly dim H(V, Q (L) ) in case 0 < q
Abstract: In 1954 F. Hirzebruch [8] obtained an interesting formula which makes it possible to determine the alternating sum X]5( —z) dim H(V, J2(L)) for any complete intersection V of hypersurfaces in a complex projective space and for any k^Z where L is the analytic line bundle over V induced by hyperplanesection. He further determined dimH^y, J2) by using some vanishing theorem. In the author's knowledge, however, the genera] dim H(V, J2(Z/)) seem not to have been determined yet. In this note we shall give a formula which determines directly dim H (V, Q (L) ) in case 0<^q<^dim V, by using the theory of isolated singularity. (See Theorem 2.3.1, Corollary 2.3.1.) Part I is concerned with the general theory of isolated singularity and is of preparatory nature. The readers who have known the standard of the theory (e.g. Greuel [4]) may bypass it after they become familiar with the terminology and notations. Part II begins with the study of C*-actions over isolated singularities. The main theme there is to compute the characters of the representations of C* over various cohomology groups attached to the singularities, we apply it to the cones associated with algebraic manifolds and prove the required formula finally. The almost all results obtained in this paper have already been announced in [11], [12].
TL;DR: It is argued that the Griffiths singularity which has been proven to exist in dilute Ising magnets is an artifact of the thermodynamic limit procedure for finite macroscopically large samples, the behavior will be regular in most experimental samples while minute sample-dependent corrections, whose observability is uncertain, may appear in an exceedingly small fraction of real samples.
Abstract: It is argued that the Griffiths singularity which has been proven to exist in dilute Ising magnets is an artifact of the thermodynamic limit procedure For finite macroscopically large samples, the behavior will be regular in most experimental samples while minute sample-dependent corrections, whose observability is uncertain, may appear in an exceedingly small fraction of the real samples The Griffiths singularity may at most appear by averaging these effects over an extremely large set of samples
TL;DR: In this article, a second-order, discontinuity-fitting, finite-difference approach was used to determine the flow field resulting from the interaction of a moving planar shock wave with a compression corner.
Abstract: The unsteady, two-dimensional flowfield resulting from the interaction of a moving planar shock wave with a compression corner is determined using a second-order, discontinuity-fitting, finite-difference approach. The time-dependent Euler equations are transformed to normalize the distance between the body and peripheral shock and to include the existing self-similar property of the flow. The resulting set of partial differential equations in conservation-law form is then solved in a time-dependent fashion using MacCormack's scheme. The vortical singularity, which lies on the body surface, and the single reflected shock are both treated as discontinuities in the numerical procedure. The results of the numerical simulation compare quite favorably with existing experimental interferograms and yield better flowfield resolution than previous first-order, shock-capturing, numerical solutions.
TL;DR: An analytical global solution to the Einstein field equations for the case of Bianchi IV symmetry is constructed in this paper, which is axisymmetric, has a singularity in the finite past, is asymptotically flat as t..-->.. infinity, does not admit the presence of a perfect fluid source, and must have zero cosmological constant.
Abstract: An analytical global solution to the Einstein field equations for the case of Bianchi IV symmetry is constructed. It is axisymmetric, has a singularity in the finite past, is asymptotically flat as t ..-->.. infinity, does not admit the presence of a perfect fluid source, and must have zero cosmological constant.
TL;DR: In this article, the asymptoptic structure of the singularity in expansive free interactions is derived and comparison with the numerical computations shows good agreement. But the results are qualitatively the same as those in supersonic interacting flows.
Abstract: Transonic free interactions for compressive and expansive boundary‐layer flows are studied numerically and analytically. The results are qualitatively the same as those in supersonic interacting flows. However, it is found that the upstream decay is either algebraic or exponential, depending on whether the transonic interaction parameter is zero or not. The asymptoptic structure of the singularity in expansive free interactions is derived and comparison with the numerical computations shows good agreement.
TL;DR: In this paper, the authors investigated the nature of the singular stress field developed at the point where a crack intersects the bonded interface of a bimaterial full-plane by the complex variable method.
Abstract: The nature of the singular stress field developed at the point where a crack intersects the bonded interface of a bimaterial full-plane was investigated by the complex variable method. The crack lips were assumed to be under homogeneous stress, displacement or mixed boundary conditions. The order of the singularity was shown to be dependent on both the geometry and the four elastic constants of the two materials of the composite. Graphs showing the variation of the stress singularity with the aforementioned parameters were given. Valuable results indicating how the stress singularity depends on the geometry, the elastic constants and the boundary conditions of the cracked composite full-plane were derived.