Showing papers on "Singularity published in 1984"
TL;DR: In this paper, a graph theory of single spin-flip kinetic Ising models is developed and applied to a class of spin models with strongly cooperative dynamics, and self-consistent approximations for the spin time correlation function are presented.
Abstract: A graph theory of single-spin-flip kinetic Ising models is developed and applied to a class of spin models with strongly cooperative dynamics. Self-consistent approximations for the spin time correlation function are presented. One of the dynamical models exhibits a glass transition with no underlying thermodynamic singularity. The approximation for the time correlation function predicts a critical temperature, below which small fluctuations from equilibrium in the thermodynamic limit cannot relax in a finite amount of time.
558 citations
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
01 Jan 1984
TL;DR: In this article, the existence of a singularity at some finite moment of time t = 0 in the past was shown to be a fundamental difficulty in the analysis of the Friedmann-Robertson-Walker cosmological model, and it is shown that the Friedman solution cannot be analytically continued to the region t < 0 where there is no space and no time.
Abstract: The standard Friedmann-Robertson-Walker cosmological model which is widely accepted now provides a good description of the modern state of our Universe, but it encounters a fundamental difficulty, namely, the existence of a singularity at some finite moment of time t = 0 in the past. The Friedman solution cannot be analytically continued to the region t < 0 where, strictly speaking, there is no space and no time. So, in the scope of classical general relativity, it is in principle impossible to ask what was the state of our Universe before t = 0. Thus a bound to the human cognition is set here. The consideration of more general inhomogeneous and anisotropic cosmological models does not improve the situation because, as is well known, these models are also singular.1
126 citations
TL;DR: In this article, it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions, which appear to form a fractal with positive Hausdorff-Besicovitch dimension.
Abstract: For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
113 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: In this paper, a general method for the formulation of min-max structural design problems directly via a formal procedure is provided, which is substantiated on the basis of generalized Kuhn-Tucker theory.
97 citations
TL;DR: In this article, the authors investigated the existence of singularity in the design space of a two-degree-of-freedom system and provided necessary and sufficient conditions for local and global optimality.
93 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: In this article, a unified treatment of the natural mode representations for induced currents and scattered fields is obtained by use of fundamental concepts regarding causality and superposition, and the implications of these results on natural resonance target identification schemes are discussed.
Abstract: A unified treatment of the natural mode representations for induced currents and scattered fields is obtained by use of fundamental concepts regarding causality and superposition. The transient scattered field response is shown to have the form of a constant coefficient complex exponential sum only in the "late-time," after the last driven reponse is received from the object. Prior to this, the "early.time" response is found to be due to direct physical optics fields as well as a sum of temporally modulated natural modes produced by the progressive illumination of the incident wavefront. Alternate representations and s -plane behaviors are considered. The implications of these results on natural resonance target identification schemes are discussed.
88 citations
TL;DR: In this paper, it was shown that the chiral-invariant vacuum is unstable for a color, fourth-component vector power-like potential rsup..cap alpha.. (0, 0), but breaks down for rsup.cap α..> or =3 (number of spatial dimensions) due to severe infrared singularities.
Abstract: Using the Bogoliubov-Valatin variational method, we show that the chiral-invariant vacuum is unstable for a color, fourth-component vector powerlike potential rsup..cap alpha.. (0 ..0, but breaks down for ..cap alpha..> or =3 (number of spatial dimensions) due to severe infrared singularities. If the confining potential possesses a spin-spin piece, there are critical values of its strength, depending on the power ..cap alpha.., beyond which the stability of the chiral-invariant vacuum is restored. In the case of the harmonic oscillator ..cap alpha.. = 2, the gap equation reduces to a nonlinear second-order differential equation. We find (besides the usual chiral degeneracy) an infinite number of solutions breaking chiral symmetry, higher in energy as the number of their nodes increases. We compute the expectation value of psi-barpsi and the mass gapmore » for the new vacuum, the lowest solution in energy. The infrared singularity of the massless fermion self-energy is removed for the stable broken solution.« less
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.
TL;DR: In this article, a procedure that introduces additional generalized stresses and strains so that the spurious singular modes are eliminated and the consistency of the resulting finite difference equations is not impaired is presented.
TL;DR: In this article, the crack-tip stress singularity and complete field solutions are derived for edge delaminated, angle-ply c omposites subjected to uniform axial extension for illustrative purposes.
Abstract: ion has caused severe concern in the design and analysis of advanced composite materials and structures. Due to its complex nature, very limited knowledge for the problem is currently available. It involves not only geometric and material discontinuities b ut also inherently coupled mode I, I1 and 111 fracture in the layered anisotropic system. Based on complex-variable stress potentials in the a nisotropic elasticity theory and eigenfunction expansion, exact o rders of the cracktip stress singularity and complete field solutions are obtained. Results are given for edge delaminated, angle-ply c omposites subjected to uniform axial extension for illustrative purposes. Effects of geometric, lamination, and crack variables are determined.
TL;DR: In this article, the singularity structure of the equations of motion associated to integrable two-dimensional Hamiltonians with second integrals of order higher than 2 was examined, and it was shown that the integrability is associated to a singularity expansion of the weak painleve type.
Abstract: We examine the singularity structure of the equations of motion associated to integrable two‐dimensional Hamiltonians with second integrals of order higher than 2. We show in these specific examples that the integrability is associated to a singularity expansion of the ‘‘weak‐Painleve’’ type. New cases of integrability are discovered, with still higher‐order integrals which are explicitly computed.
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.
TL;DR: In this article, the authors considered the representation of the electromagnetic field scattered by a perfectly conducting finite-extent scatterer immersed in a lossless medium as a singularity expansion and constructed explicit time domain representations that are counterpart to the Laplace domain representation.
Abstract: The representation of the electromagnetic field scattered by a perfectly conducting finite-extent scatterer immersed in a lossless medium as a singularity expansion is considered. While the analytic properties of the temporal Laplace transform of the surface currents residing on such an object have received a great deal of attention, the properties of the scattered fields have not. It is shown that the representation of the transform of the scattered field must include an exponential entire function except for observation points in the forward-scattered direction. Explicit time domain representations that are counterpart to the Laplace domain representation are constructed and are shown to embody, in the early time, temporal variation besides that of the damped sinusoidal factors intrinsic to the singularity expansion. An important practical consequence of this more complicated time variation arises in connection with the application of the singularity expansion for target classification purposes and is commented upon herein.
TL;DR: In this article, the authors studied the Potts model on a diamond hierarchical lattice with random interactions and calculated the position and exponents of the random fixed point which appears when the specific heat exponent ap of the pure system becomes positive.
Abstract: We study the Potts model on a diamond hierarchical lattice with random interactions. Using weak disorder expansions, we calculate analytically the position and the exponents of the random fixed point which appears when the specific heat exponent ap of the pure system becomes positive. At ap = 0, we find how the logarithmic singularity is modified by the disorder. Lastly we suggest that this model should present Griffiths-like singularities.
TL;DR: In this article, an exact set of solutions of the magnetohydrodynamic equations describing flows with homogeneous deformations in the vicinity of magnetic zero lines and having a singularity is presented.
TL;DR: In this article, the geometry of a bifurcation diagram in the base of a versal deformation of a singularity is studied for single singularities on a manifold with boundary.
Abstract: The geometry of a bifurcation diagram in the base of a versal deformation of a singularity is studied for single singularities on a manifold with boundary. In particular, vector fields and groups of diffeomorphisms are studied which are defined in a neighborhood of a bifurcation diagram as are stratification of a bifurcation diagram and decomposition of singularities.
TL;DR: In this paper, the authors discuss the initial value problem in one-dimensional linear visco-elasticity with a step-jump in the initial data and show that if the memory kernel is sufficiently smooth on (infinity), the solution exhibits discontinuities propagating along characteristics and a (higher order) stationary discontinuity at the position of the original step jump.
Abstract: : This document discusses the initial value problem in one-dimensional linear visco-elasticity with a step-jump in the initial data. If the memory kernel is sufficiently smooth on (infinity), the solution exhibits discontinuities propagating along characteristics and a (higher order) stationary discontinuity at the position of the original step-jump. For a singular memory kernel, the propagating waves are smoothed in a manner depending on the nature of the singularity in the kernel, but the stationary discontinuity remains. Also discussed are the effects of these phenomena on the regularity of solutions with arbitrary initial data.
TL;DR: In this article, the singularity of the dyadic Green's function is taken into account in the calculation of the effective permittivity functions, and correlation functions for the random medium with different scatterer constituents and size distributions are derived.
Abstract: The strong fluctuation theory is applied to the study of electromagnetic wave scattering from a layer of random discrete scatterers. The singularity of the dyadic Green’s function is taken into account in the calculation of the effective permittivity functions. The correlation functions for the random medium with different scatterer constituents and size distributions are derived. Applying the dyadic Green’s function for a two‐layer medium and using the bilocal and distorted Born approximations, the first and the second moments of the fields are then calculated. Both the backscattering and bistatic scattering coefficients are obtained, and the former is shown to match favorably with experimental data obtained from snow fields.
01 Jan 1984
TL;DR: An integral equation technique for the Neumann problem of finding a function Φ satisfying ΔΦ = 0 with prescribed values of ∂Φ∂n on the boundary is described in this article.
Abstract: An integral equation technique for the Neumann problem of finding a function Φ satisfying ΔΦ = 0 with prescribed values of ∂Φ∂n on the boundary is described. Fourier representation of the potential Φ on the boundary with respect to two angle-like variables transforms the integral equation to an infinite set of linear equations for the Fourier coefficients of Φ. The singularity of the Green's function is treated by a regularization method: a function with the same singularity is subtracted and its analytically calculated Fourier-transform is added to the Fourier transformed integral equation. A computer code named NESTOR is developed. Applications include studies of toroidal magnetic vacuum fields and calculation of the vacuum field contribution for the 3D free-boundary equilibrium problem.
TL;DR: In this article, the structure of discrete-time linear systems with stationary inputs in the geometric framework of splitting subspaces is studied, and the relation between models with and without noise in the observation channel is investigated.
Abstract: From a conceptual point of view, structural properties of linear stochastic systems are best understood in a geometric formulation which factors out the effects of the choice of coordinates. In this paper we study the structure of discrete-time linear systems with stationary inputs in the geometric framework of splitting subspaces set up in the work by Lindquist and Picci. In addition to modifying some of the realization results of this work to the discrete-time setting, we consider some problems which are unique to the discrete-time setting. These include the relations between models with and without noise in the observation channel, and certain degeneracies which do not occur in the continuous-time case. One type of degeneracy is related to the singularity of the state transition matrix, another to the rank of the observation noise and invariant directions of the matrix Riccati equation of Kalman filtering. We determine to what extent these degeneracies are properties of the output process. The geometric framework also accommodates infinite-dimensional state spaces, and therefore the analysis is not limited to finite-dimensional systems.
25 Jun 1984
TL;DR: In this paper, the authors prove local and global existence theorems for a model equation in nonlinear viscoelasticity, where the memory function has a singularity.
TL;DR: In this article, it was shown that strong causality is violated arbitrarily close to ℐ+ from the singularity, which does not satisfy the strong curvature condition regardless of whether causality was violated or not.
Abstract: In this paper we prove that if there is a naked singularity, then there will be some null geodesic, reaching ℐ+ from the singularity, which does not satisfy the strong curvature condition regardless of whether causality is violated or not. Assuming that a naked singularity is a strong curvature singularity only sufficiently far to the future, we prove that strong causality is violated arbitrarily close to ℐ+.
TL;DR: In this paper, the authors considered the approximate evaluation of a fixed Lebesgue integrable function by quadrature rules of the form σ σ √ √ n √ k(x)f(x)-dx.
Abstract: This paper considers the approximate evaluation of $\int _a^b k(x)f(x)dx$, where k is a fixed Lebesgue integrable function, by quadrature rules of the form $\Sigma _{i = 0}^{m_n } w_{in} f(x_{in} )$. Normally rules of this kind are used only for smooth functions f, and any singularities are incorporated into k. Here, however, we allow f to have a singularity, either at an endpoint or in the interior. General convergence properties are established for a wide class of product integration rules. More detailed results are established for the class of rules which are exact if f belongs to a specified family of piecewise polynomials.
TL;DR: In contrast to the Stokes paradox for flow past an isolated cylinder, if either type of singularity, with suitably chosen strength and location, is present, there can exist a flow which is uniform at infinity as discussed by the authors.
Abstract: A study is made of Stokes flows in which a line rotlet or stokeslet is in the presence of a circular cylinder in a viscous fluid. In contrast to the Stokes Paradox for flow past an isolated cylinder, it is shown that if either type of singularity, with suitably chosen strength and location, is present, there can exist a flow which is uniform at infinity. A similar phenomenon can occur when two equal cylinders rotate with equal and opposite angular velocities, and the flow pattern is then such that there is a closed streamline enclosing both cylinders.