Showing papers on "Singularity published in 1993"

Universal pinching of 3D axisymmetric free-surface flow

Abstract: We consider the viscous motion of an axisymmetric column of fluid with a free surface. The Navier-Stokes equation forms a singularity as the height of the fluid neck goes to zero. Close to pinchoff, the solutions have a scaling form characterized by a set of universal exponents. The shape of the neck and its velocity field is described by scaling functions, which we predict without adjustable parameters.

On classification of N =2 supersymmetric theories

Abstract: We find a relation between the spectrum of solitons of massiveN=2 quantum field theories ind=2 and the scaling dimensions of chiral fields at the conformal point. The condition that the scaling dimensions be real imposes restrictions on the soliton numbers and leads to a classification program for symmetricN=2 conformal theories and their massive deformations in terms of a suitable generalization of Dynkin diagrams (which coincides with the A-D-E Dynkin diagrams for minimal models). The Landau-Ginzburg theories are a proper subset of this classification. In the particular case of LG theories we relate the soliton numbers with intersection of vanishing cycles of the corresponding singularity; the relation between soliton numbers and the scaling dimensions in this particular case is a well known application of Picard-Lefschetz theory.

A simple approach to solve boundary-value problems in gradient elasticity

Abstract: We outline a procedure for obtaining solutions of certain boundary value problems of a recently proposed theory of gradient elasticity in terms of solutions of classical elasticity. The method is applied to illustrate, among other things, how the gradient theory can remove the strain singularity from some typical examples of the classical theory.

Evidence for a Singularity of the Three Dimensional, Incompressible Euler Equations

Abstract: Three-dimensional, incompressible Euler calculations of the interaction of perturbed anti-parallel vortex tubes using a variety of smooth initial profiles in bounded domains with bounded initial vorticity is discussed. It will be shown that trends towards either exponential, non-singular growth of the peak vorticity or power law, singular behavior can be strongly dependent on details of the initial conditions. A numerical method that uses symmetries and additional resolution in the direction and location of maximum coin-pression is used to simulate periodic boundary conditions in all directions. For the initial condition that yields singular type behavior the growth of the peak vorticity roughly obeys (t c − t)−1 and the growth of the strain along the vorticity at this point obeys (t c − t)−γwhere γ ≈ 1.

Singularities of the X-ray transform and limited data tomography in R 2 and R 3

Abstract: Given a function f, the author specifies the singularities of f that are visible in a stable way from limited X-ray tomographic data. This determines which singularities of f can be stably recovere...

Naked singularities in spherically symmetric inhomogeneous Tolman-Bondi dust cloud collapse

Abstract: We investigate here the occurrence and nature of a naked singularity for the inhomogeneous gravitational collapse of Tolman-Bondi dust clouds. It is shown that the naked singularities form at the center of the collapsing cloud in a wide class of collapse models, which includes the earlier cases considered by Eardley and Smarr and Christodoulou. This class also contains self-similar as well as non-self-similar models. The structure and strength of this singularity are examined, and the question is investigated as to when a nonzero measure set of nonspacelike trajectories could be emitted from the singularity, as opposed to isolated trajectories coming out. It is seen that the weak energy condition and positivity of energy density ensures that the families of nonspacelike trajectories come out of the singularity. The curvature strength of the naked singularity is examined, which provides an important test for its physical significance. This is done in terms of the strong curvature condition, which ensures that all the volume forms must be crushed to zero size in the limit of approach to the singularity, and, also, the divergence of the Kretschmann scalar $\mathcal{K}={R}^{\mathrm{abcd}}{R}_{\mathrm{abcd}}$ is pointed out. We show that the class considered here contains subclasses of solutions which admit strong curvature naked singularities in either of the senses stated above. The conditions are discussed for the naked singularity to be globally naked. An implication for the fundamental issue of the final fate of gravitational collapse is that naked singularities need not be considered as artifacts of geometric symmetries of space-time such as self-similarity, but arise in a wide range of gravitational collapse scenarios once the inhomogeneities in the matter distribution are taken into account. It is argued that a physical formulation for the cosmic censorship may be evolved which avoids the features above. Possibilities in this direction are suggested while indicating that the analysis presented here should be useful for any possible rigorous formulation of the cosmic censorship hypothesis.

Cosmological theory without singularities

Abstract: A theory of gravitation is constructed in which all homogeneous and isotropic solutions are nonsingular, and in which all curvature invariants are bounded All solutions for which curvature invariants approach their limiting values approach de Sitter space The action for this theory is obtained by a higher-derivative modification of Einstein's theory We expect that our model can easily be generalized to solve the singularity problem also for anisotropic cosmologies

Complete surfaces of constant mean curvature-1 in the hyperbolic 3-space

Abstract: In the study of minimal surfaces in the euclidean 3-space, the Weierstrass representation plays an important role. Bryant [Br] showed that an analogue of the Weierstrass-representation formula holds for surfaces of constant mean curvature-i in the hyperbolic 3-space X3. In this article we abbreviate the term "constant mean curvature-i" as CMC-1. Like minimal surfaces in the euclidean space, the hyperbolic Gauss map of CMC-1 surfaces is defined as a holomorphic map to C U {oo}. However, in contrast to the euclidean case, the hyperbolic Gauss map of a CMC-1 surface may not be extended across the ends even if the total Gaussian curvature is finite. We call a complete CMC-1 surface, whose Gauss map can be extended across all of its ends, a CMC-1 surface of regular ends. In this article we produce an explicit tool to construct CMC-1 surfaces of regular ends. In Section 2 we show that such surfaces are constructed by solving some ordinary differential equations with regular singularity. In our CMC-1 category, Ossermann's inequality is not expected and the Cohn-Vossen inequality is the best possible one. We show in Section 4 that the equality of the Cohn-Vossen inequality never holds for complete CMC-1 surfaces in XH3. In Section 5 we give a necessary and sufficient condition that a regular end of a CMC-1 surface be embedded. In Section 6 we classify complete CMC-1 surfaces of genus 0 with two regular ends. Our classification contains new examples. Furthermore, in Section 7, we construct several new CMC-1 surfaces with regular embedded ends. Each of these examples has a nontrivial deformation, which is mentioned in Section 3. It should be remarked that our construction does not work for surfaces with irregular ends. But there is another construction: By perturbing minimal surfaces in the euclidean 3-space, the authors constructed CMC-1 surfaces, all of whose ends are irregular (see [UY1]).

Experimentally observed extended saddle point singularity in the energy spectrum of YBa2Cu3O6.9 and YBa2Cu4O8 and some of the consequences

Abstract: Angle-resolved photoemission experiments show that the high temperature superconductors YBa 2 Cu 3 O 6.9 and YBa 2 Cu 4 O 8 have a spectral density which exhibits an “extended” saddle point singularity along the Γ-Y symmetry direction, centered on the Y point. This, in turn, leads to a van Hove singularity in the density of states with a divergence stronger than the well-known logarithmic one. We examine the consequences of this singularity in several limiting cases on the superconducting transition temperature, the ratio 2Δ(0)/ T c , and the isotope effect. We conclude that this singularity alone, although it can possibly lead to sufficiently high transition temperatures and a vanishingly small isotope effect, does not explain the ratio of 2Δ(0)/ T c ∼6 observed experimentally.

Limiting configurations allowed by the energy conditions

Abstract: The energy conditions of general relativity are satisfied by all experimentally detected fields. We discuss their interpretation and application to charged spheres. It is found that they prevent the existence of naked singularities, and demand that the effective gravitational mass be everywhere non-negative. We focus on the emergence of limiting configurations-sources of the Reissner-Nordstrom field that have vanishing effective mass everywhere within the sphere. These configurations have a number of interesting features. Among them we find that, near the center, the limiting form of the equation of state isρ+3p=0. Notably this is the only equation of state consistent with the existence of zero-point electromagnetic field, and it has been considered in different contexts, in discussions of cosmic strings and in derivations of (3+1) properties of matter from (4+1) geometry. The consistency of these configurations with the Einstein-Maxwell equations is shown by means of explicit examples. These configurations can be interpreted as due to selfinteracting gravitational effects of the zero-point electromagnetic field.

Kinetics of a one‐dimensional granular medium in the quasielastic limit

Abstract: The dynamics of a one‐dimensional granular medium has a finite time singularity if the number of particles in the medium is greater than a certain critical value. The singularity (‘‘inelastic collapse’’) occurs when a group of particles collides infinitely often in a finite time so that the separations and relative velocities vanish. To avoid the finite time singularity, a double limit in which the coefficient of restitution r approaches 1 and the number of particles N becomes large, but is always below the critical number needed to trigger collapse, is considered. Specifically, r→1 with N∼(1−r)−1. This procedure is called the ‘‘quasielastic’’ limit. Using a combination of direct simulation and kinetic theory, it is shown that a bimodal velocity distribution develops from random initial conditions. The bimodal distribution is the basis for a ‘‘two‐stream’’ continuum model in which each stream represents one of the velocity modes. This two‐stream model qualitatively explains some of the unusual phenomena seen in the simulations, such as the growth of large‐scale instabilities in a medium that is excited with statistically homogeneous initial conditions. These instabilities can be either direct or oscillatory, depending on the domain size, and their finite‐amplitude development results in the formation of clusters of particles.

Free energy of particle insertion

Abstract: In free energy simulations involving the creation of a new particle in a condensed medium, there is a well-known ‘origin singularity’ when the particle is first introduced, unless the coupling of the particle with its surroundings is increased very slowly. This singularity is analysed theoretically for a liquid that interacts with the new particle through a soft-sphere or a Lennard-Jones potential, using a virial expansion of the free energy derivative. For a soft-sphere potential u(r) = λα Ar-n (where λ is a coupling constant), we show rigorously that the derivative of the free energy with respect to λ is infinite at the origin if and only if α < n/3. This confirms a previous, approximate, prediction based on scaled particle theory. For a Lennard-Jones potential u(r) = λα Ar -12 + λβ Br -6, such that α ˇ- 2β, the derivative of the free energy is again infinite if and only if α < n/3. If the volume of the solute is proportional to the coupling constant, there is no singularity.

Singularity, complexity, and quasi-integrability of rational mappings

Abstract: We investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models, by exploiting their nature of birational transformations of projective spaces. We give an algorithmic analysis of the structure of invariants of such mappings. We discuss some characteristic conditions for their (quasi)-integrability, and in particular its links with their singularities (in the 2-plane). Finally, we describe some of their propertiesqua dynamical systems, making contact with Arnol'd's notion of complexity, and exemplify remarkable behaviours.

Singular Stress Field Near the Corner of Jointed Dissimilar Materials

Abstract: In this paper, the characteristics of the stress field near a corner of jointed dissimilar materials are studied as a plane problem. It is found that the order of singularity is dependent not only on the elastic constants of materials and the local geometry of corner, but also on the deformation mode. The dependence of the order of singularity was established for the case of mode I and the case of mode II. An explicit closed-form expression is given for the singular stress field at the close vicinity of the corner, in which the stress field is expressed as a sum of the symmetric state with a stress singularity of 1/r 1-λ1 and the skew symmetric state with a stress singularity of 1/r 1-λ2 . When both λ1 and λ2 are real the singular stress field around the point singularity is defined in terms of two constants K 1 , λ1 , K 11 , λ2 , as in the case of crack problems.

The analysis of space-time singularities.

Abstract: The theorems of Hawking and Penrose show that space-times are likely to contain incomplete geodesics. Such geodesics are said to end at a singularity if it is impossible to continue the space-time and geodesic without violating the usual topological and smoothness conditions on the space-time. In this book the different possible singularities are defined, and the mathematical methods needed to extend the space-time are described in detail. The results obtained (many appearing here for the first time) show that singularities are associated with a lack of smoothness in the Riemann tensor. While the Friedmann singularity is analysed as an example, the emphasis is on general theorems and techniques rather than on the classification of particular exact solutions.

Existence of infinitely many smooth, static, global solutions of the Einstein / Yang-Mills equations

Abstract: We prove the existence of infinitely-many globally defined singularity- free solutions, to the EYM equations with SU(2) gauge group. The solutions are indexed by a coupling constant, have distinct winding numbers, and their corresponding Einstein metrics decay at infinity to the flat Minkowski metric. Each solution has a finite (ADM) mass; these masses are derived from the solutions, and are not arbitrary constants.

Critical behavior of the aperiodic quantum Ising chain in a transverse magnetic field

Abstract: We consider the quantum spin-1/2 Ising chain in a uniform transverse magnetic field, with an aperiodic sequence of ferromagnetic exchange couplings. This system is a limiting anisotropic case of the classical two-dimensional Ising model with an arbitrary layered modulation. Its formal solution via a Jordan-Wigner transformation enables us to obtain a detailed description of the influence of the aperiodic modulation on the singularity of the ground-state energy at the critical point. The key concept is that of thefluctuation of the sums of any number of consecutive couplings at the critical point. When the fluctuation isbounded, the model belongs to the “Onsager universality class” of the uniform chain. The amplitude of the logarithmic divergence in the specific heat is proportional to the velocity of the fermionic excitations, for which we give explicit expressions in most cases of interest, including the periodic and quasiperiodic cases, the Thue-Morse chain, and the random dimer model. When the couplings exhibit anunbounded fluctuation, the critical singularity is shown to be generically similar to that of the disordered chain: the ground-state energy has finite derivatives of all orders at the critical point, and an exponentially small singular part, for which we give a quantitative estimate. In themarginal case of a logarithmically divergent fluctuation, e.g., for the period-doubling sequence or the circle sequence, there is a negative specific heat exponentα, which varies continuously with the strength of the aperiodic modulation.

Singularity formation during Rayleigh-Taylor instability

Abstract: During the motion of a fluid interface undergoing Rayleigh-Taylor instability, vorticity is generated on the interface baronclinically. This vorticity is then subject to Kelvin-Helmholtz instability. For the related problem of evolution of a nearly flat vortex sheet without density stratification (and with viscosity and surface tension neglected), Kelvin-Helmholtz instability has been shown to lead to development of curvature singularities in the sheet. In this paper, a simple approximate theory is developed for Rayleigh-Taylor instability as a generalization of Moore's approximation for vortex sheets. For the approximate theory, a family of exact solutions is found for which singularities develop on the fluid interface. The resulting predictions for the time and type of the singularity are directly verified by numerical computation of the full equations. These computations are performed using a point vortex method, and singularities for the numerical solution are detected using a form fit for the Fourier components at high wavenumber. Excellent agreement between the theoretical predictions and the numerical results is demonstrated for small to medium values of the Atwood number A, i.e. for A between 0 and approximately 0.9. For A near 1, however, the singularities actually slow down when close to the real axis. In particular, for A = 1, the numerical evidence suggests that the singularities do not reach the real axis in finite time.

Theoretical and numerical treatment of surface integrals involving the free-space Green's function

Abstract: This paper deals with the problem of the calculation of surface integrals for electromagnetic scattering in the case of the widely popular double-triangular basis functions first introduced by Rao, Wilton, and Glisson (1982). An entire set of formulas is obtained which overrides the difficulties inherent to the singularity of the integrands, and results showing the stability, accuracy, and efficiency of the methods developed are reported in an application of the method of moments to the case of perfectly conducting surfaces and computation of near field in domains including the surface itself. Furthermore, the authors provide insight as regards the capability of triangular basis functions to model near field patterns. >

Lower bounds on the number of scattering poles

Abstract: (1993). Lower bounds on the number of scattering poles. Communications in Partial Differential Equations: Vol. 18, No. 5-6, pp. 847-857.