Showing papers on "Singularity published in 1987"

Towards a unified asymptotic theory for autoregression

Abstract: SUMMARY This paper develops an asymptotic theory for a first-order autoregression with a root near unity. Deviations from the unit root theory are measured through a noncentrality parameter. When this parameter is negative we have a local alternative that is stationary; when it is positive the local alternative is explosive; and when it is zero we have the standard unit root theory. Our asymptotic theory accommodates these possibilities and helps to unify earlier theory in which the unit root case appears as a singularity of the asymptotics. The general theory is expressed in terms of functionals of a simple diffusion process. The theory has applications to continuous time estimation and to the analysis of the asymptotic power of tests for a unit root under a sequence of local alternatives.

A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals

Abstract: Almost all general purpose boundary element computer packages include a curved geometry modelling capability. Thus, numerical quadrature schemes play an important role in the efficiency of programming the technique. The present work discusses this problem in detail and introduces efficient means of computing singular or nearly singular integrals currently found in two-dimensional, axisymmetric and three-dimensional applications. Emphasis is given to a new third degree polynomial transformation which was found greatly to improve the accuracy of Gaussian quadrature scheme's within the near-singularity range. The procedure can easily be implemented into existing BE codes and presents the important feature of being self-adaptive, i.e. it produces a variable lumping of the Gauss stations toward the singularity, depending on the minimum distance from the source point to the element. The self-adaptiveness of the scheme also makes it inactive when not useful (large source distances) which makes it very safe for general usage.

On the solution of integral equations with strongly singular kernels

Abstract: Some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m ,m greater than or equal 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t-x) sup -m , terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.

Fracture of nonhomogeneous materials

Abstract: The nonhomogeneous materials considered in this work are of a class whose elastic moduli are specified by continuous and generally differentiable functions of the spatial coordinates. The elastic stress and displacement fields near a crack tip in a two-dimensional nonhomogeneous cracked body are derived utilizing an extension of the Wiliams eigenfunction expansion technique. The nature of the stress and strain singularity is ascertained to be precisely of the same form as the well-known inverse square root stress singularity near a crack tip in a homogeneous material, independent of the functional form of the elastic moduli variation. A new quasipath-independent integral has been generated which proves useful for computing the energy release rate and mixed-mode stress intensity factors in nonhomogeneous cracked bodies. The integral is used in conjunction with finite element analysis for purposes of computing stress intensity factors. Numerical results are compared with certain exact solutions which are available for nonhomogeneous cracked bodies. Cracked composite bodies have traditionally been modeled and analyzed as possessing discontinuous elastic moduli, but are treated here as having rapid, but smooth variations of the material properties.

The dimension spectrum of some dynamical systems

Abstract: We analyze the dimension spectrum previously introduced and measured experimentally by Jensen, Kadanoff, and Libchaber Using large-deviation theory, we prove, for some invariant measures of expanding Markov maps, that the Hausdorff dimensionf(α) of the set on which the measure has a singularity α is a well-defined, concave, and regular function In particular, we show that this is the case for the accumulation of period doubling and critical mappings of the circle with golden rotation number We also show in these particular cases that the functionf is universal

The wave structure of monochromatic electromagnetic radiation

Abstract: The paper considers a general field of electromagnetic waves of a single frequency and identifies the salient structurally stable features of the three-dimensional pattern of polarization. The approach is geometrical rather than analytical, and it differs from previous treatments of this kind by being applicable even when the constituent plane waves are travelling in all directions. Lines and surfaces exist where the electric or magnetic vibration ellipse is singular. The field is divided into right-handed and left-handed regions by \`T surfaces', the electric and magnetic T surfaces not being coincident. Lying in the T surfaces are \`L$^T$ lines' where the vibration is linear, and cutting through the T surfaces are `C$^T$ lines' where the vibration is circular. Both kinds of lines are surrounded by characteristic patterns of vibration ellipses, which provide a singularity index, $\pm$ 1 for L$^T$ and $\pm \frac{1}{2}$ for C$^T$. The analysis is applicable in a cavity, but a loss-free resonating cavity represents a highly degenerate case.

Numerical simulations of nonlinear axisymmetric flows with a free surface

Abstract: A numerical method is developed for nonlinear three-dimensional but axisymmetric free-surface problems using a mixed Eulerian-Lagrangian scheme under the assumption of potential flow. Taking advantage of axisymmetry, Rankine ring sources are used in a Green's theorem boundary-integral formulation to solve the field equation; and the free surface is then updated in time following Lagrangian points. A special treatment of the free surface and body intersection points is generalized to this case which avoids the difficulties associated with the singularity there. To allow for long-time simulations, the nonlinear computational domain is matched to a transient linear wavefield outside. When the matching boundary is placed at a suitable distance (depending on wave amplitude), numerical simulations can, in principle, be continued indefinitely in time. Based on a simple stability argument, a regriding algorithm similar to that of Fink & Soh (1974) for vortex sheets is generalized to free-surface flows, which removes the instabilities experienced by earlier investigators and eliminates the need for artificial smoothing. The resulting scheme is very robust and stable.For illustration, three computational examples are presented: (i) the growth and collapse of a vapour cavity near the free surface; (ii) the heaving of a floating vertical cylinder starting from rest; and (iii) the heaving of an inverted vertical cone. For the cavity problem, there is excellent agreement with available experiments. For the wave-body interaction calculations, we are able to obtain and analyse steady-state (limit-cycle) results for the force and flow field in the vicinity of the body.

An elliptic equation with singular nonlinearity

Abstract: The purpose of this paper is to study the problem (P) −Δu+u−α=f in Ω, u=0 on ∂Ω, u−α∈L1(Ω), u>0 in Ω, where Ω is a bounded smooth open set of RN, f≥0, f∈L1(Ω) and 0<α<1. When α≤0 this problem corresponds to a monotone semilinear equation for which the existence and uniqueness of solutions are well known. On the other hand, semilinear equations with nonmonotone perturbations have also been considered in the literature by many different methods. The originality of our study comes from the singularity of the nonlinearity in (P).

Local mesh refinement multilevel techniques

Abstract: The purpose of this study is to provide criteria for optimizing meshsizes near singularities and develop fast and flexible multigrid methods for creating the nonuniform grids, their difference equations and their solutions. For simplicity, the Poisson problem is studied, with singularities introduced either in the forcing terms (algebraic singularities or sources) or in the shape of the boundaries (re-entrant corners). Local refinements are created by multigrid structures in which some extra finer levels cover increasingly narrower neighborhoods of the singularity, as proposed in [6]. The main innovations here are: (1) Extra local relaxation sweeps near structural singularities (such as re-entrant corners) are employed to restore the asymptotic convergence rates to their values in regular (e.g. infinite) domains. (2) An exchange-rate algorithm ($\lambda $-FMG) is introduced to maintain linear dependence of solution time on number of gridpoints. With these two algorithmic modifications, and with the optima...

Naked singularities in self-similar spherical gravitational collapse

Abstract: We present general-relativistic solutions of self-similar spherical collapse of an adiabatic perfect fluid. We show that if the equation of state is soft enough (\ensuremath{\Gamma}-1\ensuremath{\ll}1), a naked singularity forms. The singularity resembles the shell-focusing naked singularities that arise in dust collapse. This solution increases significantly the range of matter fields that should be ruled out in order that the cosmic-censorship hypothesis will hold.

Casimir effect around a cone

Abstract: The vacuum average of the energy density of a free, massless scalar field around a conical flux-tube singularity in d+1 space-time dimensions is calculated. A complex contour method is employed and the results involve higher-order Bernoulli polynomials. The formulas might have applications to cosmic strings.

Some Characterizations of Strange Sets

Abstract: A thermodynamic formalism is exhibited that is the canonical version of Halseyet al.'s microcanonical formulation. This formalism is applied to a four-scale Cantor set and it is shown that the singularity spectrum fails to uniquely encode the underlying dynamics.

Weakly dispersive spectral theory of transients, part I: Formulation and interpretation

Abstract: Dispersive effects in transient propagation and scattering are usually negligible over the high frequency portion of the signal spectrum, and for certain configurations, they may be neglected altogether. The source-excited field may then be expressed as a continuous spatial spectrum of nondispersive time-harmonic local plane waves, which can be inverted in closed form into the time domain to yield a fundamental field representation in terms of a spatial spectrum of transient local plane waves. By exploiting its analytic properties, one may evaluate the basic spectral integral in terms of its singularities-real and complex, time dependent and time independent-in the complex spectral plane. These singularities describe distinct features of the propagation and scattering process appropriate to a given environment. The theory is developed in detail for the generic local plane wave spectra representative of a broad class of two-dimensional propagation and diffraction problems, with emphasis on physical interpretation of the various spectral contributions. Moreover, the theory is compared with a similar approach that restricts all spectra to be real, thereby forcing certain wave processes into a spectral mold less natural than that admitting complex spectra. Finally, application of the theory is illustrated by specific examples. The presentation is divided into three parts. Part I, in this paper, deals with the formulation of the theory and the classification of the singularities. Parts II and III, to appear subsequently, contain the evaluation and interpretation of the spectral integral and the applications, respectively.

Images, Numerical Analysis of Singularities and Shock Filters

Abstract: This work is concerned primarily with establishing a natural mathematical framework for the Numerical Analysis of Singularities, a term which we coined for this new evolving branch of numerical analysis. The problem of analyzing singular behavior of nonsmooth functions is implicitly or explicitly ingrained in any successful attempt to extract information from images. The abundance of papers on the so called Edge Detection testifies to this statement. We attempt to make a fresh start by reformulating this old problem in the rigorous context of the Theory of Generalized Functions of several variables with stress put on the computational aspects of essential singularities. We state and prove a variant of the Divergence Theorem for discontinuous functions which we call Fundamental Theorem of Edge Detection, for it is the backbone of the advocated here numerical analysis based on the estimates of contribution5 furnished by the essential singularities of functions. We further extend this analysis to arbitrary order singularities by utilization of the Miranda''s calculus of tangential derivatives. With this machinery we are able to explore computationally the internal geometry of singularities including singular, i.e., nonsmooth, singularity boundaries. This theory give5 rise to singularity detection scheme called "rotating thin masks" which is applicable to arbitrary order n-dimensional essential singularities. In the particular implementation we combined first-order detector with derived here various curvature detectors. Preliminary experimental results are presented. We also derive a new class of nonlinear singularity detection schemes based on tensor products of distributions. Finally, a novel computational approach to the problem of image enhancement is presented. We call this construction the Shock Filters, since it is founded on the nonlinear PDE''s whose solutions exhibit formation of discontinuous profiles, corresponding to shock waves in gas dynamics. An algorithm for experimental Shock Filter, based on the upwind finite difference scheme is presented and tested on the one and two dimensional data.

On reciprocal properties of the Caudrey-Dodd-Gibbon and Kaup-Kupershmidt hierarchies

Abstract: New hierarchies of nonlinear evolution equations are introduced which are linked by reciprocal transformations to the Caudrey-Dodd-Gibbon and Kaup-Kupershmidt sequences. Invariance under a Mobius transformation of the singularity manifold equations for these sequences leads to a novel generic invariance property of the new systems. The latter have as base members Kawamoto-type equations. Explicit auto-Backlund transformations for the Caudry-Dodd-Gibbon and Kaup-Kuperschmidt hierarchies are generated via a reciprocal property.

Traction BIE solutions for flat cracks

Abstract: The paper deals with the numerical solution techniques for the traction boundary integral equation (BIE), which describes the opening (and sliding) displacements of the surface of the traction loaded crack or arbitrary planform embedded in an elastic infinite body (buried crack problem). The traction BIE is a singular integral equation of the first kind for the displacement gradients. Its solution poses a number of numerical problems, such as the presence of derivatives of the unknown function in the integral equation, the modeling of the crack front displacement gradient singularity, and the regularization of the equation's singular kernels. All of the above problems have been addressed and solved. Details of the algorithm are provided. Numerical results of a number of crack configurations are presented, demonstrating high accuracy of the method.

Singularity prevention and broken Lorentz symmetry

Abstract: It is shown that, if the local gauge symmetry of general relativity is broken, the singularity theorems may be evaded, and in particular the cosmological singularity of the standard model may be prevented, even if the gravitational sources satisfy the strong energy condition.

The surface singularity method applied to partially cavitating hydrofoils

Abstract: The surface singularity or boundary integral method is formulated numerically for the problem of the fully nonlinear potential flow past a partially cavitating hydrofoil. An iterative scheme is employed to locate the cavity surface. Upon convergence the exact boundary conditions are satisfied on all portions of the foil-cavity boundary. The effects of hydrofoil section thickness and camber on cavity volume are investigated. The results are compared with those generated by a numerical linear theory, which includes the effect of section thickness, and with Tulin and Hsu's "short cavity" theory. Both section thickness and camber are shown to have significant effects on cavity volume.

Linear theory of rotating fluids using spherical harmonics part I: Steady flows

Abstract: It is shown that a systematic development of physical quantities using spherical harmonics provides analytical solutions to a whole class of linear problems of rotating fluids. These solutions are regular throughout the whole domain of the fluid and are not much affected by the equatorial singularity of steady boundary layers in spherical geometries. A comparison between this method and the one based on boundary layer theory is carried out in the case of the steady spin-up of a fluid inside a sphere.

Implications of boundary singularities in complex geometries

Abstract: Viscoelastic flow problems are singular in Weissenberg number whenever the stress for the corresponding Newtonian flow becomes infinite at a boundary point (a re-entrant corner, for example); this is the apparent reason that the critical Weissenberg number in numerical computation decreases with increasing refinement for sufficiently fine meshes. The explicit form of the singularity for the second order fluid shows that viscoelastic stresses reach physically-unrealistic values over length scales at which the continuum hypothesis remains valid, and non-integrable stresses can result. The present study points to the need for new formulations (constitutive models and/or boundary conditions) for the description of viscoelastic flows near boundary discontinuities.

The semigroup of values of a curve singularity with several branches

Abstract: We give an explicit description of the semigroup of values of a plane curve singularity with several branches in terms of the usual invariants of the equisingularity type in the sense of Zariski. The main tool is the set of elements called maximals, specially the absolute and the relative ones. First, we describe the semigroup in terms of the relative maximals and these ones in terms of the absolute maximals by means of a symmetry property which generalizes the well known property of symmetry for the singularities with only one branch. Then the absolute maximals are described in terms of the theory of maximal contact of higher genus developed by Lejeune.

An improved pencil-of-functions method and comparisons with traditional methods of pole extraction

Abstract: Prony's method is usually used for the determination of poles from experimental data in the singularity expansion method (SEM) literature, It is the aim here to show that this is not the best method and that Jain's pencil-of-functions (POF) method can be generalized to provide a better method. An example of such a method is given.

Electromagnetic inverse profiling: Theory and numerical implementation

Abstract: Introduction General introduction DIRECT SCATTERING FREQUENCY-DOMAIN TECHNIQUES Introduction Scattering by a homogeneous, lossy dielectric slab: singularity expansion method Appendix: Influence of the conductivity on the location of the poles Scattering by an inhomogeneous, lossy dielectric slab: singularity expansion method Appendix: The natural modes of a symmetric Epstein layer Scattering by an inhomogeneous, lossy dielectric slab in between two homogeneous, lossless half-spaces: Fourier inversion Scattering by a lossy, radially inhomogeneous dielectric circular cylinder: singularity expansion method and Fourier inversion Appendix: Mean-square convergence of the angular Fourier series as a function of and t Scattering by a lossy, radially inhomogeneous dielectric circular cylinder: complementary interpretation of singularity expansion method Appendix: The residue of at a pole Conclusions References TIME-DOMAIN TECHNIQUES Introduction General aspects of the marching-on-in-time method Appendix: A relaxation method for the solution of an ill-conditioned system of linear equations Scattering by an inhomogeneous, lossy dielectric slab Scattering by a perfectly conducting cylinder Appendix: Source representation for two-dimensional electromagnetic-field quantities Appendix: Determination of and its space derivatives Scattering by an inhomogeneous, lossy dielectric cylinder Conclusions References IDENTIFICATION A PRONY-TYPE METHOD Introduction Prony's algorithm Application to a slab signal Conclusions References INVERSE-PROFILING FREQUENCY-DOMAIN TECHNIQUES Introduction General aspects of the one-dimensional inverse-scattering problem The one-dimensional inverse-scattering problem for low contrast and oblique incidence The one-dimensional inverse-scattering problem for a lossless dielectric slab Conclusions References TIME-DOMAIN TECHNIQUES Introduction Born-type iterative procedure with a vacuum as the background medium Appendix: Smoothing procedure Optimization approach Born-type iteractive procedure with the reference medium as the background medium Conclusions References Subject index

On the solution of integral equations with a generalized cauchy kernel

Abstract: In this paper a certain class of singular integral equations that may arise from the mixed boundary value problems in nonhomogeneous materials is considered. The distinguishing feature of these equations is that in addition to the Cauchy singularity, the kernels contain terms that are singular only at the end points. In the form of the singular integral equations adopted, the density function is a potential or a displacement and consequently the kernel has strong singularities of the form (t-x) sup-2, x sup n-2 (t+x) sup n, (n or = 2, 0x,tb). The complex function theory is used to determine the fundamental function of the problem for the general case and a simple numerical technique is described to solve the integral equation. Two examples from the theory of elasticity are then considered to show the application of the technique.

Generalised cosmic strings

Abstract: It is shown that two-dimensional and timelike 'elementary' quasi-regular singularities provide a generalised version of the unusual cosmic strings. The properties of such generalised cosmic strings are investigated and it is shown in particular that they are totally geodesic submanifolds on which the mass density mu is constant.