Kinds Of Singularities Research Articles

Singularity analysis of quasi-linear DAEs

A reduction method is introduced to explore the quasi-linear DAE. On the basis of reduction, three kinds of singularities of quasi-linear DAE are discussed.

Multi-Fractal Spectral Analysis of the 1987 Stock Market Crash

The multifractal model of asset returns captures the volatility persistence of many financial time series. Its multifractal spectrum computed from wavelet modulus maxima lines provides the spectrum of irregularities in the distribution of market returns over time and thereby of the kind of uncertainty or randomness in a particular market. Changes in this multifractal spectrum display distinctive patterns around substantial market crashes or drawdowns. In other words, the kinds of singularities and the kinds of irregularity change in a distinct fashion in the periods immediately preceding and following major market drawdowns. This paper focuses on these identifiable multifractal spectral patterns surrounding the stock market crash of 1987. Although we are not able to find a uniquely identifiable irregularity pattern within the same market preceding different crashes at different times, we do find the same uniquely identifiable pattern in various stock markets experiencing the same crash at the same time. Moreover, our results suggest that all such crashes are preceded by a gradual increase in the weighted average of the values of the Lipschitz regularity exponents, under low dispersion of the multifractal spectrum. At a crash, this weighted average irregularity value drops to a much lower value, while the dispersion of the spectrum of Lipschitz exponents jumps up to a much higher level after the crash. Our most striking result, therefore, is that the multifractal spectra of stock market returns are not stationary. Also, while the stock market returns show a global Hurst exponent of slight persistence 0.5

Elastic-viscoplastic fields near the tip of a propagating crack under anti-plane shear

The elastic-viscoplastic model proposed by Bingham was used to analyse the stress and strain surrounding the tip of a propagating crack under antiplane shear. The proper displacement pattern was given; the asymptotic equations were derived and solved numerically. The analysis and calculation show that for smaller viscosity the crack-tip possesses logarthmic singularity, and for larger viscosity it possesses power-law singularity. In critical case, the two kinds of singularity are consistent with each other. The result revealed the important role of viscosity for crack-tip field.

Green’s function of anisotropic plates with unrestricted coupling and degeneracy, part 1: the infinite plate

Green’s function of the infinite domain is determined for all types of nondegenerate and degenerate anisotropic plates with bending–extension coupling, and for various kinds of singularities including (i) concentrated forces and moments in both in-plane and normal directions and (ii) dislocations of the tangential and normal displacements and of the midplane slopes. Explicit analytical expressions of Green’s function are obtained in terms of the zeroth-order and higher-order eigenvectors. Higher-order eigenvectors occur in the various degenerate cases and may significantly affect the angular variation of Green’s function. Some familiar types of laminates, including all isotropic laminates and symmetric quasi-isotropic laminates, are degenerate or extra-degenerate. Since all previous works on Green’s functions of laminates assumed distinct eigenvalues, the results contain no higher-order eigenvectors. They are only valid in the relatively simple nondegenerate case, and are not applicable to any type of degenerate laminate.

On the analysis of active reflector supporting manipulator for the large spherical radio telescope

This paper presents a solution to the problem of active reflector supporting manipulator. A new type of spatial three degrees-of-freedom (DOF) parallel manipulator combining two degrees rotation and one degree translation is proposed. The inverse kinematics problem is described in closed form and the velocity equation of the parallel manipulator is given. The forward kinematics problem is investigated by number analysis method. Three kinds of singularities are presented. The position and orientation workspace for the manipulator are defined and analyzed. The manipulator accuracy and the active reflector fitting accuracy are researched systematically. The analysis results prove that the manipulator can be used as the supporting manipulator of spherical radio telescope. And the research builds the theory basis for dimensional design, trajectory planning and control of this type of manipulator.

A combined analytical–numerical method for treating corner singularities in viscous flow predictions

AbstractA combined analytical–numerical method based on a matching asymptotic algorithm is proposed for treating angular (sharp corner or wedge) singularities in the numerical solution of the Navier–Stokes equations. We adopt an asymptotic solution for the local flow around the angular points based on the Stokes flow approximation and a numerical solution for the global flow outside the singular regions using a finite‐volume method. The coefficients involved in the analytical solution are iteratively updated by matching both solutions in a small region where the Stokes flow approximation holds. Moreover, an error analysis is derived for this method, which serves as a guideline for the practical implementation. The present method is applied to treat the leading‐edge singularity of a semi‐infinite plate. The effect of various influencing factors related to the implementation are evaluated with the help of numerical experiments. The investigation showed that the accuracy of the numerical solution for the flow around the leading edge can be significantly improved with the present method. The results of the numerical experiments support the error analysis and show the desired properties of the new algorithm, i.e. accuracy, robustness and efficiency. Based on the numerical results for the leading‐edge singularity, the validity of various classical approximate models for the flow, such as the Stokes approximation, the inviscid flow model and the boundary layer theory of varying orders are examined. Although the methodology proposed was evaluated for the leading‐edge problem, it is generally applicable to all kinds of angular singularities and all kinds of finite‐discretization methods. Copyright © 2004 John Wiley & Sons, Ltd.

On trigonal non-Gorenstein curves with zero Maroni invariant

By means of canonical models we answer some questions on trigonal non-Gorenstein curves with zero Maroni invariant: the number of non-Gorenstein points, the kind of such singularities, possible canonical models, uniqueness and number of base points of a g 1 3 on the curve.

Low-Speed Aerodynamics, Second Edition

Low-Speed Aerodynamics, Second Edition

Canonical theory of spherically symmetric spacetimes with cross-streaming null dusts

The Hamiltonian dynamics of two-component spherically symmetric null dust is studied with regard to the quantum theory of gravitational collapse. The components--the ingoing and outgoing dusts--are assumed to interact only through gravitation. Different kinds of singularities, naked or "clothed", that can form during collapse processes are described. The general canonical formulation of the one-component null-dust dynamics by Bicak and Kuchar is restricted to the spherically symmetric case and used to construct an action for the two components. The transformation from a metric variable to the quasilocal mass is shown to simplify the mathematics. The action is reduced by a choice of gauge and the corresponding true Hamiltonian is written down. Asymptotic coordinates and energy densities of dust shells are shown to form a complete set of Dirac observables. The action of the asymptotic time translation on the observables is defined but it has been calculated explicitly only in the case of one-component dust (Vaidya metric).

Invariant Curves and Focal Points in a Lyness Iterative Process

We investigate the properties of recurrence of the type [Formula: see text], known as Lyness iterations from [Lyness, 1942, 1945, 1961] and recently analyzed by several authors in the case a > 0, see e.g. [Kocic et al., 1993; Csornyei & Laczkovich, 2000]. We reconsider Lyness recurrences at the light of some recent results on iterated maps with denominator, given in [Bischi et al., 1999a], where new kinds of singularities, such as focal points and prefocal curves, have been defined. In this paper, in particular, we give an answer to one of the open problems proposed in [Kocic & Ladas, 1993, pp. 141] concerning the dynamic behavior of Lyness recurrences for a < 0. We also give some new results in the case a > 0, and we improve a previous result on Lyness "periodic recurrences".

Analysis and dimensional design of a novel hybrid machine tool

This paper presents a four degrees of freedom (DOF) hybrid machine tool based on a novel planar 3-DOFs parallel manipulator and a long movement of the worktable. Closed-form solutions are developed for both the inverse and direct kinematics about the parallel manipulator. Three kinds of singularities are presented. Based on the formulation of condition number and the definition of the position workspace and orientation workspace, the effects of the design parameters on the dexterity and workspace were discussed. The change ratios of condition number were defined and used as the evaluated indicator to analyze the effects of the design parameters on the performances. A dimensional design approach that could satisfy the requirement of motion platform realizing orientation capability, dexterity and workspace was proposed. A novel 4-DOFs hybrid machine tool was manufactured. The methodology presented in this paper can be of great help in the design, trajectory planning, control and application of such devices.

Analysis of a novel cylindrical 3-DoF parallel robot

In this paper, a novel three degrees of freedom (DoFs), that are two degrees of translational freedom and one degree of rotational freedom, parallel robot is proposed. The parallel robot consists of a base plate, a movable platform, and three connecting legs. Because the locus of any point on the moving platform is the subset of a cylinder if z is specified, the robot is called a cylindrical parallel robot. The inverse and forward kinematics problems are described in closed forms. And the velocity equation of the new parallel robot is given. Three kinds of singularities are presented. The workspace for the robot is analyzed systematically, especially, a new index to evaluate the mobility (In this paper, mobility means rotational capability.) of the moving platform of the robot is defined and discussed in detail. The parallel robot has wide application in the fields of industrial robots, simulators, microrobots and parallel machine tools.

Algebraic classification of rational CR structures on topological 5-sphere with transversal holomorphic S-1-action in C-4

Let X be a compact connected CR manifold in ℂN. X is called a rational CR manifold if its geometric genus pg(X) is equal to zero. In this paper we classify all rational CR structures on a topological 5–sphere with transverse S1–action in ℂ4 up to algebraic equivalence. In fact the main theorem of this paper gives a classification of all 3–dimensional isolated rational weighted homogeneous singularities with links homeomorphic to topological 5–sphere. These are the simplest kind of singularities in some sense.

M theory, G2-manifolds and four-dimensional physics

M theory on a manifold of G2-holonomy is a natural framework for obtaining vacua with four large spacetime dimensions and \U0001d4a9 = 1 supersymmetry. In order to obtain, within this framework, the standard features of particle physics, namely non-Abelian gauge groups and chiral fermions, we consider G2-manifolds with certain kinds of singularities at which these features reside. The aim of this paper is to describe in detail how the above picture emerges. Along the way we will see how interesting aspects of strongly coupled gauge theories, such as confinement, receive relatively simple explanations within the context of M theory. We conclude with a brief discussion of the particle physics features of these vacua of M theory.

Interactions of a sub-surface crack with a center of dilatation in an elastic half-plane

This paper examines the interaction between a crack parallel to the free surface of an elastic half-plane and an internal center of dilatation. The problem is decomposed into two auxiliary problems. When the center of dilatation approaches the crack tip, two kinds of singularity are analytically obtained. If the overburden stress and the friction on crack surface are neglected, both modes I and II stress intensity factors ( K I and K II) are induced at the crack tips. The maximum of K I and K II occurs when the center of dilatation is located in front of the crack tips. The tensile cracking is likely to be prohibited by the overburden stress, while shear cracking remains possible even including the effects of both overburden and friction on the crack surface.

Singularity analysis of geometric constraint systems

Singularity analysis is an important subject of the geometric constraint satisfaction problem. In this paper, three kinds of singularities are described and corresponding identification methods are presented for both under-constrained systems and over-constrained systems. Another special but common singularity for under-constrained geometric systems, pseudo-singularity, is analyzed. Pseudo-singularity is caused by a variety of constraint matching of under-constrained systems and can be removed by improving constraint distribution. To avoid pseudo-singularity and decide redundant constraints adaptively, a differentiation algorithm is proposed in the paper. Its correctness and efficiency have been validated through its practical applications in a 2D/3D geometric constraint solver CBA.

NS Open Strings with B Fields and Their Interactions with NS Closed Strings

We consider interactions of a NS open string with first important states of a NS closed string, i.e., a closed-string tachyon and a graviton, where both ends of the NS open string are attached on a D-brane, and a constant background B field is lying along directions parallel to the D-brane world volume. Contrary to general expectations, there are no constraints on these vertex operators coming from the B field. However, we point out that these vertex operators have singularities at both ends of the NS open string when external momenta take some values. These kinds of singularities essentially come from the Dirichlet boundary conditions along directions transverse to the D-brane world volume.

Multiple-Camera Tracking of Rigid Objects

In this paper we describe a method for tracking rigid objects using one or several cameras. The tracking process consists of aligning a 3-D model representation of an object with image contours by measuring and minimizing the image error between predicted model points and image contours. The tracker behaves like a visual servo loop where the internal and external camera parameters are updated at each new image acquisition. We study in detail the Jacobian matrix associated with this minimization process in the presence of both point-to-point and point-to-contour matches. We establish the minimal number of matches that are needed as well as the singular configurations leading to a rank-deficient Jacobian matrix. We find a mathematical link between the point-to-point and point-to-contour cases. Based on this link we show that the latter has the same kind of singularities as the former. Moreover, we study multiple-camera configurations which optimize the robustness of the method in the presence of single-camera singularities, bad, noisy, or missing data. Extensive experiments done with a complex ship part and with up to three cameras validate the method. In particular we show that the tracker may well be used as a camera calibration procedure.

Singularities in crystalline curvature flows

This paper discusses the behaviour of polygonal convex curves in the plane moving under crystalline curvature flows, in which the speed of motion of each edge is determined by a function of its length. The behaviour depends on the rate of growth of the speed as the length of the edge approaches zero: For slow growth — including the homogeneous case where speed is inversely proportional to a power α ∈ (0, 1) of the length — there are always solutions for which the enclosed area approaches zero while the length remains positive. If α > 1, then all solutions are asymptotic to homothetically contracting solutions, and if α = 1 then there is a range of different kinds of singularity that occur. 1. Crystalline curvature flows Several authors have considered crystalline curvature flows of polygonal curves in the plane, since their introduction in [T]. We refer the reader to [TCH] and [AG] for a discussion of the geometric and physical motivation for such flows. For present purposes we consider only convex curves, although the flows can be defined much more generally. In this case the flows can be defined in the following way: Let γ be a closed convex N -sided polygon in the plane, and label the edges γ0, . . . , γN−1 in an anticlockwise order. Let θi ∈ S = R/2πZ be the angle of the exterior normal of γi, and let `i be the length of γi. Moving γ by a crystalline curvature flow consists of finding a continuous family of polygonal curves γ(t) starting from γ so that each edge keeps the same direction but moves in the outward normal direction with speed vi determined by its length: (1) vi(t) = gi(`i). Here gi is a smooth function defined on (0,∞) which is monotone increasing for each i. This paper mostly concerns contraction flows, for which gi 0 and fi is a positive real number for each i. 1991 Mathematics Subject Classification. 53C44, 52A10, 34C11. 1

Differential Groupoids and Their Application to the Theory of Spacetime Singularities

The transformation groupoid Γ = $$\overline {OM} $$ × G, where $$\overline {OM} $$ is the total space of the generalized frame G-bundle over spacetime with a singular boundary, is not a Lie groupoid but a differential groupoid, i.e., a smooth groupoid in the category of structured spaces. We define this concept and use it to investigate spacetimes with various kinds of singularities. Any differential transformation groupoid can be represented by an algebra of operators on a bundle of Hilbert spaces defined on the groupoid fibers. This algebra reflects the structure of a given fiber even if it is a fiber over a singularity. It is also shown that any spacetime with singularities can be regarded as a noncommutative space. Its geometry is done in terms of a noncommutative algebra defined on the corresponding differential transformation groupoid. We focus on the structure of “malicious singularities” such as the ones appearing in the beginning and in the end of the closed Friedman universe.