Kinds Of Singularities Research Articles

The Special Chao and Singularity in a Simple Traffic Model

We have proposed a modified traffic model in which a single car moves through a sequence of traffic lights controlled by a step function instead of a sine function. We have found the complex behaviors in this simple traffic model and given three kinds of bifurcation diagrams. We have found that in this model there are chaotic and complex periodic motions, as well as special singularity. In this paper, we will introduce two special kinds chao which is different from general chao and the singularities of the simple traffic model. There are three different kinds of singularity which are single singularity, unsymmetic and symmetic double singularity.

Twisted genera of symmetric products

We give a new proof of formulae for the generating series of (Hodge) genera of symmetric products X (n) with coefficients, which hold for complex quasi-projective varieties X with any kind of singularities and which include many of the classical results in the literature as special cases. Important specializations of our results include generating series for extensions of Hodge numbers and Hirzebruch’s χ y -genus to the singular setting and, in particular, generating series for intersection cohomology Hodge numbers and Goresky-MacPherson intersection cohomology signatures of symmetric products of complex projective varieties. Our proof applies to more general situations and is based on equivariant Kunneth formulae and pre-lambda structures on the coefficient theory of a point, $${\bar{K}_0(A(pt))}$$ , with A(pt) a Karoubian $${\mathbb{Q}}$$ -linear tensor category. Moreover, Atiyah’s approach to power operations in K-theory also applies in this context to $${\bar{K}_0(A(pt))}$$ , giving a nice description of the important related Adams operations. This last approach also allows us to introduce very interesting coefficients on the symmetric products X (n).

Spectral singularities of a complex spherical barrier potential and their optical realization

The mathematical notion of a spectral singularity admits a physical interpretation as a zero-width resonance. It finds an optical realization as a certain type of lasing effect that occurs at the threshold gain. We explore spectral singularities of a complex spherical barrier potential and study their realization as transverse spherical electromagnetic waves emitted by a gain medium with a spherical geometry. In particular, for a typical dye laser material, we obtain a lower bound on the size of the gain medium for the occurrence of this kind of spectral singularities.

Singularity Elimination of Parallel Mechanism by Redundant Actuations

Singularity is a basic problem for parallel mechanism design because of its many negative influences on the mechanism performances. Traditionally, three kinds of singularities are suitable for analysis of the parallel mechanisms. By considering both the common and redundant actuation parallel mechanisms, general kinematic model is used to obtain input factor matrix and output factor matrix. Based on ranks of the two matrices, the singularity type becomes easy to judge. And the second kind of singularity can be reduced or eliminated by adding special redundant actuation sub-chains. A two-degree-of-freedom parallel mechanism is provided to illustrate the method about how to create the needed redundant actuation sub-chains.

The influence of object orientation and shading on pictorial relief of Lambertian surfaces

We studied the influence of object orientation and lighting direction on pictorial relief of 2D images of generic 3D objects. We used two objects, namely a globally convex object with a furrow in it and a globally convex object with a dimple in it. Participants adjusted a local surface attitude probe at 200-250 different locations in the image such that it seems to lie on the pictorial surface. In a 2×2 orthogonal design we manipulated the object orientation and the lighting direction. Lighting direction was always with respect to the view frame. The orientations were sufficiently altered to be qualitatively different from each other; that is, different kinds of shading singularities and contour singularities were present in the images. It was found that changes in lighting condition induced systematic changes on the settings for both orientations of the objects. First, the results showed that the reconstructed depth range was altered by a change in lighting direction. Second, we looked at the parabolic lines in the surface reconstructions; parabolic lines are curves on a surface where at least one of the principle curvatures equals zero. It was found that the parabolic lines shift due to a change in light direction. This is a particularly interesting finding since the actual location of the parabolic lines on the 3D object is invariant for changes in light direction and object orientation.

Kinematics Analysis of a Quadruped Robot

This paper presents the kinematics model of a complex quadruped robot. Initially, the so-called position analysis problem is approached for both the inverse and the direct kinematics problems. All equations and geometric constraints of the system are presented. Finally a singularity analysis of closed-chain kinematics is done, pointing out which configurations should be used or avoided for a desired behavior of the robot. The singularity analysis was done by considering three different kinds of singularities: direct, inverse and combined.

Pricing Double-Barrier Options Using the Boundary Element Method

A numerical method to price double-barrier options with moving barriers is proposed. Using the so-called Boundary Element Method, an integral representation of the double-barrier option price is derived in which two of the integrand functions are not given explicitly but must be obtained solving a system of Volterra integral equations of the first kind. This system of equations is affected by several kinds of singularities, therefore it is first regularized and then solved using a low-order finite element method based on product integration. Several numerical experiments are carried out showing that the method proposed is extraordinarily fast and accurate, also when the barriers are not differentiable functions. Moreover the numerical algorithm presented in this paper performs significantly better than the finite difference approach.

Interior design of a two-dimensional semiclassical black hole

We look into the inner structure of a two-dimensional dilatonic evaporating black hole. We establish and employ the homogenous approximation for the black-hole interior. Two kinds of spacelike singularities are found inside the black hole, and their structure is investigated. We also study the evolution of spacetime from the horizon to the singularity.

Rational ODEs with Movable Algebraic Singularities

A class of second‐order rational ordinary differential equations, admitting certain families of formal algebraic series solutions, is considered. For all solutions of these equations, it is shown that any movable singularity that can be reached by analytic continuation along a finite‐length curve is an algebraic branch point. The existence of these formal series expansions is straightforward to determine for any given equation in the class considered. We apply the theorem to a family of equations, admitting different kinds of algebraic singularities. As a further application we recover the known fact for generic values of parameters that the only movable singularities of solutions of the Painlevé equations PII–P VI are poles.

Surfaces and Fronts with Harmonic-mean Curvature One in Hyperbolic Three-space

We show a global representation formula for a certain kind of Weingarten surface in hyperbolic three-space, which is based on the formula due to Gálvez, Martínez and Milán. As an application of the representation formula, we also investigate surfaces with harmonic-mean curvature one (HMC-1 surfaces). We allow them to have certain kinds of singularities, and discuss some global properties.

Currents and flat chains associated to varifolds, with an application to mean curvature flow

We prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature flow.

A Very Fast and Accurate Boundary Element Method for Options with Moving Barrier and Time-Dependent Rebate

A numerical method to price options with moving barrier and time-dependent rebate is proposed. In particular, using the so-called Boundary Element Method, an integral representation of the barrier option price is derived in which one of the integrand function is not given explicitly but must be obtained solving a Volterra integral equation of the first kind. This equation is affected by several kinds of singularities, some of which are removed using a suitable change of variables. Then the transformed equation is solved using a low-order finite element method based on product integration. Several numerical experiments are carried out showing that the method proposed is extraordinarily fast and accurate. In particular a high level of accuracy is achieved also when the initial price of the underlying asset is close to the barrier, when the barrier and the rebate are not differentiable functions, or when the option's maturity is particularly long. Finally the numerical method proposed in this paper performssignificantly better than binomial and trinomial lattices that are commonly employed in barrier option pricing.

Experimental study on phase vortices of speckles and their propagation properties

By recording the interference patterns of the speckle fields and the reference beam with the charge coupled device （CCD）, and by using the digital Fourier transform technique, the amplitude and the phase distribution of the speckle fields are experimentally extracted. We find that at the tangential points of the zero curves of the real and the imaginary parts, a new kind of speckle phase singularities may appear. Different from the conventional singularities at the zero crossings of the real and imaginary parts with the monotonically spiral change of phase, this new kind of singularities has the property that the phase undergoes an increase and then a decrease around the singular point with nearly a symmetric distribution. We introduce the concept of quasi twin phase vortices to explain the formation of the new kind of phase vortices. Based on a theoretical study of the longitudinal autocorrelation function of the speckle intensities, we experimentally observe the propagation of the phase vortices of speckles. It is found that in planes at different propagation distances but within a longitudinal correlation length, the real and the imaginary parts of the complex speckle field vary considerably, but the position of the phase vortices and the angle of the zero crossings of the real and imaginary parts remain unchanged.

Mobility, Constraint Singularity and Isotropy of the 3- Translational Parallel Mechanism

The non-overconstrained 3-degrees of freedom(DOF) translational parallel mechanism(TPM) has received much attention due to its advantages in reduced cost of fabrication and assembly. Researches are being conducted in the area of type synthesis, kinematic analysis and dimensional synthesis. Mobility, constraint singularity and isotropy of a 3-R non-overconstrained TPM are studied, where P denote the prismatic pair, R the revolute pair and the overline indicates the same axis direction of the kinematic pair. The different arrangements of the three limbs affect the kinematic performance of this kind of TPM. First, the mobility analysis, actuation selection, and the constraint singularity of the general 3-R TPM are conducted based on screw theory. For a general 3-R TPM, the three prismatic pairs cannot be chosen as actuators and two kinds of constraint singularities are identified. In the first constraint singularity, the moving platform has four instantaneous DOFs. In the second constraint singularity, the moving platform has five instantaneous DOFs. Then, an orthogonal 3-R TPM is proposed, which can be actuated by three prismatic pairs and has no constraint singularities. Further, the forward and inverse kinematic analysis of the orthogonal TPM are presented. The input-output equations of the orthogonal TPM are totally decoupled. The full isotropy of the orthogonal TPM is proved by establishing the Jacobian matrix , which is an identity 3×3 diagonal matrix in the whole workspace. The orthogonal3-R TPM has great potential in application like fast pick-and-place manipulator, parallel machine and micro-motion manipulator.

Singular perturbations generating complexification phenomena for elliptic shells

This paper deals with elliptic shell problems using the Koiter shell model. When the shell is well-inhibited, the limit membrane problem satisfies the Shapiro–Lopatinskii condition and we have a classical singular perturbation problem. In a previous paper, the existence of two kinds of singularities was put in a prominent position for this kind of problem. Conversely, for ill-inhibited shells (when a part of the boundary is free), the limit problem does not satisfy the Shapiro–Lopatinskii condition. Complexification phenomenon appears when the thickness approaches zero, leading to large oscillations corresponding to a new kind of instability on the free boundary. To complete the theoretical analysis, numerical simulations are performed with a finite element software coupled with an anisotropic adaptive mesh generator. This enables us to visualize precisely the singularities and the instabilities predicted by the theory with only a small number of elements.

Combined use of sensor data and structural knowledge processed by Bayesian network: Application to a railway diagnosis aid scheme

This article describes a hybrid diagnosis system based on the combined use of sensor data (local information) and structural knowledge (global information). The approach is illustrated on an application that involves the detection of broken rail for railway infrastructure. Recently, there have been a large number of attempts to solve diagnosis problems by mixing low-level and high-level data. The inherent difficulty of combining different levels of data is offset by the benefits that accrue from additional knowledge: prior information can improve the understanding of sensor data. To introduce the application context, the paper describes first the importance of rail diagnosis. The technological solutions for broken rail detection are then listed. Since the used sensor is able to detect other kind of rail defects or singularities, the defect classification phase also involves the problem of distinguishing between real (broken rail) and false defects (rail singularities). With the help of prior information extracted from an infrastructure database, a Bayesian network is designed to infer the probabilities of membership of real or false defect classes on the basis of previous decisions. The performance of the combined use of these probabilities and sensor processing are finally presented. They demonstrate the advantage of the approach based on both a local description of the defect and a description of its environment.

Complete rank theorem of advanced calculus and singularities of bounded linear operators

Let E and F be Banach spaces, f: U ⊂ E → F be a map of Cr (r ⩾ 1), x0 ∈ U, and ft (x0) denote the FreLechet differential of f at x0. Suppose that f′(x0) is double split, Rank(f′(x0)) = ∞, dimN(f′(x0)) > 0 and codimR(f′(x0)) s> 0. The rank theorem in advanced calculus asks to answer what properties of f ensure that f(x) is conjugate to f′(x0) near x0. We have proved that the conclusion of the theorem is equivalent to one kind of singularities for bounded linear operators, i.e., x0 is a locally fine point for f′(x) or generalized regular point of f(x); so, a complete rank theorem in advanced calculus is established, i.e., a sufficient and necessary condition such that the conclusion of the theorem to be held is given.

DESINGULARIZATIONS OF CALABI-YAU 3-FOLDS WITH CONICAL SINGULARITIES. II. THE OBSTRUCTED CASE

This is the second of two papers studying Calabi–Yau 3-folds with conical singularities and their desingularizations. In our first paper [Y.-M. Chan, Quart. J. Math. 57 (2006), 151–181] we constructed the desingularization of the conically singular manifold M0 by gluing an asymptotically conical (AC) Calabi–Yau 3-fold Y into M0 at the singular point, thus obtaining a 1-parameter family of compact, non-singular Calabi–Yau 3-folds Mt for small t > 0. During the gluing process one may encounter a kind of cohomological obstruction to defining a 3-form Ωt on Mt which interpolates between the 3-form Ω0 on M0 and the scaled 3-form t3 ΩY on Y if the rate λ at which the AC Calabi–Yau 3-fold Y converges to the Calabi–Yau cone is equal to − 3. The first paper [3] studied the simpler case λ < −3 where there is no obstruction. This paper extends the result in the first one by considering a more complicated situtation when λ = −3. Assuming the existence of singular Calabi–Yau metrics on compact complex 3-folds with ordinary double points, our result in this paper can be applied to repairing such kinds of singularities, which is an analytic version of Friedman's result giving necessary and sufficient conditions for smoothing ordinary double points.

On the Essential Instabilities Caused by Fractional‐Order Transfer Functions

The exact stability condition for certain class of fractional‐order (multivalued) transfer functions is presented. Unlike the conventional case that the stability is directly studied by investigating the poles of the transfer function, in the systems under consideration, the branch points must also come into account as another kind of singularities. It is shown that a multivalued transfer function can behave unstably because of the numerator term while it has no unstable poles. So, in this case, not only the characteristic equation but the numerator term is of significant importance. In this manner, a family of unstable fractional‐order transfer functions is introduced which exhibit essential instabilities, that is, those which cannot be removed by feedback. Two illustrative examples are presented; the transfer function of which has no unstable poles but the instability occurred because of the unstable branch points of the numerator term. The effect of unstable branch points is studied and simulations are presented.

Vanishing Cycles in Complex Symplectic Geometry

We study the vanishing cycles on the Milnor fibre of a holomorphic map germ with special kind of non-isolated singularities which appear in symplectic geometry. We show, under assumptions given in the text, that the Lefschetz vanishing cycles freely generate the corresponding homology group of the Milnor fibre. We study two examples of applications: involutive systems in symplectic geometry (as characteristic varieties of $\hbar$-differential equations are of this type) and ajoint orbits in Lie algebras.