In this paper we discuss the relationship between the moving planes of a rational parametric surface and the singular points on it. Firstly, the intersection multiplicity of several planar curves is introduced. Then we derive an equivalent definition for the order of a singular point on a rational parametric surface. Based on the new definition of singularity orders, we derive the relationship ...

In the process of phase unwrapping for an image obtained by an interferometer or in-line holography, noisy image data may pose difficulties. Traditional phase unwrapping algorithms used to estimate a two-dimensional phase distribution include much estimation error, due to the effect of singular points. This paper introduces an accurate phase-unwrapping algorithm based on three techniques: a rot...

Singular points of random matrix-valued analytic functions are a common generalization of eigenvalues of random matrices and zeros of random polynomials. The setting is that we have an analytic function of z taking values in the space of n× n matrices. Singular points are those (random) z where the matrix becomes singular, that is, the zeros of the determinant. This notion was introduced in the...

Let V be a projective hypersurface of fixed degree and dimension which has only isolated singular points. We show that, if the sum of the Milnor numbers at the singular points of V is large, then V cannot have a point of large multiplicity, unless V is a cone. As an application, we give an affirmative answer to a conjecture of Dimca and Papadima.

Let p ≥ 5 be a prime number and let Fp be a finite field. In this work, we determine the number of rational points on singular curves Ea : y = x(x − a) over Fp for some specific values of a. Keywords—Singular curve, elliptic curve, rational points.

In this article, we study elliptic boundary-value problems, depending on a real parameter, in domains with conical points. We present asymptotic formulas for solutions near singular points, as linear combinations of special singular functions and regular functions. These functions and the coefficients of the linear combination are regular with respect to the parameter.

An upper estimate for the Alexander polynomial of an algebraic curve is obtained, which sharpens Libgober’s estimate in terms of the local polynomials at the singular points of the curve: only those singular points may contribute to the Alexander polynomial of the curve which are in excess of the hypothesis of Nori’s vanishing theorem.

We have suggested the numerical schemes of collocation methods for approximative solution of singular integrodifferential equations with kernels of Cauchy type. The equations are defined on the arbitrary smooth closed contours of complex plane. The researched methods are based on Fejér points. Theoretical background of collocation methods has been obtained in Generalized Hölder spaces. Singular...

Ever since Higman’s ground-breaking work $1, there have been many rank 3 characterizations of the classical symplectic, unitary and orthogonal groups and geometries. The purpose of this paper is to add another such characterization to that pile. Take a finite symplectic, unitary or orthogonal geometry having a totally singular (projective) plane. Let R be the number of singular points # x ortho...