scholarly journals Conical singular points and vector fields

2020 ◽  
Vol 65 (4) ◽  
pp. 649-661
Author(s):  
Sergei N. Burian ◽  
Keyword(s):  
Singular Point ◽  
Tangent Space ◽  
Tangent Vector ◽  
Common Point ◽  
Singular Points ◽  
Configuration Spaces ◽  
Dimensional Torus ◽  
Smooth Curves ◽  
Theoretical Approaches ◽  
First Case

In this article, several examples of mechanical systems which configuration spaces are smooth manifolds with a unique singular point are considered. Configuration spaces are the following: two smooth curves with a common point (or tangent) on the two-dimensional torus, four smooth curves on the four-dimensional torus with a common point, twodimensional cone (cusp) in the space R6. The main problem in the article is the calculation of (co)tangent space at a singular point by using different theoretical approaches. Outside of the singular point, the motion could be described in the frames of classical mechanics. But in the neighborhood of the singular points the terms like “tangent vector” and “cotangent vector” must have new conceptual definitions. In this article, the approach of differential spaces is used. Two differential structures for the modeling conical singular point are studied in order to construct (co)tangent space at singular points: locally-constants functions near to the cone vertex and the algebra of the restrictions of smooth functions in the comprehensive Euclidean space on the cone. In the first case, tangent and cotangent spaces at the singular points are zero. In the second case, the value of the functions on the cotangent bundle is constant on the cotangent layer under the singular point.

On the removability of isolated singular points for elliptic equations involving variable exponent

2016 ◽  
Vol 5 (2) ◽  
Author(s):  
Yongqiang Fu ◽  
Yingying Shan
Keyword(s):  
Singular Point ◽  
Elliptic Equations ◽  
Variable Exponent ◽  
Singular Points ◽  
Sufficient Condition ◽  
Variable Exponents ◽  
Isolated Singularities

AbstractIn this paper, we study the problem of removable isolated singularities for elliptic equations with variable exponents. We give a sufficient condition for removability of the isolated singular point for the equations in

scholarly journals A Customized Semantic Segmentation Network for the Fingerprint Singular Point Detection

2020 ◽  
Vol 10 (11) ◽  
pp. 3868
Author(s):  
Jiong Chen ◽  
Heng Zhao ◽  
Zhicheng Cao ◽  
Fei Guo ◽  
Liaojun Pang
Keyword(s):  
Singular Point ◽  
Semantic Segmentation ◽  
Singular Points ◽  
Detection Methods ◽  
Orientation Field ◽  
Global Features ◽  
Fingerprint Image ◽  
Detection Rates ◽  
Fingerprint Classification ◽  
Point Detection

As one of the most important and obvious global features for fingerprints, the singular point plays an essential role in fingerprint registration and fingerprint classification. To date, the singular point detection methods in the literature can be generally divided into two categories: methods based on traditional digital image processing and those on deep learning. Generally speaking, the former requires a high-precision fingerprint orientation field for singular point detection, while the latter just needs the original fingerprint image without preprocessing. Unfortunately, detection rates of these existing methods, either of the two categories above, are still unsatisfactory, especially for the low-quality fingerprint. Therefore, regarding singular point detection as a semantic segmentation of the small singular point area completely and directly, we propose a new customized convolutional neural network called SinNet for segmenting the accurate singular point area, followed by a simple and fast post-processing to locate the singular points quickly. The performance evaluation conducted on the publicly Singular Points Detection Competition 2010 (SPD2010) dataset confirms that the proposed method works best from the perspective of overall indexes. Especially, compared with the state-of-art algorithms, our proposal achieves an increase of 10% in the percentage of correctly detected fingerprints and more than 16% in the core detection rate.

Physical aspects of the relaxation model in two-phase flow

1990 ◽  
Vol 428 (1875) ◽  
pp. 379-397 ◽  
Keyword(s):  
Phase Space ◽  
Singular Point ◽  
Shock Waves ◽  
Saddle Points ◽  
Singular Points ◽  
Relaxation Model ◽  
Two Phase ◽  
Point Patterns ◽  
All Solutions ◽  
Topological Pattern

The paper explores the potential of the homogeneous relaxation model (HRM) as a basis for the description of adiabatic, one-dimensional, two-phase flows. To this end, a rigorous mathematical analysis highlights the similarities and differences between this and the homogeneous equilibrium model (HEM) emphasizing the physical and qualitative aspects of the problem. Special attention is placed on a study of dispersion, characteristics, choking and shock waves. The most essential features are discovered with reference to the appropriate and convenient phase space Ω for HRM, which consists of pressure P , enthalpy h , dryness fraction x , velocity w , and length coordinate z . The geometric properties of the phase space Ω enable us to sketch the topological pattern of all solutions of the model. The study of choking is intimately connected with the occurrence of singular points of the set of simultaneous first-order differential equations of the model. The very powerful centre manifold theorem allows us to reduce the study of singular points to a two-dimensional plane Π , which is tangent to the solutions at a singular point, and so to demonstrate that only three singular-point patterns can appear (excepting degenerate cases), namely saddle points, nodal points and spiral points. The analysis reveals the existence of two limiting velocities of wave propagation, the frozen velocity a f and the equilibrium velocity a e . The critical velocity of choking is the frozen speed of sound. The analysis proves unequivocally that transition from ω < a f to w > a f can take place only via a singular point. Such a condition can also be attained at the end of a channel. The paper concludes with a short discussion of normal, fully dispersed and partly dispersed shock waves.

Indicating the Singular Point in First-Order Optical Flow Fields

Perception ◽  
1996 ◽  
Vol 25 (1_suppl) ◽  
pp. 80-80
Author(s):  
A M L Kappers ◽  
S F Te Pas ◽  
J J Koenderink ◽  
J Dentener
Keyword(s):  
Singular Point ◽  
Flow Field ◽  
Optical Flow ◽  
Stimulus Presentation ◽  
Singular Points ◽  
First Order ◽  
Optical Flow Field ◽  
Visible Part ◽  
Presentation Time ◽  
Random Dot Patterns

We investigated the accuracy with which subjects can indicate the singular point in a first-order optical flow field. This singular point might be important in navigation and orientation. The stimuli were expanding or rotating sparse random-dot patterns consisting of 80 dark dots on a light background. The stimulus window was circular with a diameter of 20 deg arc. The singular point could be at one of 48 different locations. Subjects had to indicate the location of this singular point with a cursor, while fixating in the centre of the stimulus. Presentation time was unlimited, though each dot had a limited lifetime (114 ms) to avoid density cues. Both veridicality and reproducibility for our subjects increased with increasing values of expansion or rotation in a nonlinear way. We did not find any systematic differences between expansion and rotation. When we blocked either the outer rim or the central part of the stimulus, performance remained the same for singular points that were within the visible part of the stimulus. For singular points outside this visible part, the reproducibility also remained the same, but subjects tended to locate the singular points closer to the rim of the visible part of the stimulus.

Summability Tests for Singular Points

1972 ◽  
Vol 15 (4) ◽  
pp. 525-528 ◽  
Author(s):  
F. W. Hartmann
Keyword(s):  
Singular Point ◽  
Singular Points ◽  
Radius Of Convergence ◽  
Image Position

King [5] devised two tests for determining when z = 1 is a singular point of the function f(z) defined by1having radius of convergence equal to one. The point z = 1 and radius of convergence one may be chosen without loss of generality.

scholarly journals Identification of discrete chaotic maps with singular points

2001 ◽  
Vol 6 (3) ◽  
pp. 147-156
Author(s):  
P. G. Akishin ◽  
P. Akritas ◽  
I. Antoniou ◽  
V. V. Ivanov
Keyword(s):  
Neural Network ◽  
Singular Point ◽  
Numerical Scheme ◽  
Chaotic Maps ◽  
Accurate Determination ◽  
Singular Points ◽  
Test Model ◽  
Simple Test ◽  
Chebyshev Neural Network

We investigate the ability of artificial neural networks to reconstruct discrete chaotic maps with singular points. We use as a simple test model the Cusp map. We compare the traditional Multilayer Perceptron, the Chebyshev Neural Network and the Wavelet Neural Network. The numerical scheme for the accurate determination of a singular point is also developed. We show that combining a neural network with the numerical algorithm for the determination of the singular point we are able to accurately approximate discrete chaotic maps with singularities.

Fingerprint Singular Point Detection Using Orientation Field Reliability

2011 ◽  
Vol 403-408 ◽  
pp. 4499-4506 ◽  
Author(s):  
Ravinder Kumar ◽  
Pravin Chandra ◽  
M. Hanmandlu
Keyword(s):  
Singular Point ◽  
Detection Rate ◽  
Singular Points ◽  
Identification System ◽  
Fingerprint Matching ◽  
Orientation Field ◽  
Point Detection ◽  
Field Reliability ◽  
Fingerprint Alignment ◽  
Novel Algorithm

Singular point detection is the most important step in Automatic Fingerprint Identification System (AFIS) and is used in fingerprint alignment, fingerprint matching, and particularly in classification. The computation of orientation field of a fingerprint can be verified by computing orientation field reliability. The most unreliable portion in orientation field can be the possible location of singular points. In this paper we have proposed a novel algorithm for detecting singular points using reliability of the fingerprint orientation field. Experimental results show that the proposed algorithm accurately detects singular points (core and delta) with the detection rate of 92.6 %.

New Double Bifurcation of Nilpotent Focus

2021 ◽  
Vol 31 (04) ◽  
pp. 2150053
Author(s):  
Feng Li ◽  
Hongwei Li ◽  
Yuanyuan Liu
Keyword(s):  
Hopf Bifurcation ◽  
Singular Point ◽  
Limit Cycles ◽  
Second Order ◽  
Singular Points ◽  
Weak Focus ◽  
Bifurcation Phenomenon ◽  
Planar Systems ◽  
Nilpotent Singular Point

In this paper, a new bifurcation phenomenon of nilpotent singular point is analyzed. A nilpotent focus or center of the planar systems with 3-multiplicity can be broken into two complex singular points and a second order elementary weak focus. Then, two more limit cycles enclosing the second order elementary weak focus can bifurcate through the multiple Hopf bifurcation.

scholarly journals LIMIT CYCLE BIFURCATIONS FROM CENTERS OF SYMMETRIC HAMILTONIAN SYSTEMS PERTURBED BY CUBIC POLYNOMIALS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350043 ◽  
Author(s):  
ZHAOPING HU ◽  
BIN GAO ◽  
VALERY G. ROMANOVSKI
Keyword(s):  
Singular Point ◽  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Standard Form ◽  
Melnikov Function ◽  
Singular Points ◽  
Computationally Efficient ◽  
Cubic Polynomials ◽  
Nilpotent Center

We study cubic near-Hamiltonian systems obtained by perturbing a symmetric cubic Hamiltonian system with two symmetric singular points. First, following [Han, 2012], we develop a method to study the analytical property of the Melnikov function near the origin for near-Hamiltonian system having the origin as its elementary center or nilpotent center. A computationally efficient algorithm based on the method is established to systematically compute the coefficients of the Melnikov function. Then, we consider the symmetric singular points and present the conditions for one of them to be an elementary center or a nilpotent center. Under the condition for the singular point to be a center, we obtain the standard form of the Hamiltonian system near the center. Moreover, perturbing the symmetric cubic Hamiltonian systems by cubic polynomials we study limit cycles bifurcating from the center. Finally, perturbing the symmetric Hamiltonian system by symmetric cubic polynomials, we consider the number of limit cycles near one of the symmetric centers of the symmetric near-Hamiltonian system, which is the same as that of another center.

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