Quadrarcs, St. Peter's, and the Colosseum

1970 ◽  
Vol 63 (3) ◽  
pp. 209-215
Author(s):  
N. T. Gridgeman
Keyword(s):  
Abrupt Change ◽  
Applied Mathematics ◽  
Closed Curve ◽  
Closed Curves ◽  
The Everyday ◽  
Isolated Points ◽  
Axis Of Symmetry ◽  
Regular Closed

1. Ovals. An oval is a regular closed curve in the plane, having at least one axis of symmetry and continuons curvature. However, if this last condition is re laxed, the definition will admit a class of ovals (in the everyday sense of the word— perhaps they should formally be called pseudoövals) that are important in applied mathematics. These are the composite closed curves made up of symmetrically disposed circular ares joined “normally,” that is, where tangent and normal are common, so that the abrupt change of eurvature is not apparent. We may regard such an oval as having an evolute that is not itself a curve but a set of isolated points-discrete centers of curvature. Ovals of this kind have long been familiar to architects and engineers. We shall now discuss the commonest type, the quadrarc, and afterwards look at two celebrated architectural examples.

Nowhere Dense Subsets of Metric Spaces with Applications to Stone-Cech Compactifications

1972 ◽  
Vol 24 (4) ◽  
pp. 622-630 ◽  
Author(s):  
Jack R. Porter ◽  
R. Grant Woods
Keyword(s):  
Metric Space ◽  
Hausdorff Space ◽  
Metric Spaces ◽  
Dense Subset ◽  
Locally Compact ◽  
Completely Regular ◽  
Isolated Points ◽  
Remote Points ◽  
Topological Boundary ◽  
Regular Closed

Let X be a metric space. Assume either that X is locally compact or that X has no more than countably many isolated points. It is proved that if F is a nowhere dense subset of X, then it is regularly nowhere dense (in the sense of Katětov) and hence is contained in the topological boundary of some regular-closed subset of X. This result is used to obtain new properties of the remote points of the Stone-Čech compactification of a metric space without isolated points.Let βX denote the Stone-Čech compactification of the completely regular Hausdorff space X. Fine and Gillman [3] define a point p of βX to be remote if p is not in the βX-closure of a discrete subset of X.

On the Maximum Curvature of Closed Curves in Negatively Curved Manifolds

2015 ◽  
Vol 58 (4) ◽  
pp. 713-722
Author(s):  
Simon Brendle ◽  
Otis Chodosh
Keyword(s):  
Maximum Principle ◽  
Geodesic Curvature ◽  
Sharp Inequality ◽  
Closed Curve ◽  
Closed Curves ◽  
The Maximum Principle ◽  
Curved Manifolds ◽  
Maximum Curvature ◽  
Negatively Curved ◽  
Simply Connected

AbstractMotivated by Almgren’s work on the isoperimetric inequality, we prove a sharp inequality relating the length and maximum curvature of a closed curve in a complete, simply connected manifold of sectional curvature at most −1. Moreover, if equality holds, then the norm of the geodesic curvature is constant and the torsion vanishes. The proof involves an application of the maximum principle to a function defined on pairs of points.

A New Positioning System for Traffic Accident Scene Investigation Integrated with RFID and WSN

2014 ◽  
Vol 602-605 ◽  
pp. 2495-2499
Author(s):  
Hong Song ◽  
Jun Tang ◽  
Jian Xiong Ming ◽  
Wei Jian Duan ◽  
Zhi Yong Yin
Keyword(s):  
Traffic Accident ◽  
Reference Points ◽  
End Points ◽  
Closed Curves ◽  
New Radio ◽  
Efficiency And Effectiveness ◽  
Isolated Points ◽  
Scene Investigation ◽  
Location Map ◽  
Accident Scene

In traffic accident scene investigation, the traditional measuring and positioning methods cannot meet current requirement for efficiency and effectiveness. Which means the Investigators need use tape to measure the distance between accident factors and reference points, and then mark the corresponding elements -- dots, straight lines, curves, closed curves and so on - on the diagram for traffic accident scene investigation. A new radio location system integrated with RFID and WSN has been established to solve the problem. The function of RFID is to clearly mark the ID, attribute and affiliation; WSN to acquire locations of factors from the sensors; ZIGBEE to transmit the combined data (RFID and WSN data) to computers through networks. Using special software to automatically processing the received data for a visualized location map, including the following processes of A) locating survey marks with the geometric algorithm. B) Defining the types of survey marks, such as isolated points, line end points, curve end points, intermediate points etc. C) Classifying the survey marks, such as front wheels, rear wheels, skid marks, body locations, position of fallout etc.

scholarly journals Monotone homotopies and contracting discs on Riemannian surfaces

2018 ◽  
Vol 10 (02) ◽  
pp. 323-354 ◽  
Author(s):  
Gregory R. Chambers ◽  
Regina Rotman
Keyword(s):  
Finite Volume ◽  
Minimal Surfaces ◽  
Analogous Result ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Riemannian Surface ◽  
Closed Curves ◽  
Minimal Hypersurfaces ◽  
Riemannian Surfaces ◽  
Complete Manifolds

A monotone homotopy is a homotopy composed of simple closed curves which are also pairwise disjoint. In this paper, we prove a “gluing” theorem for monotone homotopies; we show that two monotone homotopies which have appropriate overlap can be replaced by a single monotone homotopy. The ideas used to prove this theorem are used in [G. R. Chambers and Y. Liokumovich, Existence of minimal hypersurfaces in complete manifolds of finite volume, arXiv:1609.04058] to prove an analogous result for cycles, which forms a critical step in their proof of the existence of minimal surfaces in complete non-compact manifolds of finite volume. We also show that, if monotone homotopies exist, then fixed point contractions through short curves exist. In particular, suppose that [Formula: see text] is a simple closed curve of a Riemannian surface, and that there exists a monotone contraction which covers a disc which [Formula: see text] bounds consisting of curves of length [Formula: see text]. If [Formula: see text] and [Formula: see text], then there exists a homotopy that contracts [Formula: see text] to [Formula: see text] over loops that are based at [Formula: see text] and have length bounded by [Formula: see text], where [Formula: see text] is the diameter of the surface. If the surface is a disc, and if [Formula: see text] is the boundary of this disc, then this bound can be improved to [Formula: see text].

Taming Wild Simple Closed Curves with Monotone Maps

1972 ◽  
Vol 24 (5) ◽  
pp. 768-788
Author(s):  
W. S. Boyd ◽  
A. H. Wright
Keyword(s):  
Simple Closed Curve ◽  
Closed Curve ◽  
Closed Curves ◽  
Monotone Map ◽  
Monotone Maps

Hempel [6, Theorem 2] proved that if S is a tame 2-sphere in E3 and f is a map of E3 onto itself such that f|S is a homeomorphism and f(E3 - S) = E3- f(S), then f(S) is tame. Boyd [4] has shown that the converse is false; in fact, if S is any 2-sphere in E3, then there is a monotone map f of E3 onto itself such that f |S is a homeomorphism, f(E3 — S) = E3 — f(S), and f(S) is tame.It is the purpose of this paper to prove that the corresponding converse for simple closed curves in E3 is also false. We show in Theorem 4 that if J is any simple closed curve in a closed orientable 3-manifold M3, then there is a monotone map f : M3 → S3 such that f |J is a homeomorphism, f(J) is tame and unknotted, and f(M3 - J) = S3 - f(J).In Theorem 1 of § 2, we construct a cube-with-handles neighbourhood of a simple closed curve in an orientable 3-manifold.

THE MELLIN CENTRAL PROJECTION TRANSFORM

2017 ◽  
Vol 58 (3-4) ◽  
pp. 256-264
Author(s):  
JIANWEI YANG ◽  
LIANG ZHANG ◽  
ZHENGDA LU
Keyword(s):  
Radial Direction ◽  
Mellin Transform ◽  
Closed Curve ◽  
Chinese Characters ◽  
Central Projection ◽  
Affine Invariant ◽  
Closed Curves ◽  
Affine Invariants ◽  
Invariant Features ◽  
Special Case

The central projection transform can be employed to extract invariant features by combining contour-based and region-based methods. However, the central projection transform only considers the accumulation of the pixels along the radial direction. Consequently, information along the radial direction is inevitably lost. In this paper, we propose the Mellin central projection transform to extract affine invariant features. The radial factor introduced by the Mellin transform, makes up for the loss of information along the radial direction by the central projection transform. The Mellin central projection transform can convert any object into a closed curve as a central projection transform, so the central projection transform is only a special case of the Mellin central projection transform. We prove that closed curves extracted from the original image and the affine transformed image by the Mellin central projection transform satisfy the same affine transform relationship. A method is provided for the extraction of affine invariants by employing the area of closed curves derived by the Mellin central projection transform. Experiments have been conducted on some printed Chinese characters and the results establish the invariance and robustness of the extracted features.

CONSTRUCTION OF INVARIANTS OF CURVES AND FRONTS USING WORD THEORY

2010 ◽  
Vol 19 (09) ◽  
pp. 1205-1245
Author(s):  
NOBORU ITO
Keyword(s):  
Infinite Sequence ◽  
Closed Curve ◽  
Closed Curves

We construct an infinite sequence of invariants for surface curves by using the word theory introduced by Turaev. We modify these invariants to various classes of curves: plane closed curves, long curves and fronts. For a plane closed curve, for example, these modified invariants contain Arnold's basic invariants. We also discuss how curves are classified by these modified invariants.

A New Class of Two-Dimensional Chaotic Maps with Closed Curve Fixed Points

2019 ◽  
Vol 29 (07) ◽  
pp. 1950094 ◽  
Author(s):  
Haibo Jiang ◽  
Yang Liu ◽  
Zhouchao Wei ◽  
Liping Zhang
Keyword(s):  
Fixed Points ◽  
Nonlinear Function ◽  
Closed Curve ◽  
Computer Search ◽  
Two Dimensional ◽  
Closed Curves ◽  
New Class ◽  
Dynamical Behaviors ◽  
The Mathematical Model ◽  
Search Program

This paper constructs a new class of two-dimensional maps with closed curve fixed points. Firstly, the mathematical model of these maps is formulated by introducing a nonlinear function. Different types of fixed points which form a closed curve are shown by choosing proper parameters of the nonlinear function. The stabilities of these fixed points are studied to show that these fixed points are all nonhyperbolic. Then a computer search program is employed to explore the chaotic attractors in these maps, and several simple maps whose fixed points form different shapes of closed curves are presented. Complex dynamical behaviors of these maps are investigated by using the phase-basin portrait, Lyapunov exponents, and bifurcation diagrams.

scholarly journals Normal closures of powers of Dehn twists in mapping class groups

1992 ◽  
Vol 34 (3) ◽  
pp. 314-317 ◽  
Author(s):  
Stephen P. Humphries
Keyword(s):  
Class Group ◽  
Simple Closed Curve ◽  
Isotopy Class ◽  
Dehn Twist ◽  
Closed Curve ◽  
Mapping Class ◽  
Class Groups ◽  
Closed Curves ◽  
Dehn Twists ◽  
Definition Of

LetF = F(g, n)be an oriented surface of genusg≥1withn<2boundary components and letM(F)be its mapping class group. ThenM(F)is generated by Dehn twists about a finite number of non-bounding simple closed curves inF([6, 5]). See [1] for the definition of a Dehn twist. Letebe a non-bounding simple closed curve inFand letEdenote the isotopy class of the Dehn twist aboute. LetNbe the normal closure ofE2inM(F). In this paper we answer a question of Birman [1, Qu 28 page 219]:Theorem 1.The subgroup N is of finite index in M(F).

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