scholarly journals Area Properties of Strictly Convex Curves

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 391 ◽  
Author(s):  
Dong-Soo Kim ◽  
Young Ho Kim ◽  
Yoon-Tae Jung
Keyword(s):  
Partial Answer ◽  
Level Curve ◽  
Conic Sections ◽  
Curve Segment ◽  
Level Curves ◽  
Strictly Convex ◽  
Convex Curves ◽  
Constant Area ◽  
Characterization Theorems

We study functions defined in the plane E 2 in which level curves are strictly convex, and investigate area properties of regions cut off by chords on the level curves. In this paper we give a partial answer to the question: Which function has level curves whose tangent lines cut off from a level curve segment of constant area? In the results, we give some characterization theorems regarding conic sections.

Sets of Polynomials Orthogonal Simultaneously on Four Ellipses

1968 ◽  
Vol 20 ◽  
pp. 1281-1294
Author(s):  
Ruth Goodman
Keyword(s):  
The Other ◽  
Level Curve ◽  
Entire Family ◽  
Level Curves ◽  
Concentric Circles

It has been shown by Walsh (3) and Szegö (2) that if a set of polynomials is orthogonal on both of two distinct curves, then one curve is a level curve of the other. Szegö (2) has determined all sets of polynomials which are orthogonal simultaneously on an entire family of level curves. There are five essentially different sets, two of which are orthogonal on concentric circles, and three of which are orthogonal on confocal ellipses. Merriman (1) has shown that the orthogonality of a set of polynomials on both of two concentric circles is sufficient to guarantee their orthogonality on the entire family of circles.

A new sea-level curve from Nova Scotia: evidence for a rapid acceleration of sea-level rise in the late mid-Holocene

1995 ◽  
Vol 32 (12) ◽  
pp. 2071-2080 ◽  
Author(s):  
D. B. Scott ◽  
K. Brown ◽  
E. S. Collins ◽  
F. S. Medioli
Keyword(s):  
South Carolina ◽  
Nova Scotia ◽  
Sea Level ◽  
Late Holocene ◽  
Atlantic Coast ◽  
Level Curve ◽  
Global Response ◽  
Level Curves ◽  
Ice Melt ◽  
Vertical Extent

A new late Holocene sea-level curve is presented from the Atlantic coast of Nova Scotia. Contrary to earlier data from the same area, this curve starts at 4400 sidereal years before present (BP) and shows a rapid acceleration between 4400 and 3800 BP, which coincides with a similar acceleration already reported from the Northumberland Strait (Nova Scotia) and an oscillation observed in South Carolina. Comparing the two Nova Scotia curves suggests that the acceleration lasts just over 1000 years and has a vertical extent of 10 m. One puzzling fact is that the 10 m vertical extent in Nova Scotia is 8 m more than the same event measured in South Carolina and it cannot be accounted for simply by postglacial isostatic depression, since that occurs on a much longer time scale. A closer examination of most of the sea-level curves from northeastern North America reveals that either the record is missing from this interval or it is inconsistent. We suggest that this acceleration is part of a global response that coincides with the end of the mid-Holocene warming period, possibly indicating a lag response between warming and ice melt.

scholarly journals Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces

2021 ◽  
Vol 61 (1) ◽  
Author(s):  
Tomasz Adamowicz ◽  
Giona Veronelli
Keyword(s):  
Harmonic Functions ◽  
Geodesic Curvature ◽  
Integral Curvature ◽  
Level Curve ◽  
Singular Surfaces ◽  
Constant Gaussian Curvature ◽  
Conical Singularities ◽  
Logarithmic Convexity ◽  
Level Curves ◽  
Laplace Type

AbstractWe investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as well as with singular Alexandrov surfaces (also called surfaces with bounded integral curvature), a class which includes for instance surfaces with conical singularities and surfaces of CAT(0) type. Moreover, we study the geodesic curvature of the level curves and of the steepest descent for harmonic functions on surfaces with non-necessarily constant Gaussian curvature K. Such geodesic curvature functions turn out to satisfy certain Laplace-type equations and inequalities, from which we infer various maximum and minimum principles. The results are complemented by a number of growth estimates for the derivatives $$L'$$ L ′ and $$L''$$ L ′ ′ of the length of the level curve function L, as well as by examples illustrating the presentation. Our work generalizes some results due to Alessandrini, Longinetti, Talenti, Ma–Zhang and Wang–Wang.

On singular curves with Gaussian curvature bounds

Analysis ◽  
2018 ◽  
Vol 38 (3) ◽  
pp. 127-136
Author(s):  
Hao Fang ◽  
Weifeng Wo
Keyword(s):  
Gaussian Curvature ◽  
Short Note ◽  
Geodesic Curvature ◽  
Curvature Bounds ◽  
Strictly Convex ◽  
Convex Curves

Abstract In this short note, we consider piece-wise smooth strictly convex curves in {\mathbb{R}^{2}} with prescribed singular angles. Given some geodesic curvature bounds, we give the sharp length estimates.

On strictly convex curves and linear monotonicity

1961 ◽  
Vol 65 (3) ◽  
pp. 213-219 ◽  
Author(s):  
Fr. Fabricius-Bjerre
Keyword(s):  
Strictly Convex ◽  
Convex Curves

Lagrange’s theorem, convex functions and Gauss map

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 321-334
Author(s):  
Miodrag Mateljevic ◽  
Miloljub Albijanic
Keyword(s):  
Convex Functions ◽  
Gauss Map ◽  
Unique Property ◽  
The Other ◽  
Concave Functions ◽  
Strictly Convex ◽  
Convex Curves ◽  
Lagrange Theorem ◽  
Darboux Theorem

As one of the main results we prove that if f has Lagrange unique property then f is strictly convex or concave (we do not assume continuity of the derivative), Theorem 2.1. We give two different proofs of Theorem 2.1 (one mainly using Lagrange theorem and the other using Darboux theorem). In addition, we give a few characterizations of strictly convex curves, in Theorem 3.5. As an application of it, we give characterization of strictly convex planar curves, which have only tangents at every point, by injective of the Gauss map. Also without the differentiability hypothesis we get the characterization of strictly convex or concave functions by two points property, Theorem 4.2.

Sea-level Data for Parts of the Bering-Chukchi Shelves of Beringia from 19,000 to 10,000 14C yr B.P.

1984 ◽  
Vol 21 (3) ◽  
pp. 317-325 ◽  
Author(s):  
Dean A. McManus ◽  
Joe S. Creager
Keyword(s):  
Sea Level ◽  
Land Plants ◽  
Level Curve ◽  
Sea Level Changes ◽  
Level Curves ◽  
Marine Plants ◽  
Time Period ◽  
Level Data ◽  
The Pacific ◽  
Regional Sea Level

Sea-level changes in Beringia are especially significant because they affect the migration of land plants and animals between Asia and North America, and marine plants and animals between the Pacific and Arctic oceans. Previous studies of cores from the Bering and Chukchi shelves produced sea-level curves. Evaluation of these data suggests that nine of the radiocarbon-dated estimates of sea-level position are most reliable for the time period 19,000 to 10,000 yr B.P. The trend of these nine points is proposed as the basis for a regional sea-level curve for central Beringia. Constraints on the data must be noted, however, by anyone using them.

scholarly journals LSTM-Enabled Level Curve Tracking in Scalar Fields Using Multiple Mobile Robots

2021 ◽  
Author(s):  
Kunj J. Parikh ◽  
Wencen Wu
Keyword(s):  
Kalman Filter ◽  
Mobile Robots ◽  
Sensor Network ◽  
Control Strategy ◽  
Scalar Fields ◽  
Level Curve ◽  
Mobile Sensor ◽  
Mobile Sensor Network ◽  
Level Curves ◽  
Curve Tracking

Abstract In this work, we investigate the problem of level curve tracking in unknown scalar fields using a limited number of mobile robots. We design and implement a long short term memory (LSTM) enabled control strategy for a mobile sensor network to detect and track desired level curves. Based on the existing work of cooperative Kalman filter, we design an LSTM-enhanced Kalman filter that utilizes the sensor measurements and a sequence of past fields and gradients to estimate the current field value and gradient. We also design an LSTM model to estimate the Hessian of the field. The LSTM enabled strategy has some benefits such as it can be trained offline on a collection of level curves in known fields prior to deployment, where the trained model will enable the mobile sensor network to track level curves in unknown fields for various applications. Another benefit is that we can train using larger resources to get more accurate models, while utilizing a limited number of resources when the mobile sensor network is deployed in production. Simulation results show that this LSTM enabled control strategy successfully tracks the level curve using a mobile multi-robot sensor network.

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