On Planar Continuous Families of Curves

1969 ◽  
Vol 21 ◽  
pp. 513-530 ◽  
Author(s):  
Tudor Zamfirescu
Keyword(s):  
Convex Body ◽  
General Setting ◽  
The Other ◽  
Continuous Family ◽  
Families Of Curves ◽  
Family Of Curves ◽  
Planar Convex

In a recent paper (3), Grünbaum has found a general and unifying setting for a number of properties of some special lines associated with a planar convex body. Besides various interesting results, two conjectures are stated and two kinds of convexity and polygonal connectedness are introduced.In the present paper, we shall prove one of Grünbaum's conjectures (§ 3, Theorem 1); we consider the other in § 4 and establish some related results in §§ 5 and 6. Six-partite problems are studied in this general setting (§ 7) and a question raised by Ceder (2) is answered. We give a generalization of the notion of a continuous family of curves in § 8, and discuss some new kinds of connectedness in § 9.

Continuous Families of Curves

1966 ◽  
Vol 18 ◽  
pp. 529-537 ◽  
Author(s):  
Branko Grünbaum
Keyword(s):  
Fixed Point ◽  
Jordan Curve ◽  
Convex Sets ◽  
General Setting ◽  
Jordan Curve Theorem ◽  
Planar Line ◽  
Families Of Curves ◽  
Planar Convex ◽  
Free Involution

The present paper is an attempt to find the unifying principle of results obtained by different authors and dealing—in the original papers—with areabisectors, chords, or diameters of planar convex sets, with outwardly simple planar line families, and with chords determined by a fixed-point free involution on a circle. The proofs in the general setting seem to be simpler and are certainly more perspicuous than many of the original ones. The tools required do not transcend simple continuity arguments and the Jordan curve theorem. The author is indebted to the referee for several helpful remarks.

Mapping and depth ordering of residual gravity sources

Geophysics ◽  
1993 ◽  
Vol 58 (10) ◽  
pp. 1408-1416 ◽  
Author(s):  
Jacira F. Beltrão ◽  
João B. C. Silva
Keyword(s):  
Apparent Density ◽  
A Priori ◽  
Depth Interval ◽  
Constant Density ◽  
Vertical Coordinate ◽  
Density Distributions ◽  
Families Of Curves ◽  
Density Maps ◽  
Family Of Curves ◽  
Priori Information

Apparent density maps are derived from observed residual Bouguer maps under the assumptions that the sources are restricted to a depth interval and that the density distribution is not a function of the vertical coordinate. If the depth interval containing the sources is known, the computed apparent density maps are very close to the true density distributions. Even when the depth interval of the sources is not known, but the sources have constant density, their horizontal extent may be delineated by the half‐maximum contours of the apparent densities. We developed an interpretation method using a family of curves based on source density, depth to top of sources, and source thickness. The main advantage in the use of these families of curves is the broad range of possible interpretations involving the above‐mentioned parameters. Rough semiquantitative interpretations are possible in the absence of any further a priori information; more refined semiquantitative interpretations require additional quantitative knowledge of one parameter and just a lower or upper bound for another parameter. Finally, quantitative interpretations are possible only with additional quantitative knowledge of two of the three parameters involved.

scholarly journals Retrieving convex bodies from restricted covariogram functions

2007 ◽  
Vol 39 (3) ◽  
pp. 613-629 ◽  
Author(s):  
Gennadiy Averkov ◽  
Gabriele Bianchi
Keyword(s):  
Convex Body ◽  
Lebesgue Measure ◽  
Dimensional Space ◽  
Convex Bodies ◽  
Baire Category ◽  
The Body ◽  
Planar Case ◽  
Large Classes ◽  
Negative Results ◽  
Planar Convex

The covariogram gK(x) of a convex body K ⊆ Ed is the function which associates to each x ∈ Ed the volume of the intersection of K with K + x, where Ed denotes the Euclidean d-dimensional space. Matheron (1986) asked whether gK determines K, up to translations and reflections in a point. Positive answers to Matheron's question have been obtained for large classes of planar convex bodies, while for d ≥ 3 there are both positive and negative results. One of the purposes of this paper is to sharpen some of the known results on Matheron's conjecture indicating how much of the covariogram information is needed to get the uniqueness of determination. We indicate some subsets of the support of the covariogram, with arbitrarily small Lebesgue measure, such that the covariogram, restricted to those subsets, identifies certain geometric properties of the body. These results are more precise in the planar case, but some of them, both positive and negative ones, are proved for bodies of any dimension. Moreover some results regard most convex bodies, in the Baire category sense. Another purpose is to extend the class of convex bodies for which Matheron's conjecture is confirmed by including all planar convex bodies possessing two nondegenerate boundary arcs being reflections of each other.

scholarly journals Cheeger Sets for Rotationally Symmetric Planar Convex Bodies

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Antonio Cañete
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Rotationally Symmetric ◽  
Planar Convex ◽  
Cheeger Sets

AbstractIn this note we obtain some properties of the Cheeger set $$C_\varOmega $$ C Ω associated to a k-rotationally symmetric planar convex body $$\varOmega $$ Ω . More precisely, we prove that $$C_\varOmega $$ C Ω is also k-rotationally symmetric and that the boundary of $$C_\varOmega $$ C Ω touches all the edges of $$\varOmega $$ Ω .

scholarly journals Analysis of simultaneous inpainting and geometric separation based on sparse decomposition

2021 ◽  
pp. 1-50
Author(s):  
Van Tiep Do ◽  
Ron Levie ◽  
Gitta Kutyniok
Keyword(s):  
Missing Data ◽  
Convergence Analysis ◽  
General Setting ◽  
The Other ◽  
Natural Images ◽  
Sparse Decomposition ◽  
Main Application ◽  
Geometric Characteristics ◽  
Cluster Coherence ◽  
Minimization Approach

Natural images are often the superposition of various parts of different geometric characteristics. For instance, an image might be a mixture of cartoon and texture structures. In addition, images are often given with missing data. In this paper, we develop a method for simultaneously decomposing an image to its two underlying parts and inpainting the missing data. Our separation–inpainting method is based on an [Formula: see text] minimization approach, using two dictionaries, each sparsifying one of the image parts but not the other. We introduce a comprehensive convergence analysis of our method, in a general setting, utilizing the concepts of joint concentration, clustered sparsity, and cluster coherence. As the main application of our theory, we consider the problem of separating and inpainting an image to a cartoon and texture parts.

scholarly journals Kinetic Analysis of the Permeability of Human Erythrocytes to NH4Cl

1968 ◽  
Vol 51 (4) ◽  
pp. 579-587 ◽  
Author(s):  
F. R. Hunter
Keyword(s):  
Kinetic Analysis ◽  
Human Erythrocytes ◽  
The Other ◽  
A Value ◽  
Straight Line ◽  
Family Of Curves ◽  
Series Of Experiments

A densimeter technique was used to make a kinetic analysis of the rate of swelling of human erythrocytes suspended in 1% NaCl following successive additions of NH4Cl. Two series of experiments were performed, one in the absence of and the other in the presence of 6 x 10-4 M NaHCO3. An analysis of the data using Widdas's equations gave a family of curves in each instance. When LeFevre's equation was used with a value of ø = 1.3 isotones, a straight line was obtained with both series of data. It is concluded that this system shows carrier kinetics.

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