Area properties associated with a convex plane curve

Dong-Soo Kim; Young Ho Kim; Hyeong-Kwan Ju; Kyu-Chul Shim

doi:10.1515/gmj-2016-0027

Area properties associated with a convex plane curve

Georgian Mathematical Journal ◽

10.1515/gmj-2016-0027 ◽

2017 ◽

Vol 24 (3) ◽

pp. 429-437

Author(s):

Dong-Soo Kim ◽

Young Ho Kim ◽

Hyeong-Kwan Ju ◽

Kyu-Chul Shim

Keyword(s):

Characteristic Property ◽

Plane Curve ◽

Characterization Theorem ◽

Convex Curve ◽

Convex Plane ◽

Convex Curves ◽

Necessary And Sufficient ◽

Locally Convex

AbstractArchimedes knew that for a point P on a parabola X and a chord AB of X parallel to the tangent of X at P, the area of the region bounded by the parabola X and chord AB is four thirds of the area of the triangle {\bigtriangleup ABP}. Recently, the first two authors have proved that this fact is the characteristic property of parabolas.In this paper, we study strictly locally convex curves in the plane {{\mathbb{R}}^{2}}. As a result, generalizing the above mentioned characterization theorem for parabolas, we present two conditions, which are necessary and sufficient, for a strictly locally convex curve in the plane to be an open arc of a parabola.

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Affine-Distance symmetry sets

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-14422 ◽

2003 ◽

Vol 93 (2) ◽

pp. 247

Author(s):

P. J. Giblin ◽

P. A. Holtom

Keyword(s):

Plane Curve ◽

Plane Curves ◽

Convex Curve ◽

Euclidean Case ◽

Generic Structure ◽

Symmetry Set ◽

Convex Curves ◽

Euclidean Symmetry

The affine distance symmetry set (ADSS) of a plane curve is an affinely invariant analogue of the euclidean symmetry set (SS) [7], [6]. We list all transitions on the ADSS for generic 1-parameter families of plane curves. We show that for generic convex curves the possible transitions coincide with those for the SS but for generic non-convex curves, further transitions occur which are generic in 1-parameter families of bifurcation sets, but are impossible in the euclidean case. For a non-convex curve there are also additional local forms and transitions which do not fit into the generic structure of bifurcation sets at all. We give computational and experimental details of these.

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Some results on normal GCR-lightlike submanifolds of indefinite nearly Kaehler manifolds

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819500373 ◽

2019 ◽

Vol 16 (03) ◽

pp. 1950037

Author(s):

Megha ◽

Sangeet Kumar

Keyword(s):

Sufficient Conditions ◽

Characterization Theorem ◽

Kaehler Manifold ◽

Kaehler Manifolds ◽

Holomorphic Bisectional Curvature ◽

Necessary And Sufficient ◽

Nearly Kaehler Manifold ◽

Lightlike Submanifolds ◽

Lightlike Submanifold ◽

Normal Formula

The purpose of this paper is to study normal [Formula: see text]-lightlike submanifolds of indefinite nearly Kaehler manifolds. We find some necessary and sufficient conditions for an isometrically immersed [Formula: see text]-lightlike submanifold of an indefinite nearly Kaehler manifold to be a normal [Formula: see text]-lightlike submanifold. Further, we derive a characterization theorem for holomorphic bisectional curvature of a normal [Formula: see text]-lightlike submanifold of an indefinite nearly Kaehler manifold.

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Interpolation in Separable Frechet Spaces with Applications to Spaces of Analytic Functions

Canadian Journal of Mathematics ◽

10.4153/cjm-1975-116-x ◽

1975 ◽

Vol 27 (5) ◽

pp. 1110-1113 ◽

Cited By ~ 2

Author(s):

Paul M. Gauthier ◽

Lee A. Rubel

Keyword(s):

Sufficient Conditions ◽

Fréchet Space ◽

Complex Numbers ◽

Frechet Space ◽

Fréchet Spaces ◽

Continuous Linear Functional ◽

Interpolating Sequence ◽

Necessary And Sufficient ◽

Locally Convex ◽

Spaces Of Analytic Functions

Let E be a separable Fréchet space, and let E* be its topological dual space. We recall that a Fréchet space is, by definition, a complete metrizable locally convex topological vector space. A sequence {Ln} of continuous linear functional is said to be interpolating if for every sequence {An} of complex numbers, there exists an ƒ ∈ E such that Ln(ƒ) = An for n = 1, 2, 3, … . In this paper, we give necessary and sufficient conditions that {Ln} be an interpolating sequence. They are different from the conditions in [2] and don't seem to be easily interderivable with them.

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ON TWO CONJECTURES CONCERNING CONVEX CURVES

International Journal of Mathematics ◽

10.1142/s0129167x05003260 ◽

2005 ◽

Vol 16 (10) ◽

pp. 1157-1173

Author(s):

V. SEDYKH ◽

B. SHAPIRO

Keyword(s):

Real Line ◽

The Other ◽

Convex Curve ◽

Convex Curves ◽

Projective Curves ◽

Tangent Developable ◽

Special Case

In this paper we recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the first nontrivial case of curves in ℝP3. Namely, we show that (i) the tangent developable of any convex curve in ℝP3has degree 4 and (ii) construct an example of 4 tangent lines to a convex curve in ℝP3such that no real line intersects all four of them. The question (discussed in [4, 18]) whether the second conjecture is true in the special case of rational normal curves still remains open.

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Projections and embeddings of locally convex operator spaces and their duals

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100062228 ◽

1984 ◽

Vol 96 (2) ◽

pp. 321-323 ◽

Cited By ~ 3

Author(s):

Jan H. Fourie ◽

William H. Ruckle

Keyword(s):

Dual Space ◽

Sufficient Conditions ◽

Linear Operators ◽

Locally Convex Spaces ◽

Continuous Linear ◽

Operator Spaces ◽

Complemented Subspace ◽

Necessary And Sufficient ◽

Locally Convex ◽

Continuous Linear Operators

AbstractLet E, F be Hausdorff locally convex spaces. In this note we consider conditions on E and F such that the dual space of the space Kb (E, F) (of quasi-compact operators) is a complemented subspace of the dual space of Lb (E, F) (of continuous linear operators). We obtain necessary and sufficient conditions for Lb(E, F) to be semi-reflexive.

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Evolution of non-simple closed curves in the area-preserving curvature flow

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210517000269 ◽

2017 ◽

Vol 148 (3) ◽

pp. 659-668 ◽

Cited By ~ 1

Author(s):

Xiao-Liu Wang ◽

Wei-Feng Wo ◽

Ming Yang

Keyword(s):

Global Solution ◽

Blow Up ◽

Curvature Flow ◽

Convex Curve ◽

Closed Curves ◽

Area Preserving ◽

Locally Convex

The convergence and blow-up results are established for the evolution of non-simple closed curves in an area-preserving curvature flow. It is shown that the global solution starting from a locally convex curve converges to an m-fold circle if the enclosed algebraic area A0 is positive, and evolves into a point if A0 = 0.

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AREA OF TRIANGLES ASSOCIATED WITH A STRICTLY LOCALLY CONVEX CURVE

Honam Mathematical Journal ◽

10.5831/hmj.2015.37.1.41 ◽

2015 ◽

Vol 37 (1) ◽

pp. 41-52 ◽

Cited By ~ 1

Author(s):

Dong-Soo Kim ◽

Dong Seo Kim ◽

Hyun Seon Bae ◽

Hye-Jung Kim

Keyword(s):

Convex Curve ◽

Locally Convex

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Global Schauder decompositions of locally convex spaces

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-15032 ◽

2007 ◽

Vol 101 (1) ◽

pp. 65

Author(s):

Milena Venkova

Keyword(s):

Convex Space ◽

Entire Functions ◽

Locally Convex Space ◽

Locally Convex Spaces ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient ◽

Locally Convex ◽

Convex Spaces

We define global Schauder decompositions of locally convex spaces and prove a necessary and sufficient condition for two spaces with global Schauder decompositions to be isomorphic. These results are applied to spaces of entire functions on a locally convex space.

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