scholarly journals A four vertex theorem for strictly convex space curves

1993 ◽  
Vol 46 (1-2) ◽  
pp. 119-126 ◽  
Author(s):  
Juan J. Nu�o Ballesteros ◽  
M. Carmen Romero Fuster
Keyword(s):  
Convex Space ◽  
Strictly Convex ◽  
Space Curves ◽  
Strictly Convex Space ◽  
Vertex Theorem

An n-Vertex Theorem for Convex Space Curves

1985 ◽  
Vol 37 (2) ◽  
pp. 217-237
Author(s):  
Tibor Bisztriczky
Keyword(s):  
Topological Space ◽  
Euclidean Plane ◽  
Convex Surface ◽  
Space Curves ◽  
Straight Line ◽  
History Of ◽  
General Convex ◽  
Vertex Theorem ◽  
The Subject ◽  
Three Space

The classical four-vertex theorem states that a simple closed convex C2 curve in the Euclidean plane has at least four vertices (points of extreme curvature). This theorem has many generalizations with regard to both the curve and the topological space and for a history of the subject, we refer to [4] and [1]. The particular generalization of concern, credited to H. Mohrmann, is the following n-vertex theorem.Let a simple closed C3 curve on a closed convex surface be intersected by a suitable plane in n points. Then the curve has at least n inflections (vertices).The closed convex surface in the preceding is defined as having at most two points in common with any straight line. Presently, we extend this result to curves on more general convex surfaces in a real projective three-space P3.

Inflectional Convex Space Curves

1984 ◽  
Vol 36 (3) ◽  
pp. 537-549 ◽  
Author(s):  
Tibor Bisztriczky
Keyword(s):  
Convex Hull ◽  
Convex Space ◽  
Intersection Point ◽  
List Type ◽  
Multiple Points ◽  
Space Curves ◽  
Euclidean Three Space ◽  
Set Of Points ◽  
Three Space ◽  
Regular Closed

Let Φ be a regular closed C2 curve on a sphere S in Euclidean three-space. Let H(S)[H(Φ) ] denote the convex hull of S[Φ]. For any point p ∈ H(S), let O(p) be the set of points of Φ whose osculating plane at each of these points passes through p.1. THEOREM ([8]). If Φ has no multiple points and p ∈ H(Φ), then |0(p) | ≧ 3[4] when p is [is not] a vertex of Φ.2. THEOREM ( [9]). a) If the only self intersection point of Φ is a doublepoint and p ∈ H(Φ) is not a vertex of Φ, then |O(p)| ≧ 2.b) Let Φ possess exactly n vertices. Then(1)|O(p)| ≦ nforp ∈ H(S) and(2)if the osculating plane at each vertex of Φ meets Φ at exactly one point, |O(p)| = n if and only if p ∈ H(Φ) is not vertex.

On the Four-vertex theorem for space curves

1986 ◽  
Vol 27 (2) ◽  
pp. 166-174 ◽  
Author(s):  
Tibor Bisztriczky
Keyword(s):  
Space Curves ◽  
Vertex Theorem

scholarly journals The four-vertex theorem for a certain type of space curves

1937 ◽  
Vol 43 (10) ◽  
pp. 737-742 ◽  
Author(s):  
W. C. Graustein ◽  
S. B. Jackson
Keyword(s):  
Space Curves ◽  
Vertex Theorem

Particle and Astro-physics Challenge Kant’s Phenomenolism

1998 ◽  
pp. 323-333
Author(s):  
Lawrence H. Starkey
Keyword(s):  
General Relativity ◽  
Black Hole ◽  
Physical Basis ◽  
Primary Sources ◽  
One Dimension ◽  
Space Curves ◽  
Slowing Down ◽  
Parity Conservation ◽  
Charge Parity ◽  
Einstein’S General Relativity

For two centuries Kant's first Critique has nourished various turns against transcendent metaphysics and realism. Kant was scandalized by reason's impotence in confronting infinity (or finitude) as seen in the divisibility of particles and in spatial extension and time. Therefore, he had to regard the latter as subjective and reality as imponderable. In what follows, I review various efforts to rationalize Kant's antinomies-efforts that could only flounder before the rise of Einstein's general relativity and Hawking's blackhole cosmology. Both have undercut the entire Kantian tradition by spawning highly probable theories for suppressing infinities and actually resolving these perplexities on a purely physical basis by positing curvatures of space and even of time that make them reëntrant to themselves. Heavily documented from primary sources in physics, this paper displays time’s curvature as its slowing down near very massive bodies and even freezing in a black hole from which it can reëmerge on the far side, where a new universe can open up. I argue that space curves into a double Möbius strip until it loses one dimension in exchange for another in the twin universe. It shows how 10-dimensional GUTs and the triple Universe, time/charge/parity conservation, and strange and bottom particle families and antiparticle universes, all fit together.

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