Parabolic Differentiation

Lane, N. D.

doi:10.4153/cjm-1963-057-4

Cited by 5 publications

(9 citation statements)

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“…As a result, the classification of the do-differentiable points is rather more complicated than in the cases (i)-(iv), given in 2. However, the subsequent development still yields results parallel to those in the above cases [3], [4], [13].…”

Section: Let 12 Be Independent If P ~ M E Then E W {P} and Independentsupporting

confidence: 51%

Independence structures in the geometry of orders

Lane

Scherk²,

Turgeon

1987

Aeq. Math.

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Dedicated to Professor Otto Haupt with best wishes on his lOOth birthday1. An Arc [A curve] is the homeomorphic image of a segment [of the projective line].The geometry of orders of arcs and curves is developed in [2]. In a fundamental domain G, a set #g of curves and arcs K is given. The geometry of orders studies the structure of a given arc B, the basic arc, by means of the set #g. This set #g of characteristic curves has a dimension, the fundamental number k = k(#g). In the local theory, a point p on B, the base point, is introduced, and the behaviour of B near p is examined. For details we refer to the first pages of [2]. However, the following concepts are central: The #g-order of B is the number sup ] K n B I. The #g-order ofp is the #g-order Ke~" of a sufficiently small neighbourhood of p on B. The point p is #g-singular if this number is greater than k.2. Within the framework of the geometry of orders, various mathematicians have done direct differential geometry. Choosing some #g, they defined differentiability with respect to #g and classified the #g-differentiable points. Then they proved theorems on their #g-order. Their sets #g and fundamental domains G were

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Section: Let 12 Be Independent If P ~ M E Then E W {P} and Independentsupporting

confidence: 51%

Independence structures in the geometry of orders

Lane

Scherk²,

Turgeon

1987

Aeq. Math.

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“…The limit straight line X is the ordinary tangent of A at p. (ii) If A is differentiable, the non-degenerate, non-tangent parabolas through an interior point p of A all intersect A at p or all of them support (2,Theorem 2).…”

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confidence: 99%

“…(iii) Let A -p C ï*, s £ A -p. The two parabolas 7ri(<£; S) and 7r 2 (</>; S) of <t> C T at an end point p which pass through s can be numbered in such a way that lim 7T 2 (0; s) exists and is equal to the limit of the double ray through s with the vertex p (2,Lemma 17).…”

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confidence: 99%

“…(viii) The non-osculating tangent parabolas of A at an interior point all support A there (2,Theorem 6).…”

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confidence: 99%

“…(ix) Let p be an end point of A. If 0 is one of the subfamilies of r defined above and 0 ^ a, then the parabolas 7Ti(<£; s) of <£ through s £ A, i = 1, 2, converge to a double ray on X with the vertex p as s -> /> (2,Theorem 7).…”

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confidence: 99%

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Arcs of Parabolic Order Four

Lane

1964

Can. j. math.

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This paper is concerned with some of the properties of arcs in the real affine plane which are met by every parabola at not more than four points. Many of the properties of arcs of parabolic order four which we consider here are analogous to the corresponding properties of arcs of cyclic order three in the conformai plane which are described in (1). The paper (2), on parabolic differentiation, provides the background for the present discussion.In Section 2, general tangent, osculating, and superosculating parabolas are introduced. The concept of strong differentiability is introduced in Section 3; cf. Theorem 1. Section 4 deals with arcs of finite parabolic order, and it is proved (Theorem 2) that an end point p of an arc A of finite parabolic order is twice parabolically differentiable.

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Polynomial differentiability, order and characteristic

Gupta

Lane

1973

J Geom

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Parabolic Differentiation

Cited by 5 publications

References 1 publication

Independence structures in the geometry of orders

Independence structures in the geometry of orders

Arcs of Parabolic Order Four

Polynomial differentiability, order and characteristic

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