A least upper bound for the number of singular points on normal arcs and curves of cyclic order four

1978 ◽  
Vol 7 (1) ◽  
Author(s):  
G. Spoar
Keyword(s):  
Upper Bound ◽  
Singular Points ◽  
Cyclic Order ◽  
Order Four

On Singular Points of Normal Arcs of Cyclic Order Four

1974 ◽  
Vol 17 (3) ◽  
pp. 391-396 ◽  
Author(s):  
G. Spoar ◽  
N. D. Lane
Keyword(s):  
Singular Points ◽  
Cyclic Order ◽  
Inversive Plane ◽  
Order Four

In [5] N. D. Lane and P. Scherk discuss arcs in the conformai (inversive) plane which are met by every circle at not more than three points; i.e., arcs of cyclic order three. This paper is concerned with the analysis of normal arcs of cyclic order four in the conformai plane.

Arcs of Parabolic Order Four

1964 ◽  
Vol 16 ◽  
pp. 321-338 ◽  
Author(s):  
N. D. Lane
Keyword(s):  
Affine Plane ◽  
Cyclic Order ◽  
Proved Theorem ◽  
Present Discussion ◽  
The Real ◽  
End Point ◽  
Real Affine ◽  
Order Four ◽  
Strong Differentiability

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.

scholarly journals Quantitative Regularity for p-Minimizing Maps Through a Reifenberg Theorem

2021 ◽  
Author(s):  
Mattia Vedovato
Keyword(s):  
Riemannian Manifolds ◽  
Upper Bound ◽  
Type Theorem ◽  
Regularity Result ◽  
Singular Points ◽  
Singular Set ◽  
Minkowski Content ◽  
Minimizing Maps ◽  
Quantitative Regularity ◽  
Appl Math

AbstractIn this article we extend to arbitrary p-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case $$p=2$$ p = 2 . We first show that the set of singular points of such a map can be quantitatively stratified: we classify singular points based on the number of almost-symmetries of the map around them, as done in Cheeger and Naber (Commun Pure Appl Math 66(6): 965–990, 2013). Then, adapting the work of Naber and Valtorta (Ann Math (2) 185(1): 131–227, 2017), we apply a Reifenberg-type Theorem to each quantitative stratum; through this, we achieve an upper bound on the Minkowski content of the singular set, and we prove it is k-rectifiable for a k which only depends on p and the dimension of the domain.

scholarly journals Differentiable curves of cyclic order four

1982 ◽  
Vol 102 (1) ◽  
pp. 209-220 ◽  
Author(s):  
Gary Spoar
Keyword(s):  
Cyclic Order ◽  
Order Four

scholarly journals On a certain class of Linear Substitutions with Common Invariants, and an Associated Substitution of Order Four

1912 ◽  
Vol 31 ◽  
pp. 54-70
Author(s):  
D. G. Taylor
Keyword(s):  
Cyclic Order ◽  
Root Of Unity ◽  
Image Position ◽  
Order Four

The determinanteach row of wliich contains the same n elements in the same cyclic order, with ′a1 always in the leading diagonal, is the product of n linear factors, which we shall write as followswhere ρ is any primitive nth root of unity.

scholarly journals Coloring Properties of Mixed Cycloids

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1539
Author(s):  
György Dósa ◽  
Nicholas Newman ◽  
Zsolt Tuza ◽  
Vitaly Voloshin
Keyword(s):  
Upper Bound ◽  
Cyclic Order ◽  
Discrete Structures ◽  
Mixed Hypergraph

In this paper, we investigate partitions of highly symmetrical discrete structures called cycloids. In general, a mixed hypergraph has two types of hyperedges. The vertices are colored in such a way that each C-edge has two vertices of the same color, and each D-edge has two vertices of distinct colors. In our case, a mixed cycloid is a mixed hypergraph whose vertices can be arranged in a cyclic order, and every consecutive p vertices form a C-edge, and every consecutive q vertices form a D-edge in the ordering. We completely determine the maximum number of colors that can be used for any p≥3 and any q≥2. We also develop an algorithm that generates a coloring with any number of colors between the minimum and maximum. Finally, we discuss the colorings of mixed cycloids when the maximum number of colors coincides with its upper bound, which is the largest cardinality of a set of vertices containing no C-edge.

Binary diagrams of mesomorphous compounds exhibiting singular points

1978 ◽  
Vol 3 ◽  
pp. 381-386 ◽  
Author(s):  
F. Hardouin ◽  
G. Sigaud ◽  
M.-F. Achard ◽  
H. Gasparoux
Keyword(s):  
Singular Points
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