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 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.

Union and Extension of Arcs of Cyclic Order Three

1968 ◽  
Vol 20 ◽  
pp. 629-638
Author(s):  
K. D. Singh ◽  
N. D. Lane
Keyword(s):  
Related Problem ◽  
Present Note ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Cyclic Order ◽  
Inversive Plane ◽  
Necessary And Sufficient

In (2) Lane and Scherk discussed differentiate points of arcs in the conformai (inversive) plane. Arcs A3 of cyclic order three were discussed in (3; 4). In the present note we give necessary and sufficient conditions for the union of two A3's to be an A3 (Theorem 1), and for an A3 to be extensible to a larger one (Theorem 2). The related problem of extending arcs in projective n-space was dealt with by Haupt in (1) and Sauter in (5; 6).

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

Effect of proximity of the Fermi level to singular points in the band structure on the kinetic and lattice properties of metals and alloys

1988 ◽  
Vol 154 (3) ◽  
pp. 525 ◽  
Author(s):  
V.P. Antropov ◽  
Valentin G. Vaks ◽  
M.I. Katsnel'son ◽  
V.G. Koreshkov ◽  
A.I. Likhtenshtein ◽  
...  
Keyword(s):  
Band Structure ◽  
Fermi Level ◽  
Metals And Alloys ◽  
Singular Points
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