The volume of a lattice polyhedron

1963 ◽  
Vol 59 (4) ◽  
pp. 719-726 ◽  
Author(s):  
I. G. Macdonald
Keyword(s):  
Coordinate System ◽  
Simplicial Complex ◽  
Jordan Curve ◽  
Affine Plane ◽  
Unit Area ◽  
The Real ◽  
Real Affine ◽  
Image Position ◽  
Lattice Polyhedron

Let L be the lattice of all points with integer coordinates in the real affine plane R2 (with respect to some fixed coordinate system). Let X be a finite rectilinear simplicial complex in R2 whose 0-simplexes are points of L. Suppose X is pure and the frontier Ẋ of X is a Jordan curve; then there is a well-known formula for the area of X in terms of the number of points of L which lie in X and Ẋ respectively, namelywhere L(X) (resp. L(Ẋ)) is the number of points of L which lie in X (resp. Ẋ), and μ(X) is the area of X, normalized so that a fundamental parallelogram of L has unit area.

Parabolic Differentiation

1963 ◽  
Vol 15 ◽  
pp. 546-562 ◽  
Author(s):  
N. D. Lane
Keyword(s):  
Affine Plane ◽  
Proved Theorem ◽  
The Real ◽  
Real Affine

The purpose of this paper is the study of parabolically differentiable points of arcs in the real affine plane. In Section 2, two different definitions of convergence of a family of parabolas are given and it is observed (Theorem 1) that these are equivalent. In Section 3, tangent parabolas at a point p of an arc A are discussed and it is proved (Theorem 2) that all the non-degenerate non-tangent parabolas of A through p intersect A at p or that all of them support. In Section 4, osculating parabolas are introduced and the condition that an arc be twice parabolically differentiable at a point p is stated.

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 A $q$-Queens Problem. I. General Theory

10.37236/4093 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Seth Chaiken ◽  
Christopher R. H. Hanusa ◽  
Thomas Zaslavsky
Keyword(s):  
General Theory ◽  
Affine Plane ◽  
Exact Formula ◽  
Weighted Graphs ◽  
Variable Size ◽  
Ehrhart Theory ◽  
The Real ◽  
Straight Line ◽  
Real Affine ◽  
Inside Out

By means of the Ehrhart theory of inside-out polytopes we establish a general counting theory for nonattacking placements of chess pieces with unbounded straight-line moves, such as the queen, on a polygonal convex board. The number of ways to place $q$ identical nonattacking pieces on a board of variable size $n$ but fixed shape is (up to a normalization) given by a quasipolynomial function of $n$, of degree $2q$, whose coefficients are polynomials in $q$. The number of combinatorially distinct types of nonattacking configuration is the evaluation of our quasipolynomial at $n=-1$. The quasipolynomial has an exact formula that depends on a matroid of weighted graphs, which is in turn determined by incidence properties of lines in the real affine plane. We study the highest-degree coefficients and also the period of the quasipolynomial, which is needed if the quasipolynomial is to be interpolated from data, and which is bounded by some function, not well understood, of the board and the piece's move directions.

Indecomposable quadratic bundles of rank 4n over the real affine plane

1983 ◽  
Vol 71 (3) ◽  
pp. 643-653 ◽  
Author(s):  
M. Ojanguren ◽  
Raman Parimala ◽  
R. Sridharan
Keyword(s):  
Affine Plane ◽  
The Real ◽  
Real Affine

Conical Differentiation

1964 ◽  
Vol 16 ◽  
pp. 169-190 ◽  
Author(s):  
N. D. Lane ◽  
K. D. Singh
Keyword(s):  
Projective Plane ◽  
Affine Plane ◽  
Parameter Family ◽  
Real Projective Plane ◽  
The Real ◽  
Parabolic Case ◽  
Real Affine

This paper follows naturally a note on parabolic differentiation (2) in which the parabolically differentiable points in the real affine plane were discussed. In the parabolic case, the four-parameter family of parabolas in the affine plane led to three differentiability conditions. In the present paper, the five-parameter family of conies in the real projective plane gives rise to four differentiability conditions and a point of an arc in the projective plane is called conically differentiable if these four conditions are satisfied. The differentiable points are classified by the nature of their families of osculating conies, superosculating conies, and their ultraosculating conies.

Moulton Affine Hjelmslev Planes

1978 ◽  
Vol 21 (2) ◽  
pp. 135-142 ◽  
Author(s):  
Catharine Baker
Keyword(s):  
Affine Plane ◽  
Local Rings ◽  
The Real ◽  
Zero Divisors ◽  
Real Affine ◽  
The One

Desarguesian affine Hjelmslev planes (D.A.H. planes) were introduced by Klingenberg in [1] and generalized by Lane and Lorimer in [2]. D.A.H. planes are coordinatized by affine Hjelmslev rings (A.H. rings) which are local rings whose radicals are equal to their sets of two-sided zero divisors and whose principal right ideals are totally ordered. In [5], ordered D.A.H. planes were defined and the induced orderings of their A.H. rings were discussed. In this note an ordered non-Desarguesian A.H. plane is constructed from an arbitrary ordered D.A.H. plane. The existence of such planes ensures that the discussion of ordered non-Desarguesian A.H. planes by J. Laxton in [3] is meaningful. The basic idea employed is essentially the same as the one used in the construction of the classical Moulton plane from the real affine plane (cf. [4])

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

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