scholarly journals Some infinite families of finite incidence-polytopes

1990 ◽  
Vol 55 (1) ◽  
pp. 60-73 ◽  
Author(s):  
Asia Ivić Weiss
Keyword(s):  
Finite Incidence

scholarly journals BRUCK NETS AND PARTIAL SHERK PLANES

2017 ◽  
Vol 104 (1) ◽  
pp. 1-12
Author(s):  
JOHN BAMBERG ◽  
JOANNA B. FAWCETT ◽  
JESSE LANSDOWN
Keyword(s):  
Affine Plane ◽  
Finite Geometries ◽  
Affine Planes ◽  
Incidence Structures ◽  
Finite Affine ◽  
Odd Order ◽  
Finite Incidence ◽  
Desarguesian Affine Plane

In Bachmann [Aufbau der Geometrie aus dem Spiegelungsbegriff, Die Grundlehren der mathematischen Wissenschaften, Bd. XCVI (Springer, Berlin–Göttingen–Heidelberg, 1959)], it was shown that a finite metric plane is a Desarguesian affine plane of odd order equipped with a perpendicularity relation on lines and that the converse is also true. Sherk [‘Finite incidence structures with orthogonality’, Canad. J. Math.19 (1967), 1078–1083] generalised this result to characterise the finite affine planes of odd order by removing the ‘three reflections axioms’ from a metric plane. We show that one can obtain a larger class of natural finite geometries, the so-called Bruck nets of even degree, by weakening Sherk’s axioms to allow noncollinear points.

Colouring finite incidence structures

1986 ◽  
Vol 2 (1) ◽  
pp. 339-346 ◽  
Author(s):  
J. Csima ◽  
Z. Füredi
Keyword(s):  
Incidence Structures ◽  
Finite Incidence

An interactive program for finite incidence structures

1980 ◽  
Vol 14 (2) ◽  
pp. 174-181
Author(s):  
Marilena Barnabei ◽  
Giorgio Nicoletti
Keyword(s):  
Interactive Program ◽  
Incidence Structures ◽  
Finite Incidence

Finite Incidence Structures with Orthogonality

1967 ◽  
Vol 19 ◽  
pp. 1078-1083 ◽  
Author(s):  
F. A. Sherk
Keyword(s):  
Incidence Structure ◽  
Affine Planes ◽  
Axiom A ◽  
Incidence Structures ◽  
Finite Incidence

An incidence structure consists of two sets of elements, called points and blocks, together with a relation, called incidence, between elements of the two sets. Well-known examples are inversive planes, in which the blocks are circles, and projective and affine planes, in which the blocks are lines. Thus in various examples of incidence structures, the blocks may have various interpretations. Very shortly, however, we shall impose a condition (Axiom A) which ensures that the blocks behave like lines. In anticipation of this, we shall refer to the set of blocks as the set of lines. Also, we shall employ the usual terminology of incidence, such as “lies on,” “passes through,” “meet,” “join.” etc.

Hermite–Birkhoff interpolation by Hermite–Birkhoff splines

1981 ◽  
Vol 88 (3-4) ◽  
pp. 195-201 ◽  
Author(s):  
T. N. T. Goodman
Keyword(s):  
Unique Solution ◽  
Incidence Matrix ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Birkhoff Interpolation ◽  
Piecewise Polynomials ◽  
Finite Incidence ◽  
Derivatives Of

SynopsisWe consider interpolation by piecewise polynomials, where the interpolation conditions are on certain derivatives of the function at certain points, specified by a finite incidence matrix E. Similarly the allowable discontinuities of the piecewise polynomials are specified by a finite incidence matrix F. We first find necessary conditions on (E, F) for the problem to be poised, that is to have a unique solution for any given data. The main result gives sufficient conditions on (E, F) for the problem to be poised, generalising a well-known result of Atkinson and Sharma. To this end we prove some results involving estimates of the numbers of zeros of the relevant piecewise polynomials.

scholarly journals Characterizations of Finite Projective and Affine Spaces

1969 ◽  
Vol 21 ◽  
pp. 64-75 ◽  
Author(s):  
William M. Kantor
Keyword(s):  
Affine Space ◽  
Incidence Structure ◽  
Principal Result ◽  
List Type ◽  
Affine Spaces ◽  
Finite Projective Spaces ◽  
Finite Incidence ◽  
Finite Incidence Structure ◽  
Positive Integers ◽  
Projective And Affine Spaces

A well-known result of Dembowski and Wagner (4) characterizes the designs of points and hyperplanes of finite projective spaces among all symmetric designs. By passing to a dual situation and approaching this idea from a different direction, we shall obtain common characterizations of finite projective and affine spaces. Our principal result is the following.Theorem 1.A finite incidence structure is isomorphic to the design of points and hyperplanes of a finite projective or affine space of dimension greater than or equal to4if and only if there are positive integers v, k, and y, with μ> 1and(μ– l)(v — k) ≠ (k—μ)2such that the following assumptions hold.(I)Every block is on k points, and every two intersecting blocks are on μ common points.(II)Given a point and two distinct blocks, there is a block containing both the point and the intersection of the blocks.(III)Given two distinct points p and q, there is a block on p but not on q.(IV)There are v points, and v– 2 ≧k>μ.

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