BRUCK NETS AND PARTIAL SHERK PLANES

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.

Hermite–Birkhoff interpolation by Hermite–Birkhoff splines

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.

Characterizations of Finite 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.