Showing papers on "Affine transformation published in 1969"

Characterizations of Finite Projective and Affine Spaces

Abstract: 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 to4 if and only if there are positive integers v, k, and y, with μ > 1 and (μ – l)(v — k) ≠ (k — μ)2 such 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.

Principal homogeneous spaces for finite group schemes

Abstract: Here, a Galois A-object is exactly what geometers mean by a principal homogeneous space (PHS) for Spec A over Spec R, the Extzar refers to extensions of sheaves in the Zariski topology over R, and AD is the linear R-dual of A. We wish to give a new, short proof of this theorem having two advantages: It applies even when Spec R is replaced by an arbitrary prescheme, and it explains why only the Zariski topology is needed. The price paid is that a certain amount of machinery is used, the proof we give being less explicit than the original one. If 7r: Y-)X is an affine morphism of preschemes, we shall say that Y is locally a projective module over X if and only if for every affine open U in X, r(7r-(U), oy) is a projective r(u, ox) module. As an example, if X is noetherian and Y is finite and flat over X, then Y is locally a projective module over X.

Structures of the Affine Families of Switching Functions

Abstract: The equivalence relation that partitions switching functions into affine families leads to a natural graphic representation of the families. From the graphs, it is possible to derive information on the self-symmetries of switching functions and the relative symmetries of functions that are in the same family. A catalog of the structures of the 39 families of four-variable functions appears in the paper. The catalog and, more generally, the theory of affine equivalence are useful tools for the design of logic networks that contain exclusive-OR modules among the set of primitive building blocks.

Boolean Near-Rings

Abstract: In this paper we introduce the concept of Boolean near-rings. Using any Boolean ring with identity, we construct a class of Boolean near-rings, called special, and determine left ideals, ideals, factor near-rings which are Boolean rings, isomorphism classes, and ideals which are near-ring direct summands for these special Boolean near-rings. Blackett [6] discusses the near-ring of affine transformations on a vector space where the near-ring has a unique maximal ideal. Gonshor [10] defines abstract affine near-rings and completely determines the lattice of ideals for these near-rings. The near-ring of differentiable transformations is seen to be simple in [7], For near-rings with geometric interpretations, see [1] or [2].

Relativistische Bewegungsprobleme. IV Rotator‐Spinteilchen in schwachen Gravitationsfeldern

Abstract: The classical spin particle formerly described by a pole-dipole-like mass distribution is kinematicly realized by a non-complanar threeleg which is orthogonal to the particles worldline. Only twelve of the ten conservation laws following from LORENTZ-invariance and the nine conservation laws following from invariance against arbitrary affine leg transformations are linearly independent, corresponding to the particles twelve degrees of freedom. The interaction of the mass distribution with a weak gravitational field is introduced via the energy-momentum conservation laws, and for the affine rotator the interaction is introduced by means of an appropriate Lagrangian, the two descriptions being equivalent. The canonical formalism for the affine rotator is developed in some detail.

An application of a numerical criterion of ampleness

Abstract: It is proven that, if an affine segment of a complete surface contains all its singularities, the surface is then projective, and the complement to this affine segment is cut by a hyperplane upon some projective embedding of the surface.

Ergodic theory of G-spaces

Abstract: The thesis is in the form of three papers. In Paper I, affine transformations of a locally compact group are considered. In Part I, it is assumed that the group is not compact: it is shown that an affine transformation of an abelian or connected group cannot be ergodic unless the transformation is of one exceptional type. An attempt is made to obtain stronger conditions than non-ergodicity. Part II deals with compact groups: it is shown that an affine transformation of a compact group is ergodic if and only if it has a dense orbit. For a connected group, alternative conditions are given. In particular, it is shown that an affine transformation of a Lie group cannot be ergodic unless the group is a torus. Papers II and III are concerned with the entropy theory of a measure-preserving transformation. The entropy of a transformation T (denoted by h(T)) was introduced by Kolmogorov in 1958 (and later modified by Sinai) as a 'non-spectral invariant': two transformations cannot be isomorphic unless they have the same entropy. Zero entropy has a special significance. In general, every transformation has a unique part with zero entropy; if this part is trivial, the transformation is said to have completely positive entropy. It is very useful to know that a transformation has completely positive entropy: such transformations are mixing of all orders and invertible transformations are Kolmogorov automorphisms. Paper II considers the question of completely positive entropy when the measure space of the transformation T is a G-space for a compact separable group G. T is required to commute with G-action: T.g = g.T for all g in G, where o is a group endomorphism of G onto G. TS(G) denotes the induced transformation on the space of G-orbits. It is proved that if T is weakly mixing (has a continuous spectrum) and TS(G) has completely positive entropy, then T has completely positive entropy. This theorem 'lifts' the property of completely positive entropy from the orbit space to the fundamental space. In Paper III, it is shown that under suitable conditions (which are not in fact very restrictive) h(T) = h(TS(G)) + h(o)