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affine plane. That is, Artin starts with the affine plane axioms, given in terms of points and lines, and proceeds to construct a field associated with a given affine plane. In this paper we initiate the study of generalized affine planes ana consider the problem of associating an algebraic structure with these geometries. (Roughly speaking, a generalized affine plane is a geometry in which it is possible for two points to be incident with more than one line.) In Section 1 we introduce generalized affine planes and investigate some of their basic properties. In Section 2, we adjoin an additional axiom (uniform axiom) and investigate some properties of our new geometry. We now (Theorem 2.2) associate a near-ring to each generalized affine plane. The algebraic structure of this near-ring is considered in Section 3. As a result we find that one can distinguish the affine planes from the generalized affine planes by considering these associated near-rings.

A convex body is said to have constant diagonal if and only if the main diagonal of the circumscribed boxes has constant length. It is shown that ann-dimensional convex body,n≧3, is the affine image of a body of constant breadth if and only if it has constant diagonal. Affine images of bodies of constant breadth are also characterized by the property that the orthogonal projection on each hyperplane is the affine image of a body of constant breadth in that hyperplane.

In I, extended affine geometry obtained by extending the affine group parameters to functions of coordinates is established and is then realized in the differentiable manifolds. In II, a connections is introduced into the extended affine principal fibre bundle of the extended structure group.

An important characteristic of an affine space is that each of its hyperplanes is itself an affine space of one dimension lower. This property plays an essential part in the analysis of the structure of affine spaces. The purpose of this paper is to develop an analogous structure theory for general affine designs. This enables the introduction of a concept of dimension for affine designs and also yields techniques for constructing affine designs from smaller ones. The initial sections introduce the required basic results and terminology. In w is proved the main decomposition theorem which demonstrates how an affine design may be constructed with a given decomposition into sets of smaller affine designs. The results of this paper form part of my doctoral thesis at the University of London. To my supervisor Professor D.R. Hughes, I am indebted for invaluable assistance and guidance.

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.

In this paper the method of infinitesimal transformations of coordinates, used by Weyl to determine conditions that a function of the tensors gik and φi, and certain of their derivatives, should be a scalar density, is applied (with certain modifications so as to give tensor relations) to functions of and . It is known that in order that such a function should be a scalar density it must be a homogeneous function, of degree ½n, of , and this must of course be deducible from the equations found by the infinitesimal transformations. In view of the part which these equations may play, as “equations of energy,” etc., in purely affine field theories, it seems desirable that the connection should be explicitly shown, and this is done in § 3.

Formulae (27) and (28) which make it possible to estimate the minimum and maximum value of the deformation in length in affine transformation in space have been derived. Both values are determined by means of the elements of the given transformation matrixA without having to form the productAA T, or compute the corresponding eigenvalues explicitly. The derived formulae hold true under the assumption that both extreme values do not differ from each other too much.

If Ec are independent identically distributed random vari- ables with values 0 and 1, each with probability 2 then P( E zkk = O)(N-312 k=--N 7r

By Maleev's theorem a quasivariety of algebras is determined up to rational equivalence by its underlying concrete category (that is category with underlying set functor), so one loses nothing ff one identifies a quasivariety with its underlying concrete category. In this paper we show tha t the concrete category ~ of convex subsets of real affine spaces and restrictions to convex subsets of affine maps is an axiomatic quasivariety, and we describe the algebraic structure of ~. F~rrz OSTERMA~N and Jt)RGV, N SCHMIDT have shown in [4] tha t the concrete category 9 /of real affine spaces and affme maps is a variety, and have described the algebraic structure. Their result is stated in section 1. The axiomatisation of convex subsets of affine spaces as algebras is ~ v e n in section 2 and proved in section 3. The final section is devoted to a general discussion of the quasivariety ~ and the var iety generated by it. I n particular we show tha t !~ has only one nontrivial proper subvariety, namely the variety of semilattices. In the following affine spaces will always be affine spaces over the real numbers R. We remark tha t OSTERMA~N and SCHMIDT proved their result for affine spaces over more general rings.

This paper is concerned with the analysis and synthesis of the transition graphs of affine transformations on vector spaces over finite fields. A brief summary on the graphs of linear transformations is first presented. The graphs of affine transformations are then analyzed by means of the vector space structures and the divisibility of polynomials. It is shown that the graph of an affine transformation differs from that of its linear part only in the cycle lengths. The structures as well as the conditions for the existence of the affine cycles are pointed out. A method for evaluating the affine cycles is presented. Then the synthesis of graphs by affine transformations is considered. The synthesis techniques developed are similar to those for linear transformations. The connection between affine graphs and the realizability of linear machines is also pointed out and illustrated. With these results, it is shown that the synthesis of autonomous machines can be greatly simplified.

A affine dynamical system is a dynamical system whose attainability map is affine. It is shown that over a reflexive Banach space the class of affine dynamical systems is equivalent to the class of linear differential systems.

The class of almost affine functions is investigated. A function is called almost-affine if, on any segment contained in its domain of definition (supposed convex), it is either constant or monotonic.

The quantity called mp-vector is introduced of which components are transformed in the same way as the partial derivatives ∂ s x i/∂uα1... up to p-th order under tke transformations of coordinate system (x) and of parameter system (u). As the transformation is non-linear with respect to the components for p>1, its linear representation is given. There are stated some generalizations of mp-vector and some properties of the first variation of an mp-vector. A connection for mp-vector is defined which is named an extended affine connection, because it reduces to an affine connection for p=1.

An ordered plane [1, p. 177 ff.] is a set of undefined entities called points, together with an undefined relation of intermediacy, satisfying the following axioms: (The symbol [ABC] means ″B is between A and C″.)

where a is variable in V, A is fixed in V, and T is a linear transformation of V. The affine transformation is nonsingular if and only if T is nonsingular. A few familiar facts [1 ] are: 1. The nonsingular affine transformations form the "affine group" under iteration. 2. The translations, i.e. those A with T= I, form a normal subgroup of the affine group. 3. The quotient group of the affine group modulo the translation group is isomorphic to the group of nonsingular linear transformations. 4. If V is finite-dimensional with basis (a, , * , a.), let

The closed wedges in C(X) (the space of real continuous functions on a compact Hausdorff space X) which are also inf-lattices have been characterized by Choquet and Deny (2); see also (5). The present note extends their result to certain wedges of affine continuous functions on a Choquet simplex, the generalization being in the same spirit as the generalization of the Kakutani- Stone theorem obtained by Edwards in (4).I should like to thank my supervisor, Dr D. A. Edwards, for suggesting this problem and for his subsequent help. I am also grateful to the referee for correcting several slips.