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Showing papers on "Affine transformation published in 1974"



Journal ArticleDOI
TL;DR: The definition of a projector under a semileast square inverse of a complex matrix is given in this article, where the same concept can also be defined in terms of projectors under seminorms.

63 citations


Journal ArticleDOI
TL;DR: In this paper, a modified holomony group was proposed for affine structures on real 2-dimensional torus T, where the affine structure on T is a maximal atlas whose coordinate transformations belong to the universal covering group of the identity component of A(2).
Abstract: In these two papers we intend to study the space of all affine structures on the real 2-dimensional torus T, a problem suggested by C. Ehresmann in 1936, or more specifically by S. S. Chern in one of his lectures and attacked by N. H. Kuiper [6] among others. An affine structure on a mainfold is a maximal atlas whose coordinate transformations belong to the affine transformation group A(ri) on the affine space. Our main purpose is to describe the set {Γ} of all affine structures on T module the group Diίf \\T2]e\\ here Diff [T ],, is the group of all diίfeomorphisms of T which induce the identity on the fundamental group π^T). The space {r}/Diff[T2]e, equipped with an appropriate topology, is regarded as an affine version of the Teichmϋller space. In the usual case the holonomy group H of an affine structure on a mainfold is defined as a subgroup of the affine transformation group A(n) up to the conjugate class. In this work, however, we construct a modified holomony group if* for an affine structure so that in the case of 2-dimensional affine torus the group H* is a subgroup ofA(2)e, the universal covering group of the identity component of A(2). We do this in such a way that the modified holonomy group H* is mapped onto the usual holonomy group by the projection mapping. With this modification of the holonomy group the first main result in the paper could be summarized as follows (Theorem 3.3 and 4.15): the affine structures on T are completely determined by their modified holonomy groups #*. Carrying out the determination of holonomy groups Jΐ*, we describe the space {Γh} IΌiff[T 2]e of the homogeneous affine structures on T . As Y. Matsushima [7] discusses for complex tori in a somewhat different way, we show the following (Theorem 3.10): the space {ΓA}/DiίF[Γ 2]β is an affine algebraic or, more precisely a 4-dimensional quadratic cone in R without singularities variety, except at the vertex, the vertex itself corresponding to the natural affine

59 citations




Journal ArticleDOI
TL;DR: In this paper, it was shown that a strongly n-uniform affine Hjelmslev plane A has an extension to projective Hjelslev planes if and only if the parallelism of A is even.
Abstract: An affine Hjelmslev plane is a near affine Hjelmslev plane with a parallelism. It is proved that every strongly n-uniform near affine Hjelmslev plane possesses an even parallelism and, if n > 2, uneven parallelisms as well. Secondly, we prove that a strongly n-uniform affine Hjelmslev plane A has an extension to a strongly n-uniform projective Hjelmslev plane if and only if the parallelism of A is even. The above results are applied to yield the first known examples of affine Hjelmslev planes which possess no extensions to projective Hjelmslev planes.

15 citations




Book ChapterDOI
01 Jan 1974

8 citations


Journal ArticleDOI
TL;DR: Two generalizations of affine Hjelmslev planes in which the parallel axiom is not required to hold are made, and Integer invariants are obtained for the finite planes in these new classes.

6 citations





01 Jan 1974
TL;DR: In this paper, a method was devised, using an affine transformation, to relate the aerial photographs or topographic maps to the tapes, which can be used for the registration of two tapes for the same area and for the geometric correction of images.
Abstract: During the development of a project to estimate wheat production, it became necessary to pull data, corresponding to particular fields in a test site, off an ERTS computer compatible tape. Aerial photographs and topographic maps were on hand for the test site. A method was devised, using an affine transformation, to relate the aerial photographs or topographic maps to the tapes. One can thereby access data on the tape corresponding to regions covered by only a few pixels. The theory can be used for the registration of two tapes for the same area and for the geometric correction of images.


Journal ArticleDOI
TL;DR: In this article, the existence of projective plane pathological with respect to the collineation groups of its sub and quotient planes was shown to be expressable in the following theorem.
Abstract: In [7] the author showed the existence of projective plane pathological with respect to the collineation groups of its sub and quotient planes. Similar pathologies are obtainable with respect to collineation groups of associated affine planes. (i.e. the affine planes obtained by distinguishing a line as the line at infinity) as expressable in the following theorem.

Journal ArticleDOI
TL;DR: In this paper, the authors prove two results about the continuous maps F, from the space of d-dimensional convex bodies K of ℝ d into the non-empty compact sets of K, which are subadditive and invariant by affine permutations.
Abstract: We prove two results about the continuous maps F, from the space of d-dimensional convex bodies K of ℝ d into the space of non-empty compact sets of ℝ d , which are subadditive and invariant by affine permutations.

Journal ArticleDOI
TL;DR: In this article, the problem of associating an algebraic structure with affine planes has been studied, and it has been shown that one can distinguish the affine plane from the generalized plane by considering these associated near-rings.
Abstract: 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.




Journal ArticleDOI
TL;DR: Jellett's theorem about the resolution of the space, A(X) of continuous affine functions on a compact ahoquet simplex X into a direct sum is generalized to simplex spaces.
Abstract: Jellett's theorem about the resolution of the space, A(X) of continuous affine functions on a compact ahoquet simplex X into a direct sum is generalized to simplex spaces. A new characteristic property of the space A(X), dual tol1(Γ), is given.