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Showing papers on "Affine transformation published in 1972"


Journal ArticleDOI
TL;DR: Given a point to set mapf on a simplex with certain conditions, an algorithm for computing fixed points is described, which operates by following the fixed point as an initially affine function is deformed towardsf.
Abstract: Given a point to set mapf on a simplex with certain conditions, an algorithm for computing fixed points is described. The algorithm operates by following the fixed point as an initially affine function is deformed towardsf.

317 citations



Journal ArticleDOI
TL;DR: In this article, the authors studied the control of affine hereditary differential systems with initial data in the space $M^2 $ and proved the existence and characterization of the optimal feedback operator for the system.
Abstract: This paper is concerned with two aspects of the control of affine hereditary differential systems. They are (i) the theory of various types of controllability and observability for such systems and (ii) the problem of optimal feedback control with a quadratic cost. The study is undertaken within the framework of hereditary differential systems with initial data in the space $M^2 $ (cf. Delfour and Mitten [6], [7]). The main result of this paper is the existence and characterization of the optimal feedback operator for the system.

144 citations


Journal ArticleDOI
TL;DR: A theorem for certain categories that generalizes Ramsey's Theorem is established, strong enough to establish G-C and imply the Ramsey theorem for n-parameter sets.

134 citations


Journal ArticleDOI
TL;DR: In this paper, the rank of an incidence matrix of all points vs. all e-spaces of a finite d-dimensional projective or affine space is the number of points of the geometry, where 1 <_e<-d-1.
Abstract: It is welt-known that the rank of each incidence matrix of all points vs. all e-spaces of a finite d-dimensional projective or affine space is the number of points of the geometry, where 1 <_e<-d-1 (see [1], p. 20). In this note we shall generalize this fact: Theorem. Let 0 <-e < f <= d-e-1, and let Me, l be an incidence matrix of all e-spaces vs. all f-spaces of PG(d, q) or AG(d, q). Then the rank of M~,f is the number of e-spaces of the geometry. We shall only prove the theorem in the case of AG(d, q). The projective case is similar and simpler. We note that the same proof shows that,./f 1 < e < f < d-e , an incidence mat~'i-r of all e-sets vs. all f-sets of a set of d points has rank (de) The relevant definitions are found in [t], w167 1.3 and 1,4. The dimension of a subspace X of a projective space will be denoted dim(X). The empty subspace has dimension-1,

106 citations


Journal ArticleDOI
TL;DR: In this article, the notions of controllability and observability for an affine abstract system defined in a Hilbert space with initial data, controls and observations also belonging to Hilbert spaces were systematically studied.
Abstract: This paper systematically studies the notions of controllability and observability for an affine abstract system defined in a Hilbert space with initial data, controls and observations also belonging to Hilbert spaces. Necessary and sufficient conditions are obtained in that framework and the duality property is studied. This theory can find applications in the study of “boundary controllability” and “boundary observability” for parabolic partial differential equations. Specific results have been obtained for affine hereditary differential systems defined in the $M^2 $-space framework (cf. Delfour and Mitter [1], [2], [5]).

63 citations



Journal ArticleDOI
TL;DR: In this article, a generalized version of the Ramsey theorem for finite affine spaces is presented, which is sufficiently general to include as special cases the finite vector space analog to Ramsey's theorem.
Abstract: In this paper we present a Ramsey theorem for certain categories which is sufficiently general to include as special cases the finite vector space analog to Ramsey’s theorem (conjectured by Gian-Carlo Rota), the Ramsey theorem for n-parameter sets [21, as well as Ramsey’s theorem itself [4, 61. The Ramsey theorem for finite affine spaces is obtained here simultaneously with that for vector spaces. That these two are equivalent was already known [5, II, and the arguments previously used to show that the affine theorem implies the projective theorem are also special cases of the results of this paper.

26 citations


Journal ArticleDOI
TL;DR: The Baer subplane as mentioned in this paper is the largest possible proper subplane of a projective or an affine plane, and it is defined as a configuration of projective and affine planes with an improper line.
Abstract: Let π be a projective or an affine plane ; a configuration C of π is a subset of points and a subset of lines in π such that a point P of C is incident with a line I of C if and only if P is incident with I in π. A configuration of a projective plane π which is a projective plane itself is called a projective subplane of π, and a configuration of an affine plane π’ which is an affine plane with the improper line of π‘ is an affine subplane of π‘. Let π be a finite projective (respectively, an affine) plane of order n and π 0 a projective (respectively, an affine) subplane of π of order n0 different from π; then n0 ≦ . If n0 = , then π 0 is called a Baer subplane of π. Thus, Baer subplanes are the “biggest” possible proper subplanes of finite planes.

22 citations



Journal ArticleDOI
TL;DR: In this paper, an analogous structure theory for general affine designs is developed, which enables the introduction of a concept of dimension for affine spaces and also yields techniques for constructing affine structures from smaller ones.
Abstract: 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.



Journal ArticleDOI
TL;DR: In this article, it was shown that the Jacobson radical of a ring is the intersection of its maximal left (or right) ideals, where k is the algebraic closure of k. Since k is perfect, the coradical of fc(g)fc A is k®kR.
Abstract: PROOF. The Jacobson radical of a ring is the intersection of its maximal left (or right) ideals. Dually the coradical of a coalgebra C over a field is identical with the socle of C as a right (or left) C-comodule. Since k is perfect, the coradical of fc(g)fc A is k®kR, where k is the algebraic closure of k. Hence we can assume that k — k. Moreover A can be assumed to be finitely generated as a fc-algebra. Let Vi9 i = 1, 2 be two finite dimensional right A-comodules. Becauce G(A°) = Algfc(̂ 4., k) is dense in A* = Hom^A, k) [6, Lem. 3.6], V* is a semisimple A-comodule iff Vi is a semisimple left G(A°)-module. Hence by the remark above if V4 are semisimple, then Vι ® V2 is also semi-simple. This means that R (x) R is a semisimple right A-comodule. Since the multiplication μ: A (x) A —> A is a right A-comodule map, R R is contained in R. Clearly R is stable under the antipode of A. Hence R is a sub-Hopf algebra of A.






Journal ArticleDOI
TL;DR: In this paper, the description of all linear subspaces and the automorphism group of the affine n-space over a planar nearfield is given. But the description is restricted to linear subspace.
Abstract: Two proceeding papers were devoted to the description of all linear subspaces and of the automorphism group of the “affine” n-space over a planar nearfield.

Journal ArticleDOI
TL;DR: In this paper, it was shown that there are 2-sided non-abelian finite incidence-groups with abelian affine kernel, which are derivations of finite incidence groups.
Abstract: In the first part a derivation method for incidence-groups is developed. The connections to the usual derivations of near-rings are shown in part 2 and examples are constructed. We characterize those finite slit incidence-groups with special affine kernel which are derivations of abelian incidence-groups. Applying these results to special classes of finite incidence-groups we show that there are 2-sided non-abelian finite incidence-groups with abelian affine kernel. The class of derivations of abelian finite incidence-groups contains all splitting a-2-sided incidence-groups with abelian kernel and no splitting kernel-2-sided non-2-sided incidence-group with abelian kernel. In the last part the a-2-sided incidence-groups are algebraically described. The results were partly communicated at the “Conference on Geometry” in March 1971 at the University of Waterloo, Ontario, Canada.

Journal ArticleDOI


Book ChapterDOI
01 Jan 1972

Journal ArticleDOI
01 Feb 1972
TL;DR: In this article, a necessary and sufficient condition that a simply connected affine symmetric space M such that K coincides with the linear isotropy group dHp at some point p in M is given.
Abstract: Let K be a subgroup of the general linear group GL(n). The author found a necessary and sufficient condition that there exist an «-dimensional simply connected affine symmetric space M such that K coincides with the linear isotropy group of all affine automorphisms of M at some point in M. Let M he an w-dimensional manifold with affine connection, A(M) the group of all affine automorphisms of M, HB the subgroup of A(M) consisting of all elements of A(M) which fix a point p in M, and dHv the linear isotropy group determined by Hp. Let V be an «-dimensional vector space, GL(n) the general linear group of V, and K a subgroup of GL(n). We shall find a necessary and sufficient condition that there exists a simply connected affine symmetric space M such that K coincides with the linear isotropy group dHp at some point p in M. We discussed similar problems for a Riemannian symmetric space [6]. First of all we shall prove the following: Lemma. Let T be a tensor in V®V*®V*®V* which satisfies the following conditions. (1) T,jkl = — T.jtk, (2) r.m + r.klj + r.lik = o, / 3\ -TU 'T'A 'T'A fi T-A nri T-A T-l _ A \D) i -hmn1 ikl l -jmn1 -hkl * -kmn1 -jhl L -tinn1 -ikh — u> where T.'m are the components of T. Then there is an affine symmetric space whose curvature tensor at some point of it coincides with T. Proof. We integrate the following differential equations. dcofdt = da' + akô4, dô^/dt = T.kjla¡of, with initial conditions (w')(=0=0, ((74)i=0=0. The solutions ¿ô*, œ'k are linear forms in da1, • • ■ , da" whose coefficients are integral functions of t, a1, • • ■ , an. If we set r=l and replace a1 by x\ we have forms tü'(jc, dx), w)(x, dx). Since the determinant of the Received by the editors June 16, 1971. AMS 1970 subject classifications. Primarv 53C35.