Showing papers on "Affine transformation published in 1982"
TL;DR: In this article, a new type of spread is constructed by combining geometric, group theoretic and matrix methods, which correspond to affine translation planes, and new Kerdock sets are obtained having various interesting properties.
Abstract: In an orthogonal vector space of type $\Omega ^ + ( 4n,q )$, a spread is a family of $q^{2n - 1} + 1$ totally singular $2n$-spaces which induces a partition of the singular points; these spreads are closely related to Kerdock sets. In a $2m$-dimensional vector space over $GF ( q )$, a spread is a family of $q^m + 1$ subspaces of dimension m which induces a partition of the points of the underlying projective space; these spreads correspond to affine translation planes. By combining geometric, group theoretic and matrix methods, new types of spreads are constructed and old examples are studied. New Kerdock sets and new translation planesare obtained having various interesting properties.
138 citations
01 Jan 1982
89 citations
TL;DR: The pythagoras number (P(A) as discussed by the authors is the smallest number n^oo such that any sum of squares in a ring can be expressed as a sum of at most n squares in A. The number P (A) is an interesting, but very delicate, arithmetic invariant of the ring A; the explicit computation of P(A), is, in general, a difficult task.
Abstract: For a (commutative) ring A, thepythagoras number, P(A), ofA is the smallest number n^oo such that any sum of squares in A can be expressed äs a sum of at most n squares in A. For instance, P(f?) = l, P(Fq) = 2 (if 2\q), and, by Lagrange's Theorem, P (Z) = P (G) = 4. The number P (A) is an interesting, but very delicate, arithmetic invariant of the ring A; the explicit computation of P (A) is, in general, a difficult task. Given a ring, it is often far from easy even just to decide if P (A) is finite of infinite. For some results on P(A) in the recent literature, see, for example, [P], [P2], [CEP], [HJ], [R], [PeJ, [Pe2], [EL], [Br], [Pr], etc. In some of these papers, the invariant P (A) has appeared under an assortment of other names: for instance, "Pfister dimension" in [R], "Quadratstufe" in [Pe2], and "reduced height" in [HJ] and [L]. In this paper, following [Br], [Pr], we shall call P(A) the pythagoras number and hope that, in the future, other mathematicians will adopt the same terminology.
84 citations
TL;DR: In this article, it was shown that a sufficiently large class of incentive games with perfect or partial dynamic information admits an optimal incentive scheme that is affine in the available information, and explicit expressions for these affine incentive schemes are obtained.
Abstract: Through a geometric approach, it is shown that a sufficiently large class of incentive (Stackelberg) problems with perfect or partial dynamic information admits optimal incentive schemes that are affine in the available information. As a byproduct of the analysis, explicit expressions for these affine incentive schemes are obtained, and the general results are applied to two different classes of Stackelberg game problems with partial dynamic information.
79 citations
Journal Article•
74 citations
TL;DR: In this article, the authors assume that G is real semisimple throughout this paper and study the double coset decomposition H\\G/P for affine symmetric spaces, where G is a connected Lie group, σ an involutive automorphism of G, and H a subgroup of G satisfying (Gσ)oczHc:Gσ where Gσ = {xeG\\σ(x) = x} and σ 0 is the identity component of Gσ.
Abstract: Let G be a connected Lie group, σ an involutive automorphism of G and H a subgroup of G satisfying (Gσ)oczHc:Gσ where Gσ = {xeG\\σ(x) = x} and (Gσ)0 is the identity component of Gσ. Then the triple (G, H9 σ) is called an affine symmetric space. We assume that G is real semisimple throughout this paper. Let P be a minimal parabolic subgroup of G. Then the double coset decomposition H\\G/P is studied in [3] and [4]. Let P' be an arbitrary parabolic subgroup of G containing P. Then we have a canonical surjection
69 citations
01 Jan 1982
TL;DR: In this paper, the invertibility of multivariable non-linear control systems is studied and the necessary and sufficient conditions for invertability are derived. But the authors focus on the controllability of the control system.
Abstract: Abstract The paper deals with the invertibility of multivariable non-linear control systems. By using the recently developed theory on controlled invariant and controllability distributions necessary and sufficient conditions for invertibility are derived.
65 citations
TL;DR: In this article, the invertibility of multivariable non-linear control systems is studied and the necessary and sufficient conditions for invertability are derived. But the authors focus on the controllability of the control system.
62 citations
TL;DR: In this article, it was shown that a rank three projective module over a smooth affine 3-fold with prescribed Chern classes has a free direct summand of rank one (Cor. 2.4).
Abstract: On any smooth affine surface over an algebraically closed field, it is easy to construct rank two projective modules with given determinant as the first Chern class and an arbitrary rational equivalence class of a zero cycle as the second Chern class. We will briefly indicate this method in Section 1. Here we construct projective modules over smooth affine 3-folds with prescribed Chern classes and of the right rank. That is to say, we construct rank two projective modules with prescribed first and second Chem classes and rank three projective modules with prescribed first, second and third Chern classes (Theorem 2.1). As a corollary, using Roitman's theorem on torsion in zero cycles and Suslin's cancellation theorem, one gets that a rank three projective module over a smooth affine threefold with top Chem class zero has a free direct summand of rank one (Cor. 2.4). For example, over a rational threefold any projective module splits into a free module and a rank two projective module. Thus we obtain necessary and sufficient conditions in terms of appropriate Chern classes for modules to be efficiently generated. For example, on an affine rational 3-fold any line bundle is 3-generated and any rank two projective module is 4-generated. We also prove that any maximal ideal over a smooth affine variety of dimension d is the zero of a section of a rank d projective module (Theorem 3.1). For an affine 3-fold, if A3(X) = 0, we prove the validity of the Eisenbud-Evans estimate for finitely generated modules. We do not know the answers for most of these questions in higher dimensions. Sometimes we assume that the characteristic of the ground field is different from 2, 3 and 5 due to technical reasons.**
48 citations
TL;DR: In this paper, the equations of balance of mass, inertia, momentum and generalized moment of momentum for materials with affine structure are derived as consequences of the invariance under change of observer of a postulated law of energy balance.
Abstract: The equations of balance of mass, inertia, momentum and generalized moment of momentum for materials with affine structure are derived as consequences of the invariance under change of observer of a postulated law of energy balance.
TL;DR: In this article, it was shown that an affine nonlinear Hamiltonian system is "controllable" if and only if it is "observable", in the sense that strong accessibility implies local weak observability and vice versa.
Abstract: It is shown that an affine nonlinear Hamiltonian system is "controllable" if and only if it is "observable," in the sense that strong accessibility implies local weak observability and vice versa. Furthermore, it is shown that a nonminimal Hamiltonian system can be reduced to a locally weakly observable and strongly accessible system, in such a way that the reduced system is again Hamiltonian.
TL;DR: In this paper, the authors combine several results on related (or conjugate) connections, defined on banachable fiber bundles, and set up a machinery, which permits to study various transformations of linear connections.
Abstract: Combining several results on related (or conjugate) connections, defined on banachable fibre bundles, we set up a machinery, which permits to study various transformations of linear connections. Global and local methods are applied throughout. As an application, we get an extension of the classical affine transformations to the context of infinite-dimensional vector bundles. Another application shows that, realising the ordinary linear differential equations (in Banach spaces) as connections, we get the usual transformations of (equivalent) equations. Thus, some classical results on differential equations, such as the Theorem of Floquet, can have a “geometric” interpretation.
TL;DR: Finite affine planes are constructed admitting nonabelian sharply point-transitive collineation groups and are of two sorts: dual translation planes and planes of type II1 derived from them as discussed by the authors.
Abstract: Finite affine planes are constructed admitting nonabelian sharply point-transitive collineation groups These planes are of two sorts: dual translation planes, and planes of type II1 derived from them
TL;DR: In this paper, the connection between linear transitive groups of collineations and the algebraic description of projective (or affine) planes has been established for Laguerre-planes.
Abstract: Analogous to the wellknown results about the connection between linear transitive groups of collineations and the algebraic description of projective (or affine) planes, we give some statements for Laguerre-planes. In particular we use automorphisms with fixpoints, which induce in the residual plane of a fixpoint dilatations, translations, shears or reflections. We apply these methods in order to characterize certain ovoidal Laguerre-planes.
01 Jan 1982
TL;DR: The perturbation analyses done in this research verify the viability of using the parameters of a process model as a feature vector in a pattern recognition scheme.
Abstract: A method for the extraction of features for pattern recognition by system identification is presented. A test waveform is associated with a parameterized process model (PM) which is an inverse filter. The structure of the PM corresponds to the redundant information in a waveform, and the parameter values correspond to the discriminatory information. The PM used in this research is a linear predictive system whose parameters are the linear predictive coefficients (LPC's). This technique is applied to feature extraction of electrocardiograms (ECG's) for differential diagnosis. The LPC's are calculated for each ECG and used as a feature vector in a hypergeometric affine N-space spanned by the LPC's. The efficacy of this feature extraction technique is tested by three different perturbation methods, namely noise, matrix distortion, and a newly developed method called directed distortion. Both the Euclidean and Itakura distances between feature vectors in N-space are shown in increase with increasing perturbation of the template waveform. The monotonic behavior of a distance measure is a necessary attribute of a valid feature space. Thus the perturbation analyses done in this research verify the viability of using the parameters of a process model as a feature vector in a pattern recognition scheme.
TL;DR: A polynomial algorithm for the weighted 1-center problem (indeed minimization of the ratio of convex quadratic and an affine function over a polyhedral set) is presented.
Dissertation•
01 Jan 1982
TL;DR: In this article, it was shown that this result is still valid in arbitrary projective spaces; this provides a different and shorter proof of [3] in the finite case, and it is shown that every copolar space fully embedded in a finite projective space PG(n, q ), with q >, is the copolar spaces arising from a symplectic polarity.
TL;DR: In this article, a relatively simple algebraic derivation of the maximum likelihood estimates is given, when these estimates are based on single observed vectors from each of the $n$ populations and an independent estimate of the common covariance matrix.
Abstract: In a linear (or affine) functional model the principal parameter is a subspace (respectively an affine subspace) in a finite dimensional inner product space, which contains the means of $n$ multivariate normal populations, all having the same covariance matrix. A relatively simple, essentially algebraic derivation of the maximum likelihood estimates is given, when these estimates are based on single observed vectors from each of the $n$ populations and an independent estimate of the common covariance matrix. A new derivation of least squares estimates is also given.
14 Jun 1982
TL;DR: In this article, the authors study the derivation of optimal incentive schemes in two-agent stochastic decision problems with a hierarchical decision structure, in a general Hilbert space setting, where the agent at the top of the hierarchy is assumed to have access to the value of other agent's decision variable as well as to some common and private information.
Abstract: In this paper we study the derivation of optimal incentive schemes in two-agent stochastic decision problems with a hierarchical decision structure, in a general Hilbert space setting. The agent at the top of the hierarchy is assumed to have access to the value of other agent's decision variable as well as to some common and private information, and the second agent's loss function is taken to be strictly convex. In this set-up, it is shown that there exists, under some fairly mild structural restrictions, an optimal incentive policy for the first agent, which is affine in the dynamic information and generally nonlinear in the static (common and private) information. Certain special cases are also discussed and a numerical example is solved.