Showing papers on "Affine transformation published in 1975"

Automorphisms of affine surfaces. i

Abstract: We study the group of automorphisms Aut(X) of an affine surface X which can be made complete by adding a zigzag. This study is based on the computation of the action of Aut(X) on a certain tree ΔX associated with the surface X. Our results are used to give a description of forms of the surface X and of algebraic subgroups of Aut(X).Bibliography: 15 items.

On equimultiple subvarieties of algebroid hypersurfaces.

Abstract: We deal with a finite projection πv of a hypersurface V in (r + 1)-space onto an affine r-space Ar and with the critical variety Δ of πv in Ar. We show that if Δ is equimultiple along a smooth subvariety W0, then W0 admits a unique lifting to a subvariety W of V (necessarily isomorphic to W0) and that also V is then equimultiple along W.

On Finite Non-Commutative Affine Spaces

Abstract: We consider incidence structures (cf. DEMBOWSKI [10,p.1]) consisting of a non-void set of elements called points, certain subsets of points called lines, an operation sometimes called join, mapping every ordered pair of different points surjectively onto the set of all lines and finally a binary relation on the lines called parallelism, all these things with additional properties gained in a natural way. The most known structures of this type are the well-known affine spaces (e.g. in the sense of Tamaschke [20,21]) but also many other examples have been developed in recent times (see e.g. Dembowski [10], Pickert [18], Sperner [19], Arnold [6], Wille [26], also Andre [1], Bachmann [8]).

Affine Barbilian-Ebenen II

Abstract: In part I we proved that every affine Barbilian plane is up to isomorphisms an affine geometry over a Z-ring R as defined in the introducing abstract there. Now we carry out that conversely all affine ring geometries over a Z-ring are affine Barbilian planes and represent their automorphisms algebraically as semilinear bijections. Finally, we present several classification theorems as, for instance, that the class of Desarguesian affine planes coinzides with the class of affine Barbilian planes, satisfying the additional axiom that two different points are always non-neighboured. The weaker condition that there is always exactly one line passing through two different points corresponds with the fact that the underlying ring is a right Bezoutring. There is at most one line passing through two different points iff the corresponding ring R has no zero-divisors.

Optimal policies for identification of stochastic linear systems

Abstract: The problem of designing closed-loop policies for identification of multiinput-multioutput linear discrete-time systems with random time-varying parameters is considered in this paper using a Bayesian approach. A sensitivity index gives a measure of performance for the closed-loop laws. The computation of the optimal laws is shown to be nontrivial, an exercise in stochastic control, but open-loop, affine, and open-loop feedback optimal inputs are shown to yield tractable problems. Numerical examples are given. For time-invariant systems, the criterion considered is shown to be related to the trace of the information matrix associated with the system.

Coordinate theorems for affine Hjelmslev planes

Abstract: It is shown that an affine Hjelmslev plane ℋ is a translation plane if and only if each of its coordinate biternary rings B=〈k, T, T0, 0, 1〉 are linear. Addition and multiplication in the ternary ring 〈k, T, 0, 1〉 are defined by a+b=T(a, 1, b) and a·b= =T(a, b, 0), respectively, and it is proved that every biternary ring of a translation plane has the additional properties that 〈k,+〉 is an abelian group 〈k, +, ·〉 is right distributive, and T(a, 1, b)=T0(a, 1, b). Moreover, if a single linear biternary ring of ℋ has these three properties, then ℋ is a translation plane. It is shown that a translation plane is Desarguesian if and only if it has a linear biternary ring such that T=T0 and 〈k, +, ·〉 is an affine Hjelmslev ring. Hessenberg’s theorem for affine Hjelmslev planes is proved, and a special configurational condition which is equivalent to the commutativity of multiplication in each biternary ring is introduced.

Harmonic and relatively affine mappings

Abstract: Let (M, F) be a manifold of dimension n with symmetric affine connection F, and (TV, F) a manifold of dimension p with symmetric affine connection F, where n,p>2. Let there be given a differentiable mapping f:M-*N which we denote sometimes by /: (M, F) —> (N,F). Manifolds, mappings and geometric objects which we discuss in this paper are assumed to be of differentiability class C°°. Take coordinate neighborhoods {U x} of M and {U,y} of N in such a way that f(U) C U, where (x) = (x\ x, , x) and (y) = (y, y, , y) are local coordinates of M and N respectively. The indices h, /, /, k, I, m, r, s, t run over the range {1,2, , ri\, and the indices a, β, γ, δ, λ, μ, v the range {T, 2, , p). The summation convention will be used with respect to these two systems of indices. Suppose that /: (M, F) —> (TV, F) is represented by equations

Transitive affine transformations on groups.

Abstract: An affine transformation T on a group G is an automorphism followed by a translation; T is transitive if for each JC, y E. G there is an integer n such that T (x) = y. All groups with transitive affine transformations are determined: the infinite cyclic and infinite dihedral group are the only infinite examples; while the finite examples are semi-direct products of certain odd-order groups by a cyclic, dihedral or quaternion 2-group. The automorphism groups of the above groups are described, and the automorphisms which occur as parts of transitive affine transformations are given.

A Kalman-Bucy Filtering Theory for Affine Hereditary Differential Equations

Abstract: There have been a number of different approaches to the filtering problem for delay systems, for example, Kwakernaak [12], Kushner and Barnea [11], Kailath [9] and Lindquist [13). The main theoretical contribution to the problem is by Lindquist, who proves a duality theorem between estimation and control for stochastic systems with time delay, using the (nonrandom) theory of linear functional differential equations as expounded by Halanay, Hale, Banks et al neatly avoiding the Riccati equation which occurs in the Kalman-Bucy theory. This paper incorporates a more direct approach and generalizes the Kalman-Bucy filtering theory for a class of linear delay equations, along the lines of Kwakernaak in [12]. This is done by formulating the problem as one in the abstract Hilbert space M2 introduced by Delfour and Mitter in their theory of affine hereditary differential equations in [7], and using a similar approach to that in “Infinite Dimensional Filtering” [4].