Showing papers on "Affine transformation published in 1980"

Parameter space design of robust control systems

Abstract: Find a state or output feedback with fixed gains such that nice stability (defined by a region in the eigenvalue plane) is robust with respect to large plant parameter variations, sensor failures, and quantization effects in the controller. Keep the required magnitude of control inputs small in this design. A tool for tackling such problems by design in the controller parameter space K is introduced. Pole placement is formulated as an affine map from the space P of characteristic polynomial coefficients to the K space. This allows determining the regions in the K space, which place all eigenvalues in the desired region in the eigenvalue plane. Then tradeoffs among a variety of different design specifications can be made in K space. The use of this tool is illustrated by the design of a crane control system. Several open research problems result from this approach: graphical computer-aided design of robust systems, algebraic robustness conditions, and algorithms for iterative design of robust control systems.

An analogue of Bäcklund's theorem in affine geometry

Abstract: It is well-known that there is a correspondence between solutions of the Sine-Gordon equation (SGE) -~—,~Hr= sin q> dx dt Y and the surfaces of constant curvature -1 in R (see below). The classical Backlund transformation of such surfaces furnishes a way to generate new solutions of the SGE from a given solution. This has received much attention in recent studies of the soliton solutions of the SGE, and the technique has been used successfully in the study of other non-linear evolution equations. In the first section of this paper we present a simple derivation of the classical Backlund theorem and its applications by using the method of moving frames. Our main result concerns affine minimal surfaces. They arise as the solution of the variation problem for affine area. The corresponding Euler-Lagrange equation is a fourth order partial differential equation. In §2,we develop the basic properties of affine minimal surfaces.In §3 we study the transformation of affine surfaces by realizing them as the focal surfaces of a line congruence. The natural conditions that the congruence be a W-congruence and that the affine normals at corresponding points be parallel lead to the conclusion that both surfaces are affine minimal. This is the content of Theorem 4, the main result of our paper. As in the classical case, the Theorem leads to the construction of new affine minimal surfaces from a given one by the solution of a completely integrable system of first order partial differential equations. 1. The classical Backlund theorem and its consequences. Let M be a surface in R. We choose a local field of orthonormal frames v\, v O3 be the dual coframe of Vi, v2, #3. We can write dx 2^0ava a s Here and throughout this paper we shall agree on the index ranges (1.2) 1 £ij,k£2, 1 £a,s,r H 3. The structure equations of R are *Work done under partial support of NSF grant MCS74-23180. **Work done under partial support of NSF Grant MCS76-01692. Received by the editors on April 13, 1978. Copyright © 1980 Rocky Mountain Mathematics Consortium 105 106 S. S. CHERN AND C. L. TERNG dda = 2] Os A Osa, 6as + Osa = 0 ddas = 2 0«r A 0ri8. Restricting these forms to the frames defined above, we have (1.4) 03 = 0 and hence (1.5) 0 =

Hyperbolic affine hyperspheres

Abstract: A locally strongly convex hypersurface in the affine space Rn + 1 is called an affine hypersphere if the affine normals (§ 1) through each point of the hypersurface either all intersect at one point, called its center, or else are all mutually parallel It is called elliptic, parabolic or hyperbolic according to whether the center is, respectively, on the concave side of the hypersurface, at infinity or on the convex side This class of hypersurfaces was first studied systematically by W Blaschke ([1]) in the frame of affine geometry In his paper [3] E Calabi redefined it and proposed a problem of determining all complete hyperbolic affine hyperspheres and raised a conjecture that these hypersurfaces are asymptotic to the boundary of a convex cone and every non-degenerate cone V determines a hyperbolic affine hypersphere, asymptotic to the boundary of V, uniquely by the value of its mean curvature

Collinearity-preserving functions between Desarguesian planes

Abstract: Using concepts from valuation theory, we obtain a characterization of all collinearity-preserving functions from one affine or projective Desarguesian plane into another. The case in which the planes are projective and the range contains a quadrangle has been treated previously in the literature. Our results permit one or both planes to be affine and include cases in which the range contains a triangle but no quadrangle. A key theorem is that, with the exception of certain embeddings defined on planes of order 2 and 3, every collinearity-preserving function from one affine Desarguesian plane into another can be extended to a collinearity-preserving function between enveloping projective planes.

Collinearity-preserving functions between Desarguesian planes

Abstract: Using concepts from valuation theory, we obtain a characterization of all collinearity-preserving functions from one affine or projective Desarguesian plane into another. The case in which the planes are projective and the range contains a quadrangle has been treated previously in the literature. Our results permit one or both planes to be affine and include cases in which the range contains a triangle but no quadrangle. A key theorem is that, with the exception of certain embeddings defined on planes of order 2 and 3, every collinearity-preserving function from one affine Desarguesian plane into another can be extended to a collinearity-preserving function between enveloping projective planes.

Team decision theory for linear continuous-time systems

Abstract: This paper develops a team decision theory for linear-quadratic (LQ) continuous-time systems. First, a counterpart of the well-known result of Radner on quadratic static teams is obtained for two-member continuous-time LQ static team problems when the statistics of the random variables involved are not necessarily Gaussian. An iterative convergent scheme is developed, which in the limit yields the optimal team strategies. For the special case of Gaussian distributions, the team-optimal solution is affine in the information available to each DM, and for the further special case when the team cost function does not penalize the intermediate values of state, the optimal strategies can be obtained by solving a Liapunov type time-invariant matrix equation. This static theory is then extended to LQG continuous-time dynamic teams with sampled observations under the one-step-delay observation sharing pattern. The unique solution is again affine in the information available to each DM, and further, it features a certainty-equivalence property.

Path Integrals for Affine Variables

Abstract: The phase-space path-integral quantization of systems subject to p > 0 is studied. The nonexistence of a q representation is circumvented by adopting affine variables rather than canonical ones, and the phase-space path integral attains a meaningful interpretation in terms of affine coherent states. A Lagrangian path-integral quantization finds its proper interpretation in terms of a two-dimensional system rather than an obvious one-dimensional one. Throughout, the formalism is applied to an elementary model of quantum gravity; nonetheless, the results for this model may have implications for the full theory of quantum gravity.

Mapping image properties into shape constraints: skewed symmetry, affine-transformable patterns, and the shape-from-texture paradigm

Abstract: In this paper we demonstrate two new approaches to deriving three-dimensional surface orientation information (“shape”) from two-dimensional image cues. The two approaches are the method of affine-transformable patterns and the shape-from-texture paradigm. They are introduced by a specific application common to both: the concept of skewed symmetry. Skewed symmetry is shown to constrain the relationship of observed distortions in a known object regularity to a small subset of possible underlying surface orientations. Besides this constraint, valuable in its own right, the two methods are shown to generate other surface constraints as well. Some applications are presented of skewed symmetry to line drawing analysis, to the use of gravity in shape understanding, and to global shape recovery.

Traversing Large Pieces of Linearity in Algorithms that Solve Equations by Following Piecewise-Linear Paths

Abstract: Simplicial or fixed-point algorithms trace piecewise-linear paths to approximate solutions of systems of nonlinear equations. We present improved methods that can traverse several simplices simultaneously. The technique applies particularly to functions with special structure, but also generally, since large pieces are always induced by the artifical affine function. The special structures considered are separability and partial separability.

Affine manifolds and solvable groups

Abstract: Let M be a compact affine manifold. Thus Af has a distinguished atlas whose coordinate changes are locally in Aff(&), the group of affine automorphisms of Euclidean w-space E. Assume M is connected and without boundary. The universal covering M of M has an affine immersion D: M—+E which is unique up to composition with elements of Aff(E). Corresponding to D there is a homomorphism a: n —> Aff (2?), where n is the group of deck transformations of Af, such that/) is equivariant for a. Set a(7r) = T. LetZ: Aff(E)~-* GL(E) be the natural map.

Affine Barbilian structures

Abstract: W.Leissner has characterized, by geometric axioms, the affine Barbilian planes over a Z-ring (i.e, a ring with 1 such that ab=1 ⇒ ba=1) [10].The aim of the present paper is to characterize correspondingly the affine Barbilian planes over an arbitrary ring with 1. First we shall deal with the translation Barbilian planes, which generalize Leissner's parallelodromic planes [11]. The paper concludes with a study of the kernel of the translation Barbilian plane.

Collineation groups irreducible on the components of a translation plane

Abstract: We investigate finite translation planes of odd dimension over their kernels in which the translation complement induces on each component l a permutation group whose order is divisible by a p-primitive divisor. Using results of this investigation, we show that rank 3 affine planes of odd dimension over their kernels are either generalized Andre planes or semi-field planes. A similar result is given for translation planes having a collineation group which is doubly transitive on each affine line; besides the above two possibilities, there is a third possibility; the plane has order 27, the translation complement is doubly transitive on $$\mathfrak{U}$$ , and SL(2, 13) is contained in the translation complement. We also consider translation planes of odd dimension over their kernels which have a collineation group isomorphic to SL(2, w) with w prime to 5 and the characteristic, and having no affine perspectivity. We show that such planes have order 27, the prime power w=13, and the given group together with the translations forms a doubly transitive collineation group on {ie153-1}. This indicates quite strongly that the Hering translation plane of order 27 is unique with respect to the above properties.

The little Desarguesian theorem for algebras in modular varieties

Abstract: 0. Introduction. In [2] R. Freese and B. Jonsson have shown that the congruence lattices of algebras in modular varieties are in fact arguesian. Since the congruence lattice of an algebra W may be thought of as the (abstract) projective geometry of W it is natural to ask whether there might be an affine analogue to this result. An affine geometry, Kongruenzklassengeometrie by name, was introduced and investigated by R. Wille in [7]. Simply take the elements of W as points and every congruence class as a subspace. In this general setting there is no transition nor even a connection known between these two geometries. Thus it may be surprising that there is an affine analogue to the above-mentioned result of Freese and Jonsson. Namely, as we are about to prove, the "Little Desarguesian Theorem" is true in the Kongruenzklassengeometrie of algebras in modular varieties.

Notes on infinitesimal variations of submanifolds

Abstract: Publisher Summary Submanifolds under consideration are compact and orientable and using integral formulas, some global results on infinitesimal isometric, affine and conformal variations of the submanifolds, are obtained. Infinitesimal variations, infinitesimal affine variations and infinitesimal conformal variations are discussed. If the variation is conformal and R is a constant, the infinitesimal variation is homothetic. If an infinitesimal affine variation of a compact orientable submanifold of a Rieiizannian manifold satisfies then the variation is an isometry.

Strain and stress field calculations for precipitates and dislocation loops using affine transformations

Abstract: Starting from the covariant formulation of the linear theory of elasticity the elastic strain and stress fields inside and outside precipitates are calculated using appropriate affine coordinates. Explicit integral formulae well suited for numerical evaluation are derived for ellipsoidal, cylindrical, polyhedral, and plate-like particles as well as rhomboidal and elliptical dislocation loops, allowing for an arbitrary homogeneous stress-free strain and elastic anisotropy. The application of the results in precipitate structure analysis by diffraction contrast image matching will be considered in a subsequent paper.