Showing papers on "Affine transformation published in 1983"

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.

Optimal stationary control for dynamic systems with Markov perturbations

Abstract: Affine continuous and discrete-time dynamic systems with homogeneous jump Markov perturbations are considered and the existence of an optimal stationary control under a quadratic cost is discussed. In order to solve this problem some new stability results for linear systems with Markov perturbations are given.

On a question of Quillen

Abstract: Let R be a regular local ring, and F a regular parameter of R. Quillen asked whether every projective R f -module is free. We settle this question when R is a regular local ring of an affine algebra over a field k. Further, if R has infinite residue field, we show that projective modules over Laurent polynomial extensions of R f are also free.

Invertibility of nonlinear analytic single-input systems

Abstract: Necessary and sufficient conditions for the invertibility of nonlinear analytic single-input systems are derived in this note, extending previous work on the class of systems affine in control. A construction of a left-inverse system is also included.

Surfaces of class vii0 and affine geometry

Abstract: This paper gives a complete and detailed proof of the theorem to the effect that the only surfaces of class VII0 with b2=0 are the Inoue-Kodaira surfaces. Besides that, it contains several results on manifolds with affine structures whose holonomy groups are commutative; in particular, the general case is reduced to the case when the holonomy group is diagonal. Bibliography: 16 titles.

On the relationship between global and local controllability

Abstract: Given a pair(X0,D), whereX0 is a vector field andD is a family of vector fields on a manifoldM, we can define a system affine in the control, i.e., the new family of all the vector fields of the formX0 +uX, whereX ∈D andu is a real parameter. Such a system will be called globally controllable if each state is reachable from each other, whenever unbounded values foru are allowed. It is proved that a system affine in the control is globally controllable if and only if in any set of a suitable partition ofM there exist points locally controllable with bounded values foru. Further, it is proved that, under more restrictive assumptions, global controllability implies the existence of points locally controllable at a fixed time with bounded values foru. In the case of simply connected manifolds, a full equivalence among all the forms of controllability considered here is obtained.

Embedding Semi-Algebraic Spaces

Abstract: Let R be any real closed field (which we take to be totally ordered) and let A" denote n-dimensional affine R-space. An affine semi-algebraic set is a subset of some A" which has a finite description in terms of polynomial equalities and inequalitities. A semi-algebraic space is an object obtained by pasting finitely many semi-algebraic sets together along open semi-algebraic subsets and may be thought of as the semi-algebraic version of an integral separated R-scheme of finite type. Semi-algebraic spaces were introduced in [2] by Delfs and Knebusch and continue to be the objects of intensive and fruitful investigation by these authors. The question which we address is "Can every semi-algebraic space be embedded in some affine space?" This question was raised in [2] and the answer is provided by

A hybrid asymptotic-finite element method for stiff two-point boundary value problems*

Abstract: An accurate and efficient numerical method has been developed for a nonlinear stiff second order two-point boundary value problem. The scheme combines asymptotic methods with the usual solution techniques for two-point boundary value problems. A new modification of Newton's method or quasilinear- ization is used to reduce the nonlinear problem to a sequence of linear problems. The resultant linear problem is solved by patching local solutions at the knots or equivalently by projecting onto an affine subset constructed from asymptotic expansions. In this way, boundary layers are naturally incorporated into the approximation. An adaptive mesh is employed to achieve an error of O(1/N2)+O(/e). Here, N is the number of intervals and e << is the singular perturbation parameter. Numerical computations are presented.

Local similarity manifolds

Abstract: A local similarity manifold is defined as a locally affine manifold for which the transition functions of an affine atlas are similarity transformations inRn. The main result of this paper is that, for n≧3, the compact local similarity manifolds (which are not locally Euclidean) are given by the formula M=(Rn{0} G, where G is a group of covering transformations such that G={ht0k¦h e H, k eZ, H being a finite orthogonal group without fixed points inRn{0},and t0 being some conformal linear transformation ofRn which commutes with H.

Blocking Sets in Affine Planes

Abstract: Publisher Summary This chapter discusses the blocking sets in affine planes. AG(2,q) (PG(2,q)) denotes the affine (projective) plane over GF(q) the finite field of order q. π is considered as a finite projective plane of order n. A blocking set in π is a subset S of the points of π satisfying the two conditions: (1) each line of π contains one point in S and (2) each line of π contains one point not in S. S is irreducible if no proper subset of S is also a blocking set. This is equivalent to saying that each point P of S lies on at least one tangent to S. The chapter discusses the derivation of new bounds on the size of irreducible blocking sets in the classical projective planes.

Affine Structure and Invariant Policies for Dynamic Programs

Abstract: This paper studies a family of sequential decision processes whose affine structure is shown to imply that attention can be restricted to policies that select the same decision for all states. A sequence of n invariant policies (decisions) is shown to be optimal when the planning horizon is n epochs. When the planning horizon is infinite, added structure is imposed, and a stationary invariant policy is shown to be optimal. Computational methods and examples are included.

Gravity, supergravities and integrable systems

Abstract: Around 1968 three wonderful concepts emerged in different places and in seemingly unrelated domains of mathematical physics. They are the Kac-Moody algebras (among them the “affine” Kac-Moody algebras are related to current algebras and to gauge groups over one-dimensional “space-times”), the method of inverse scattering (for nonlinear partial differential equations in two-dimensional space-times), and finally the dual string model which is a two-dimensional field theory describing extended particles moving in a space-time of dimension 26 (10 or 2 if one dresses the string with internal degrees of freedom). In the last two years it was realized that gravity and supergravities provide a three-legged bridge between them and this revived hopes (at least with the author) of breaking the 2-dimensionality constraint for the integrability of interesting nonlinear problems. We shall not here discuss the Yang-Mills self-duality equations for lack of space ; they effectively are reduced to two-dimensions by considering the anti-self-dual null 2-planes. After reviewing the known connections between the 3 concepts listed above, we shall present the table of internal Lie symmetries of the Poincare (super)- gravities in various numbers of dimensions. Finally, we shall see that a Kac-Moody group (affine type I) plays important roles as a) transformation group of solutions, b) parameter space where fields take their values, c) phase-space.

Affine Geometries Obtained from Projective Planes and Skew Resolutions on AG(3,q)

Abstract: Publisher Summary This chapter presents a construction of AG(n+l,q) from the projective plane PG(2,qn). This construction can be applied to construct the skew resolutions of the lines in AG(n,q) and pairs of orthogonal resolutions. It is shown that there exists a skew resolution of the lines in AG(3,q) by constructing a resolution of the lines in PG(3,q). A skew resolution in AG(n,q) along with the natural resolution of lines in AG(n,q) obtained from parallelism form a pair of orthogonal resolutions. The finite affine geometry AG(n,q) is usually obtained from a vector space over a Galois field. A finite affine plane is more simply defined to be a (q2,q,l)-BIBD. It is well known that an affine plane has a unique resolution (parallelism).

An affine invariant theory and its application in computational geometry

Abstract: In this paper, an affine invariant theory of parametric curves is first introduced into computational geometry by the authors. We prove a basie theorem that for a parametric curve of degreen, there are m(n-m) - 2 intrinsic affine invariants in an m-dimensional affine hyperspace. We further perfect the theory which is of importance to application in industries, and obtain, among other things, sufficient conditions for being convex everywhere on the plane Bezier curve of degreen; elassification of the plane cubic Bezier curves; distribation of inflexion-points of the plane quartic Bezier curve; cusps and double points for a class of the plane quintic parametric curves. Part of these results has been put into use in the shipbuilding, aeronautic, and automobile industry.

Affine Maczyński logics on compact convex sets of states

Abstract: Sets of affine functions satisfying Maczynski orthogonality postulate and defined on compact convex sets of states are examined. Relations between affine Maski logics and Boolean algebras when the set of states is a Bauer simplex (classical mechanics, some models of nonlinear quantum mechanics) are studied. It is shown that an affine Maczynski logic defined on a Bauer simplex is a Boolean algebra if it is a sublattice of a lattice consisting of all bounded affine functions defined on the simplex.

The effect of affine transformation on redundancy analysis

Abstract: Canonical redundancy analysis provides an estimate of the amount of shared variance between two sets of variables and provides an alternative to canonical correlation. The proof that the total redundancy is equal to the average squared multiple correlation coefficient obtained by regressing each variable in the criterion set on all variables in the predictor set is generalized to the case in which there are a larger number of criterion than predictor variables. It is then shown that the redundancy for the criterion set of variables is invariant under affine transformation of the predictor variables, but not invariant under transformation of the criterion variables.

On a Characterization of the Grassmann Spaces Associated with an Affine Space

Abstract: Affine Grassmann spaces are defined and investigated. Under suitable assumptions any such space is proved to be isomorphic to the incidence structure (G h (IA), F h (IA), where IA is an affine space, G h (IA) is the family of its h-di-mensional subspaces and F h (IA) is the family formed by pencils of h-dimensional subspaces, the incidence relation being the usual set-theoretic inclusion.

Resistance value and field distribution of rectangular anisotropic resistive region

Abstract: High accurate resistance values and field distributions for a rectangular anisotropic resistive region are presented by using an affine transformation and a Schwarz-Christoffel transformation. Interesting comparison between the field distribution in a Hall generator and the present result is also presented.

Near rings associated with Sperner spaces

Abstract: In this paper we introduce Sperner spaces with operators, denoted by TSO, and associate to each such space a near-ring. The associated near-ring provides a generalization of the kernel of an affine translation space. We obtain a characterization of those TSO for which the associated near-ring is a near-field. Special attention is given to finite TSO's with cyclic monoids of operators.

Sufficient conditions for optimal time-delay systems with applications to functionally constrained control problems

Abstract: Sufficient conditions for affine time-delay systems with state-space constraints are derived, and it is shown that the results are applicable to a control problem involving a novel type of functional constraint. The discrete-time version of sufficiency is also obtained. Numerical examples are given to compare the functional control constraints with the conventional pointwise control constraint.

Orbits on real affine symmetric spaces Part I: the infinitesimal case

Abstract: An exposition is given of the infinitesimal orbit theory on real affine symmetric spaces. Main references are the results by Kostant and Rallis on complex symmetric spaces and the treatment by Varadarajan of the theory of orbits under the adjoint group. Partial results are due to Oshima and Matsuki. In a final chapter we consider the problem of the existence of invariant measures on the orbits in symmetric spaces.

On Certain Linear Congruence Class Geometries

Abstract: Summary A subset of the points of a finite affine desarguesian geometry, equipped with the induced subspace structure and the induced parallelism, is called a representable geometry. The question is answered as to which repre-sentable geometries are congruence class geometries in the sense of Wille, i.e. allow “enough” dilatations to generate the induced subspace structure and the parallelism. Each geometry induced on a subset of the points of an affine desarguesian geometry satisfies the Exchange Axiom for subspaces, and hence is linear, i.e. every line is determined uniquely by any two of its points. Thus, using results of Wille, Pasini and Herzer on finite linear congruence class geometries, one easily obtains an answer to the above question (Theorem 3.2.5). Moreover, the problem is solved as to which finite congruence class geometries have the same points and subspaces (but possibly a different parallelism) as a representable geometry (Theorem 3.2.6). Corollary 3.2.7 is a consequence of this theorem, stating that a finite congruence class geometry satisfies the Exchange Axiom, if all its subplanes do.

A Graphic Characterization of the Lines of an Affine Space

Abstract: A set of axioms is given which characterizes the incidence structure (R(A),F(A)) where R(A) denotes the family of lines in an affine space A, and F(A) the planar pencils.