Showing papers on "Affine transformation published in 1976"

Affine relationships among variables of a program

Abstract: Several optimizations of programs can be performed when in certain regions of a program equality relationships hold between a linear combination of the variables of the program and a constant. This paper presents a practical approach to detecting these relationships by considering the problem from the viewpoint of linear algebra. Key to the practicality of this approach is an algorithm for the calculation of the "sum" of linear subspaces.

On the weight enumeration of weights less than 2.5d of Reed—Muller codes

Abstract: Let Pr be the set of all polynomials of degree r in m variables over GF(2). Polynomial ƒ in Pr is said to be affine equivalent to polynomial g in Pr, if ƒ is transformable to g by an invertible affine transformation of the variables. Any polynomial of weight less than 2m−r+1 + 2m−r−1 in Pr is shown to have a simple structure. By using this fact, we find out a set of representative polynomials such that any polynomial of weight less than 2m−r+1 + 2m−r−1 in Pr is affine equivalent to one and only one polynomial of the set. By counting the number of polynomials which are affine equivalent to each representative polynomial in the set, we derive explicit formulas for the enumerators of all weights less than 2.5d of Reed—Muller codes, where d is the minimum weight.

On linear planes

Abstract: A linear plane over a ground field k is an algebraic surface in affine 3-space over k which is biregular to the affine plane and whose equation is linear in one of the three variables of the 3-space. In this note we give a concrete description of a linear plane over a field of characteristic zero, thereby proving it to be an embedded plane, i.e. we show that by an automorphism of the affine 3-space, it can be transformed to a coordinate plane.

Error location apparatus and methods

Abstract: Location of multiple symbols in error is enhanced by using affine polynomial-based apparatus and methods. An algebraic trace function 0 (vector) provides a test for whether or not the location polynomial has roots in the field of interest. The selected roots then combine with previously calculated error-indicating syndromes to indicate symbols in error. Informational states developed for error location are employed to generate error patterns. Preferred apparatus includes a sequenced and buffered special purpose finite field processor, either of the random logic, array

Simple birational extensions of two dimensional affine rational domains

Abstract: © Foundation Compositio Mathematica, 1976, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Extreme affine transformations

Abstract: We classify the extreme points of the compact convex set of affine maps of IR n which map into itself the closed unit ball. This work is a preliminary step towards solving the problem of finding the extreme points of the compact convex set of affine maps of theN×N density matrices (dynamical maps of anN-level system) and forn=3 furnishes the solution of the problem in the simplest case of a two-level system.

Discrete and continuous bodies with affine structure

Abstract: The equations of motion of an affine body are derived (also for the case of variable mass) evidencing in particular the role of the generalized moment of momentum of internal forces. Successively, local dynamic and thermodynamic equations of balance for a continuous body with an affine microstructure are proposed, also for the case when such a body is one of the constituents in a mixture.

On the transformation of diffusion processes into the Feller process

Abstract: Due to the recently revived interest toward the diffusion process proposed by Feller in the population dynamics context, the class of Kolmogorov equations that can be transformed into a similar equation possessing linear infinitesimal variance and affine drift is determined. The underlying space-time transformation is explicitly provided, and a few examples are discussed.

Point processes on spaces of flats and other homogeneous spaces

Abstract: The present paper is primarily concerned with the structure of certain random sets of flats in n -dimensional Euclidean space R n . To be more specific, let be the space of k -dimensional affine subspaces of R n (to be called k -flats below), equipped with its natural topology ((14), p. 132). is a topological homogeneous space under the action of the group of Euclidean displacements in R n and our purpose is to give conditions under which a stochastic point process on , stationary under , is a doubly stochastic Poisson process. The results will be seen to apply equally well to abstract spaces sharing some of the structural properties of .

A combinatorial perspective on the Radon convexity theorem

Abstract: Matroid-theoretic methods are employed to compute the number of complementary subsets of points of a set S whose convex hulls intersect (a number Radon proved to be nonzero when S has an affine dependency). This number is shown to be an invariant only of the dependence structure of S. Strict bounds are given depending on the cardinality and dimension of S and the number is related to other matroid invariants.

Über die affin vollständigen, endlich erzeugbaren Moduln

Abstract: A universal algebraA is calledk-affine complete, if any function of the Cartesian powerAk intoA, which is compatible with all congruence relations ofA, is a polynomial function.A is called affine complete, if it isk-affine complete for every integerk. In this paper, all affine complete finitely generated modules are characterized. Moreover, the paper contains some results on functions compatible with all congruence relations of an algebra, and on affine complete algebras in general.

A Note on Tangentially Transitive Affine Planes

Abstract: Let $1 be a finite affine translation plane. If 21 contains a subplane, collineation group pair (2I0, A) such that: * A leaves each point of 2t0 fixed and acts transitively on the points of / \\ { / n 2I0}, for / a line of 21 with / n 2l0 a line of 2l0, then 21 is said to be tangentially transitive with respect to 2l0. Jha in his thesis [3] has proved that if 21 is tangentially transitive with respect to 2I0, then 2t has square order p 2m and, with the possible exception of p = 16, 2I0 has order p. Furthermore if 2I0 has order p , then 21 is a generalised Hall plane. In this note we show that up to isomorphism there is exactly one translation plane of order 16, tangentially transitive with respect to a subplane of order 2.-This completes the exceptional case in Jha's theorem. This translation plane has also been constructed by Lorimer [5], but its existence is implicit in the work of Johnson [4], since it is also tangentially transitive with respect to subplanes of order 4 (cf. 3.1) and as such is a generalised Hall plane. I should like to mention that the main result of this note has also been independently proved by N. L. Johnson and T. G. Ostrom.

Some Aspects of the Geometry of First-Quantized Theories

Abstract: An alternative method of identifying base-space and fiber tensors is presented and is used to describe interacting Dirac and Yang-Mills fields with “internal affine symmetry.” In a separate construction, the cosmological constant is derived from the fiber bundle structure of a generalized theory of Poincare gauge symmetry. Mathematical foundations of the computations are discussed.

Transversally affine foliations

Abstract: Let ℱ be a smooth foliation of codimension p on a smooth manifold Mm. We can define ℱ by an atlas of coordinate charts (U, (x, y)), called leaf charts, where (x, y): U → Rm−p × Rp are coordinate functions for which the leaves of ℱ are given by y1 constant,…,yp constant, in U. Clearly, on the overlap of two such leaf charts (U, (x, y)) and (U′, (x′, y′)) we have a coordinate transformation of the formIf y′ is always affine in y, i.e.where and Bi are constants, we shall say that ℱ is a transversally affine foliation. This notion is, in a sense, dual to that of affine foliation, see [2], in which x′ is affine in x and each leaf has an induced flat affine structure.

Locally dual affine geometries

Abstract: Publisher Summary This chapter describes finite geometries that are locally dual affine, that is, in which every subplane is a dual affine plane. A geometry is a finite set of points and a collection of point sets called lines. It is required that lines contain at least two points, that distinct lines have at most one point in common, and that any two points are connected by some path consisting of intersecting lines. A subgeometry is a subset containing all of the lines passing through any two of its points; a plane is a subgeometry generated by two intersecting lines. Two classical geometries can be made from a vector space V of dimension n over the Galois field GF(q). A slightly less classical geometry can be defined on V if V is endowed with a nonzero alternative bilinear form f. The points and lines of the symplectic geometry (relative to f) are one-dimensional subspaces of V that are not in the radical of f and the two-dimensional subspaces on which f is not identically zero. This geometry is determined by the dimension n of V and the dimension k of the radical of f, and so is denoted by Sp(n,k,q).

Caractérisations d'espaces de Pasch-Peano

Abstract: Characterizations of Pasch-Peano spaces. We show that complete convexity spaces (Bryant and Webster, Cantwell) of dimension at least three are convex subsets of real affine spaces. The proof uses an infinite-dimensional analogue of a result of de Groot and de Vries on convex sets in projective spaces.

On nullity distributions

Abstract: The nullity concept of Riemannian manifolds is extended to affine manifolds. Results obtained by Chern and Kuiper and Maltz on Riemannian manifolds are generalized to affine manifolds. A structure theorem for affine symmetric spaces is obtained. Finally, the nullity concept is generalized to study the partial integrability of certain geometric structures.

On Translation Planes with Affine Central Collineations, II

Abstract: This article, as the name implies, is a continuation of [9]. In that article the second author investigates finite translation planes containing both affine dations and affine homologies. (See the beginning of Section 2 for definitions.) Such translation planes are called Ei7-planes. In [9] the author restricted himself to translation planes of characteristic p ≧ 5. The main reasons for this were that Ostrom's and Hering's theorem [13; 4] on affine dations excluded the case p = 3 and the conclusions were easier to interpret geometrically when p ≧ 5 (as opposed to the case p = 2). Since then Ostrom [17] has settled the case p = 3.