Plane transformations in a complex setting III: similarities

This is the third part of a study of plane transformations described in a complex setting. After the study of homotheties, translations, rotations and reflections, we proceed now to the study of plane similarities, either direct or inverse. Their group theoretical properties are described, and their action on classical geometrical objects is studied.     

Plane transformations in a complex setting II: isometries

This paper is the second part of a study of plane transformations using a complex setting. The first part was devoted to homotheties and translations, now attention is turned towards plane isometries. The group theoretic properties of plane isometries are easy to derive and images of classical geometrical objects by these transformations are easily computed in this environment.     

On non-singular transformations of a measure space. I

We consider a Lebesgue measure space (M, , m). By an automorphism of (M, , m) we mean a bi-measurable transformation of (M, , m) that together with its inverse is non-singular with respeot to m. We study an equivalence relation between these automorphisms that we call the weak equivalence. Two automorphisms S and T are weakly equivalent if there is an automorphism U such that for almost all x M U maps the S-orbit of x onto the T-orbit of U x. Ergodicity, the existence of a finite invariant measure, the existenoe of a -finite infinite invariant measure, and the non-existence of such measures are invariants of weak equivalenoe. In this paper and in its sequel we solve the problem of weak equivalenoe for a class of automorphisms that comprises all ergodic automorphisms that admit a -finite invariant measure, and also certain ergodic automorphisms that do not admit such a measure.     

Nonlinear coordinate transformations for unconstrained optimization I. Basic transformations

In this two-part article, nonlinear coordinate transformations are discussed to simplify unconstrained global optimization problems and to test their unimodality on the basis of the analytical structure of the objective functions. If the transformed problems are quadratic in some or all the variables, then the optimum can be calculated directly, without an iterative procedure, or the number of variables to be optimized can be reduced. Otherwise the analysis of the structure can serve as a first phase for solving unconstrained global optimization problems.The first part treats real-life problems where the presented technique is applied and the transformation steps are constructed. The second part of the article deals with the differential geometrical background and the conditions of the existence of such transformations.The paper was presented at the II. IIASA Workshop on Global Optimization, Sopron (Hungary), December 9–14, 1990.     

Parametric Bäcklund transformations I: Phenomenology

We begin an exploration of parametric Bäcklund transformations for hyperbolic Monge-Ampère systems. (The appearance of an arbitrary parameter in the transformation is a feature of several well-known completely integrable PDEs.) We compute invariants for such transformations and explore the behavior of four examples, two of which are new, in terms of their invariants, symmetries, and conservation laws. We prove some preliminary results and indicate directions for further research.

     

Holomorphically projective transformations on complex Riemannian manifold

The aim of the paper is to prove that if a complex Riemannian manifold with holomorphic characteristic connection is holomorphically projective equivalent to a locally symmetric space then it is a complex Riemannian manifold of pointwise constant holomorphic characteristic sectional curvature.Dedicated to N.K. Stephanidis on the occasion of his 65 th birthday.     

On birational transformations of pairs in the complex plane

This article deals with the study of the birational transformations of the projective complex plane which leave invariant an irreducible algebraic curve. We try to describe the state of the art and provide some new results on this subject.      

Colouring Arcwise Connected Sets in the Plane I

Let ? be the family of finite collections ? where ? is a collection of bounded, arcwise connected sets in ℝ² which for any S, T∈? where S∩T≠∅, it holds that S∩T is arcwise connected. We investigate the problem of bounding the chromatic number of the intersection graph G of a collection ?∈?.  Assuming G is triangle-free, suppose there exists a closed Jordan curve C⊂ℝ² such that C intersects all sets of ? and for all S∈?, the following holds: (i) S∩(C∪int (C)) is arcwise connected or S∩int (C)=∅. (ii) S∩(C∪ext (C)) is arcwise connected or S∩ext (C)=∅.  Here int(C) and ext (C) denote the regions in the interior, resp. exterior, of C. Such being the case, we shall show that χ(?) is bounded by a constant independent of ?. Revised: December 3, 1998     

Minimal-energy splines: I. Plane curves with angle constraints

This is the first in a series of papers on minimal-energy splines. The paper is devoted to plane minimal-energy splines with angle constraints. We first consider minimal-energy spline segments, then general minimal-energy spline curves. We formulate problems for minimal-energy spline segments and curves, prove the existence of solutions, justify the Lagrange multiplier rules, and obtain some nice properties (e.g., the infinite smoothness). Finally, we report our computational experience on minimal-energy splines.     

I. Wards theory in an abstract setting

Annali di Matematica Pura ed Applicata (1923 -) -     

On triangularization of complex matrices by two-dimensional unitary transformations

A theoretical discussion is presented concerning convergence difficulties in applying Jacobi-like methods for triangularization of general complex matrices.This work was performed under Contract DA-ARO(D)-31-124-G 388, Basic Research in Numerical Analysis, a grant given to the University of Texas by the Army Research Office (Durham).     

解整数线性规划中选择割平面的一个方法

本指出了[4]的一个错误。用最速下降规则，给出了选择割平面的一个新方法。一个复杂的例子说明，该方法有一定的实用价值。     

Old wine in fractal bottles I: Orthogonal expansions on self-referential spaces via fractal transformations

Our results and examples show how transformations between self-similar sets may be continuous almost everywhere with respect to measures on the sets and may be used to carry well known notions from analysis and functional analysis, for example flows and spectral analysis, from familiar settings to new ones. The focus of this paper is on a number of surprising applications including what we call fractal Fourier analysis, in which the graphs of the basis functions are Cantor sets, discontinuous at a countable dense set of points, yet have good approximation properties. In a sequel, the focus will be on Lebesgue measure-preserving flows whose wave-fronts are fractals. The key idea is to use fractal transformations to provide unitary transformations between Hilbert spaces defined on attractors of iterated function systems.     

Normals in a Minkowski Plane

In this paper we will establish bounds on the average number of normals through a point in a convex body in a Minkowski plane for certain classes of convex bodies. Also, a related Euler relation is discussed.     

Reducing complex matrices to condensed forms by unitary congruence transformations

We show that any n × n conjugate-normal matrix can be brought by a unitary congruence transformation to block-tridiagonal form with the orders of the consecutive diagonal blocks not exceeding 1, 2, 3, ..., respectively. The proof is constructive; namely, a finite process is described that implements the reduction to the desired form. Sufficient conditions are indicated for the orders of the diagonal blocks to stabilize. In this case, the condensed form is a band matrix.