Vertices of planar curves under the action of linear transformations

The number of vertices of a smooth Jordan curve with nowhere vanishing curvature can change under the action of a nonsingular real linear transformation. We examine the bifurcation set in the space of linear transformations for the number of vertices on the image curve, showing that generally there is a codimension-one set of linear transformations making an arbitrary point into a vertex, and obtaining conditions that the point be capable of being transformed into a higher vertex. We demonstrate that there is always an open set of linear transformations such that the image curves have at least six vertices. Received 23 July 1999; revised 17 March 2000.     

ON THE LEFT-DEFINITENESS OF S-HERMITIAN SYSTEMS OF DIFFERENTIAL EQUATIONS

Abstract

Initiated by Schneider and Niessen's conjecture that left-definite S-Hermitian systems are essentially canonical, a certain class of transformations of such systems is considered, and criteria for the left-definiteness are derived. The results seem to disprove the conjecture.     

Optimal transformations for scalar sequences

Summary We consider transformations which accelerate convergence in some specified classes of convergent sequences. As an asymptotic measure of acceleration we introduce the order of transformation. We find a sharp upper bound on the order and show the explicit form of transformations of maximal order. We consider also the efficiency of transformations for fast convergent sequences. As a special case we find that the Germain-Bonne version of Richardson extropolation has maximal order for linearly convergent sequences.     

Some generalizations of the Euler-Knopp transformation

Summary The purpose of this paper is to construct a generalization of the Euler-Knopp transformation. Using this, one may recover previously known transformations, derive new transformations useful for numerical calculations and derive generating functions and other formulas of theoretical interest involving well-known functions.This work was supported by the Applied Mathematical Science Research Subprogram of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38     

Solving linear ordinary differential equations by exponentials of iterated commutators

Summary A sequence of transformations of a linear system of ordinary differential equations is investigated. It is shown that these transformations produce new systems which represent progressively smaller perturbations of the original set of equations.The transformations are implemented as a basis of a numerical method. Order, stability and error control of this method are analyzed. Numerical examples demonstrate the potential of this approach.     

Remarks on t-transformations of measures and convolutions

A family of transformations of probability measures is constructed, and used to define transformations of convolutions. The relations between moments and cumulants of a measure and its transformation are presented. For transformed classical and free convolutions the central limit measures and the Poisson type limit measures are computed. Families of non-commutative random variables are constructed, which are associated to these central limit measures. They provide examples of “position operators” which act on the Interacting Fock Spaces.     

Acceleration methods based on convergence tests

Summary In this paper we propose an acceleration method based on a general convergence test and depending on an auxiliary sequence (x _n). For different choices of (x _n) we obtain some known and some new transformations and for each one sufficient conditions for acceleration are given.     

A class of generalized Ornstein transformations with the weak mixing property

It is shown that there is a class of Ornstein transformations, distinct from the Ornstein mixing class, which are almost surely weak mixing     

Symmetries in the stochastic calculus of variations

Summary. Given a stochastic action integral we define a notion of invariance of this action under a family of one parameter space-time transformations and a notion of prolonged transformations which extend the existing analogs in classical calculus of variations. We prove that a family of prolonged transformations leaves the action integral invariant if and only if it leaves invariant the heat equation associated to it as well as the structure of the extremals. We then prove a stochastic version of Noether theorem: to each family of transformations leaving the action invariant (or symmetries) we can associate a function which gives a martingale when taken along a process minimizing the action under endpoint constraints. Received: 29 June 1996 / In revised form: 19 July 1996     

Computing theCS decomposition of a partitioned orthonormal matrix

Summary This paper describes an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns.This work was supported by the Air Force Office of Scientific Research under Contract No. AFOSR-82-0078     

Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions

In this paper we develop an integer-affine classification of three-dimensional multistory, completely empty convex marked pyramids. We apply it to obtain the complete lists of compact two-dimensional faces of multidimensional continued fractions lying in planes at integer distances 2, 3, 4, …  to the origin. The faces are considered up to the action of the group of integer-linear transformations.     

Fast Givens rotations for orthogonal similarity transformations

Summary Fast Givens rotations with half as many multiplications are proposed for orthogonal similarity transformations and a matrix notation is introduced to describe them more easily. Applications are proposed and numerical results are examined for the Jacobi method, the reduction to Hessenberg form and the QR-algorithm for Hessenberg matrices. It can be seen that in general fast Givens rotations are competitive with Householder reflexions and offer distinct advantages for sparse matrices.     

Frobenius type theorems in the noncommutative case ∗

The classification of semilinear transformations preserving the determinant over a noncommutative local ring is given. For local Ore domains we obtain a classification of Adjamagbo determinant preservers. The developed technique allows us to classify Dieudonné determinant preservers for division rings which may be infinite dimensional over the center.     

Corrigendum

In [] the authors constructed, for each r∈ (0,1), a function from a homogeneous tree of degree 2t to the open unit disk in the complex plane mapping the root of the tree to 0 and any pair of neighbors to points at hyperbolic distance r. Correspondingly, there is a free group of rank t of Möbius transformations such that the vertices of the tree are mapped to the orbit of 0, and the edges to geodesic arcs. One of the principal results of [] is to show that this function is an embedding of the tree when cos(π /2t)≤ r<1. This corrigendum fills a gap in the argument.     

On a class of Jacobi-like procedures for diagonalising arbitrary real matrices

Summary A new class of elementary matrices is presented which are convenient in Jacobi-like diagonalisation methods for arbitrary real matrices. It is shown that the presented transformations possess the normreducing property and that they produce an ultimate quadratic convergence even in the case of complex eigenvalues. Finally, a quadratically convergent Jacobi-like algorithm for real matrices with complex eigenvalues is presented.     

Finitary set endofunctors are alg-universal

A category is said to be alg-universal, if every category of universal algebras can be fully embedded into it. We prove here that the category of finitary endofunctors of the category Set is alg-universal. We also present an example of a proper class of accessible set functors with no natural transformations between them (except the obvious identities). Presented by J. Adámek. The research was supported by the Grant Agency of the Czech Republic under the grant No. 201/06/0664 and by the project of Ministry of Education under the No. MSM 0021620839. Received February 14, 2006; accepted in final form April 2, 2006.     

A distance on curves modulo rigid transformations

We propose a geometric method for quantifying the difference between parametrized curves in Euclidean space by introducing a distance function on the space of parametrized curves up to rigid transformations (rotations and translations). Given two curves, the distance between them is defined as the infimum of an energy functional which, roughly speaking, measures the extent to which the jet field of the first curve needs to be rotated to match up with the jet field of the second curve. We show that this energy functional attains a global minimum on the appropriate function space, and we derive a set of first-order ODEs for the minimizer.