Showing papers on "Invariant (mathematics) published in 1982"

A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

Abstract: Let M be a compact Riemannian manifold with a fixed conformal structure. Then we introduce the concept of conformal volume of M in the following manner. For each branched conformal immersion q9 of M into the unit sphere S n, we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of S". We let Vc(n, qg) be the max imum volume of these branched immersions. The conformal volume of M is defined to be the infimum of V.(n, q0) where qo ranges over all branched conformal immersions of M into the unit sphere S". In this paper, we study the case when M is a compact surface and we call the conformal volume of M to be the conformal area of M. We demonst ra te that this conformal invariant is non-trivial. In fact, we prove that if there exists a minimal immersion of M into S" where coordinate functions are first eigenfunctions, then the conformal area of M is given by the area of M with respect to the induced metric. This enables us to compute the conformal area for several surfaces. For example, the conformal area of R P z is 6n and the conformal area of the square torus is 27c 2. We believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces. We demonstrate this claim by applying the concept of conformal area to two different branches of surface theory. The first application is to study the total curvature of a compact surface in R". This problem has a long history. Fenchel and Fary [9] proved that for a closed curve o in R", Slk[>27c where k is its curvature. Then Milnor [12]

Geometric Theory of Dynamical Systems : An Introduction

Abstract: 1 Differentiable Manifolds and Vector Fields.- 0 Calculus in ?n and Differentiable Manifolds.- 1 Vector Fields on Manifolds.- 2 The Topology of the Space of Cr Maps.- 3 Transversality.- 4 Structural Stability.- 2 Local Stability.- 1 The Tubular Flow Theorem.- 2 Linear Vector Fields.- 3 Singularities and Hyperbolic Fixed Points.- 4 Local Stability.- 5 Local Classification.- 6 Invariant Manifolds.- 7 The ?-lemma (Inclination Lemma). Geometrical Proof of Local Stability.- 3 The Kupka-Smale Theorem.- 1 The Poincare Map.- 2 Genericity of Vector Fields Whose Closed Orbits Are Hyperbolic.- 3 Transversality of the Invariant Manifolds.- 4 Genericity and Stability of Morse-Smale Vector Fields.- 1 Morse-Smale Vector Fields Structural Stability.- 2 Density of Morse-Smale Vector Fields on Orientable Surfaces.- 3 Generalizations.- 4 General Comments on Structural Stability. Other Topics.- Appendix: Rotation Number and Cherry Flows.- References.

Ergodic Properties of Invariant Measures for Piecewise Monotonic Transformations

Abstract: During the last decade a lot of research has been done on onedimensional dynamics. Different kinds of invariant measures for certain classes of piecewise monotonic transformations have been considered. In most of these cases the Perron-Frobenius-operator plays an important role. In this paper we try to unify these different examples, which are discussed in detail below. First we give a discription of the results proved in this paper.

Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study

Abstract: We consider a two-parameter family of maps of the plane to itself. Each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point. For certain parameter values there is a Hopf bifurcation to an invariant circle, which is smooth for parameter values in a neighborhood of the bifurcation point. However, computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point. This paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type.

Invariant Measures for the Zero Range Process

Abstract: On a countable set of sites $S$, the zero range process is constructed when the stochastic matrix $p(x, y)$ determining the one particle motion satisfies a mild assumption. The set of invariant measures for this process is described in the following two cases: a) The system is attractive and $p(x, y)$ is recurrent. b) The system is attractive, $p(x, y)$ corresponds to a simple random walk on the integers and the rate at which particles leave any site is bounded.

Rotation-invariant digital pattern recognition using circular harmonic expansion.

Abstract: A new method has been developed for rotation-invariant pattern recognition. One component of the circular harmonic expansion of the target is used in the preparation of the reference. Correlations between the input and reference objects are accomplished by FFT and multiplication in the frequency domain. In an experience with targets from an image with 192 × 192 pixels, target orientations were detected with an accuracy of ~0.1°. This method is also suitable for optical implementation.

Stochastic stability in some chaotic dynamical systems

Abstract: For a certain class of piecewise monotonic transformations it is shown using a spectral decomposition of the Perron-Frobenius-operator ofT that invariant measures depend continuously on 3 types of perturbations: 1) deterministic perturbations, 2) stochastic perturbations, 3) randomly occuring deterministic perturbations. The topology on the space of perturbed transformations is derived from a metric on the space of Perron-Frobenius-operators.

A direct approach to finding exact invariants for one‐dimensional time‐dependent classical Hamiltonians

Abstract: For a classical Hamiltonian H=(1/2) p2+V(q,t) with an arbitrary time‐dependent potential V(q,t), exact invariants that can be expressed as series in positive powers of p, I(q,p,t)=∑∞n=0pnfn(q,t), are examined. The method is based on direct use of the equation dI/dt=∂I/∂t +[I,H] =0. A recursion relation for the coefficients fn(q,t) is obtained. All potentials that admit an invariant quadratic in p are found and, for those potentials, all invariants quadratic in p are determined. The feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed. The systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation.

The Arveson Extension Theorem and coanalytic models

Abstract: We develop techniques which allow one to describe in simple terms the set of operators on Hilbert space of the form M* (∞) |M, where M is multiplication by z on a Hilbert space of analytic functions satisfying certain technical assumptions, M* (∞) is the direct sum of a countably infinite number of copies of M*, andM is invariant for M* (∞). One of the main ingredients in our technique is the Arveson Extension Theorem and this paper illustrates the great power and tractability of that theorem in a concrete setting.

CHAPTER VII – Differential Invariants

Abstract: Publisher Summary This chapter focuses on the theory and applications of differential invariants of Lie transformation groups, which can be infinite. It discusses the concepts of a differential invariant and special transformation of a differential equation, the so-called group stratification. The central theorem is about the finiteness of bases for differential invariants of arbitrary Lie groups. In the proposed variant Tresse's theorem is completely established in infinitesimal language by the means of invariant differential operators. An equation E is called automorphic relative to the given group G if all its solutions are situated on the orbit of one of them. In other words, from one given solution U, it is possible to find all solutions by applying to U the transformations of the group G. Classical problems of the construction and classification of automorphic systems provide excellent examples of the applications of the general theory, and these play an important role in the problems of invariant transformations for differential equations.

Residual Bounds on Approximate Eigensystems of Nonnormal Matrices

Abstract: For nonnormal matrices the norms of the residuals of approximate eigenvectors are not by themselves sufficient information to bound the error in the approximate eigenvalue. It is sufficient however to give a bound on the distance to the nearest matrix for which the given approximations are exact. This result is extended to cover approximate invariant subspaces and their residuals.The theorems are used to derive a useful set of criteria for terminating the two-sided Lanczos algorithm.The study begins with a list of error bounds for eigenvalues of nonnormal matrices.

Some remarks on Birkhoff and Mather twist map theorems

Abstract: A recent result of J. Mather [1] about the existence of quasi-periodic orbits for twist maps is derived from an appropriately modified version of G. D. Birkhoff's classical theorem concerning periodic orbits. A proof of Birkhoff's theorem is given using a simplified geometric version of Mather's arguments. Additional properties of Mather's invariant sets are discussed.

A Pathological Area Preserving C ∞ Diffeomorphism of the Plane

Abstract: The pseudocircle P is an hereditarily indecomposable planar continuum. In particular, it is connected but nowhere locally connected. We construct a COO area preserving diffeomorphism of the plane with P as a minimal set. The diffeomorphism f is constructed as an explicit limit of diffeomorphisms conjugate to rotations about the origin. There is a welldefined irrational rotation number for fjP even though fIP is not even semiconjugate to a rotation of S1. If we remove the requirement that our diffeomorphisms be area preserving, we may alter the example so that P is an attracting minimal set. The complexity of a dynamical system is reflected in part by its invariant sets. We consider here a simple dynamical system, the action of a diffeomorphism on R2. Pathology abounds in the compact connected subsets of R2, and we show that this pathology will occur in the minimal sets of diffeomorphisms, even if we restrict ourselves to-those which are C? and area preserving. We choose the pseudocircle P (defined below) as our model of extreme pathology. Its key feature is that it is hereditarily indecomposable. Indecomposable means that P cannot be written as the union of two proper compact connected subsets, and hereditarily indecomposable means that every compact connected subset of P is indecomposable. (P is, for instance, nowhere locally connected.) What makes P tractable in spite of this behavior, is that Bing's construction of pseudocircles [B2] is both simple and malleable. It is an infinite construction allowing choices at each stage. In light of [F], the resulting space is, to a great extent, independent of the choices. Using an infinite limit, we construct an embedding of P in R2 and a C area preserving diffeomorphism f: R2 -+ R2 with P as a minimal set. (One may take the domain of f to be an annulus A2.) There are two other features of f that are worth mentioning. P is defined as the intersection of annuli P =nn=l1 A,, where each inclusion An+, A, is a homotopy equivalence. It therefore makes sense to speak of a rotation number for f and indeed f has a well-defined irrational rotation number. Nonetheless, f is not semiconjugate to a homeomorphism of S1. Second, if one is willing to consider C?? diffeomorphisms which do not preserve area, f is easily perturbed to f': R2 R2 with P as an attracting minimal set. It is especially relevant that f is area preserving, as there is a long history of interest in area preserving diffeomorphisms of a surface. In particular, Birkhoff [Bir] (see also the recent work of Mather [M]) studied invariant sets which were the frontiers of invariant, open, simply connected regions, and gave criterion forcing Received by the editors April 28, 1981. 1980 Mathematics Subject Classification. Primary 58F99, 28D05. @ 1982 American Mathematical Society 0002-9939/81/0000-1068/S02.00 163 This content downloaded from 157.55.39.165 on Thu, 14 Jul 2016 06:06:00 UTC All use subject to http://about.jstor.org/terms

Fractal Dimensionality of Levy Processes

Abstract: We determine the fractal dimensionality D of the trajectories of a class of translationally invariant Markov processes. We also provide two simple operational measures to estimate D.

Selecting Among Probability Distributions Used in Reliability

Abstract: A method is given for selecting the member of a collection of families of distributions that best fits a set of observations. This method requires a noncensored set of observations. The families considered include the exponential, gamma, Weibull, and lognormal. A selection statistic is proposed that is essentially the value of the density function of a scale transformation maximal invariant. Some properties of the selection procedures based on these statistics are stated, and results of a simulation study are reported. A set of time-to-failure data from a textile experiment is used as an example to illustrate the procedure, which is implemented by a computer program.

Birth-and-death processes on the integers with phases and general boundaries

Abstract: The invariant probability distribution is found for a class of birth-and-death processes on the integers with phases and one or two boundaries. The invariant vector has a matrix geometric form and is found by solving a non-linear matrix equation and then finding an invariant probability distribution on the boundary states. Levy's concept of watching a Markov process in a subset is used to naturally decouple the computation of distributions on the boundary and interior states.

Geometric properties and invariant manifolds of the Riccati equation

Abstract: This short paper considers the general Riccati matrix differential equation. It reviews and extends results on the characterization and existence of equilibrium solutions, establishes that the Riccati equation has at most one stable equilibrium solution as t \rightarrow \infty or t \rightarrow -\infty , and confines the region of attraction to this unique stable equilibrium solution by identifying and utilizing linear and nonlinear invariant manifolds of the Riccati equation.

Borsuk-Ulam Theorems for Arbitrary S1 Actions and Applications.

Abstract: : An S to the first power version of the Borsuk-Ulam Theorem is proved for a situation where Fix S to the first power may be nontrivial. The proof is accomplished with the aid of a new relative index theory. Applications are given to intersection theorems and the existence of multiple critica points is established for a class of functionals invariant under an S to the first power symmetry. Minimax arguments from the calculus of variations serve as an important tool in establishing the existence of nonlinear vibrations of discrete mechanical systems as modelled by Hamilton's equations. In these arguments one obtains the solutions of the differential equations as critical points of an associated Lagrangian by minimaxing the Lagrangian over appropriate classes of sets. Intersection theorems such as are proved in this paper play a crucial role in this process. In addition to obtaining some intersection theorems this report illustrates their use by proving an existence theorem for multiple critical points of the functional invarianet under an S to the first power symmetry group.

On integrating the techniques of direct methods and isomorphous replacement. I. The theoretical basis

Abstract: Recent advances in direct methods are applied to structurally isomorphous pairs; in particular the probabilistic theory of the three-phase structure invariant in P1 is worked out. The neighborhood principle plays the central role. Specifically, the six-magnitude first neighborhood of each of the four kinds of three-phase structure invariant is defined and the joint probability distribution of the corresponding six structure factors is derived. This distribution leads to the conditional probability distribution of each kind of three-phase structure invariant, assuming as known the six magnitudes in its first neighborhood. In the favorable case that the variance of the distribution happens to be small, one obtains a reliable estimate (0 or π of the structure invariant [the neighborhood principle: Hauptman (1975). Acta Cryst.. A31, 680-687].

On integrating the techniques of direct methods with anomalous dispersion. I. The theoretical basis

Abstract: The recently secured mathematical formalism of direct methods is here generalized to the case that the atomic scattering factors are arbitrary complex numbers, thus including the special case that one or more anomalous scatterers are present. Once again the neighborhood concept plays the central role. Final results from the probabilistic theory of the two- and three-phase structure invariants are briefly summarized. In particular, the conditional probability distribution of the three-phase structure invariant, given the six magnitudes |E| in its first neighborhood, is described. The distribution yields an estimate for the three-phase structure invariant which is particularly good in the favorable case that the variance of the distribution happens to be small (the neighborhood principle). Particularly noteworthy is the fact that, in sharp contrast to all earlier work, the estimate is unique in the whole range 0 to 2π. An example shows that the method is capable of yielding unique estimates for tens of thousands of three-phase structure invariants with unprecedented accuracy, even in the macromolecular case. The clear implication is that the fusion of the traditional techniques of direct methods with anomalous dispersion, which is described here, will facilitate the solution of those crystal structures which contain one or more anomalous scatterers.