774 research outputs found
Best finite constrained approximations of one-dimensional probabilities
This paper studies best finitely supported approximations of one-dimensional
probability measures with respect to the -Kantorovich (or transport)
distance, where either the locations or the weights of the approximations'
atoms are prescribed. Necessary and sufficient optimality conditions are
established, and the rate of convergence (as the number of atoms goes to
infinity) is discussed. In view of emerging mathematical and statistical
applications, special attention is given to the case of best uniform
approximations (i.e., all atoms having equal weight). The approach developed in
this paper is elementary; it is based on best approximations of (monotone)
-functions by step functions, and thus different from, yet naturally
complementary to, the classical Voronoi partition approach.Comment: To appear in J. Approx. Theor
On the Gap between Random Dynamical Systems and Continuous Skew Products
AMS 2000 subject classification: primary 37-02, 37B20, 37H05; secondary 34C27, 37A20.We review the recent notion of a nonautonomous dynamical system (NDS), which has been introduced as an abstraction of both random dynamical systems and continuous skew product flows. Our focus is on fundamental analogies and discrepancies brought about by these two classes
of NDS. We discuss base dynamics mainly through almost periodicity and almost automorphy, and we emphasize the importance of these concepts for NDS which are generated by differential and difference equations. Nonautonomous dynamics is presented by means of representative examples. We also mention several natural yet unresolved questions
Fundamental Flaws in Feller's Classical Derivation of Benford's Law
Feller's classic text 'An Introduction to Probability Theory and its
Applications' contains a derivation of the well known significant-digit law
called Benford's law. More specifically, Feller gives a sufficient condition
("large spread") for a random variable to be approximately Benford
distributed, that is, for to be approximately uniformly
distributed modulo one. This note shows that the large-spread derivation, which
continues to be widely cited and used, contains serious basic errors. Concrete
examples and a new inequality clearly demonstrate that large spread (or large
spread on a logarithmic scale) does not imply that a random variable is
approximately Benford distributed, for any reasonable definition of "spread" or
measure of dispersionComment: 7 page
On Almost Automorphic Dynamics in Symbolic Lattices
1991 Mathematics Subject Classification. Primary Primary 37B10, 37A35, 43A60; Secondary
37B20, 54H20.We study the existence, structure, and topological entropy of almost automorphic arrays in symbolic lattice dynamical systems. In particular we show that almost automorphic arrays with arbitrarily large entropy are typical in symbolic lattice dynamical systems. Applications to pattern formation and spatial chaos in infinite dimensional lattice systems are considered,
and the construction of chaotic almost automorphic signals is discussed.The first author was supported by a Max Kade Postdoctoral Fellowship (at Georgia Tech). The second author was partially supported by DFG grant Si 801 and CDSNS, Georgia Tech. The third author was partially supported by NSF Grant DMS-0204119
Rigorous error bounds for RK methods in the proof of chaotic behaviour
AbstractComplicated dynamical systems can be rigorously analysed by means of Conley index theory. Due to its partly numerical nature such an analysis necessitates bounds on the truncation and the round-off error. These are provided for explicit RK methods in the form of iteration schemes ready-made for applications. The presentation is aimed to simplify error bounds already available so that different error sources can be clearly overlooked. As an immediate application, a computer-assisted analysis elucidates the intricate dynamics of a simple mechanical system
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