Let T be a measure-preserving transformation acting on a probability space (a, Sr, P). This T induces a linear isometry V on &(a, F, P) defined by V(f) = f o T. The analysis of T by using properties of V is referred to as spectral analysis. It is often profitable to consider the action of V on certain closed linear invariant subspaces 9 of L,(O, ST, P). In this note we make the following assump...

In a recent paper [1], the authors suggest a novel Riemannian framework for comparing shapes. In this framework, a simple closed surface is represented by a field of metric tensors and curvatures. A product Riemannian metric is developed based on the L norm on symmetric positive definite matrices and scalar fields. Taken as a quotient space under the group of volume-preserving diffeomorphisms, ...

The metric entropy of absolute convex hulls of sets in Hilbert spaces is studied for the general case when the metric entropy of the sets is arbitrary. Under some regularity assumptions, the results are sharp.

We show that any non-degenerate vector field u in L∞(Ω,RN), where Ω is a bounded domain in RN , can be written as u(x) = ∇1H(S(x),x) for a.e. x ∈Ω, where S is a measure preserving point transformation on Ω such that S2 = I a.e (an involution), and H : RN ×RN → R is a globally Lipschitz anti-symmetric convex-concave Hamiltonian. Moreover, u is a monotone map if and only if S can be taken to be t...

We characterize the points that satisfy Birkhoff’s ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space is not Martin-Löf random, there is a computable measure-preserving transformation and a computable set that witness that x is not typical with respect to the...

Characterization of 2 nd order local image structure by a 6-D vector (or jet) of Gaussian derivative measurements is considered. We consider the affect on jets of a group of transformations – affine intensity-scaling, image rotation and reflection, and their compositions – that preserve intrinsic image structure. We show how this group stratifies the jet space into a system of orbits. Consideri...

We study the realization of conformal symmetry in the QHE as part of the W∞ algebra. Conformal symmetry can be realized already at the classical level and implies the complexification of coordinate space. Its quantum version is not unitary. Nevertheless, it can be rendered unitary by a suitable modification of its definition which amounts to taking proper care of the quantum measure. The conseq...

This paper concerns the smoothness of Tauberian constants of maximal operators in the discrete and ergodic settings. In particular, we define the discrete strong maximal operator M̃S on Z by M̃Sf(m) := sup 0∈R⊂Rn 1 #(R ∩ Zn) ∑ j∈R∩Zn |f(m+ j)|, m ∈ Z, where the supremum is taken over all open rectangles in R containing the origin whose sides are parallel to the coordinate axes. We show that the a...

This article gives an exposition of the theory of arithmetic motivic measure, as developed by J. Denef and F. Loeser. 1. Preliminary Concepts There is much that is odd about motivic measure if it is judged by measure theory in the sense of twentieth century analysis. It does not fit neatly with the tradition of measure in the style of Hausdorff, Haar, and Lebesgue. It is best to view motivic me...