The knowledge of the metric entropy of precompact subsets of operators on finite dimensional Euclidean space is important in particular in the probabilistic methods developped by E. D. Gluskin and S. Szarek for constructing certain random Banach spaces. We give a new argument for estimating the metric entropy of some subsets such as the Grassmann manifold equipped with natural metrics. Here, th...

Abstract. We establish that for q ≥ 1, the class of convex combinations of q translates of a smooth probability density has local doubling dimension proportional to q. The key difficulty in the proof is to control the local geometric structure of mixture classes. Our local geometry theorem yields a bound on the (bracketing) metric entropy of a class of normalized densities, from which a local e...

Morphological connected set filters for extraction of filamentous details from medical images are developed. The advantages of these filters are that they are shape preserving and do not amplify noise. Two approaches are compared: (i) multi-scale filtering (ii) single-step shape filtering using connected set (or attribute) thinnings. The latter method highlights all filamentous structure in a s...

0. Introduction. Throughout this paper T will denote a measure preserving transformation on a cr-finite infinite measure space (X, (B, m) which is point isomorphic with the Lebesgue measure space of the real line. Unless otherwise stated, T will be one-one. Equations involving functions or sets will always be interpreted modulo sets of measure zero. T is said to be ergodic if T~1E = E, ££(B, im...

The construction of a quantal Kolmogorov entropy that tends to the classical Kolmogorov entropy in the limit h -t 0 is discussed. The approach is to use sets of functions rather than disjoint partitions and involves the use of pseudo-differential operators.

Almost every, essentially: Given a Lebesgue measure space (X,B, μ), a property P (x) predicated of elements of X is said to hold for almost every x ∈ X, if the set X \ {x : P (x) holds} has zero measure. Two sets A,B ∈ B are essentially disjoint if μ(A ∩B) = 0. Conservative system: Is an infinite measure preserving system such that for no set A ∈ B with positive measure are A,T−1A,T−2A, . . . p...

The present paper is intended as a first step toward the establishment of a general theory of finitely subadditive outer measures. First, a general method for constructing a finitely subadditive outer measure and an associated finitely additive measure on any space is presented. This is followed by a discussion ofthe theory ofinner measures, their construction, and the relationship oftheir prop...

Let α be an irrational number and the trasformation Tx 7→ x+ α mod 1, x ∈ [0, 1), represent an irrational rotation of the unit circle. We construct an ergodic and completely squashable smooth real extension, i.e. we find a real analytic or k time continuously differentiable real function F such that for every λ 6= 0 there exists a commutor Sλ of T such that F ◦Sλ is T -cohomologous to λφ and th...