On entropy-preserving stochastic averages

Entropy, Combinatorial Dimensions and Random Averages

In this article we introduce a new combinatorial parameter which generalizes the VC dimension and the fat-shattering dimension, and extends beyond the function-class setup. Using this parameter we establish entropy bounds for subsets of the n-dimensional unit cube, and in particular, we present new bounds on the empirical covering numbers and gaussian averages associated with classes of functio...

Entropy-Preserving Transformation Method

Quantifying Computational Effort Required for Stochastic Averages.

We propose a measure that quantifies the effort needed to compute a given stochastic average, considered in the context of molecular modeling. This "difficulty index" is defined in terms of CPU time, observed uncertainty, and a characteristic scale for the quantity being computed. This metric provides a focus for optimization and comparison of methods and, if applied broadly, could aid understa...

Local Entropy Averages and Projections of Fractal Measures

We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of self-similarity under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We apply this to prove the following conjecture of Furstenberg: if X,Y ⊆ [0, 1] are closed and invariant, respectively, under ×m mod 1 and ×n mod 1, where m,n are not...

A Local Entropy Minimum Principle for Deriving Entropy Preserving Schemes

Abstract. The present work deals with the establishment of stability conditions of finite volume methods to approximate weak solutions of the general Euler equations to simulate compressible flows. In oder to ensure discrete entropy inequalities, we derive a new technique based on a local minimum principle to be satisfied by the specific entropy. Sufficient conditions are exhibited to satisfy t...