We prove that an extension of an invertible, multiply-recurrent infinite measure preserving transformation is also multiply-recurrent.

Given two sets of quantum states {A1, . . . , Ak} and {B1, . . . , Bk}, represented as sets as density matrices, necessary and sufficient conditions are obtained for the existence of a physical transformation T , represented as a trace-preserving completely positive map, such that T (Ai) = Bi for i = 1, . . . , k. General completely positive maps without the trace-preserving requirement, and un...

converge a.e. to a finite limit, where fs is the characteristic function of the set E. G. D. Birkhoff's Ergodic Theorem asserts this conclusion if T is measure-preserving, in the sense that m(T~1E)=m(E) for each measurable set E. The same conclusion can be asserted under somewhat more general circumstances. We shall say that T admits a finite, equivalent, invariant measure if there is a finite ...

1 Ergodic theory References for this section: CFS]. 1. The basic setting of ergodic theory: a measure-preserving transformation T of a probability space (X; B; m). Usually we assume T is invertible. (More generally, measure-preserving means R f T = R f; equivalently, m(T ?1 (A)) = mA.) How many measure spaces are there? Standard Borel spaces: any Borel subset of a complete, separable metric spa...

Using the ratio ergodic theorem for a measure preserving transformation in a σ-finite measure space we give a straightforward proof of Derriennic’s reverse maximal inequality for the supremum of ergodic ratios.

In this paper, we consider an extension of the commonly used poverty gap measure that may be used to address he question of whether the elimination of poverty is a feasible objective, given sufficient political commitment. The resources potentially available to address poverty may be measured by the total amount by which the incomes of the non-poor exceed the same poverty line. The ratio of the...

We characterize a special class of unitary operators that preserve orthonormal wavelets. In the process we also prove that symmetric wavelet sets cover the real line.

One of the four well-known series of simple Lie algebras of Cartan type is the series of Lie algebras of Special type, which are divergence-free Lie algebras associated with polynomial algebras and the operators of taking partial derivatives, connected with volume-preserving diffeomorphisms. In this paper, we determine the structure space of the divergence-free Lie algebras associated with pair...

We present new explicit volume-preserving methods based on splitting for polynomial divergence-free vector fields. The methods can be divided in two classes: methods that distinguish between the diagonal part and the off-diagonal part and methods that do not. For the methods in the first class it is possible to combine different treatments of the diagonal and off-diagonal parts, giving rise to ...

This paper introduces a new measure of quality for terrain surface simplification aiming at preserving the slope of the surface, as well as a new simplification algorithm which satisfies this measure. Experimental results and comparisons with other simplification algorithms show the effectiveness of proposed algorithm according to some quality measures.