3,321 research outputs found
Entropy and set cardinality inequalities for partition-determined functions
A new notion of partition-determined functions is introduced, and several
basic inequalities are developed for the entropy of such functions of
independent random variables, as well as for cardinalities of compound sets
obtained using these functions. Here a compound set means a set obtained by
varying each argument of a function of several variables over a set associated
with that argument, where all the sets are subsets of an appropriate algebraic
structure so that the function is well defined. On the one hand, the entropy
inequalities developed for partition-determined functions imply entropic
analogues of general inequalities of Pl\"unnecke-Ruzsa type. On the other hand,
the cardinality inequalities developed for compound sets imply several
inequalities for sumsets, including for instance a generalization of
inequalities proved by Gyarmati, Matolcsi and Ruzsa (2010). We also provide
partial progress towards a conjecture of Ruzsa (2007) for sumsets in nonabelian
groups. All proofs are elementary and rely on properly developing certain
information-theoretic inequalities.Comment: 26 pages. v2: Revised version incorporating referee feedback plus
inclusion of some additional corollaries and discussion. v3: Final version
with minor corrections. To appear in Random Structures and Algorithm
Directional complexity and entropy for lift mappings
We introduce and study the notion of a directional complexity and entropy for
maps of degree 1 on the circle. For piecewise affine Markov maps we use
symbolic dynamics to relate this complexity to the symbolic complexity. We
apply a combinatorial machinery to obtain exact formulas for the directional
entropy, to find the maximal directional entropy, and to show that it equals
the topological entropy of the map. Keywords: Rotation interval, Space-time
window, Directional complexity, Directional entropy;Comment: 19p. 3 fig, Discrete and Continuous Dynamical Systems-B (Vol. 20, No.
10) December 201
The law of series
We prove a general ergodic-theoretic result concerning the return time
statistic, which, properly understood, sheds some new light on the common sense
phenomenon known as {\it the law of series}. Let \proc be an ergodic process on
finitely many states, with positive entropy. We show that the distribution
function of the normalized waiting time for the first visit to a small cylinder
set is, for majority of such cylinders and up to epsilon, dominated by the
exponential distribution function . This fact has the following
interpretation: The occurrences of such a "rare event" can deviate from
purely random in only one direction -- so that for any length of an
"observation period" of time, the first occurrence of "attracts" its
further repetitions in this period
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