In this note we show that the entropy of a skew product action of a countable amenable group satisfies the classical formula of Abramov and Rokhlin.

Abstract. In this note, it is shown that if f : Fq → F n q is any function and A = (A1, . . . , An) is uniformly distributed over F n q , then the average over (k1, . . . , kn) ∈ Fq of the Rényi (and hence, of the Shannon) entropy of f(A)+(k1A1, . . . , knAn) is at least about log 2 (q) − n bits. In fact, it is shown that the average collision probability of f(A) + (k1A1, . . . , knAn) is at mo...

Shannon introduced his concept of entropy as a measure of the uncertainty as to the outcome of some event. If this is the case, then learning about a system should be reflected by a real decrease in the entropy of the variable that is the target of learning. I examined a robot that learned to navigate in a simple arena, and analyzed the entropy of the variables for action, state, and reward. En...

Thermodynamics is an exact macroscopic theory with no known exceptions. It has a long history of development that includes many famous Christian scientists. Recent developments in theology have occasionally invoked thermodynamic quantities or concepts in ways that are not consistent with either good science or theology. The present article presents a brief introduction to both classical and sta...

This paper grounds the critique of the reduction of regions in a country , not only in its geographical and social context but also in its entropic space. The various recent plans leading to the reduction of the number of regions in metropolitan France are discussed, based on the mere distribution in the number of municipalities in the plans and analyzed according to various distribution laws. ...

Conscious experience is awash with underlying relationships. Moreover, for various brain regions such as the primary visual cortex, the system is bias toward some states over other states and, when representing this bias using a probability distribution, the system can define expected quantities. The mathematical theory in the present paper provides a link between these facts. Under the require...

In a simplified vision the universe is made by two main ingredients: atoms and bits. Within this paradigm the shape of things is just the content of information associated with each thing. In this paper we briefly discuss the role of information in defining shapes and specifically address the connection between information, entropy and energy, with the aim of showing that changing shape of a ph...

We provide an upper bound for the amount of information a human translator adds to an original text, i.e., how many bits of information we need to store a translation, given the original. We do this by creating a Bilingual Shannon Game that elicits character guesses from human subjects, then developing models to estimate the entropy of those guess sequences.

We perform an asymptotic analysis of the NSB estimator of entropy of a discrete random variable. The analysis illuminates the dependence of the estimates on the number of coincidences in the sample and shows that the estimator has a well defined limit for a large cardinality of the studied variable. This allows estimation of entropy with no a priori assumptions about the cardinality. Software i...