The generalized entropies of Rényi inspire new measures for estimating signal information and complexity in the time–frequency plane. When applied to a time–frequency representation (TFR) from Cohen’s class or the affine class, the Rényi entropies conform closely to the notion of complexity that we use when visually inspecting time–frequency images. These measures possess several additional int...

It has long been conjectured that the entropy of quantum fields across boundaries scales as the boundary area. This conjecture has not been easy to test in spacetime dimensions greater than four because of divergences in the von Neumann entropy. Here we show that the Rényi entropy provides a convergent alternative, yielding a quantitative measure of entanglement between quantum field theoretic ...

Consider the partition function S μ( ) associated in the theory of Rényi dimension to a finite Borel measure μ on Euclidean d-space. This partition function S μ( ) is the sum of the q-th powers of the measure applied to a partition of d-space into d-cubes of width . We further Guérin’s investigation of the relation between this partition function and the Lebesgue Lp norm (Lq norm) of the convol...

Sandwiched (quantum) α-Rényi divergence has been recently defined in the independent works of Wilde et al. (arXiv:1306.1586) and Müller-Lennert et al (arXiv:1306.3142v1). This new quantum divergence has already found applications in quantum information theory. Here we further investigate properties of this new quantum divergence. In particular we show that sandwiched α-Rényi divergence satisfie...

This article compares a family of methods for characterizing neural feature selectivity using natural stimuli in the framework of the linear-nonlinear model. In this model, the spike probability depends in a nonlinear way on a small number of stimulus dimensions. The relevant stimulus dimensions can be found by optimizing a Rényi divergence that quantifies a change in the stimulus distribution ...

Inequalities are at the heart of mathematical and statistical theory. No inequality is completely perfect, but the Hájek-Rényi inequality, which is the main subject of this thesis, is arguably the closest to absolute perfection of all the inequalities within all theories of probability. It has many applications in proving limit theorems, and examples of these are presented in this thesis. The s...

Consider the partition function S μ( ) associated in the theory of Rényi dimension to a finite Borel measure μ on Euclidean d-space. This partition function S μ( ) is the sum of the q-th powers of the measure applied to a partition of d-space into d-cubes of width . We further Guérin’s investigation of the relation between this partition function and the Lebesgue Lp norm (Lq norm) of the convol...

The problem of coding labeled trees has been widely studied in the literature and several bijective codes that realize associations between labeled trees and sequences of labels have been presented. k-trees are one of the most natural and interesting generalizations of trees and there is considerable interest in developing efficient tools to manipulate this class, since many NP-Complete problem...

We investigate the large-time asymptotics of nonlinear diffusion equations ut = ∆u p in dimension n ≥ 1, in the exponent interval p > n/(n+ 2), when the initial datum u0 is of bounded second moment. Precise rates of convergence to the Barenblatt profile in terms of the relative Rényi entropy are demonstrated for finite-mass solutions defined in the whole space when they are re-normalized at eac...

In this lecture we will introduce the Erdös-Rényi model of random graphs. Erdös and Rényi did not introduce them in an attempt to model any graphs found in the real world. Rather, they introduced them because they are the source of a lot of interesting mathematics. In fact, these random graphs have many properties that we do not know how to obtain through any efficient explicit construction of ...