Rényi entropies of aperiodic dynamical systems

In this paper we continue the study of Rényi entropies of measure-preserving transformations started in [22]. We have established there that for ergodic transformations with positive entropy, the Rényi entropies of orderq, q ∈ ℝ, are equal to either plus infinity (q < 1), or to the measure-theoretic (Kolmogorov-Sinai) entropy (q ≥ 1). The answer for non-ergodic transformations is different: the Rényi entropies of orderq > 1 are equal to the essential infimum of the measure-theoretic entropies of measures forming the decomposition into ergodic components. Thus, it is possible that the Rényi entropies of orderq > 1 are strictly smaller than the measure-theoretic entropy, which is the average value of entropies of ergodic components. This result is a bit surprising: the Rényi entropies are metric invariants, which are sensitive to ergodicity. The proof of the described result is based on the construction of partitions with independent iterates. However, these partitions are obtained in different ways depending onq: forq > 1 we use a version of the well-known Sinai theorem on Bernoulli factors for the non-ergodic transformations; forq < 1 we use the notion of collections of independent sets in Rokhlin-Halmos towers and their properties.     

The Bohman‐Frieze process near criticality

The Erd?s‐Rényi process begins with an empty graph on n vertices, with edges added randomly one at a time to the graph. A classical result of Erd?s and Rényi states that the Erd?s‐Rényi process undergoes a phase transition, which takes place when the number of edges reaches n/2 (we say at time 1) and a giant component emerges. Since this seminal work of Erd?s and Rényi, various random graph models have been introduced and studied. In this paper we study the Bohman‐Frieze process, a simple modification of the Erd?s‐Rényi process. The Bohman‐Frieze process also begins with an empty graph on n vertices. At each step two random edges are presented, and if the first edge would join two isolated vertices, it is added to a graph; otherwise the second edge is added. We present several new results on the phase transition of the Bohman‐Frieze process. We show that it has a qualitatively similar phase transition to the Erd?s‐Rényi process in terms of the size and structure of the components near the critical point. We prove that all components at time t_c ? ? (that is, when the number of edges are (t_c ? ?)n/2) are trees or unicyclic components and that the largest component is of size Ω(?^‐2log n). Further, at t_c + ?, all components apart from the giant component are trees or unicyclic and the size of the second‐largest component is Θ(?^‐2log n). Each of these results corresponds to an analogous well‐known result for the Erd?s‐Rényi process. Our proof techniques include combinatorial arguments, the differential equation method for random processes, and the singularity analysis of the moment generating function for the susceptibility, which satisfies a quasi‐linear partial differential equation. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Rényi’s residual entropy: A quantile approach

In the present paper, we introduce a quantile based Rényi’s entropy function and its residual version. We study certain properties and applications of the measure. Unlike the residual Rényi’s entropy function, the quantile version uniquely determines the distribution.     

On the independence number of the Erdős‐Rényi and projective norm graphs and a related hypergraph

The Erd?s‐Rényi and Projective Norm graphs are algebraically defined graphs that have proved useful in supplying constructions in extremal graph theory and Ramsey theory. Their eigenvalues have been computed and this yields an upper bound on their independence number. Here we show that in many cases, this upper bound is sharp in the order of magnitude. Our result for the Erd?s‐Rényi graph has the following reformulation: the maximum size of a family of mutually non‐orthogonal lines in a vector space of dimension three over the finite field of order q is of order q^3/2. We also prove that every subset of vertices of size greater than q^2/2 + q^3/2 + O(q) in the Erd?s‐Rényi graph contains a triangle. This shows that an old construction of Parsons is asymptotically sharp. Several related results and open problems are provided. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 113–127, 2007     

The convergence of the Rényi entropy of the normalized sums of IID random variables

We prove that the Rényi entropy of order $\alpha$ ($\alpha >1$) of the normalized sums of IID random variables with continuous differentiable density is convergent to the Rényi entropy of order $\alpha$ of the standard Gaussian distribution, and obtain the corresponding rates of convergence.     

Deux caractérisations de la mesure d'équilibre d'un endomorphisme de P k (C)

Résumé. — Soit μ la mesure d'équilibre d'un endomorphisme de P ^k (C). Nous montrons ici qu'elle est son unique mesure d'entropie maximale. Nous construisons directement μ comme distribution asymptotique des préimages de tout point hors d'un ensemble exceptionnel algébraique.

---

— Let μ be the equilibrium measure of an endomorphism of P ^k (C). We show that it is its unique measure of maximal entropy. We build μ directly as the distribution of premiages of any point outside an algebraic exceptional set.

---

---

Manucsrit re?u le 30 novembre 2000.     

Orthogonal Polarity Graphs and Sidon Sets

Determining the maximum number of edges in an n‐vertex C₄‐free graph is a well‐studied problem that dates back to a paper of Erd?s from 1938. One of the most important families of C₄‐free graphs are the Erd?s‐Rényi orthogonal polarity graphs. We show that the Cayley sum graph constructed using a Bose‐Chowla Sidon set is isomorphic to a large induced subgraph of the Erd?s‐Rényi orthogonal polarity graph. Using this isomorphism, we prove that the Petersen graph is a subgraph of every sufficiently large Erd?s‐Rényi orthogonal polarity graph.     

Mixing Limit Theorems of Maximum of Absolute Sums and their Convergence Rates

Conditioned, in the sense of Rényi (Acta Math. Acad. Sci. Hungar. 9, 215–228 1958), limit theorem in the L_p-norm of the maximum of absolute sums of independent identically distributed random variables is established and its exact rate of convergence is given. The results are equivalent to establishing analogous results for the supremum of random functions of partial sums defined on C[0,1], i.e., the invariance principle. New methodologies are used to prove the results that are profoundly different from those used in Rényi (Acta Math. Acad. Sci. Hungar. 9, 215–228, 1958) and subsequent authors in proving the conditioned central limit theorem for partial sums.     

Phase transitions for modified Erdős–Rényi processes

A fundamental and very well studied region of the Erdős–Rényi process is the phase transition at m∼n/2 edges in which a giant component suddenly appears. We examine the process beginning with an initial graph. We further examine the Bohman–Frieze process in which edges between isolated vertices are more likely. While the positions of the phase transitions vary, the three processes belong, roughly speaking, to the same universality class. In particular, the growth of the giant component in the barely supercritical region is linear in all cases.     

On deformations of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> in compact Lie groups

We study some properties of the varieties of deformations of free groups in compact Lie groups. In particular, we prove a conjecture of Margulis and Soifer about the density of non-virtually free points in such variety, and a conjecture of Goldman on the ergodicity of the action of Aut(F _n ) on such variety when n ≥ 3. The author was partially supported by NSF grant DMS-0404557, BSF grant 2004010, and the ‘Finite Structures’ Marie Curie Host Fellowship, carried out at the Alfréd Rényi Institute of Mathematics in Budapest.     

Local Topological Toughness and Local Factors

We localize and strengthen Katona’s idea of an edge-toughness to a local topological toughness. We disprove a conjecture of Katona concerning the conection between edge-toughness and factors. For the topological toughness we prove a theorem similar to Katona’s 2k-factor-conjecture, which turned out to be false for his edge-toughness. We prove, that besides this the topological toughness has nearly all known nice properties of Katona’s edge-toughness and therefore is worth to be considered. Research supported by the “Mathematics in Information Society” project carried out by Alfréd Rényi Institute of Mathematics - Hungarian Academy of Sciences, in the framework of the European Community’s “Confirming the International Role of Community Research” programme. Research supported by the Ministry of Education OTKA grant OTKA T 043520.     

Flows and functional inequalities for fractional operators

This paper collects results concerning global rates and large time asymptotics of a fractional fast diffusion equation on the Euclidean space, which is deeply related with a family of fractional Gagliardo–Nirenberg–Sobolev inequalities. Generically, self-similar solutions are not optimal for the Gagliardo–Nirenberg–Sobolev inequalities, in strong contrast with usual standard fast diffusion equations based on non-fractional operators. Various aspects of the stability of the self-similar solutions and of the entropy methods like carré du champ and Rényi entropy powers methods are investigated and raise a number of open problems.     

Transformations on density operators and on positive definite operators preserving the quantum Rényi divergence

In a certain sense we generalize the recently introduced and extensively studied notion called quantum Rényi divergence (also called ”sandwiched Rényi relative entropy”) and describe the structures of corresponding symmetries. More precisely, we characterize all transformations on the set of density operators which leave our new general quantity invariant and also determine the structure of all bijective transformations on the cone of positive definite operators which preserve the quantum Rényi divergence.     

Random self-affine multifractal Sierpinski sponges in \mathbbRd{\mathbb{R}^d}

In this paper we study the Hausdorff and packing dimensions and the Rényi dimensions of random self-affine multifractal Sierpinski sponges in \mathbbR^d{\mathbb{R}^{d}}.     

The Blackwell and Dubins Theorem and Rényi’s Amount of Information Measure: Some Applications

This survey paper provides first for an overview of how quantum-like concepts could be used in macroscopic environments like economics. The paper then argues for the use of the concept of a quantum mechanical wave function as an ‘information wave function’. A rationale is provided on why such interpretation is reasonable. After having defined the ‘information wave function’, Ψ(q), we argue how | Ψ(q)| ² can be interpreted as a Radon-Nikodym derivative. We consider how we can connect, using the | Ψ(q)| ², the Blackwell and Dubins (Ann. Math. Stat. 33:882–886, 1961) Theorem with Rényi’s (Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1961) measure of quantity of information. We also define ‘ambiguity of information’ and ‘multi-sourced information’.     

Quelques observations sur lesL 2-resolvantes

Cet article a deux parties. Dans la première nous allons étudier les normes des opérateurs d'uneL ²-résolvante. Dans la deuxième partie on va associer à chaqueL ²-résolvante (G _α)_α>0 un processus (f _α)_α>0 (l'idée de considérer ce processus appartient à I. Cuculescu). En ce qui concerne ce processus (f _α)_α>0 on va montrer que si (G _α)_α>0 est uneL ²-résolvante qui correspond à un espace de Dirichlet fortement régulier sur (E, ℬ, μ) oùE est un espace métrique compact, ℬ est le corp borélien engendré par les ouverts deE et μ est une probabilité (ou une mesure finie), alors l'ensemble des zeros de chaquef _α est un ensemble fermé avec l'intérieur vide.     

Recurrence Relationships for the Mean Number of Faces and Vertices for Random Convex Hulls

This paper studies the convex hull of n random points in R^d\mathsf{R}^{d} . A recently proved topological identity of the author is used in combination with identities of Efron and Buchta to find the expected number of vertices of the convex hull—yielding a new recurrence formula for all dimensions d. A recurrence for the expected number of facets and (d−2)-faces is also found, this analysis building on a technique of Rényi and Sulanke. Other relationships for the expected count of i-faces (1≤i<d) are found when d≤5, by applying the Dehn–Sommerville identities. A general recurrence identity (see (3) below) for this expected count is conjectured.     

Mixing normal approximations of vectors of sums and maximum sums

Summary The conditioned central limit theorem for the vector of maximum partial sums based on independent identically distributed random vectors is investigated and the rate of convergence is discussed. The conditioning is that of Rényi (1958,Acta Math. Acad. Sci. Hungar.,9, 215–228). Analogous results for the vector of partial sums are obtained. University of Petroleum and Minerals