Renato M. Capocelli

Bounds on the redundancy of Huffman codes (Corresp.)

New upper bounds on the redundancy of Huffman codes are provided. A bound that for 2/9 \leq P_{1} \leq 0.4 is sharper than the bound of Gallager, when the probability of the most likely source letter P_{1} is the only known probability is presented. The improved bound is the tightest possible for 1/3 \leq P_{1} \leq 0.4 . Upper bounds are presented on the redundancy of Huffman codes when the extreme probabilities P_{1} and P_{N} are known.

On some inequalities and generalized entropies: A unified approach

Abstract Some inequalities useful in information theory are considered and extended for various generalized entropies. The extension is done under a unified approach. Some possible applications of the generalized entropies as well as of the inequalities are discussed. Some other properties, relationships and inequalities among the entropies along with concavity, pseudoconcavity and Schur concavity are also given.

Fibonacci facts and formulas

We investigate several methods of computing Fibonacci numbers quickly and generalize some properties of the Fibonacci numbers to degree r Fibonacci (R-nacci) numbers. Sections 2 and 3 present several algorithms for computing the traditional, degree two, Fibonacci numbers quickly. Sections 4 and 5 investigate the structure of the binary representation of the Fibonacci numbers. Section 6 shows how the generalized Fibonacci numbers can be expressed as rounded powers of the dominant root of the characteristic equation. Properties of the roots of the characteristic equation of the generalized Fibonacci numbers are presented in Section 7. Section 8 introduces several properties of the Zeckendorf representation of the integers. Finally, in Section 9 the asymptotic proportion of l’s in the Zeckendorf representation of integers is computed and an easy to compute closed formula is given.

Algorithms for Factorizing and Testing Subsemigroups

Given a finite subset Γ of a fixed, finite alphabet Σ, we construct the basis B of the minimum subsemigroup of Σ+ containing Γ, such that B has various properties. The properties we consider are that B be a uniquely decipherable, a finitely decipherable, a synchronizable, or a prefix code. The algorithm for constructing the uniquely decipherable and the finitely decipherable code B requires O(n 2 L + L 2) steps, the algorithm for constructing the synchronizable code B requires O(n L 2) steps, and the algorithm for constructing the prefix code B requires O(L 2) steps. Here n is the cardinality of Γ and L is the sum of the lengths of the words in Γ. Finally, given a synchronizable or finitely decipherable code Γ, we also show how to determine its synchronizability or decipherability delay, in O(n L) steps.

A Cybernetic Approach to Population Dynamics Modeling

A few instances of current biocybernetical problems are thoroughly investigated to shed some light on the more general problem related to the dynamics of complex interactive systems subject to random perturbations. The validity of a conjecture on the persistence of a population in a randomly varying environment is proved for a wide class of models described by first order stochastic differential equations.

Flag encodings related to the Zechendorf representation of integers

The Zeckendorf’s representation has the property that it contains no sequence of r or more consecutive ones. This property can be used for separating codewords providing efficient flag encodings that are almost optimal and free from error propagation.

The Entropy of Written Italian

The entropies to the fourth order and the word entropy, with and without space symbol (blank), of an Italian text are calculated. The obtained value of the fourth order entropy shows that a redundancy of about 53% is exhibited by the text.