scholarly journals About the Algebraic Yuzvinski Formula

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Anna Giordano Bruno ◽  
Simone Virili
Keyword(s):  
Vector Space ◽  
Topological Entropy ◽  
Characteristic Polynomial ◽  
Mahler Measure ◽  
Algebraic Proof ◽  
Open Problems ◽  
Algebraic Entropy ◽  
Rational Vector ◽  
The Common ◽  
Rational Vector Space

AbstractThe Algebraic Yuzvinski Formula expresses the algebraic entropy of an endomorphism of a finitedimensional rational vector space as the Mahler measure of its characteristic polynomial. In a recent paper, we have proved this formula, independently fromits counterpart – the Yuzvinski Formula – for the topological entropy proved by Yuzvinski in 1968. In this paper we first compare the proof of the Algebraic Yuzvinski Formula with a proof of the Yuzvinski Formula given by Lind and Ward in 1988, underlying the common ideas and the differences in the main steps. Then we describe several known applications of the Algebraic Yuzvinski Formula, and some related open problems are discussed. Finally,we give a new and purely algebraic proof of the Algebraic Yuzvinski Formula for the intrinsic algebraic entropy.

Bounded Solutions of a Functional Inequality

1982 ◽  
Vol 25 (4) ◽  
pp. 491-495 ◽  
Author(s):  
Michael Albert ◽  
John A. Baker
Keyword(s):  
Vector Space ◽  
Bounded Solution ◽  
Functional Inequality ◽  
Bounded Solutions ◽  
Rational Vector ◽  
Image Position ◽  
Rational Vector Space

AbstractIt is known that if f is a real valued function on a rational vector space V, δ > 0,1and if f is unbounded then f(x + y) = f(x)f(y) for all x, y ∊ V. In response to a problem of E. Lukacs, in this paper we study the bounded solutions of (1). For example, it is shown that if f is a bounded solution of (1) then - δ ≤ f(x) ≤ (1 + (1 + 4δ)1/2)/2 for all x ∊ V and these bounds are optimal.

scholarly journals Introduction

2018 ◽  
Vol 27 (4) ◽  
pp. 441-441
Author(s):  
PAUL BALISTER ◽  
BÉLA BOLLOBÁS ◽  
IMRE LEADER ◽  
ROB MORRIS ◽  
OLIVER RIORDAN
Keyword(s):  
Computer Science ◽  
Theoretical Computer Science ◽  
Special Issue ◽  
Open Problems ◽  
Theoretical Computer ◽  
Discrete Probability ◽  
The Common

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from the 17th to the 23rd April 2016. The lectures at this meeting focused on the common themes of Combinatorics and Discrete Probability, with many of the problems studied originating in Theoretical Computer Science. The lectures, many of which were given by young participants, stimulated fruitful discussions. The fact that the participants work in different and yet related topics, and the open problems session held during the meeting, encouraged interesting discussions and collaborations.

Milnor's conjecture on monotonicity of topological entropy: Results and questions

2014 ◽  
Author(s):  
Sebastian van Strien
Keyword(s):  
Topological Entropy ◽  
State Of The Art ◽  
Rigidity Theorem ◽  
Real Polynomial ◽  
Open Problems ◽  
Polynomial Maps ◽  
History Of ◽  
The Subject ◽  
Quadratic Case ◽  
Real Polynomial Maps

This chapter discusses Milnor's conjecture on monotonicity of entropy and gives a short exposition of the ideas used in its proof. It discusses the history of this conjecture, gives an outline of the proof in the general case, and describes the state of the art in the subject. The proof makes use of an important result by Kozlovski, Shen, and van Strien on the density of hyperbolicity in the space of real polynomial maps, which is a far-reaching generalization of the Thurston Rigidity Theorem. (In the quadratic case, density of hyperbolicity had been proved in studies done by M. Lyubich and J. Graczyk and G. Swiatek.) The chapter concludes with a list of open problems.

A geometric approach to semi-dispersing billiards (Survey)

1998 ◽  
Vol 18 (2) ◽  
pp. 303-319 ◽  
Author(s):  
D. BURAGO ◽  
S. FERLEGER ◽  
A. KONONENKO
Keyword(s):  
Topological Entropy ◽  
Riemannian Geometry ◽  
Geometric Approach ◽  
Open Problems ◽  
Periodic Trajectories ◽  
Uniform Estimates ◽  
Dispersing Billiards

We summarize the results of several recent papers, together with a few new results, which rely on a connection between semi-dispersing billiards and non-regular Riemannian geometry. We use this connection to solve several open problems about the existence of uniform estimates on the number of collisions, topological entropy and periodic trajectories of such billiards.

scholarly journals On the class of square Petrie matrices induced by cyclic permutations

2004 ◽  
Vol 2004 (31) ◽  
pp. 1617-1622
Author(s):  
Bau-Sen Du
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Topological Entropy ◽  
Characteristic Polynomial ◽  
Characteristic Polynomials ◽  
Dynamical Properties

Letn≥2be an integer and letP={1,2,…,n,n+1}. LetZpdenote the finite field{0,1,2,…,p−1}, wherep≥2is a prime. Then every mapσonPdetermines a realn×nPetrie matrixAσwhich is known to contain information on the dynamical properties such as topological entropy and the Artin-Mazur zeta function of the linearization ofσ. In this paper, we show that ifσis acyclicpermutation onP, then all such matricesAσare similar to one another overZ2(but not overZpfor any primep≥3) and their characteristic polynomials overZ2are all equal to∑k=0nxk. As a consequence, we obtain that ifσis acyclicpermutation onP, then the coefficients of the characteristic polynomial ofAσare all odd integers and hence nonzero.

Decidability and essential undecidability

1957 ◽  
Vol 22 (1) ◽  
pp. 39-54 ◽  
Author(s):  
Hilary Putnam
Keyword(s):  
Simple Proof ◽  
Arithmetic Functions ◽  
Successor Function ◽  
Open Problems ◽  
Decidable Theory ◽  
Undecidable Theory ◽  
The Common

There are a number of open problems involving the concepts of decidability and essential undecidability. This paper will present solutions to some of these problems. Specifically:(1) Can a decidable theory have an essentially undecidable, axiomatizable extension (with the same constants)?(2) Are all the complete extensions of an undecidable theory ever decidable?We shall show that the answer to both questions is in the affirmative. In answering question (1), the decidable theory for which an essentially undecidable axiomatizable extension will be constructed is the theory of the successor function and a single one-place predicate. It will also be shown that the decidability of this theory is a “best possible” result in the following direction: the theory of either of the common diadic arithmetic functions and a one-place predicate; i.e., of addition and a one-place predicate, or of multiplication and a one-place predicate, is undecidable.Before establishing the main result, it is convenient to give a simple proof that a decidable theory can have an axiomatizable (simply) undecidable extension. This is, of course, an immediate consequence of the main result; but the proof is simple and illustrates the methods that we are going to use in this paper.

Similarity Classes of Linear Transformations

2020 ◽  
Vol 87 (3-4) ◽  
pp. 148
Author(s):  
Puja Bharti ◽  
Jagmohan Tanti
Keyword(s):  
Scalar Field ◽  
Vector Space ◽  
Characteristic Polynomial ◽  
Principal Ideal ◽  
Structure Theorem ◽  
Principal Ideal Domain ◽  
Linear Transformations ◽  
Ideal Domain ◽  
Number Of Classes ◽  
Finitely Generated Modules

In this paper, we investigate the similarity classes of linear transformations on a vector space using structure theorem for finitely generated modules over a principal ideal domain. We also establish formulae to count similarity classes with a given polynomial as a characteristic polynomial and to count total number of classes when the scalar field is finite.

scholarly journals Introduction

2010 ◽  
Vol 19 (5-6) ◽  
pp. 641-641
Author(s):  
Noga Alon ◽  
Béla Bollobás
Keyword(s):  
Computer Science ◽  
Theoretical Computer Science ◽  
Special Issue ◽  
Open Problems ◽  
Theoretical Computer ◽  
Discrete Probability ◽  
The Common

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from 26 April to 2 May. This meeting focused on the common themes of Combinatorics, Discrete Probability and Theoretical Computer Science, and the lectures, many of which were given by young participants, stimulated fruitful discussions. The open problems session held during the meeting, and the fact that the participants work in different and related topics, encouraged interesting discussions and collaborations.

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