Cantor Staircases in Physics and Diophantine Approximations

1998 ◽  
Vol 08 (06) ◽  
pp. 1095-1106 ◽  
Author(s):  
M. Piacquadio Losada ◽  
S. Grynberg
Keyword(s):  
Number Theory ◽  
Quantum Mechanics ◽  
Dynamical Systems ◽  
Wide Class ◽  
Irrational Numbers ◽  
Diophantine Approximations ◽  
Universal Properties ◽  
Time Variables

For a wide class of dynamical systems the variables involved relate to one another through a Cantor staircase function. When they are time variables, the staircases have well-known universal properties that suggest a connection with certain classical problems in Number Theory. In this paper we extend some of those universal properties to certain Cantor staircases that appear in Quantum Mechanics, where the variables involved are not time variables. We also develop some connections between the geometry of these Cantor staircases and the problem of approximating irrational numbers of rational ones, classical in Number Theory.

Dynamical Systems in Number Theory

1982 ◽  
pp. 157-177
Author(s):  
I. P. Cornfeld ◽  
S. V. Fomin ◽  
Ya. G. Sinai
Keyword(s):  
Number Theory ◽  
Dynamical Systems

Expansions in Complex Bases

2007 ◽  
Vol 50 (3) ◽  
pp. 399-408 ◽  
Author(s):  
Vilmos Komornik ◽  
Paola Loreti
Keyword(s):  
Number Theory ◽  
Dynamical Systems ◽  
Greedy Algorithm ◽  
Finite Automata ◽  
Complex Case ◽  
Seminal Paper

AbstractBeginning with a seminal paper of Rényi, expansions in noninteger real bases have been widely studied in the last forty years. They turned out to be relevant in various domains of mathematics, such as the theory of finite automata, number theory, fractals or dynamical systems. Several results were extended by Daróczy and Kátai for expansions in complex bases. We introduce an adaptation of the so-called greedy algorithm to the complex case, and we generalize one of their main theorems.

Dynamical Systems, Number Theory and Applications

10.1142/9695 ◽  
2015 ◽  
Author(s):  
Thomas Hagen ◽  
Florian Rupp ◽  
Jürgen Scheurle
Keyword(s):  
Number Theory ◽  
Dynamical Systems

scholarly journals Fixed Points and Fermat: A Dynamical Systems Approach to Number Theory

2000 ◽  
Vol 107 (5) ◽  
pp. 422-428
Author(s):  
Michael Frame ◽  
Brenda Johnson ◽  
Jim Sauerberg
Keyword(s):  
Number Theory ◽  
Dynamical Systems ◽  
Fixed Points ◽  
Systems Approach

scholarly journals Elementary methods in the theory of numbers

1939 ◽  
Vol 31 ◽  
pp. xvi-xxiii
Author(s):  
S. A. Scott
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Exponential Function ◽  
Monotone Sequence ◽  
Binomial Theorem ◽  
Irrational Numbers ◽  
Early Stages ◽  
Elementary Methods ◽  
Usual Notation ◽  
Theory Of Numbers

§ 1. The importance of proving inequalities of an essentially algebraic nature by “elementary” methods has been emphasised by Hardy (Prolegomena to a Chapter on Inequalities), and by Hardy, Littlewood and Polya (Inequalities). The object of this Note is to show how some of the results in the early stages of Number Theory can be obtained by making a minimum appeal to irrational numbers and the notion of a limit. We use the elementary notion of a logarithm to a base “a” > 1, and make no appeal to the exponential function. The Binomial Theorem is only used for a positive integer index. Our minimum appeal rests in the assumption that a bounded monotone sequence tends to a limit. We adopt throughout the usual notation. Finally, it need scarcely be added that the methods employed are not claimed to be new.

scholarly journals Buonomano against Bell: Nonergodicity or nonlocality?

2017 ◽  
Vol 15 (08) ◽  
pp. 1740010 ◽  
Author(s):  
Andrei Khrennikov
Keyword(s):  
Quantum Mechanics ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Bell Inequality ◽  
Quantum Measurements ◽  
Neutron Interferometry ◽  
Bell’S Inequality ◽  
Weak Mixing ◽  
Bell's Inequality

The aim of this note is to attract attention of the quantum foundational community to the fact that in Bell’s arguments, one cannot distinguish two hypotheses: (a) quantum mechanics is nonlocal, (b) quantum mechanics is nonergodic. Therefore, experimental violations of Bell’s inequality can be as well interpreted as supporting the hypothesis that stochastic processes induced by quantum measurements are nonergodic. The latter hypothesis was discussed actively by Buonomano since 1980. However, in contrast to Bell’s hypothesis on nonlocality, it did not attract so much attention. The only experiment testing the hypothesis on nonergodicity was performed in neutron interferometry (by Summhammer, in 1989). This experiment can be considered as rejecting this hypothesis. However, it cannot be considered as a decisive experiment. New experiments are badly needed. We point out that a nonergodic model can be realistic, i.e. the distribution of hidden (local!) variables is well-defined. We also discuss coupling of violation of the Bell inequality with violation of the condition of weak mixing for ergodic dynamical systems.

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