In this paper we will show the irrationality of n √ 2 for all natural numbers n ≥ 2.

We study Lagrangian systems on a closed manifold M . We link the differentiability of Mather’s β -function with the topological complexity of the complement of the Aubry set. As a consequence, when dimM ≤ 3, the differentiability of the β -function at a given homology class is forced by the irrationality of the homology class. As an application we prove the two-dimensional case of two conjectur...

people for their evil activities. Hence, irrational thinking should be associated with increased individual sensitivity concerning unfair social events. Results of our study (N = 108) confirm this hypothesized correlation between Irrationality and Justice Sensitivity. Notably, the strongest correlation was found between low frustration tolerance, a subdimension of irrationality, and justice sen...

For simplicity, we follow the rules: k, n, p, K, N are natural numbers, x, y, e1 are real numbers, s1, s2, s3 are sequences of real numbers, and s4 is a finite sequence of elements of R. Let us consider x. We introduce x is irrational as an antonym of x is rational. Let us consider x, y. We introduce x as a synonym of x. One can prove the following two propositions: (1) If p is prime, then √ p ...

This paper is devoted to the rational approximation of automatic real numbers, that is, real numbers whose expansion in an integer base can be generated by a finite automaton. We derive upper bounds for the irrationality exponent of famous automatic real numbers associated with the Thue–Morse, Rudin–Shapiro, paperfolding and Baum–Sweet sequences. These upper bounds arise from the construction o...

Inspired by the epistemic game theory framework, I elicit subjects’ preferences over outcomes, beliefs about strategies, and beliefs about beliefs in a variety of simple games. I find that the prisoners’ dilemma and the traditional centipede game are both Bayesian games, with many non-selfish types. Many players choose strategies that are clearly inconsistent with their elicited beliefs and pre...

We apply the Padé technique to find rational approximations to h±(q1, q2) = ∞ ∑ k=1 q 1 1± q 2 , 0 < q1, q2 < 1, q1 ∈ Q, q2 = 1/p2, p2 ∈ N \ {1}. A separate section is dedicated to the special case qi = q ri , ri ∈ N, q = 1/p, p ∈ N \ {1}. In this construction we make use of little q-Jacobi polynomials. Our rational approximations are good enough to prove the irrationality of h(q1, q2) and give...

In 2006, Jonathan Sondow gave a nice geometric proof that e is irrational. Moreover, he said that a generalization of his construction may be used to prove the Cantor’s theorem. But, he didn’t do it in his paper, see [1]. So, this work will give a geometric proof to Cantor’s theorem using Sondow’s construction. After, it is given an irrationality measure to some Cantor series, for that, we gene...