O-Minimal Invariants for Discrete-Time Dynamical Systems

Termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Already for the simplest variants of linear loops the question of termination relates to deep open problems in number theory, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this article, we introduce the class of o-minimal invariants , which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel’s conjecture is transcendental number theory.

Set Theory INC# ∞# Based on Innitary Intuitionistic Logic with Restricted Modus Ponens Rule. Hyper Inductive Denitions. Application in Transcendental Number Theory. Generalized Lindemann-Weierstrass Theorem

In this paper intuitionistic set theory INC#∞# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.The Goldbach-Euler theorem is obtained without any references to Catalan conjecture. Main results are: (i) number ee is transcendental; (ii) the both numbers e + π and e − π are irrational.

NOTE ON FOURIER–STIELTJES COEFFICIENTS OF COIN-TOSSING MEASURES

It is known that the Fourier–Stieltjes coefficients of a nonatomic coin-tossing measure may not vanish at infinity. However, we show that they could vanish at infinity along some integer subsequences, including the sequence ${\{b^{n}\}}_{n\geq 1}$ where $b$ is multiplicatively independent of 2 and the sequence given by the multiplicative semigroup generated by 3 and 5. The proof is based on elementary combinatorics and lower-bound estimates for linear forms in logarithms from transcendental number theory.

On the Rationality and Transcendentality of Solutions to the Equation x^n + y^n = z^n

In this paper, the author develops 2 theorems that serve as extensions to the well-known Fermat’s Last Theorem in the field of number theory. The first proposed theorem in this paper states that if there exist numbers x, y, and an integer z such that x^n + y^n = z^n for any integer n > 2, then both/either x and/or y must be irrational. The second proposed theorem in this paper states that if there exist a number x, a transcendental number y, and an integer z such that x^n + y^n = z^n for any integer n > 2, then x must be irrational. The proposed theorems in this paper expand on the notion that if there exist numbers x, y, and an integer z such that x^n + y^n = z^n for any integer n > 2, then at least x or y is not an integer, which is stated in Fermat’s Last Theorem.

Algebraic Numbers as Product of Powers of Transcendental Numbers

The elementary symmetric functions play a crucial role in the study of zeros of non-zero polynomials in C [ x ] , and the problem of finding zeros in Q [ x ] leads to the definition of algebraic and transcendental numbers. Recently, Marques studied the set of algebraic numbers in the form P ( T ) Q ( T ) . In this paper, we generalize this result by showing the existence of algebraic numbers which can be written in the form P 1 ( T ) Q 1 ( T ) ⋯ P n ( T ) Q n ( T ) for some transcendental number T, where P 1 , … , P n , Q 1 , … , Q n are prescribed, non-constant polynomials in Q [ x ] (under weak conditions). More generally, our result generalizes results on the arithmetic nature of z w when z and w are transcendental.

Calculation of π with a needle

The number πis perhaps the most famous irrational number. This constant is equal to the ratio of the circumference of a circle to its diameter. One of the most well-known mathematical problems of antiquity, which is related to π, is how to construct by using a ruler and compasses a square which has the same area as a circle. This particular problem cannot be solved, due to the fact that π is a transcendental number, which means that it cannot be obtained as the root of a polynomial equation with rational coefficients. It was Euler in the 18th century who established the notation π.