Related papers: Linear Independence Of Some Irrational Numbers

We prove that given $\lambda \in \mathbb{R}$ such that $0 < \lambda < 1$, then $\pi(x + x^\lambda) - \pi(x) \sim \displaystyle \frac{x^\lambda}{\log(x)}$. This solves a long-standing problem concerning the existence of primes in short…

The Thue--Morse sequence $\{t(n)\}_{n\geqslant 1}$ is the indicator function of the parity of the number of ones in the binary expansion of positive integers $n$, where $t(n)=1$ (resp. $=0$) if the binary expansion of $n$ has an odd (resp.…

Given $k \ge 2$ polynomials in $d \ge 1$ variables with coefficients in a field of characteristic $0$, such that no two are linearly dependent, we show that for any integer $r$ greater than $\max\left\{k {k-1 \choose 2}, 2\right\}$, the…

This paper revisits the Pearson Chi-squared independence test. After presenting the underlying theory with modern notations and showing new way of deriving the proof, we describe an innovative and intuitive graphical presentation of this…

Let $G$ be a simple graph and let $L(G)$ denote the \emph{line graph} of $G$. A \emph{$p$-independent} set in $G$ is a set of vertices $S \subseteq V(G)$ such that the subgraph induced by $S$ has maximum degree at most $p$. The…

We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.

Conditional-independence-based discovery uses statistical tests to identify a graphical model that represents the independence structure of variables in a dataset. These tests, however, can be unreliable, and algorithms are sensitive to…

In this paper, we propose various sufficient conditions to determine if a given real number is an irrational number or a transcendental number and also apply these conditions to some interesting examples, particularly,one of them comes from…

The famous result of Lindemann and Weierstrass says that if $a_{1},a_{2},\ldots,a_{n}$ are distinct algebraic numbers, then $e^{a_{1}},e^{a_{2}},\ldots,e^{a_{n}}$ are linearly independent complex numbers over the field…

This paper considers the problem of testing whether there exists a solution satisfying certain non-negativity constraints to a linear system of equations. Importantly and in contrast to some prior work, we allow all parameters in the system…

We demonstrate how to test for conditional independence of two variables with categorical data using Poisson log-linear models. The size of the conditioning set of variables can vary from 0 (simple independence) up to many variables. We…

A linear inference is a valid inequality of Boolean algebra in which each variable occurs at most once on each side. In this work we leverage recently developed graphical representations of linear formulae to build an implementation that is…

All the already known results on self descriptive numbers, together with the demonstration of the uniqueness for bases greater than 6, are here obtained through a systematic scheme of proof and not trial and error. The proof is also…

Generalizations of linear numeration systems in which the set of natural numbers is recognizable by finite automata are obtained by describing an arbitrary infinite regular language following the lexicographic ordering. For these systems of…

In this paper we use e-values in the context of multiple hypothesis testing assuming that the base tests produce independent, or sequential, e-values. Our simulation and empirical studies and theoretical considerations suggest that, under…

The rules of d-separation provide a framework for deriving conditional independence facts from model structure. However, this theory only applies to simple directed graphical models. We introduce relational d-separation, a theory for…

Let E/Q be an elliptic curve with a fixed modular parametrization F : X_0(N) --> E and let P_1,...,P_r be Heegner points on E attached to the rings of integers of distinct quadratic imaginary field k_1,...,k_r. We prove that if the odd…

We show that two distinct singular moduli $j(\tau),j(\tau')$, such that for some positive integers $m, n$ the numbers $1,j(\tau)^m$ and $j(\tau')^n$ are linearly dependent over $\mathbb{Q}$ generate the same number field of degree at most…

Spherical and hyperspherical data are commonly encountered in diverse applied research domains, underscoring the vital task of assessing independence within such data structures. In this context, we investigate the properties of test…

Testing for independence between graphs is a problem that arises naturally in social network analysis and neuroscience. In this paper, we address independence testing for inhomogeneous Erd\H{o}s-R\'{e}nyi random graphs on the same vertex…