Related papers: Another Proof of $e^{x/y}$ being irrational
We explore Mahler numbers originating from functions $f(z)$ that satisfy the functional equation $f(z) = (A(z)f(z^d) + C(z))/B(z)$. A procedure to compute the irrationality exponents of such numbers is developed using continued fractions…
We prove an analog of Lagrange's Theorem for continued fractions on the Heisenberg group: points with an eventually periodic continued fraction expansion are those that satisfy a particular type of quadratic form, and vice-versa.
In this note we evaluate multiple integrals that play a crucial role in the theory of irrationality of zeta function
The better title is "Yet another FALSE proof of the 4-colour theorem." Please consider all versions of this paper as historical material on the way to a non-computer proof of the 4-colour theorem. Interpreted as proofs, all versions are…
We provide new results on the existence of extremal solutions for discontinuous differential equations with a deviated argument which can be either delayed or advanced. The boundary condition is allowed to be discontinuous and to depend…
In the recent development in a various disciplines of physics, it is noted the need for including the deformed versions of the exponential functions. In this paper, we consider the deformations which have two purposes: to have them like…
An algebraic method is used to study the semantics of exceptions in computer languages. The exceptions form a computational effect, in the sense that there is an apparent mismatch between the syntax of exceptions and their intended…
In this paper we construct examples of irrational behavior of multiplicities and mixed multiplicities of divisorial filtrations. The construction makes essential use of anti-positive intersection products.
We prove some statements of left- and right-continuous variants of generalized inverses of non-decreasing real functions.
In this paper, we define the deformed Euler $(s,t)$-numbers ${\rm e}_{s,t,u}$ Furthermore, we prove that ${\rm e}_{as,a^2t,u^{-1}}$ and ${\rm e}_{as,a^2t,u^{-1}}^{-1}$ are irrational numbers when $a,u\in\mathbb{Q}$ and $\vert au\vert>1$,…
In this note we show that in addition to two integers forming a Pythagorean triple, there also exist two irrational numbers in terms of which this Pythagorean triple can also be obtained. We also put forward a relation between these two…
The theory of continued fractions has been generalized to l-adic numbers by several authors and presents many differences with respect to the real case. In the present paper we investigate the expansion of rationals and quadratic…
Motivated by children inventing creatively laws on how to add or multiply fractions, we introduce four fraction operations and discover some of their properties.
Practicing mathematicians often assume that mathematical claims, when they are true, have good reasons to be true. Such a state of affairs is "unreasonable", in Wigner's sense, because basic results in computational complexity suggest that…
In this paper, we clarified the relationship between continued fractions, determinants, and identities, making it easier to apply these methods systematically in other settings. In particular, we studied finite continued fractions from the…
Evidential reasoning is cast as the problem of simplifying the evidence-hypothesis relation and constructing combination formulas that possess certain testable properties. Important classes of evidence as identifiers, annihilators, and…
Analytical expressions are derived for the number of fractions with equal numerators in the Farey sequence of order $n$, $F_n$, and in the truncated Farey sequence $F_n^{1/k}$ containing all Farey fractions below $1/k$, with $1\leq k \leq…
An analysis using classical stochastic processes is used to construct a consistent system of quantum counterfactual reasoning. When applied to a counterfactual version of Hardy's paradox, it shows that the probabilistic character of quantum…
We discuss the use of matrices for providing sequences of rationals that approximate algebraic irrationalities. In particular, we study the regular representation of algebraic extensions, proving that ratios between two entries of the…
A new simple proof of Stirling's formula via the partial fraction expansion for the tangent function is presented.