Related papers: Two $\mathfrak{b}$ or not two $\mathfrak{b}$?
We give a construction of an absolutely normal real number $x$ such that for every integer $b $ greater than or equal to $2$, the discrepancy of the first $N$ terms of the sequence $(b^n x \mod 1)_{n\geq 0}$ is of asymptotic order…
We study complex finite-dimensional Leibniz algebra bimodule over $\mathfrak{sl}_2$ that as a Lie algebra module is split into a direct sum of two simple $\mathfrak{sl}_2$-modules. We prove that in this case there are only two non-split…
We give (two) non additive languages for geometry via simple generalisations of commutative rings.
In this paper I explore a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, I prove a nonstandard version of Frostman's lemma and show that Hausdorff…
We prove a generalization of one of Lie's Theorems in the context of Lie-like algebras$^{2-nd}$.
Recently it has been shown that antibrackets may be expressed in terms of Poisson brackets and vice versa for commuting functions in the original bracket. Here we also introduce generalized brackets involving higher antibrackets or higher…
Let $\alpha$ and $\beta$ be two nonnegative integers such that $\beta < \alpha$. For an arbitrary sequence $\{a_n\}_{n\geqslant 1}$ of complex numbers, we consider the generalized Lambert series in order to investigate linear combinations…
This work deals with the notion of Newton complementary duality as raised originally in the work of the second author and B. Costa. A conceptual revision of the main steps of the notion is accomplished which then leads to a vast…
The purpose of this paper is to clarify the (non-Bayesian and Bayesian) two-envelope problems in terms of quantum language (or, measurement theory), which was recently proposed as a linguistic turn of quantum mechanics (with the Copenhagen…
For a set of nonnegative integers $A$, denote by $R_{A}(n)$ the number of unordered representations of the integer $n$ as the sum of two different terms from $A$. In this paper we partially describe the structure of the sets, which have…
Probability theory is far from being the most general mathematical theory of uncertainty. A number of arguments point at its inability to describe second-order ('Knightian') uncertainty. In response, a wide array of theories of uncertainty…
The paper summarizes the construction of pairings on some standard spectral sequences in algebraic topology.
We construct an addition and a multiplication on the set of planar binary trees, closely related to addition and multiplication on the integers. This gives rise to a new kind of (noncommutative) arithmetic theory. The price to pay for this…
In this paper, we consider the poly-Bernoulli numbers and polynomials of the second kind and presents new and explicit formulae for calculating the poly-Bernoulli numbers of the second kind and the Stirling numbers of the second kind.
The paper describes two Borel-measurable functions from a measure space into a locally convex space such that the image measure for each function is Radon but their sum is not Borel-measurable.
A new characterization of conformal transformations is given. By use of this, the general form of conformal transformation on two-dimensional Minkowski space is given and its conformal structure is analyzed.
The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…
In the paper, the authors discover an integral representation, some inequalities, and complete monotonicity of Bernoulli numbers of the second kind.
The set $\Mfib$ of fibbinary numbers is defined via a bijection between the set $\BB{N}$ of natural numbers and $\Mfib$. Since the elements of $\Mfib$ do not exhaust $\BB{N}$, the structure of the complement $\overline{\Mfib}$ of $\Mfib$ in…
An elementary but useful fact is that the numerator of the difference of two consecutive Farey fractions is equal to one. For triples of consecutive fractions the numerators of the differences are well understood and have applications to…