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An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving $-\infty$ and/or $+\infty$, so-called residuations. Based on this,…
We develop a differential theory for the polarity transform parallel to that for the Legendre transform, which is applicable when the functions studied are "geometric convex", namely convex, non-negative and vanish at the origin. This…
Recently Raayoni et al. announced various conjectures on continued fractions of fundamental constants automatically generated with machine learning techniques. In this paper we prove some of their stated conjectures for Euler number $e$ and…
We consider two classes of $q$-continued fraction whose odd and even parts are limit 1-periodic for $|q|>1$, and give theorems which guarantee the convergence of the continued fraction, or of its odd- and even parts, at points outside the…
The Fock transform recently introduced by the authors in a previous paper is applied to investigate convergence of generalized functional sequences of a discrete-time normal martingale $M$. A necessary and sufficient condition in terms of…
The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in…
We introduced a new continued fraction expansions in our previous paper. For these expansions, we show formulae of probability about incomplete quotients. Furthermore, we prove the existence of invariant measures with respect to the…
We prove an explicit formula for infinitely many convergents of Hurwitzian continued fractions that repeat several copies of the same constant and elements of one arithmetic progression, in a quasi-periodic fashion. The proof involves…
Here we construct the conformal mappings with the help of continuous fractions approximations. These approximations converge to the algebraic roots $\sqrt[N]{z}$ for $N \in \mathbb{N}$ and $z$ from the right half-plane of the complex plane.…
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…
Mixed volumes, which are the polarization of volume with respect to the Minkowski addition, are fundamental objects in convexity. In this note we announce the construction of mixed integrals, which are functional analogs of mixed volumes.…
Several conjectural continued fractions found with the help of various algorithms are published in this paper.
We study a family of generalized continued fractions, which are defined by a pair of substitution sequences in a finite alphabet. We prove that they are stammering sequences, in the sense of Adamczewski and Bugeaud. We also prove that this…
We construct an absolutely normal number whose continued fraction expansion is normal in the sense that it contains all finite patterns of partial quotients with the expected asymptotic frequency as given by the Gauss-Kuzmin measure. The…
Bifractional transformations which lead to quantities that interpolate between other known quantities, are considered. They do not form a group, and groupoids are used to described their mathematical structure. Bifractional coherent states…
The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These…
The paper extends the widely used in optimisation theory decoupling techniques to infinite collections of functions. Extended concepts of uniform lower semicontinuity and firm uniform lower semicontinuity are discussed. The main theorems…
We state and prove three general formulas allowing to transform formal finite sums into formal continued fractions and apply them to generalize certain expansions in continued fractions given by Hone and Varona.
The functional interpolation problem on a continual set of nodes by an integral continued C-fraction is studied. The necessary and sufficient conditions for its solvability are found. As a particular case, the considered integral continued…
We show several properties related to the structure of the family of classes of two-dimensional periodic continued fractions. This approach to the study of the family of classes of nonequivalent two dimexsional periodic continued fractions…