Related papers: Continued logarithm representation of real numbers
We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for practical computation in…
In this paper, we apply results on number systems based on continued fraction expansions to modular arithmetic. We provide two new algorithms in order to compute modular multiplication and modular division. The presented algorithms are…
This paper provides an introduction to the design of augmented data structures that offer an efficient representation of a mathematical sequence and fast sequential summation algorithms, which guarantee both logarithmic running time and…
We explain the algebra needed to make sense of the log signature of a path, with plenty of examples. We show how the log signature can be calculated numerically, and explain some software tools which demonstrate it.
The parametric geometry of numbers has allowed to visualize the simultaneous approximation properties of a collection of real numbers through the combined graph of the related successive minima functions. Several inequalities among…
Given a list of N numbers, the maximum can be computed in N iterations. During these N iterations, the maximum gets updated on average as many times as the Nth harmonic number. We first use this fact to approximate the Nth harmonic number…
By using the strong approximation, this paper establishes several limit results on the convergent rate of a infinite series of probabilities on the other law of iterated logarithm.
We present a comprehensive survey of constructions of the real numbers (from either the rationals or the integers) in a unified fashion, thus providing an overview of most (if not all) known constructions ranging from the earliest attempts…
In this paper we define "a continued fraction expansion of the exponential integral $E_{1}(x)$ at infinity", which is analogous to the regular continued fraction expansion of real numbers, and prove that this expansion gives the same…
Persistent homology is typically computed through persistent cohomology. While this generally improves the running time significantly, it does not facilitate extraction of homology representatives. The mentioned representatives are…
In this paper, we present several novel integral representations of Catalan's constant. We begin by deriving an initial result expressed as a double integral. Subsequently, as a consequence of this result, we establish a general theorem…
The evaluation of iterated primitives of powers of logarithms is expressed in closed form. The expressions contain polynomials with coefficients given in terms of the harmonic numbers and their generalizations. The logconcavity of these…
We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog…
We introduce a $q$-analog of the higher continued fractions introduced by the last three authors in a previous work (together with Gregg Musiker), which are simultaneously a generalization of the $q$-rational numbers of Morier-Genoud and…
In this paper we discuss some convergence and divergence properties of subsequences of logarithmic means of Walsh-Fourier series . We give necessary and sufficient conditions for the convergence regarding logarithmic variation of numbers.
The purpose of this manuscript is to derive two distinct integral representations of the generalized polylogarithm using two different techniques. The first approach involves the Dirichlet series and its Laplace representation, which leads…
For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to…
In the present paper, we establish an analytic continuation of Drinfeld logarithms by using the techniques introduced in [Fur20]. This result can be seen as an analogue of the analytic continuation of the elliptic integrals of the first…
We establish a central limit theorem for counting large continued fraction digits $(a_n)$, i.e. we count occurrences $\{a_n>b_n\}$, where $(b_n)$ is a sequence of positive integers. Our result improves a similar result by Philipp which…
A Lagrange Theorem in dimension 2 is proved, for a particular two-dimensional algorithm, with a very natural geometrical definition. Dirichlet-type properties for the convergence of the algorithm are also proved. These properties procced…