Related papers: Research on Arithmetic
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…
The structure of the Euler-Lagrange equations for a general Lagrangian theory is studied. For these equations we present a reduction procedure to the so-called canonical form. In the canonical form the equations are solved with respect to…
We construct continued fraction expansions for several families of the Laurent series in $\mathbb{Q}[[t^{-1}]]$. To the best of the author's knowledge, this is the first result of this kind since Gauss derived the continued fraction…
The gauge symmetries of a general dynamical system can be systematically obtained following either a Hamiltonean or a Lagrangean approach. In the former case, these symmetries are generated, according to Dirac's conjecture, by the first…
Arithmetical structures on graphs were first introduced in \cite{Lorenzini89}. Later in \cite{arithmetical} they were further studied in the setting of square non-negative integer matrices. In both cases, necessary and sufficient conditions…
This text grew out of my lecture notes for a 4-hours minicourse delivered on October 17 \& 19, 2016 during the research school "Applications of Ergodic Theory in Number Theory" -- an activity related to the Jean-Molet Chair project of…
The Jacobi-Stirling numbers were discovered as a result of a problem involving the spectral theory of powers of the classical second-order Jacobi differential expression. Specifically, these numbers are the coefficients of integral…
We deal with various splitting methods in algebraic logic. The word `splitting' refers to splitting some of the atoms in a given relation or cylindric algebra each into one or more subatoms obtaining a bigger algebra, where the number of…
Certain analytical expressions which "feel" the divisors of natural numbers are investigated. We show that these expressions encode to some extent the well-known algorithm of the sieve of Eratosthenes. Most part of the text is written in…
This text is reproduced with the kind permission of Fran\c{c}ois Ap\'ery. It was originally edited by Fran\c{c}ois Gu\'enard and Gilbert Leli\`evre for the book "Penser les math\'ematiques". It is the modified and abridged version of a text…
Independent to a great extent from the scientific development of the discipline, a trend for statistics has developed in France, from 1827 on. It was probably sparked by Charles Dupin's 'Carte figurative de l'instruction populaire', with…
This article presents the emergence of formal methods in theory of partial differential equations (PDE) in the french school of mathematics through Janet's work in the period 1913-1930. In his thesis and in a series of articles published…
This is a set of lecture notes that developed out of courses on the lambda calculus that I taught at the University of Ottawa in 2001 and at Dalhousie University in 2007 and 2013. Topics covered in these notes include the untyped lambda…
In the present work we redefine and generalize the action principle for dissipative systems proposed by Riewe by fixing the mathematical inconsistencies present in the original approach. In order to formulate a quadratic Lagrangian for…
The Fibonacci numbers are familiar to all of us. They appear unexpectedly often in mathematics, so much there is an entire journal and a sequence of conferences dedicated to their study. However, there is also another sequence of numbers…
In this work we study a multi-step scheme on time-space grids proposed by W. Zhao et al. [28] for solving backward stochastic differential equations, where Lagrange interpolating polynomials are used to approximate the time-integrands with…
We undertake a detailed study of the $L^2$ discrepancy of rational and irrational 2-dimensional lattices either with or without symmetrization. We give a full characterization of lattices with optimal $L^2$ discrepancy in terms of the…
In this article, the notion of a mathematical model in science is attempted to be enlightened from several points of view. In particular, it is shown that mathematical models are introduced differently and used differently in different…
A well known theorem of Lagrange states that the simple continued fraction of a real number $\alpha$ is periodic if and only if $\alpha$ is a quadratic irrational. We examine non-periodic and non-simple continued fractions formed by two…
The main point of this paper is to present a class of equations over integers that one can check if they have a solution by checking a set of inequalities. The prototype of such equations is the equations appearing in the well-known…