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David Pouvreau

Publications and source records attributed to David Pouvreau.

6 recordsLinked to original sources
arXiv2022

The alternate congruo-harmonic series Part 2 -- Accelerations of convergence

For every couple (p;q) of strictly positive integers, the `` alternate congruo-harmonic '' series parametrized by (p;q), whose general term is (-1)^k/(pk+q), converges infra-linearly and very slowly. On the basis of a generalized continued fraction expansion of the partial rest of the series, this paper elaborates a family of algorithms which accelerate its convergence. The convergence speed of the sequences generated by these algorithms are compared. A precise asymptotic analysis is conducted, which reveals the possibility to accelerate the convergence either infra-linearly (but with an infinite diversity of possible speeds), or linearly (with a convergence rate that appears universal relatively to (p;q)), or super-linearly, by means of sequences extractions. Several open problems are also discussed, which concern the relative `` performance '' of the algorithms thus built and the possible optimality of some of them.

math.CA↗
arXiv2021

Beyond Conway's concyclicity theorem: generalization and alternatives

The famous concyclicity theorem stated by John H. Conway is here reconsidered by means of a parametrisation of the associated triangular configuration with arbitrary triplets of real numbers ($α$;$β$;$γ$). This theorem, thus corresponding to the case ($α$;$β$;$γ$)=(1;1;1), is generalized while demonstrating that there always exist an infinite family of such triplets which keeps unchanged the conclusion. The "anti-Conway" configuration corresponding to the case ($α$;$β$;$γ$)=(-1;-1;-1) is also investigated : Xavier Dussau's theorem of concurrent lines is redemonstrated and completed by another concyclicity theorem. It is also proved that there exist in general a unique triplet ($α$;$β$;$γ$)$\ne$(-1;-1;-1) which is a function of the sides of the considered triangle and which keeps unchanged the conclusion of Dussau's theorem.

math.AG↗
arXiv2021

Derived system and dual sequence of a barypolygonal sequence -- Part 1

This study continues three recent papers in which barypolygonal sequences have been defined and their properties of convergence demonstrated. Any barypolygonal sequence $\mathcal{B}$ of a finite set $\mathcal{A}$ comprising $p\ge 2$ points of any finite dimensional affine space can be used in order to define recurrently a definite sequence of barypolygonal sequences starting with $\mathcal{B}$. This sequence $(\mathcal{B}^{(m)})_{M\in N}$, called sequence of B's derivatives, is determined by real sequences that are solutions of a non linear recurrent system $(S)$: the barypolygonal derived system of $\mathcal{B}$. Each term of the sequence $(\mathcal{B}^{(m)})_{M\in N}$ converges toward a point $G_m$. The sequence $(G_m )_{m\in N}$ is the dual sequence of B. The convergence of the latter and the properties of the derived system are here investigated for any $p$ if $\mathcal{B}$ is regular and in any case if $p\in{2;3}$.

math.AG↗
arXiv2021

On the acceleration of the convergence of the "Mādhava-Leibniz series"

This paper expounds very innovative results achieved between the mid-14th century and the beginning of the 16th century by Indian astronomers belonging to the so-called "Mādhava school". These results were in keeping with researches in trigonometry: they concern the calculation of the eight of the circumference of a circle. They not only expose an analog of the series expansion of arctan(1) usually known as the "Leibniz series", but also other analogs of series expansions, the convergence of which is much faster. These series expansions are derived from evaluations of the rests of the partial sums of the primordial series, by means of some convergents of generalized continued fractions. A justification of these results in modern terms is provided, which aims at restoring their full mathematical interest.

math.HO↗
arXiv2021

Themes of parity in the valuation for integer numbers of the zeta function and of five related functions

This paper considers the problem of the valuation for integer numbers of the zeta function and of five other functions which are naturally associated to it. A relatively elementary approach is exposed, which closely connects this still partially open problem to five themes of parity: the notions of parity of a function and of parity of the degree of a polynomial are here related to the distinctions of parity concerning the natural argument of the six considered functions as well as the integer numbers of which some inverse powers are summed. The adopted method essentially aims at enabling the students in mathematics to have an entry into this problem.

math.HO↗