Related papers: The rational iteration method by Georges Lemaitre …
While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants…
In this article algorithmic methods are presented that have essentially been introduced into computer algebra systems like Mathematica within the last decade. The main ideas are due to Stanley and Zeilberger. Some of them had already been…
The Dirac delta function has solid roots in 19th century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac's discovery by over a century, and illuminating the nature of Cauchy's infinitesimals and his…
We present a simple iteration for the Lebesgue identity on partitions, which leads to a refinement involving the alternating sums of partitions.
Abraham Robinson's framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the…
We survey and develop the most elementary model of electron motion introduced by R$.$Feynman. In this game, a checker moves on a checkerboard by simple rules, and we count the turns. Feynman checkers are also known as a one-dimensional…
To our knowledge, there are two main references [9], [12] regarding the periodical solutions of multi-time Euler-Lagrange systems, even if the multi-time equations appeared in 1935, being introduced by de Donder. That is why, the central…
The calculation of the electron g-factor was carried out in 1950 by Karplus and Kroll. Seven years later, Petermann detected and corrected a serious error in the calculation of a Feynman diagram. Although it is hard to believe, neither the…
This work wishes to support various mathematical issues concerning the iterative methods with the help of new programming languages. We consider a way to show how problems in math have an answer by using different academic resources and…
One of the biggest mysteries of astrophysics is the question of how highly energetic particles in relativistic jets and cosmic rays are accelerated. Recently, it has been suggested that gravitational repulsion is the mechanism responsible…
By using ideas on complexity and randomness originally suggested by the mathematician-philosopher Gottfried Leibniz in 1686, the modern theory of algorithmic information is able to show that there can never be a "theory of everything" for…
After sketching the basic principles of renewal theory we discuss the classical Poisson process and offer two other processes, namely the renewal process of Mittag-Leffler type and the renewal process of Wright type, so named by us because…
This paper treats about one of the most remarkable achievements by Riemann, that is the symmetric form of the functional equation for {\zeta}(s). We present here, after showing the first proof of Riemann, a new, simple and direct proof of…
A well-known problem in computing some matrix functions iteratively is the lack of a clear, commonly accepted residual notion. An important matrix function for which this is the case is the matrix exponential. Suppose the matrix exponential…
Graeffe iteration was the choice algorithm for solving univariate polynomials in the XIX-th and early XX-th century. In this paper, a new variation of Graeffe iteration is given, suitable to IEEE floating-point arithmetics of modern digital…
Using the Carleman linearization technique the continuous iteration of a mapping is studied. Based on the detailed analysis of the Carleman embedding matrix the precise mathematical meaning is given to such notion. The ordinary differential…
A matrix approach to continuous iteration is proposed for general formal series. It leads, in particular, to an order{to{order iteration of the exponential function, and consequently to an algorithmic approach to tetration. Lower{order…
Consider a rational map $R$ of degree $d\geq 2$ with coefficients over the non-archimedean field $\mathbb{C}_p$, with $p$ a fixed prime number. If $R$ has a cycle of Siegel disks and has good reduction, then it was shown by Rivera-Letelier…
Vlastimil Pt\'ak's method of nondiscrete induction is based on the idea that in the analysis of iterative processes one should aim at rates of convergence as functions rather than just numbers, because functions may give convergence…
This is the translation into English of my paper published in Russian at the paper collection on cosmology at \\ 1981. The content of the book is presented in the Appendix. This is the first paper where the dynamical mechanism of phoenix…