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How Long Might We Wait at Random?Jun 14 2019In discrete time, customers arrive at random. Each waits until one of three servers is available; each thereafter departs at random. We seek the distribution of maximum line length of idle customers. Algebraic expressions obtained for the two-server scenario ... More

Arquímedes y las superficies cuádricasJun 10 2019A brief review of the history of the conic sections would not be complete without an exhaustively tolerable account of all the things related to the subject that can be found in the extensive work of the wise Archimedes. There is no strong evidence that ... More

When do we have 1 + 1 = 11 and 2 + 2 =5?Jun 05 2019This work is inspired in part by the following passage from the famous dystopian novel 1984, by George Orwell. "He wrote first in large clumsy capitals: FREEDOM IS SLAVERY. Then almost without a pause he wrote beneath it: TWO AND TWO MAKE FIVE." Here ... More

When do we have 1 + 1 = 11 and 2 + 2 =5?Jun 05 2019Jun 11 2019This work is inspired in part by the following passage from the famous dystopian novel 1984, by George Orwell. "He wrote first in large clumsy capitals: FREEDOM IS SLAVERY. Then almost without a pause he wrote beneath it: TWO AND TWO MAKE FIVE." Here ... More

The lure of conformal symmetryMay 30 2019The Clifford algebra, generated by the real (Majorana) gamma-matrices and by a hermitian gamma_5, gives room to the reductive Lie algebra u(2,2) of the conformal group extended by the u(1) helicity operator. Its unitary positive energy ladder representations, ... More

Elements of mathematics in problems. Through olympiads and circles to professionMay 24 2019This is a collection of teaching materials used in several Russian universities, schools, and mathematical circles. Most problems are chosen in such a way that in the course of the solution and discussion a reader learns important mathematical ideas and ... More

Convexity in Greek antiquityMay 21 2019We consider several appearances of the notion of convexity in Greek antiquity, more specifically in mathematics and optics, in the writings of Aristotle, and in art. The final version of this article will appear in the book `Geometry in History', ed. ... More

Measuring Mountains on the MoonMay 15 2019Following a technique of Galileo we compute the height on a mountain on the Moon. It is based on a simple observation that precisely on a half Moon day, the Earth, the Moon, and the Sun form the vertices of a right triangle with the Moon at the 90 degree ... More

Cubic equations of Babylonian mathematicsMay 14 2019In this paper I shall clarify three cubic equations of Babylonian mathematics, whose solutions have not been fully explained; BM 85200, no.6 and no.7, and YBC 4669 B2.

On the share of mathematics published by Elsevier and SpringerMay 13 2019For-profit editors such as Elsevier and Springer have been subject to sustained criticism from academics and university libraries, including calls to boycott, and discontinued subscriptions. Mathematicians have played a particularly active role in this ... More

The Strohmer and Beaver Conjecture for Gaussian Gabor Systems - A Deep Mathematical Problem (?)May 13 2019In this article we are going to discuss the conjecture of Strohmer and Beaver for Gaussian Gabor systems. It asks for an optimal sampling pattern in the time-frequency plane, where optimality is measured in terms of the condition number of the frame operator. ... More

Approximating cube roots of integers, after Heron's Metrica III.20May 09 2019Heron, in Metrica III.20-22, is concerned with the the division of solid figures - pyramids, cones and frustra of cones - to which end there is a need to extract cube roots. We report here on some of our findings on the conjecture by Taisbak in C.M.Taisbak, ... More

Ordered Line and Skew-Fields in the Desargues Affine PlaneMay 08 2019This paper introduces ordered skew fields that result from the construction of a skew field over an ordered line in a Desargues affine plane. A special case of a finite ordered skew field in the construction of a skew field over an ordered line in a Desargues ... More

Activity Theory in Didactics of Mathematics -- What is Taken As SharedMay 07 2019These few pages briefly present the way in which Activity Theory has been adopted for several years now by French researchers in didactics of mathematics and has been adapted to study the learning of school mathematics in relation with the teaching that ... More

On the Diophantine Equation 1/a + 1/b = (q+1) / pqMay 06 2019Let $p$ and $q$ be distinct primes such that $q+1 | p-1$. In this paper we find all integer solutions $a$, $b$ to the equation $1/a + 1/b = (q+1)/pq$ using only elementary methods.

Sobre Introducción al Análisis Matemático de Mario O. GonzálezMay 05 2019In this article, we use the 75th anniversary of the publication of \emph{Introducci\'on al An\'alisis Matem\'atico} by Mario O. Gonz\'alez to study its importance. Besides, we give new biographical data from his author in the previous years to the publication ... More

La Emancipación Conceptual de Número Real de la Idea de Magnitud: Una Mirada GermánicaMay 05 2019In the present article we study the decisive contributions of three members of the German mathematical school to the separation of the concept of number from the physical concept of magnitude, during the second half of the XIXth century. Besides we analyze ... More

Richard Dedekind y la arquitectura del continuo aritméticoMay 04 2019It is usually considered that the structuralist tendency in mathematics began in the twentieth century, at some point after the works on set theory and obtained its spreading through the works made by the Bourbaki group. In the present paper we argument ... More

The main theorem of the Galois theory proven with ideas from the first Mémoire of GaloisApr 29 2019A proof of the main theorem of the Galois theory is presented using the main theorem of symmetric polynomials. The idea originated from studying the "M\'emoire sur les conditions de r\'esolubilit\'e des \'equations par radicaux" of Evariste Galois. The ... More

Model Theory, Arithmetic & Algebraic GeometryApr 28 2019In this pages I give an overview of the relationship between Model Theory, Arithmetic and Algebraic Geometry. The topics will be the basic ones in the area, so this is just an invitation, in the presentation of topics I mainly follow the philosophy of ... More

Enhancing logical deduction with math: the rationale behind Gardner and CarrollApr 24 2019Math is widely considered as a powerful tool and its strong appeal depends on the high level of abstraction it allows in modelling a huge number of heterogeneous phenomena and problems, spanning from the static of buildings to the flight of swarms. As ... More

Big Math and the One-Brain Barrier A Position Paper and Architecture ProposalApr 23 2019Over the last decades, a class of important mathematical results have required an ever increasing amount of human effort to carry out. For some, the help of computers is now indispensable. We analyze the implications of this trend towards "big mathematics", ... More

Introduction to Gestural Similarity in Music. An Application of Category Theory to the OrchestraApr 22 2019Mathematics, and more generally computational sciences, intervene in several aspects of music. Mathematics describes the acoustics of the sounds giving formal tools to physics, and the matter of music itself in terms of compositional structures and strategies. ... More

Mathematical MonstersApr 19 2019Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways between these two monstrous habitats. I start from Jeffrey Jerome Cohen's influential account of monster culture and explore how well mathematical monsters ... More

Arc length of function graphs via Taylor's formulaApr 15 2019We use Taylor's formula with Lagrange remainder to prove that functions with bounded second derivative are rectifiable in the case when polygonal paths are defined by interval subdivisions which are equally spaced. We discuss potential benefits for such ... More

Topics in Applied Mathematics and Nonlinear WavesApr 15 2019The selection of topics in this text has formed the core of a one semester course in applied mathematics at the Arctic University of Norway that has been running continuously since the 1970s. The class has, during its existence, drawn participants from ... More

Frobenius's last proofApr 13 2019Around about 1917, Issai Schur rediscovered the Rogers-Ramanujan identities, and proved a system of polynomial identities that imply them. Schur wrote that Georg Frobenius (his former advisor) had shown him a simple, direct proof of these polynomial identities. ... More

Free groups, covering spaces and Artin's theoremApr 13 2019May 05 2019In this expository note we provide a proof of Artin's theorem which states that the commutator subgroup of a free group on two generators is not finitely generated. The proof employs the infinite grid as in two other proofs in the literature mentioned ... More

The Dodecahedron as a Voronoi Cell and its (minor) importance for the Kepler conjectureApr 12 2019The regular dodecahedron has a 2% smaller volume than the rhombic dodecahedron which is the Voronoi cell of a fcc packing. From this point of view it seems possible that the dodecahedral aspect which is the core of the so-called dodecahedral conjecture, ... More

Una breve historia imaginariaApr 11 2019In this paper, from the historical point of view, we present short anecdotes about the development of the object that we well known as complex numbers.

Una breve historia imaginariaApr 11 2019Apr 24 2019In this paper, from the historical point of view, we present short anecdotes about the development of the object that we well known as complex numbers. We will start by introducing the oldest known evidence of the calculation of the square root of a negative ... More

Calculus, constrained minimization and Lagrange multipliers: Is the optimal critical point a local minimizer?Apr 10 2019In this short note, we discuss how the optimality conditions for the problem of minimizing a multivariate function subject to equality constraints have been dealt with in undergraduate Calculus. We are particularly interested in the 2 or 3-dimensional ... More

A commented translation of Hans Richter's early work "The isotropic law of elasticity"Apr 09 2019We provide a faithful translation of Hans Richter's important 1948 paper "Das isotrope Elastizit\"atsgesetz" from its original German version into English. Our introduction summarizes Richter's achievements.

M/M/$c$ Queues and the Poisson Clumping HeuristicApr 08 2019In continuous time, customers arrive at random. Each waits until one of $c$ servers is available; each thereafter departs at random. The distribution of maximum line length of idle customers was studied over 25 years ago. We revisit two good approximations ... More

What is a proof? What should it be?Apr 08 2019Mathematical proofs should be paired with formal proofs, whenever feasible.

Germain and Her Fearless Attempt to Prove Fermat's Last TheoremApr 07 2019Two centuries ago, Sophie Germain began to work on her "grand plan to prove the theorem of Fermat," the famous conjecture that $x^n + y^n = z^n$ is impossible for integral values of $x$, $y$, and $z$, when $n > 2$. At that time, this was an open question ... More

Germain and Her Fearless Attempt to Prove Fermat's Last TheoremApr 07 2019Apr 13 2019Two centuries ago, Sophie Germain began to work on her "grand plan to prove the theorem of Fermat," the famous conjecture that $x^n + y^n = z^n$ is impossible for integral values of $x$, $y$, and $z$, when $n > 2$. At that time, this was an open question ... More

Learning in teams: peer evaluation for fair assessment of individual contributionsApr 06 2019The "free rider" problem has long plagued pedagogies based on collaborative learning. The most common solution to the free rider problem is peer evaluation. As well other existing methods of peer evaluation include self-evaluation --- and hence are prone ... More

Deducing factoring methods through concrete materialApr 05 2019We formulate and prove a criterion for reducibility of a quadratic polynomial over the integers. The main theorem was suggested by the teaching experience with the concrete material called "the polynomial box". Through the corollaries we relate our theorem ... More

Increasingly Enumerable Submonoids of R: Music Theory as a Unifying ThemeApr 05 2019Apr 08 2019We analyze the set of increasingly enumerable additive submonoids of R, for instance, the set of logarithms of the positive integers with respect to a given base. We call them $\omega$-monoids. The $\omega$-monoids for which consecutive elements become ... More

Evidence, Proofs, and DerivationsApr 04 2019The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, ... More

A differential extension of Descartes' foundational approach: a new balance between symbolic and analog computationApr 04 2019In La G\'eom\'etrie, Descartes proposed a balance between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In modern terms, that is a balance between analog and symbolic computation. Descartes' geometric ... More

Some remarks on the first Hardy-Littlewood conjectureApr 04 2019Starting from the first Hardy-Littlewood conjecture some topics will be covered: an empirical approach to the distribution of the twin primes in classes mod(10) and a simplified proof of the Bruns theorem . Finally, it will be explored an approach based ... More

Some remarks on the first Hardy-Littlewood conjectureApr 04 2019Apr 18 2019Starting from the first Hardy-Littlewood conjecture some topics will be covered: an empirical approach to the distribution of the twin primes in classes mod(10) and a simplified proof of the Bruns theorem . Finally, it will be explored an approach based ... More

From Euclid to Riemann and BeyondApr 03 2019The purpose of this essay is to trace the historical development of geometry while focusing on how we acquired mathematical tools for describing the "shape of the universe." More specifically, our aim is to consider, without a claim to completeness, the ... More

Some Constructions of the Golden Ratio in an Arbitrary TriangleApr 02 2019We establish some new constructions of the golden ratio in an arbitrary triangle using symmedians and nine-point circle.

Operational approach to the topological structure of the physical spaceApr 02 2019Abstract axiomatic formulation of mathematical structures are extensively used to describe our physical world. We take here the reverse way. By making basic assumptions as starting point, we reconstruct some features of both geometry and topology in a ... More

Technologies for "Complete, Transparent & Interactive Models of Math" in EducationApr 01 2019A new generation of educational mathematics software is being shaped in ThEdu and other academic communities on the side of computer mathematics. Respective concepts and technologies have been clarified to an extent, which calls for cooperation with educational ... More

Isomorphic-Dilations of the skew-fields constructed over parallel lines in the Desargues affine planeMar 30 2019This paper considers dilations and translations of lines in the Desargues affine plane. A dilation of a line transforms each line into a parallel line whose length is a multiple of the length of the original line. In addition to the usual Playfair axiom ... More

Ahab's Arithmetic; or, the mathematics of Moby-DickMar 28 2019Herman Melville's novel Moby-Dick contains a surprising number of mathematical allusions. In this article we explore some of these, as well as discussing the questions that naturally follow: why did Melville choose to use so much mathematical imagery? ... More

On approximation of functions by polynomials and by entire functions of exponential typeMar 26 2019A brief overview of publications in approximation theory of functions known to the author and connected with scientific publications by V.~K.~Dzyadyk (1919--1998).

A Lighthouse Illumination ProblemMar 20 2019This paper discusses a problem that consists of $n$ "lighthouses" which are circles with radius 1, placed around a common center, equidistant at $n$ units away from the placement center. Consecutive lighthouses are separated by the same angle: $360^\circ/n$ ... More

A^1-homotopy theory and contractible varieties: a surveyMar 19 2019We survey some topics in ${\mathbb A}^1$-homotopy theory. Our main goal is to highlight the interplay between ${\mathbb A}^1$-homotopy theory and affine algebraic geometry, focusing on the varieties that are "contractible" from various standpoints.

Etude d'un dispositif d'aide a l'intention d'{é}l{è}ves en difficult{é} dans la r{é}solution d'une situation-probl{è}me math{é}matiqueMar 18 2019In this paper, we analyze an aid session tested by an elementary school teachers. This aid session has been set up by a teacher for some students with difficulties after the work in the whole class. We first show how this aid session can help the pupils, ... More

Questionnements autour de la synchronisation dans l'enseignement des math{é}matiques a des {é}l{è}ves sourdsMar 18 2019In this communication we study a device set up to school deaf pupils. We analyze some sessions of mathematics classroom in which participated these pupils. We show in particular that if all the pupils seem globally in phase, cycles of desynchronizations ... More

Using group theory in pairwise comparisons: a brief critiqueMar 17 2019Compelling evidence against a heedless group theory generalization of pairwise comparisons elements is provided by means of counter-examples and mathematical reasoning. The lack of acceptable semantics for selected groups (with negative and complex numbers) ... More

Paul Mansion (1844-1919) : more than 400 academic publications, Centenary Paul Mansion (working paper, 2019)Mar 15 2019According to the Liber Memorialis of the University of Ghent, the Belgian mathematician Paul Mansion (1844-1919) has published more than 349 academic papers and books. For our part, we were able to calculate the correct number by using the journal Das ... More

On the equivalence of Playfair's axiom to the parallel postulateMar 12 2019We show that the classical equivalence of Euclid's parallel postulate and Playfair's axiom collapses in the absence of triangle congruence. In particular, we construct a non-SAS geometry that models the Playfair axiom but not the parallel postulate.

Linear algebraic techniques for spanning tree enumerationMar 12 2019Apr 16 2019Kirchhoff's Matrix-Tree Theorem asserts that the number of spanning trees in a finite graph can be computed from the determinant of any of its reduced Laplacian matrices. In many cases, even for well-studied families of graphs, this can be computationally ... More

Divisibility Tests Unified: Stacking the Trimmings for SumsMar 12 2019Divisibility tests are algorithms that can quickly decide if one integer is divisible by another. There are many tests but most are either of the trimming or summing variety. Our goals are to present Zbikowski's family of trimming tests as one test and ... More

Algebra in Bishop's style: some major features of the book "A Course in Constructive Algebra'' by Mines, Richman, and RuitenburgMar 11 2019The book "A Course in Constructive Algebra" (1988) shows the way of understanding classical basic algebra in a constructive style similar to Bishop's Constructive Mathematics. Classical theorems are revisited, with a new flavour, and become much more ... More

Polyhedral products and features of their homotopy theoryMar 11 2019A polyhedral product is a natural subspace of a Cartesian product that is specified by a simplicial complex. The modern formalism arose as a generalization of the spaces known as moment-angle complexes which were developed within the nascent subject of ... More

The 5-Way ScaleMar 08 2019In this paper, we discuss coin-weighing problems that use a 5-way scale which has five different possible outcomes: MUCH LESS, LESS, EQUAL, MORE, and MUCH MORE. The 5-way scale provides more information than the regular 3-way scale. We study the problem ... More

A Translation of "Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen" by Heinz BachmannMar 08 2019This is a translation of Heinz Bachmann's influential paper, wherein the Bachmann-Howard ordinal is defined, and some general considerations given on systems of ordinal functions. Permission to post has been granted by the editors of Vierteljahrsschrift ... More

A data analysis of women's trails among ICM speakersMar 06 2019The International Congress of Mathematicians (ICM), inaugurated in 1897, is the greatest effort of the mathematical community to strengthen international communication and connections across all mathematical fields. Meetings of the ICM have historically ... More

In Search of Shadows: the First Topological Conference, Moscow 1935Mar 05 2019Mar 10 2019We discuss some mistakes and curiosities concerned with the celebrated First International Topological Conference in Moscow, 1935.

In Search of Shadows: the First Topological Conference, Moscow 1935Mar 05 2019We discuss some mistakes and curiosities concerned with the celebrated First International Topological Conference in Moscow, 1935.

In Search of Shadows: the First Topological Conference, Moscow 1935Mar 05 2019May 25 2019We discuss some mistakes and curiosities concerned with the celebrated First International Topological Conference in Moscow, 1935.

Ola Bratteli and his diagramsMar 04 2019This article discusses the life and work of Professor Ola Bratteli (1946--2015). Family, fellow students, his advisor, colleagues and coworkers review aspects of his life and his outstanding mathematical accomplishments.

Some facts about the life and the scientific work of the Belgian mathematician Paul Mansion (1844-1919) Centenary Paul Mansion (working paper, 2019)Feb 27 2019In this article, we are interested in the life and scientific work of the Belgian mathematician Paul Mansion. The year 2019 marks the centenary of his passing. We bring some new insights into Paul Mansion's work thanks to his scientific correspondence ... More

Geo/Geo/2 Queues and the Poisson Clumping HeuristicFeb 25 2019In discrete time, customers arrive at random. Each waits until one of two servers is available; each thereafter departs at random. We seek the distribution of maximum line length of idle customers. In the context of an emergency room (for medical treatment), ... More

Fluid-Structure Interaction for the Classroom: Speed, Accuracy, Convergence, and Jellyfish!Feb 20 2019When is good, good enough? This question lingers in approximation theory and numerical methods as a competition between accuracy and practicality. Numerical Analysis is traditionally where the rubber meets the road: students begin to use numerical algorithms ... More

A constructive Knaster-Tarski proof of the uncountability of the realsFeb 20 2019We give an uncountability proof of the reals which relies on their order completeness instead of their sequential completeness. We use neither a form of the axiom of choice nor the law of excluded middle, therefore the proof applies to the MacNeille reals ... More

Unit shapes and a wealth of calculus problemsFeb 19 2019For a given family of similar shapes, what we call a "unit shape" strongly analogizes the role of the unit circle within the family of all circles. Within many such families of similar shapes, we present what we believe is naturally and intrinsically ... More

The fundamental theorem of calculus via Taylor's theoremFeb 17 2019We use Taylor's theorem with Lagrange remainder to give a short proof of a version of the fundamental theorem of calculus for a version of the integral defined by Riemann sums with left (or right) endpoints which are equally spaced. We discuss the potential ... More

Poisson's fundamental theorem of calculus via Taylor's formulaFeb 17 2019Mar 26 2019We use Taylor's formula with Lagrange remainder to make a modern adaptation of Poisson's proof of a version of the fundamental theorem of calculus in the case when the integral is defined by Euler sums, that is Riemann sums with left (or right) endpoints ... More

Survey of Cubic Fibonacci Identities - When Cuboids Carry WeightFeb 15 2019The aim of this paper is to present a comprehensive survey of cubic Fibonacci identities, trying to uncover as many as possible. From the outset, our rationale for a very careful search on an apparently obscure problem was not only a matter of mathematical ... More

Un lemme d'analyse dont use Ibn al-Haytham en gnomonique, dioptrique et cinématique célesteFeb 08 2019In several places and for different purposes, Ibn al-Haytham has studied the variations of a well-known function. The context of these applications reveals some characteristics of Ibn al-Haytham's mathematical and physical thought. An in-depth study of ... More

English Translation of Cournot's "Exposition de la théorie des chances et des probabilités" / "Exposition of the Theory of Chances and Probabilities"Feb 08 2019In spite of the wide range of his book, Cournot did not know some essential discoveries in natural sciences (William Herschel, Daniel Bernoulli, Humboldt) and his deliberations about measurement were almost useless. But he introduced the median and a ... More

English Translation of Poisson's "Recherches sur la probabilité des jugements en matière criminelle et en matière civile" / "Researches into the Probabilities of Judgements in Criminal and Civil Cases"Feb 08 2019In spite of its title, the book mostly treats probability theory: the law of large numbers (regarded as a principle); formal definition of a random variable and law of distribution; the misnamed Cauchy distribution; functions now named after Dirac; difference ... More

A short introduction to Monstrous MoonshineFeb 07 2019This paper is an introduction to the Monstrous Moonshine correspondence aiming at an undergraduate level. We review first the classification of finite simple groups and some properties of the monster $\mathbb{M}$, and then the theory of classical modular ... More

A short introduction to Monstrous MoonshineFeb 07 2019Feb 18 2019This paper is an introduction to the Monstrous Moonshine correspondence aiming at an undergraduate level. We review first the classification of finite simple groups and some properties of the monster $\mathbb{M}$, and then the theory of classical modular ... More

Division of an angle into equal parts and construction of regular polygons by multi-fold origamiFeb 05 2019This article analyses geometric constructions by origami when up to $n$ simultaneous folds may be done at each step. It shows that any arbitrary angle can be $m$-sected if the largest prime factor of $m$ is $p\le n+2$. Also, the regular $m$-gon can be ... More

Stiefel manifolds and polygonsFeb 04 2019Polygons are compound geometric objects, but when trying to understand the expected behavior of a large collection of random polygons -- or even to formalize what a random polygon is -- it is convenient to interpret each polygon as a point in some parameter ... More

Stiefel manifolds and polygonsFeb 04 2019Apr 30 2019Polygons are compound geometric objects, but when trying to understand the expected behavior of a large collection of random polygons -- or even to formalize what a random polygon is -- it is convenient to interpret each polygon as a point in some parameter ... More

La cosmología y los matemáticos (Cosmology and mathematicians)Feb 04 2019Free translation of the original abstract in Spanish: Some of the most relevant milestones due to, or instigated by, mathematicians concerning the creation, development and advances of Cosmology as a scientific discipline are presented and discussed. ... More

The polycons: the sphericon (or tetracon) has found its familyJan 30 2019This paper introduces a new family of solids, which we call \textit{polycons}, which generalise the sphericon in a natural way. The static properties of the polycons are derived, and their rolling behaviour is described and compared to that of other developable ... More

The polycons: the sphericon (or tetracon) has found its familyJan 30 2019Mar 12 2019This paper introduces a new family of solids, which we call \textit{polycons}, which generalise the sphericon in a natural way. The static properties of the polycons are derived, and their rolling behaviour is described and compared to that of other developable ... More

An estimation method for game complexityJan 30 2019We looked at a method for estimating the complexity measure of game tree size (the number of legal games). It seems effective for a number of children's games such as Tic-Tac-Toe, Connect Four and Othello.

A simple and more general approach to Stokes' theoremJan 27 2019Oftentimes, Stokes' theorem is derived by using, more or less explicitly, the invariance of the curl of the vector field with respect to translations and rotations. However, this invariance -- which is oftentimes described as the curl being a "physical" ... More

Nonlinear diffusion processes: geometric ideas and beyondJan 25 2019This is a survey article written for the Springer's Intelligencer, in the occasion of the 2018 International Congress of Mathematicians.

An introduction to singular stochastic PDEs: Allen-Cahn equations, metastability and regularity structuresJan 22 2019These notes have been prepared for a series of lectures to be given at the Sarajevo Stochastic Analysis Winter School, from January 28 to February 1, 2019. There already exist several excellent lecture notes and reviews on the subject, such as (Hairer ... More

An introduction to the classical three-body problem: From periodic solutions to instabilities and chaosJan 22 2019The classical three-body problem arose in an attempt to understand the effect of the Sun on the Moon's Keplerian orbit around the Earth. It has attracted the attention of some of the best physicists and mathematicians and led to the discovery of chaos. ... More

Soñando con números, María Andresa Casamayor (1720-1780)Jan 18 2019Mar\'ia Andresa Casamayor, born in Zaragoza, is known as the first woman who published a scientific book in Spain. In this paper we provide answers to several of the most important questions about her unknown biography such as her birth day, the origins ... More

A Modern Retrospective on Probabilistic NumericsJan 14 2019May 05 2019This article attempts to place the emergence of probabilistic numerics as a mathematical-statistical research field within its historical context and to explore how its gradual development can be related both to applications and to a modern formal treatment. ... More

A Modern Retrospective on Probabilistic NumericsJan 14 2019This article attempts to cast the emergence of probabilistic numerics as a mathematical-statistical research field within its historical context and to explore how its gradual development can be related to modern formal treatments and applications. We ... More

A Modern Retrospective on Probabilistic NumericsJan 14 2019Mar 07 2019This article attempts to place the emergence of probabilistic numerics as a mathematical-statistical research field within its historical context and to explore how its gradual development can be related both to applications and to a modern formal treatment. ... More

AritméticaJan 14 2019This is an exposition of facts about Arithmetic with an approach via mathematical logic. In Section 1 we present Peano Arithmetic, PA, and the complete theory of $\mathbb{N}$, and we show that $\mathbb{N}$ is a prime model of the theory of $\mathbb{N}$. ... More

Definitions, notations and proofs for Bernoulli numbersJan 14 2019This is a collection of definitions, notations and proofs for the Bernoulli numbers $B_n$ appearing in formulas for the sum of integer powers, some of which can be found scattered in the large related historical literature in French, English and German. ... More

Four Fundamental Questions in Probability Theory and StatisticsJan 12 2019This study has the purpose of addressing four questions that lie at the base of the probability theory and statistics, and includes two main steps. As first, we conduct the textual analysis of the most significant works written by eminent probability ... More