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Basel problem: a physicist's solutionAug 20 2019Some time ago Wastlund reformulated the Basel problem in terms of a physical system using the proportionality of the apparent brightness of a star to the inverse square of its distance. Inspired by this approach, we give another physical interpretation ... More
Positional Voting, Doubly Stochastic Matrices, and the Braid ArrangementAug 18 2019We provide elementary proofs of results from \cite{Saari} and \cite{DEMO} regarding the possible outcomes arising from a fixed profile within the class of positional voting systems. Our arguments enable a simple and explicit construction of paradoxical ... More
Discrete and Fast Fourier Transform Made ClearAug 17 2019Fast Fourier transform was included in the Top 10 Algorithms of 20th Century by Computing in Science & Engineering. In this paper, we provide a new simple derivation of both the discrete Fourier transform and fast Fourier transform by means of elementary ... More
Exploration of another Sol Lewitt puzzle from Barry CipraAug 15 2019At MOVES 2019, Barry Cipra casually introduced a new "Sol Lewitt" puzzle to fellow conference goers. Several brainstorming sessions ensued with Barry, Peter Winkler , Donna Dietz, and other attendees. This paper is to document the puzzle and some insights ... More
How many ways to color the map of America?Aug 15 2019Although the Four Color Conjecture originated in cartography, surprisingly, there is nothing in the literature on the number of ways to color an actual geographic map with four or fewer colors. In this paper, we compute these numbers, with exponentially ... More
The strange properties of the infinite power towerAug 15 2019In this article we investigate some "unexpected" properties of the "Infinite Power Tower". \[y = f(x) = {x^{{x^{{x^{{x^ {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} }}}}}}}\] The material collected ... More
A Recreational Application of Two Integer Sequences and the Generalized Repetitious Number PuzzleAug 15 2019In this article, we give a particular recreational application of two integer sequences. These sequences are respectively the sequence A000533 and sequence A261544 in "The On-line Encyclopedia of Integer Sequences" (OEIS). The recreational application ... More
Nonlinear algebra via tensegrity structuresAug 14 2019In this paper, we discuss tensegrity from the perspective of nonlinear algebra in a manner accessible to undergraduates. We compute explicit examples and include the SAGE code so that readers can continue their own experiments and computations. The entire ... More
Moments of Maximum: Segment of AR(1)Aug 12 2019Let $X_{t}$ denote a stationary first-order autoregressive process. Consider five contiguous observations (in time $t$) of the series (e.g., $X_{1}, ..., X_{5}$). Let $M$ denote the maximum of these. Let $\rho$ be the lag-one serial correlation, which ... More
Polygons of Petrovic and Fine, algebraic ODEs, and contemporary mathematicsAug 09 2019Here, we study the genesis and evolution of geometric ideas and techniques in investigations of movable singularities of algebraic ordinary differential equations. This leads us to the work of Mihailo Petrovic on algebraic differential equations and in ... More
From lecture to active learning: Rewards for all, and is it really so difficult?Aug 06 2019We describe the evolution of a personal non-lecture active learning pedagogy developed in numerous courses at all university levels. A distinguishing feature is its tight integration of pre-class preparation, involving student reading/writing/questions ... More
The largest angle bisection procedureAug 06 2019For a given triangle $\Delta ABC$, with $\angle A\ge \angle B \ge \angle C$, the $largest\, angle \, bisection\, procedure$ consists in constructing $AD$, the angle bisector of angle $\angle A$, and replacing $\Delta ABC$ by the two newly formed triangles, ... More
Outer linear measure of connected sets via Steiner treesAug 06 2019We resurrect an old definition of the linear measure of a metric continuum in terms of Steiner trees, independently due to Menger (1930) and Choquet (1938). We generalise it to any metric space and provide a proof of a little-known theorem of Choquet ... More
A Glimpse of Arithmetic DynamicsAug 05 2019In this note, we offer a palatable introduction to the field of arithmetic dynamics. That is, we study the patterns that arise when iterating a polynomial map. This note is accessible to those who have taken an introductory proof based course and some ... More
Euler and the GammafunctionAug 05 2019We review Euler's idea on the Gammafunction. We will explain, how Euler obtained them and how Euler's ideas anticipate more modern approaches and theories.
Relationships Between Six IncirclesAug 04 2019If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into six smaller triangles. We give theorems about the relationship between the radii of the circles inscribed in these triangles.
Iterative methods for linear systems of equations: A brief historical journeyAug 02 2019This paper presents a brief historical survey of iterative methods for solving linear systems of equations. The journey begins with Gauss who developed the first known method that can be termed iterative. The early 20th century saw good progress of these ... More
Online quizzes as a form of assessment in pure mathematicsAug 02 2019First year mathematics students are often initially overwhelmed with the transition from A Level mathematics to university level mathematics, especially when being acquainted with the new language of pure mathematics. This report outlines and evaluates ... More
A parametrization of 8x8 magic squares of squares through octonionic multiplicationAug 02 2019In an analogous construction as by Euler for 4x4 matrices, a parametrization of 8x8 magic squares of squares with orthogonal rows is shown to be obtainable by extending the quaternionic method, as shown by Hurwitz, to octonions, but not possible to be ... More
A parametrization of 8x8 magic squares of squares through octonionic multiplicationAug 02 2019Aug 05 2019In an analogous construction as by Euler for 4x4 matrices, a parametrization of 8x8 magic squares of squares with orthogonal rows is shown to be obtainable by extending the quaternionic method, as shown by Hurwitz, to octonions, but not possible to be ... More
On the perimeter length determination of the eight-centered ovalAug 02 2019On the perimeter length determination of the eight-centered oval. Several studies have shown that an eight-centered oval coincides almost perfectly with the ellipse constructed on the same axes and can be considered as a representation of the latter provided ... More
Some new theorems on Pentagon and PentagramAug 02 2019We establish some new theorems on pentagon and pentagram.
Visualization of Abel's Impossibility TheoremAug 01 2019In this paper we construct a visualization of the Abel's Impossibility Theorem also known as the Abel-Ruffini Theorem. Using the canvas object in JavaScript along with the p5.js library, and given any expression that uses analytic functions and radicals ... More
Some constructions in the Mānava ŚulvasūtraAug 01 2019The M\=anava \'Sulvas\=utra, while less sophisticated than the other \'sulvas\=utras, is seen to contain some mathematical ideas and constructions not found in the other \'sulvas\=utras. Here we discuss some of these constructions and discuss their significance ... More
Pairwise Rational Points on a ParabolaJul 30 2019This paper presents a solution to the following open problem in Number Theory and Geometry: How many points can you find on the (half) parabola $y=x^2$, $x>0$, so that the distance between any pair of them is rational? This problem sounds like a geometry ... More
Mathematical Modeling as a Means to Capacity Building in 21st Century STEM CareersJul 29 2019Mathematicians have traditionally been a select group of academics that produce high-impact ideas allowing substantial results in several fields of science. Throughout the past 35 years, undergraduates enrolling in mathematics or statistics have represented ... More
Mathematical Modeling as a Means to Capacity Building in 21st Century STEM CareersJul 29 2019Aug 10 2019Mathematicians have traditionally been a select group of academics that produce high-impact ideas allowing substantial results in several fields of science. Throughout the past 35 years, undergraduates enrolling in mathematics or statistics have represented ... More
Evidence-based teaching: how do we all get there?Jul 28 2019There are compelling reasons to shift our pedagogy toward evidence-based active learning methods that substantially improve student success, and now plenty of resources to aid in that shift. These include the recent CBMS Statement on Active Learning, ... More
A Quantitative Study on Average Number of Spins of Two-Player DreidelJul 27 2019We give an excellent approximation of the average number of spins of a simplified version of a two-player version of the game Dreidel. We also make a conjecture on the average number of spins of the full version of the game.
Generalized Rascal TrianglesJul 25 2019The Rascal Triangle was introduced by three middle school students in 2010, and in this paper we describe number triangles that are generalizations of the Rascal Triangle and show that these Generalized Rascal Triangles are characterized by arithmetic ... More
Generalized Rascal TrianglesJul 25 2019Jul 27 2019The Rascal Triangle was introduced by three middle school students in 2010, and in this paper we describe number triangles that are generalizations of the Rascal Triangle and show that these Generalized Rascal Triangles are characterized by arithmetic ... More
Training future teachers in natural sciences and mathematics by means of computer simulation: a social constructivist approachJul 23 2019The monograph defines the conditions of training of future teachers in natural sciences and mathematics by means of computer simulation, developed a structural-functional model of training, selected socio-constructivist forms of organization, methods ... More
An analysis of IQ LINK$^{TM}$Jul 22 2019This is a theoretical and computational strategy exploration of the visually attractive game IQ-Link. Not all games which are visually appealing are worthy of your time as a puzzlist. This analysis gives a would-be addict some idea of what type of game ... More
A category for bijective combinatoricsJul 21 2019The category of matchings between finite sets extends to the category of cobordisms of signed sets. A chain of cobordisms that starts and ends with unsigned sets A and B yields a matching from A to B. This is a convenient way to package the involution ... More
Creating and experiencing Flipped Learning in Multivariable Calculus for EngineeringJul 20 2019This article discusses the process of creating, implementing and experiencing Flipped Learning in a Multivariable Calculus course for second year engineering students. We describe the construction of the teaching material, consisting of short videos for ... More
Using Symbolic Computation to analyze some Children's Board GamesJul 19 2019In a delightful article that recently appeared in Mathematics Magazine, David and Lori Mccune analyze the board game "Count Your Chickens!", recommended to children three and up. Alas, they use the advanced theory of Markov chains, that presupposes a ... More
Student Inquiry and the Rascal TriangleJul 17 2019Those of us who teach Mathematics for Liberal Arts (MLA) courses often underestimate the mathematical abilities of the students enrolled in our courses. Despite the fact that many of these students suffer from math anxiety and will admit to hating mathematics, ... More
19th century real analysis, forward and backwardJul 17 201919th century real analysis received a major impetus from Cauchy's work. Cauchy mentions variable quantities, limits, and infinitesimals, but the meaning he attached to these terms is not identical to their modern meaning. Some Cauchy historians work in ... More
On mathematical realism and the applicability of hyperrealsJul 16 2019Jul 30 2019We argue that Robinson's hyperreals have just as much claim to applicability as the garden variety reals. In a recent text, Easwaran and Towsner (ET) analyze the applicability of mathematical techniques in the sciences, and introduce a distinction between ... More
On mathematical realism and the applicability of hyperrealsJul 16 2019We argue that Robinson's hyperreals have just as much claim to applicability as the garden variety reals. In a recent text, Easwaran and Towsner (ET) analyze the applicability of mathematical techniques in the sciences, and introduce a distinction between ... More
Modernism, Fiction and MathematicsJul 12 2019This is an expanded version of my review of Nina Engelhardt's book "Modernism, Fiction and Mathematics", Edinburgh University Press 2018. A considerably shortened version will appear in the Notices of the AMS.
Elementary proofs of generalized continued fraction formulae for $e$Jul 12 2019In this short note we prove two elegant generalized continued fraction formulae $$e= 2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4+\ddots}}}}$$ and $$e= 3+\cfrac{-1}{4+\cfrac{-2}{5+\cfrac{-3}{6+\cfrac{-4}{7+\ddots}}}}$$ using elementary methods. The ... More
Truth, Proof, and Reproducibility: There's no counter-attack for the codelessJul 11 2019Current concerns about reproducibility in many research communities can be traced back to a high value placed on empirical reproducibility of the physical details of scientific experiments and observations. For example, the detailed descriptions by 17th ... More
Una dimostrazione diretta della legge di probabilità di Poisson (A direct proof of the Poisson probability law)Jul 09 2019The purpose of this paper is to prove directly, by an elementary method, the Poisson probability law. This proof is offered as an alternative to the more usual derivation from binomial distribution in the limit of small probabilities. The same proof is ... More
The Newton integral and the Stirling formulaJul 04 2019We present details of logically simplest integral sufficient for deducing the Stirling asymptotic formula for n!. It is the Newton integral, defined as the difference of values of any primitive at the endpoints of the integration interval. We review in ... More
MatchTheNet -- An Educational Game on 3-Dimensional PolytopesJul 04 2019We present an interactive game which challenges a single player to match 3-dimensional polytopes to their planar nets. It is open source, and it runs in standard web browsers
GeoGebra e situações que envolvem modelação numa abordagem STEAMJul 03 2019In order to implement a STEAM approach including the use of technology, namely the use of interactive mathematics software GeoGebra, in mathematics classes, in the lusophone space, the materials presented here were conceived, to be implemented in a first ... More
Rod in a train: a mechanical problem of H.Whitney, or Much Ado About Nothing (In Russian)Jul 02 2019In 1941 a mechanical problem about a rod in a moving train (there is a initial position such that rod does not touch the floor while train is moving) was published by R.Courant and H.Robbins in their popular book "What is mathematics?" and attributed ... More
Rod in a train: a mechanical problem of H.Whitney, or Much Ado About NothingJul 02 2019Jul 07 2019In 1941 a mechanical problem about a rod in a moving train (there is a initial position such that rod does not touch the floor while train is moving) was published by R.Courant and H.Robbins in their popular book "What is mathematics?" and attributed ... More
Increasing student engagement in math placement and preparationJul 01 2019Math placement is a crucial step between admission and full engagement with the university for students in STEM majors. Students' placement experiences influence not only their mathematical entry point on their degree pathway, but their perceptions of ... More
Male Under-performance in Undergraduate Engineering Mathematical Courses: Causes and Solution StrategyJul 01 2019Jul 02 2019The performance of students in Mathematics and its allied fields has been a topic of great interest for mathematical educationalists worldwide. In this paper, we study the student performance in one of the most math heavy fields-Engineering. An analysis ... More
Mentoring Undergraduate Interdisciplinary Mathematics Research Students: Junior Faculty ExperiencesJun 28 2019To be successful, junior faculty must properly manage their time in the face of expanding responsibilities. One such responsibility is supervising undergraduate research projects. Student research projects (either single or multi-student) can be undertaken ... More
Reminiscences by a student of LanglandsJun 26 2019This article gives some memories of Thomas Hales of his years at Princeton as a graduate student under Robert Langlands. It has been prepared for the book "The Genesis of Langlands' Program," edited by Dr. Julia Mueller and Dr. Freydoon Shahidi.
Editorial note to: Erwin Schrödinger, Dirac electron in the gravitational field IJun 25 2019Jul 22 2019Editorial Note with a mathematical and historical introduction to a 1932 paper by Erwin Schr\"odinger on the generalization of the Dirac equation to a curved spacetime -- to appear in the 'Golden Oldie' section of the Journal of General Relativity and ... More
Editorial note to: Erwin Schrödinger, Dirac electron in the gravitational field IJun 25 2019Jul 11 2019Editorial Note with a mathematical and historical introduction to a 1932 paper by Erwin Schr\"odinger on the generalization of the Dirac equation to a curved spacetime -- to appear in the 'Golden Oldie' section of the Journal of General Relativity and ... More
Editorial note to: Erwin Schrödinger, Dirac electron in the gravitational field IJun 25 2019Editorial Note with a mathematical and historical introduction to a 1932 paper by Erwin Schr\"odinger on the generalization of the Dirac equation to a curved spacetime -- to appear in the 'Golden Oldie' section of the Journal of General Relativity and ... More
Editorial note to: Erwin Schrödinger, Dirac electron in the gravitational field IJun 25 2019Jul 08 2019Editorial Note with a mathematical and historical introduction to a 1932 paper by Erwin Schr\"odinger on the generalization of the Dirac equation to a curved spacetime -- to appear in the 'Golden Oldie' section of the Journal of General Relativity and ... More
Broken legos and the pick-up sticks problemJun 24 2019We generalize the well-known broken stick problem in several ways, including a discrete "lego" analogue and a sequential "pick-up sticks/legos" version. The limit behavior of the broken lego problem gives a combinatorial proof of the broken stick problem. ... More
Extraction de la racine carree d'un entier naturel chez al-BaghdadiJun 19 2019Between the ninth and fifteenth centuries, several Arab mathematicians studied numerical algorithms on integers. The extraction of the square root of an integer is based on an algorithm known at least since al-Khwarizmi (died around 850) which presents ... More
Integer Representations and Trajectories of the 3x+1 ProblemJun 17 2019This paper studies certain trajectories of the Collatz function. I show that if for each odd number $n$, $n\sim 3n+2$ then every positive integer $n \in \mathbb{N}\setminus 2^{\mathbb{N}}$ has the representation $$n=\left(2^{a_{k+1}}-\sum_{i=0}^{k}{2^{a_i}3^{k-i}}\right)/ ... More
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
How Long Might We Wait at Random?Jun 14 2019Jul 06 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
The beginnings of symplectic topology in Bochum in the early eightiesJun 14 2019I outline the history and the original proof of the Arnold conjecture on fixed points of Hamiltonian maps for the special case of the torus, leading to a sketch of the proof for general symplectic manifolds and to Floer homology. This is the written version ... More
Trignometry Without GeometryJun 12 2019The development of the trigonometric functions in introductory texts usually follows geometric constructions using right triangles or the unit circle. While these methods are satisfactory at the elementary level, advanced mathematics demands a more rigorous ... 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
Arquímedes y las superficies cuádricasJun 10 2019Jun 13 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
Vortices and atoms in the Maxwellian eraJun 02 2019The mathematical study of vortices began with Herman von Helmholtz's pioneering study in 1858. It was pursued vigorously over the next two decades, largely by British physicists and mathematicians, in two contexts: Maxwell's vortex analogy for the electromagnetic ... More
A Novel Method for Drawing a Circle Tangent to Three Circles Lying on a Plane by Straightedge,Compass,and Inversion CirclesMay 31 2019In this paper, we present a novel method to draw a circle tangent to three given circles lying on a plane. Using the analytic geometry and inversion (reflection) theorems, the center and radius of the inversion circle are obtained. Inside any one of the ... 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
Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierumMay 24 2019This is the translation of Euler's Latin textbook Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum (second volume) into English.
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
Beginning Mathematical Writing AssignmentsMay 18 2019Writing assignments in any mathematics course always present several challenges, particularly in lower-level classes where the students are not expecting to write more than a few words at a time. Developed based on strategies from several sources, the ... 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
The Voronoi Cell in a saturated Circle Packing and an elementary proof of Thue's theoremMay 13 2019The famous Kepler conjecture has a less spectacular, two-dimensional equivalent: The theorem of Thue states that the densest circle packing in the Euclidean plane has a hexagonal structure. A common proof uses Voronoi cells and analyzes their area applying ... 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