# Mathematics - History and Overview Publications (50)

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## Mathematics - History and Overview Publications

Given the subjective preferences of three roommates, the rent of a 3-bedroom apartment can be divided among the rooms in such a way that the three roommates decide on pairwise distinct rooms and are not envious of each other. We give a simple combinatorial proof of the fact that the subjective preferences of only two of the roommates actually suffice to achieve this envy-free rent division using Sperner's lemma. Our proof, in particular, yields an algorithm to find the fair division of rent. Read More

We explain the solution of the following two problems: obtaining of Kepler's laws from Newton's laws (so called two bodies problem) and obtaining the forth Newton's law (the formula for gravitation) as a corollary of Kepler's laws. This small book is devoted to the scholars, who are interested in physics and mathematics. We also make a series of digressions, where explain some technique of the higher mathematics, which are used in the proofs. Read More

**Affiliations:**

^{1}ETH Zürich

**Category:**Mathematics - History and Overview

The classical quadratic formula and some of its lesser known variants for solving the quadratic equation are reviewed. Then, a new formula for the roots of a quadratic polynomial is presented. Read More

We review the theory of intrinsic geometry of convex surfaces in the Euclidean space and prove the following theorem: if the surface of a convex body K contains arbitrary long closed simple geodesics, then K is an isosceles tetrahedron. Read More

We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulation of the product of two-dimensional hyperbolic space with one-dimensional euclidean space is available at http://h2xe.hypernom. Read More

We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulations of three-dimensional hyperbolic space are available at http://h3.hypernom. Read More

The liar paradox is widely seen as not a serious problem. I try to explain why this view is mistaken. Read More

**Affiliations:**

^{1}MPI, IRMA

The origin of quasiconformal mappings, like that of conformal mappings, can be traced back to old cartography where the basic problem was the search for mappings from the sphere onto the plane with minimal deviation from conformality, subject to certain conditions which were made precise. In this paper, we survey the development of cartography, highlighting the main ideas that are related to quasiconformality. Some of these ideas were completely ignored in the previous historical surveys on quasiconformal mappings. Read More

We pay attention to those integrable Hamiltonians to which Liouville--Arnold theorem does not apply, or whose action--angle coordinates are not explicitly computable. We propose, in a rather general setting, a new notion of integrability (that we call "renormalizable") which would allow to switch to a new Hamiltonian, completely equivalent to the initial one. As a paradigmatic example, we discuss the case of the {\it two-centre problem}, whose solution is usually written in terms of elliptic integrals. Read More

Through the Poem Helen of the Nobel Prize laureate George Seferis, we question the foundations of Fuzzy Logic, which seem to appear in the literature from the time of Euripides. Read More

In this article using elementary school level Geometry we observe an alternative proof of Pythagorean Theorem from Heron's Formula. Read More

We investigate three-dimensional surfaces where the normal vector forms a constant angle with the radius vector. These surfaces naturally extend equiangular (logarithmic) spirals in the plane. Read More

A very old problem in campanology is the search for peals. The latter can be thought of as a heavily constrained sequence of all possible permutations of a given size, where the exact nature of the constraints depends on which method of ringing is desired. In particular, we consider the methods of bobs-only Stedman Triples and Erin Triples; the existence of the latter is still an open problem. Read More

This paper demonstrates how a nineteenth century Japanese votive temple problem known as a sangaku from Okayama prefecture can be solved using traditional mathematical methods of the Japanese Edo (1603-1868 CE). We compare a modern solution to a sangaku problem from Sacred Geometry: Japanese Temple Problems of Tony Rothman and Hidetoshi Fukagawa with a traditional solution of \=Ohara Toshiaki (?-1828). Our investigation into the solution of \=Ohara provides an example of traditional Edo period mathematics using the tenzan jutsu symbolic manipulation method, as well as producing new insights regarding the contextual nature of the rules of this technique. Read More

We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a natural correspondence between plane polygons and the Grassmann manifold of 2-planes in real $n$-space proposed by Allen Knutson and Jean-Claude Hausmann. This correspondence defines a natural probability measure on plane polygons. Read More

Lebesgue's dominated convergence theorem is a crucial pillar of modern analysis, but there are certain areas of the subject where this theorem is deficient. Deeper criteria for convergence of integrals are described in this article. Read More

The following work shows how the first digit frequency in a group of numbers in certain real-life situations can be explained using basic algebraic continuous real-valued functions. For instance, the first digits frequency of the numbers representing the change in human growth can be understood better by looking at the square root function in a particular way. In addition, an analysis of basic discrete functions was done by approximating a discrete function to a continuous one. Read More

This paper discusses some aspects of the history of the Paley graphs and their automorphism groups. Read More

Musical intervals in multiple of semitones under 12-note equal temperament, or more specifically pitch-class subsets of assigned cardinality ($n$-chords) are conceived as positive integer points within an Euclidean $n$-space. The number of distinct $n$-chords is inferred from combinatorics with the extension to $n=0$, involving an Euclidean 0-space. The number of repeating $n$-chords, or points which are turned into themselves during a circular permutation, $T_n$, of their coordinates, is inferred from algebraic considerations. Read More

Between 17th and 19th centuries, mathematically orientated votive tablets appeared in Shinto shrines and Buddhist temples all over Japan. Known as sangaku, they contained problems of a largely geometrical nature. In the 17th century, the Japanese mathematician Seki Takakazu developed a form of algebra known as tenzan jutsu. Read More

We survey the dimension theory of self-affine sets for general mathematical audience. The article is in Finnish. Read More

In this article, we present a geometrical proof of sum of $\cos n\varphi$ where $n$ goes from $1$ up to $m$. Although there exist some summation forms and the proofs are simple, they use complex numbers. Our proof comes from a geometrical construction. Read More

We consider loci of points such that their sum of distances or sum of squared distances to each of the sides of a given triangle is constant. These loci are inspired by Viviani's theorem and its extension. The former locus is a line segment or the whole triangle and the latter locus is an ellipse. Read More

In 1900, Macfarlane proposed a hyperbolic variation on Hamilton's quaternions that closely resembles Minkowski spacetime. Viewing this in a modern context, we expand upon Macfarlane's idea and develop a model for real hyperbolic 3-space in which both points and isometries are expressed as complex quaternions, analogous to Hamilton's famous theorem on Euclidean rotations. We use this to give new computational tools for studying isometries of hyperbolic 2- and 3-space. Read More

Transcription into modern notations of the derivation by Stirling and De Moivre of an asymptotic series for $\log(n!)$, usually called Stirling's series. The previous discovery by Wallis of an infinite product for $\pi$, and later results on the divergence of the series are also presented. We conclude that James Stirling has priority over Abraham de Moivre for Stirling's formula and Stirling's series. Read More

The limit of a sequence by the definition with $\varepsilon$ is introduced by the notion of checkmate in two moves. The idea is also extended to define the limit of a function with $\varepsilon$ and $\delta$. Read More

In teaching infinitesimal calculus we sought to present basic concepts like continuity and convergence by comparing and contrasting various definitions, rather than presenting "the definition" to the students as a monolithic absolute. We hope that this could be useful to other instructors wishing to follow this method of instruction. A poll run at the conclusion of the course indicates that students tend to favor infinitesimal definitions over epsilon, delta ones. Read More

Cauchy's method from two centuries ago for computing integrals along the real axis by passing into the complex plane is not rigorous by present-day standards. Yet when properly formulated, his original approach is simpler than modern presentations of the residue calculus. Read More

An expansion upon Donald Kunth's quarter-imaginary base system is introduced to handle any imaginary number base where its real part is zero and the absolute value of its imaginary part is greater than one. A brief overview on number bases is given as well as conversion to both positive and negative bases. Additionally gives examples for addition, subtraction, multiplication for imaginary bases and adds a division method for imaginary bases as well as mentions possible uses. Read More

In this article, we investigate how Euler might have been led to conjecture the Prime Number Theorem, based on what he knew. We also speculate on why he did not do so. Read More

The primary aim of this chapter is, commemorating the 150th anniversary of Riemann's death, to explain how the idea of {\it Riemann sum} is linked to other branches of mathematics. The materials I treat are more or less classical and elementary, thus available to the "common mathematician in the streets." However one may still see here interesting inter-connection and cohesiveness in mathematics. Read More

Polyominoes have been the focus of many recreational and research investigations. In this article, the authors investigate whether a paper cutout of a polyomino can be folded to produce a second polyomino in the same shape as the original, but now with two layers of paper. For the folding, only "corner folds" and "half edge cuts" are allowed, unless the polyomino forms a closed loop, in which case one is allowed to completely cut two squares in the polyomino apart. Read More

In this article I conduct a short review of the proofs of the area inside a circle. These include intuitive as well as rigorous analytic proofs. This discussion is important not just from mathematical view point but also because pedagogically the calculus books still today use circular reasoning to prove the area inside a circle (also that of an ellipse) on this important historical topic, first illustrated by Archimedes. Read More

We present a self-contained proof of the Gauss-Bonnet theorem for two-dimensional surfaces embedded in $R^3$ using just classical vector calculus. The exposition should be accessible to advanced undergraduate and non-expert graduate students. It may be viewed as an illustration and exercise in multivariate calculus and a motivation to go deeper into the fields of geometry and topology. Read More

We provide a new proof of Vivinai's Theorem using what George Polya calls a 'leading particular case.' Our proof highlights the role of generalization in mathematics. Read More

We discuss a couple of examples of Markov chains. This note is written primarily for school students; it is based on a lecture given by the first author at a Math Circle at NAS (www.assagames. Read More

In this paper we discuss how teaching of mathematics for middle school and high school students can be improved dramatically when motivation of concepts and ideas is done through the classical problems and the history of mathematics. This method improves intuition of students, awakens their curiosity, avoids memorizing useless formulas, and put concepts in a historical prospective. To illustrate we show how diagonalizing quadratic forms, elliptic integrals, discriminants of high degree polynomials, and geometric constructions can be introduced successfully in high school level. Read More

We discuss the art and science of producing conformally correct euclidean and hyperbolic tilings of compact surfaces. As an example, we present a tiling of the Chmutov surface by hyperbolic (2, 4, 6) triangles. Read More

In real Hilbert spaces, this paper generalizes the orthogonal groups $\mathrm{O}(n)$ in two ways. One way is by finite multiplications of a family of operators from reflections which results in a group denoted as $\Theta(\kappa)$, the other is by considering the automorphism group of the Hilbert space denoted as $O(\kappa)$. We also try to research the algebraic relationship between the two generalizations and their relationship to the stable~orthogonal~group~$\mathrm{O}=\varinjlim\mathrm{O}(n)$ in terms of topology. Read More

Following a paper in which the fundamental aspects of probabilistic inference were introduced by means of a toy experiment, details of the analysis of simulated long sequences of extractions are shown here. In fact, the striking performance of probability-based inference and forecasting, compared to those obtained by simple `rules', might impress those practitioners who are usually underwhelmed by the philosophical foundation of the different methods. The analysis of the sequences also shows how the smallness of the probability of what has been actually observed, given the hypotheses of interest, is irrelevant for the purpose of inference. Read More

Allegedly, Brouwer discovered his famous fixed point theorem while stirring a cup of coffee and noticing that there is always at least one point in the liquid that does not move. In this paper, based on a talk in honour of Brouwer at the University of Amsterdam, we will explore how Brouwer's ideas about this phenomenon spilt over in a lot of different areas of mathematics and how this eventually led to an intriguing geometrical theory we now know as mirror symmetry. Read More

This article provides a survey on some main results and recent developments in the mathematical theory of water waves. We discuss the mathematical modeling of water waves and give an overview of local and global well--posedness results for the model equations. Moreover, we present reduced models in various parameter regimes for the approximate description of the motion of typical wave profiles and discuss the mathematically rigorous justification of the validity of these models. Read More

This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program of Friedman and Simpson. We look in particular at: (i) the long arc from Poincar\'e to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak K\"onig's Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others. Read More

In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians' approach to interpreting James Gregory's expression ultimate terms in his paper attempting to prove the irrationality of pi. Read More

In this study we develop an early warning system for a large first year STEM undergraduate course and discuss the data analytics decisions at each stage. Numerous studies have provided evidence in favour of adopting early warning systems as a means of identifying at-risk students in online courses. Many of these early warning systems are in practical use and rely on data from students' engagement with Virtual Learning Environments (VLEs), also referred to as Learning Management Systems. Read More

We show the results on the history of the invention of the conjugacy $h(x)=\frac{2}{\pi}\arcsin\sqrt{x}$ of one-dimensional $[0,\, 1]\rightarrow [0,\, 1]$ maps $f(x)=4x(1-x)$ and $g(x)=1-|1-2x|$. Read More

**Affiliations:**

^{1}Inria Grenoble Rhône-Alpes

**Category:**Mathematics - History and Overview

About 160 years ago, the Italian mathematician Fa\`a di Bruno published two notes dealing about the now eponymous formula giving the derivative of any order of a composition of two functions. We reproduce here the two original notes, Fa\`a di Bruno (1855, 1857), written respectively in Italian and in French, and propose a translation in English. Read More

It is a book on geodesy and the theory of errors for the students of the cycle of geomatics, surveying and mapping. The geometric and spatial aspects of geodesy are presented in the first part of the book. The second part is about the theory of errors. Read More

In this article we provide several exact formulae to calculate the probability that a random triangle chosen within a planar region contains a given fixed point $O$ in the interior of that region or on its boundary. These formulae are in terms of only one integration over the boundary of the region of an appropriate function with respect to a distribution which depends of the point $O$. One of our formulae is a generalization of the formula in Proposition II (\cite{Kleitman}, page 226). Read More