# Mathematics - History and Overview Publications (50)

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## Mathematics - History and Overview Publications

This is a short essay on the roles of Max Dehn and Axel Thue in the formulation of the word problem for (semi-)groups, and the story of the proofs showing that the word problem is undecidable. Read More

We discuss the understanding of geometry of the circle in ancient India, in terms of enunciation of various principles, constructions, applications etc. during various phases of history and cultural contexts. Read More

In "Quartic Coincidences and the Singular Value Decomposition" by Clifford and Lachance, Mathematics Magazine, December, 2013, it was shown that if there is a midpoint ellipse(an ellipse inscribed in a quadrilateral, $Q$, which is tangent at the midpoints of all four sides of $Q$), then $Q$ must be a parallelogram. We strengthen this result by showing that if $Q$ is not a parallelogram, then there is no ellipse inscribed in $Q$ which is tangent at the midpoint of three sides of $Q$. Second, the only quadrilaterals which have inscribed ellipses tangent at the midpoint of even two sides of $Q$ are trapezoids or what we call a midpoint diagonal quadrilateral(the intersection point of the diagonals of $Q$ coincides with the midpoint of at least one of the diagonals of $Q$). Read More

Not only a review of Weintraub's Differential Forms: Theory and Practice but also a discussion of why differential forms should be taught to undergraduates and an overview of some of the other possible texts that could be used. Read More

This book introduces to the theory of probabilities from the beginning. Assuming that the reader possesses the normal mathematical level acquired at the end of the secondary school, we aim to equip him with a solid basis in probability theory. The theory is preceded by a general chapter on counting methods. Read More

**Affiliations:**

^{1}LMJL

The first part of this article intends to present the role played by Thom in diffusing Smale's ideas about immersion theory, at a time (1957) where some famous mathematicians were doubtful about them: it is clearly impossible to make the sphere inside out! Around a decade later, M. Gromov transformed Smale's idea in what is now known as the h-principle. Here, the h stands for homotopy. Read More

We present a variety of prime-generating constructions that are based on sums of primes. The constructions come in all shapes and sizes, varying in the number of dimensions and number of generated primes. Our best result is a construction that produces 6 new primes for every starting prime. Read More

A pilot survey was sent to chairs of 14 doctoral math departments asking for three types of data: (1) category on job-placements for research post-docs leaving their department in three recent years; (2) category of jobs from which their new faculty hires came in two recent years and two years a decade earlier; and (3) preparation for future careers offered by their department to their research post-docs. Eleven departments submitted data on post-docs. Of the 162 departing post-docs for whom data was supplied, 25% obtained tenure-track jobs in doctoral departments; 22% took another post-doc; and 18% were reported as "unknown/other". Read More

We provide an explicit geometric algorithm involving only ruler and compass constructions in order to specify the specular reflection point on the surface of a reflecting sphere of radius $r$ given two focal points $A$ and $B$ lying outside of it. By numerically implementing the algorithm we compute the point in question for a number of cases. We conclude by discussing how the first iteration of the algorithm constitutes a first order approximation to the real solution by providing a closed expression for it as well as the error involved in doing so, as a function of the distances of the two focal points from the origin and the angle formed between them. Read More

The Monthly has published roughly fifty papers on the $\Gamma$ function or Stirling's formula. We survey those papers (discussing only our favourites in any detail) and place them in the context of the larger mathematical literature on $\Gamma$. Read More

We explore the rational, formal and non-formal criteria of consistency, non-triviality and redundancy in the mathematical research now a days. We develop a paradigmatic discussion by analysing the different conceptions of those criteria, from the logic-formal ones to the non formal ones (but still rational criteria).We illustrate the discussion with concrete examples obtained form the mathematical reseach, particularly from the results that were published in the last 50 years in the mathematical theory of deterministic dynamical systems. Read More

Weyl's original scale geometry of 1918 ("purely infinitesimal geometry") was withdrawn by its author from physical theorizing in the early 1920s. It had a comeback in the last third of the 20th century in different contexts: scalar tensor theories of gravity, foundations of gravity, foundations of quantum mechanics, elementary particle physics, and cosmology. It seems that Weyl geometry continues to offer an open research potential for the foundations of physics even after the turn to the new millennium. Read More

The accepted lore is that Operational Research traces its roots back to the First and Second World Wars, when scientific research was used to improve military operations. In this essay we provide a different perspective on the origins of Operational Research by arguing that these are deeply intertwined with the impressive technological advances in Western Europe between the fifteenth and the sixteenth centuries. Read More

R-moulds of numerical semigroups are defined as increasing sequences of real numbers whose discretizations may give numerical semigroups. The ideal sequence of musical harmonics is an R-mould and discretizing it is equivalent to defining equal temperaments. The number of equal parts of the octave in an equal temperament corresponds to the multiplicity of the related numerical semigroup. Read More

Martha Euphemia Lofton Haynes was the first African American woman to receive a PhD in mathematics. She grew up in Washington DC, earned a bachelors degree in mathematics from Smith College in 1914, a masters in education from University of Chicago in 1930, and a doctorate in mathematics from the Catholic University of America in 1943. Haynes spent over forty-five years teaching in Washington DC from elementary and secondary level to university level. Read More

This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended Interpretation hypothesis. We highlight some applications including (1) Loeb's approach to the Lebesgue measure, (2) a radically elementary approach to the vibrating string, (3) true infinitesimal differential geometry. We explore the relation of Robinson's and related frameworks to the multiverse view as developed by Hamkins. Read More

**Affiliations:**

^{1}Gymnasium Neue Oberschule Braunschweig,

^{2}Gymnasium Neue Oberschule Braunschweig,

^{3}Technische Universität Braunschweig

The Coupon Collector's Problem is one of the few mathematical problems that make news headlines regularly. The reasons for this are on one hand the immense popularity of soccer albums (called Paninimania) and on the other hand that no solution is known that is able to take into account all effects such as replacement (limited purchasing of missing stickers) or swapping. In previous papers we have proven that the classical assumptions are not fulfilled in practice. Read More

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 fourth 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