Mathematics - History and Overview Publications (50)


Mathematics - History and Overview Publications

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

It is a collection of problems and exercises of geodesy and the theory of errors. 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


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. This formula is 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 $O$. Our formula is a generalization of the formula in Proposition II (\cite{Kleitman}, page 226). Read More

This is a tract on the art and practice of mathematical writing. Not only does the book cover basic principles of grammar, syntax, and usage, but it takes into account developments of the last twenty years that have been inspired by the Internet. There is considerable discussion of TeX and other modern writing environments. Read More

The following result, a consequence of Dumas criterion for irreducibility of polynomials over integers, is generally proved using the notion of Newton diagram: Let $f(x)$ be a polynomial with integer coefficients and $k$ be a positive integer relatively prime to the degree of $f(x)$. Suppose that there exists a prime number $p$ such that the leading coefficient of $f(x)$ is not divisible by $p$, all the remaining coefficients are divisible by $p^k$, and the constant term of $f(x)$ is not divisible by $p^{k+1}$. Then $f(x)$ is irreducible over $\mathbb{Z}$. Read More

The process of doing Science in condition of uncertainty is illustrated with a toy experiment in which the inferential and the forecasting aspects are both present. The fundamental aspects of probabilistic reasoning, also relevant in real life applications, arise quite naturally and the resulting discussion among non-ideologized, free-minded people offers an opportunity for clarifications. Read More

We discuss the general theory of realizing two-variable fuctions on slide rules (based on our paper 1977) and offer some new scales for practical use. Read More

This text is based on an invited talk at the Dedekind Symposium at Braunschweig in October 2016. It summarizes views from my recent commented edition of Dedekinds two books on the foundations of mathematics. Read More

In a recent issue of the Bulletin of the Korean Mathematical Society, Qi and Zhang discovered an interesting integral representation for the Bernoulli numbers of the second kind (these numbers are also known as Gregory's coefficients, Cauchy numbers of the first kind and reciprocal logarithmic numbers). In this short communication it is shown that this representation is a rediscovery of an old result obtained in the XIXth century by Ernst Schroder. It is also demonstrated that the same integral representation may be readily derived by means of the complex integration. Read More

Nicolas-Auguste Tissot (1824--1897) published a series of papers on cartography in which he introduced a tool which became known later on, among geographers, under the name of the "Tissot indicatrix." This tool was broadly used during the twentieth century in the theory and in the practical aspects of the drawing of geographical maps. The Tissot indicatrix is a graphical representation of a field of ellipses on a map that describes its distortion. Read More

This is an edition of the famous letter by Richard Dedekind to H. Keferstein dated 27 February 1890 Read More

Testing convergence of a series $\sum a_n$ is an important part of scientific areas. A very basic comparison test bounds the terms of $\sum a_n$ with the terms of some known convergent series $\sum b_n$ (either in the form $a_n\leq b_n$ or $a_{n+1}/a_n \leq b_{n+1}/b_n$). In $19^{th}$ century Kummer proposed a test of convergence for any positive series saying that the series $\sum a_n$ converges if and only if there is a positive series $\sum p_n$ and a real constant $c>0$ such that $p_n(a_n/a_{n+1} - p_{n+1})\geq c$. Read More

It is pointed out that the language of quotient groups and wrapped distributions allows an elementary discussion of Benford's Law, and adds arguments supporting wide-spread observability of this statistics. Read More

Frederick William Gehring was a hugely influential mathematician who spent most of his career at the University of Michigan. Gehring's major research contributions were to Geometric Function Theory, particularly in higher dimensions $\IR^n$, $n\geq 3$. This field he developed in close coordination with colleagues, primarily in Finland, over three decades 1960 -- 1990. Read More

Despite the increasing number of women graduating in mathematics, a systemic gender imbalance persists and is signified by a pronounced gender gap in the distribution of active researchers and professors. Especially at the level of university faculty, women mathematicians continue being drastically underrepresented, decades after the first affirmative action measures have been put into place. A solid publication record is of paramount importance for securing permanent positions. Read More

The election methods introduced in 1894--1895 by Phragm\'en and Thiele, and their somewhat later versions for ordered (ranked) ballots, are discussed in detail. The paper includes definitions and examples and discussion of whether the methods satisfy some properties, including monotonicity, consistency and various proportionality criteria. The relation with STV is also discussed. Read More

This article discusses the authors' experiences while teaching a basic undergraduate English medium instruction mathematics course of Single Variable Calculus by flipping the classroom. In contrast to the traditional class where the students come to the classroom unprepared, in the flipped class 101, students are prepared at home before coming into the classroom. There are many ways to flip the class, one effective way involved assigning recorded video lectures to the students that should be viewed at home before in-class sessions. Read More

As is well known, a graph is a mathematical object modeling the existence of a certain relation between pairs of elements of a given set. Therefore, it is not surprising that many of the first results concerning graphs made reference to relationships between people or groups of people. In this article, we comment on four results of this kind, which are related to various general theories on graphs and their applications: the Handshake lemma (related to graph colorings and Boolean algebra), a lemma on known and unknown people at a cocktail party (to Ramsey theory), a theorem on friends in common (to distance-regularity and coding theory), and Hall's Marriage theorem (to the theory of networks). Read More

In this expository paper written to commemorate Fibonacci Day 2016, we discuss famous relations involving the Fibonacci sequence, the golden ratio, continued fractions and nested radicals, and show how these fit into a more general framework stemming from the quadratic formula. Read More

Two perpendicular segments which divide a given triangle into 4 regions of equal area is called a quadrisection of the triangle. Leonhard Euler proved in 1779 that every scalene triangle has a quadrisection with its triangular part on the middle leg. We provide a complete description of the quadrisections of a triangle. Read More

Hilbert's Irreducibility Theorem is a cornerstone that joins areas of analysis and number theory. Both the genesis and genius of its proof involved combining real analysis and combinatorics. We try to expose the motivations that led Hilbert to this synthesis. Read More

The aim of this work is to use Napoleon's Theorem in different regular polygons, and decide whether we can prove Napoleon's Theorem is only limited with triangles or it could be done in other regular polygons that can create regular polygons. Read More

We give an elementary account of generalized Fibonacci and Lucas polynomials whose moments are Narayana polynomials of type A and type B. Read More

We provide a mathematic model for the Traditional Yin-and-Yang Double Fish Diagram which from Chinese Traditional Philosophy. Read More

One can notice that quite often difference between so-called "standard students" and "gifted" ones is not because that first are less smart, but they have different "orientation", they consider subject as a collections of rules which should not be broken. The first aim is to overcame this dogmatic point of view on this subject, and only after that possible to judge a person to be talented one or not. This is the first stage of math education, the material needs just to serve it. Read More

This is an English translation of the obituary notice by Beno Eckmann, appearing in Elemente der Mathematik 47.3 (1992) 118--122 (in German). Read More

Why are white and black piano keys in an octave arranged as they are today? This article examines the relations between abstract algebra and key signature, scales, degrees, and keyboard configurations in general equal-temperament systems. Without confining the study to the twelve-tone equal-temperament (12-TET) system, we propose a set of basic axioms based on musical observations. The axioms may lead to scales that are reasonable both mathematically and musically in any equal-temperament system. Read More

Affiliations: 1University of South Carolina, 2University of South Carolina

What initial trajectory angle maximizes the arc length of an ideal projectile? We show the optimal angle, which depends neither on the initial speed nor on the acceleration of gravity, is the solution x to a surprising transcendental equation: csc(x) = coth(csc(x)), i.e., x = arccsc(y) where y is the unique positive fixed point of coth. Read More

The "Su\`an Sh{\`u} Sh{\=u}" contains 301 instances of regular expressions for fractions. They can be "mono-dimensional" (formed with one integer name only) for unit fractions, "bidimensional" (with two integer names) for both unit and non-unit fractions, or lexicalized only for 1/3, 1/2 and 2/3. The present paper gives a complete description of the diversity of these forms. Read More

From the 16th to the 20th century, Western mathematics had a dramatic impact in China. Firstly they triggered lexical developments, e.g. Read More

Theorem. Given four various generalized circles on a plane and they don't touch in the same point. Then there are at most six generalized circles such that each of them touches each of the given. Read More

A golden-ratio-based rectangular tiling of the first quadrant of the Euclidean plane is constructed by drawing vertical and horizontal grid lines which are located at all even powers of $\phi$ along one axis, and at all odd powers of $\phi$ on the other axis. The vertices of the rectangles formed by these lines can be connected by rays starting at the origin having slopes that are odd powers of $\phi$. A refinement of this tiling results in the familiar one with horizontal and vertical grid lines at every power of $\phi$ along each axis. Read More

This article reviews the so-called "axioms" of origami (paper folding), which are elementary single-fold operations to achieve incidences between points and lines in a sheet of paper. The geometry of reflections is applied, and exhaustive analysis of all possible incidences reveals a set of eight elementary operations. The set includes the previously known seven "axioms", plus the operation of folding along a given line. Read More