Mathematics

We 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. S. G. Dani and A. Papadopoulos, Springer Verlag, 2019.

This 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 pre-class preparation and aligned worksheets for in-class dynamics, and the activities that were conducted. We discuss difficulties and key aspects to be considered while creating this material and during implementation of Flipped Learning. We present how students reacted to pre-class preparation and how in-class dynamics developed during implementation. We show results on students performance and perception when enrolling in a flipped classroom section. We present comparative results on students performance of a section taught with Flipped Learning vs a parallel section thought in the traditional expository way. We could conclude that flipped courses show similar results in passing percentage than traditionally taught courses, that student's perceptions are generally mixed, and we perceived that students repeating the course preferably do not choose flipped classes. Finally, we discussed the methodological evolution of this course converging to a mixed methodology throughout a four year period, observing that the instructors evaluation decreases in classes that were flipped. Mixed methodologies on the other hand, increased the learning experience of students resulting in an increased instructors evaluation score and higher students enrollment in the course.

Problem 1325 from the journal Crux Mathematicorum is revisited, and a new solution is presented.

In 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.

To divide a cake into equal sized pieces most people use a knife and a mixture of luck and dexterity. These attempts are often met with varying success. Through precise geometric constructions performed with the knife replacing Euclid's straightedge and without using a compass we find methods for solving certain cake-cutting problems exactly. Since it is impossible to exactly bisect a circular cake when its center is not known, our constructions need to use multiple cakes. Using three circular cakes we present a simple method for bisecting each of them or to find their centers. Moreover, given a cake with marked center we present methods to cut it into n pieces of equal size for n=3,4 and 6. Our methods are based upon constructions by Steiner and Cauer from the 19th and early 20th century.

We discuss some seemingly unrelated observations on integers, whose close or farther away neighbors show a complex of combinatorial, ordering, arithmetical or probabilistic properties, emphasizing puzzlement in more common expectations.

Leonhard Euler likely developed his summation formula in 1732, and soon used it to estimate the sum of the reciprocal squares to 14 digits --- a value mathematicians had been competing to determine since Leibniz's astonishing discovery that the alternating sum of the reciprocal odd numbers is exactly π/4 . This competition came to be known as the Basel Problem, and Euler's approximation probably spurred his spectacular solution in the same year. Subsequently he connected his summation formula to Bernoulli numbers, and applied it to many other topics, masterfully circumventing that it almost always diverges. He applied it to estimate harmonic series partial sums, the gamma constant, and sums of logarithms, thereby calculating large factorials (Stirling's series) with ease. He even commented that his approximation of π was surprisingly accurate for so little work. All this is beautifully presented in mature form in Euler's book Institutiones Calculi Differentialis. I have translated extensive selections for annotated publication as teaching source material in a book Mathematics Masterpieces; Further Chronicles by the Explorers, featuring original sources. I will summarize and illustrate Euler's achievements, including the connection to the search for formulas for sums of numerical powers. I will show in his own words Euler's idea for deriving his summation formula, and how he applied the formula to the sum of reciprocal squares and other situations, e.g., large factorials and binomial coefficients. Finally, I will discuss further mathematical questions, e.g., approximation of factorials, arising from Euler's writings.

We 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 and the use of concrete material with some well know factoring methods for quadratic polynomial with integer coeficients.

This 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. We provide elementary proofs for the original convention with B 1 =1/2 and also for the current convention with B 1 =−1/2 , using only the binomial theorem and the concise Blissard symbolic (umbral) notation.

Several continued fraction expansions for e have been produced by an automated conjecture generator (ACG) called \emph{The Ramanujan Machine}. Some of these were already known, some have recently been proved and some remain unproven. While an ACG can produce interesting putative results, it gives very limited insight into their significance. In this paper, we derive an elegant continued fraction expansion, equivalent to a result from the Ramanujan Machine, using the sequence of ratios of factorials to subfactorials or derangement numbers.