历史与综述
In a piece published in 1981, H. M. Edwards touts the benefits of reading the masters. A quarter-century later, Edwards takes seriously his own advice by publishing an encomium on Euler's Institutiones (1755). While we agree with Edwards…
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…
We show that character analysis using Fourier series is possible, at least when a mathematical character is considered. Previous approaches to character analysis are somewhat not in the spirit of harmonic analysis.
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…
The main object of this paper is to find closed form expressions for finite and infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is the number of distinct prime factors of $n$. We then derive general convergence criteria…
This note is intended primarily for college calculus students right after the introduction of the Intermediate Value Theorem, to show them how the Intermediate Value Theorem is used repeatedly and straightforwardly to prove the celebrated…
We are motivated by a problem about running: If a race was completed in an average pace of P minutes per mile, is there necessarily some mile of the race that was run in exactly P minutes? The answer is no. We explain why, and describe the…
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.com.
We use data from papers posted to the Mathematics section of the arXiv to explore the representation of women in mathematics research. We show that women are under-represented as authors of mathematics papers on the arXiv, even in…
Using the quadratic reciprocity law as the motivating example, we convey an understanding of classical reciprocity laws.
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.
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.com.
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.
The liar paradox is widely seen as not a serious problem. I try to explain why this view is mistaken.
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,…
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…
This paper discusses some aspects of the history of the Paley graphs and their automorphism groups.
We survey the dimension theory of self-affine sets for general mathematical audience. The article is in Finnish.
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…
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…