历史与综述
Mathematics has become inescapable in modern, digitized societies: there is hardly any area of life left that isn't affected by it, and we as mathematicians play a central role in this. Our actions affect what others, in particular our…
This essay explores the impact of automated proof construction on three key areas of mathematical cognition: on how we judge the role one piece of mathematics plays in another, on how we make mistakes in reasoning about mathematical…
In 1953, Enrico Fermi criticized Dyson's model by quoting Johnny von Neumann: "With four parameters I can fit an elephant, and with five I can make him wiggle his trunk." So far, there have been several attempts to fit an elephant using…
Gottlob Frege ingeniously presented a purely logical definition of the concept of number. However, one can claim that his definition is, in some way, circular, as it relies on the concept of one-to-one relation. The concept of number only…
We contribute to the zoo of dubious identities established by J.M. and P.B. Borwein in their 1992 paper, "Strange Series and High Precision Fraud" with five new entries, each of a different variety than the last. Some of these identities…
In this paper, we situate the educational movement of "Ethics in Mathematics," as outlined by the Cambridge University Ethics in Mathematics Project, in the wider area of mathematics ethics education. By focusing on the core message coming…
This work explores a possible course of evolution of mathematics in ancient times in India when there was no script, no place-value system, and no zero. Reviewing examples of time-reckoning, large numbers, sacrificial altar-making, and…
This essay examines how automation has reconfigured mathematical proof and labor, and what might happen in the future. It discusses practical standards of proof, distinguishes between prominent forms of automation in research, provides…
We present a scoping review of published literature on ethnomathematics and Indigenous mathematics as a step towards a goal to decolonize the prevailing Eurocentric view of the provenance of mathematics. Mathematical practices were…
We present in this article a general approach (in the form of recommendations and guidelines) for tackling Diophantine equation problems (whether single equations or systems of simultaneous equations). The article should be useful in…
Upon re-examining Arnold's established lemma for explaining his famous limit problem, we have determined that while the lemma itself is correct, there is a defect in the original geometric proof. In this paper, we prove the correctness of…
Given two non-zero integers $a$ and $b$ there exist integers $m$ and $n$ for which $am-bn =(a,b)$. An increasing number of mathematicians have been calling this `B\'ezout's identity', some encouraged by finding "identit\'e de B\'ezout" in…
Topology at the undergraduate level is often a theoretical mathematics course, introducing concepts from point-set topology or possibly algebraic topology. However, the last two decades have seen an explosion of growth in applied topology…
This article contains a selection of problems from the American Mathematics Competitions.
The purpose of this thesis is to convey the basic concepts of information geometry and its applications to non-specialists and those in applied fields, assuming only a first-year undergraduate background in calculus, linear algebra, and…
Prime numbers play a key role in number theory and have applications beyond Mathematics. In particular, in the Theory of Codes and also in Cryptography, the properties of prime numbers are relevant, because, from them, it is possible to…
Zero factorial, defined to be one, is often counterintuitive to students but nonetheless an interesting concept to convey in a classroom environment. The challenge is to delineate the concept in a simple and effective way through the…
We present the problems and solutions to the 12th Annual USA Junior Mathematical Olympiad.
This short note for non-experts means to demystify the tasks of evaluating the Riemann Zeta Function at non-positive integers and at even natural numbers, both initially performed by Leonhard Euler. Treading in the footsteps of G. H. Hardy…
We present the problems and solutions to the 50th Annual USA Mathematical Olympiad.