历史与综述
We present a minimal mathematical model for conducting patterns that separates geometric trajectory from temporal parametrization. The model is based on a cyclic sequence of preparation and ictus points connected by cubic Hermite segments…
W. L. Ferrar seems to have been the first mathematician to clearly draw a connection between the functional aspects of a summation formula and the behavior of the Dirichlet series underlying it. Taking a formula due to him as a starting…
In AI-rich higher education, polished written mathematics has become easier to produce than trustworthy evidence of understanding. This article develops a human-scale methodology for service mathematics, with informatics as its main running…
This note investigates the combinatorics of permutations underlying the NYT daily word game Waffle. It helps to solve Waffle games and helps to understand why some games are easy to solve while others are very hard. It shows that a perfect…
This study reconstructs the origin of a constant, here called $\Xi$ (Xi), as a primary scaling factor in Old Babylonian mathematics and astronomy. $\Xi$ arises from the practical necessity of precise measurements on the sky or a circle,…
The author was encouraged to write this review by numerous enquiries from researchers all over the world, who needed a ready-to-use algorithm for the inversion of confluent Vandermonde matrices which works in quadratic time for any values…
Linear Geometry describes geometric properties that depend on the fundamental notion of a line. In this paper we survey basic notions and results of Linear Geomery that depend on the flat hulls: flats, exchange, rank, regularity,…
In this brief article, we present the formula created by Rafael Barrett in 1903 in a note to Henri Poincar\'e, which remained unknown for decades. Discovered in the 1930s by a Uruguayan mathematician, this formula was published and analyzed…
In this methodological review, we discuss the fundamental concepts of the theory of integral invariants. This theory originated with Poincare and Cartan \cite{Koz, Kart} and was further developed by Kozlov \cite{int_K}. We demonstrate how…
The Hopf fibration mapping circles on a 3-sphere to points on a 2-sphere is well known to topologists. While the 2-sphere is embedded in 3-space, four-dimensional Euclidean space is needed to visualize the 3-sphere. Visualizing objects in…
The view that Peacock's principle of permanence has been invalidated by Hamilton's introduction of non-commutative algebras has always seemed rather odd, in light of Peacock's favorable reception of quaternions and the endorsement of his…
Recent developments show that AI can prove research-level theorems in mathematics, both formally and informally. This essay urges mathematicians to stay up-to-date with the technology, to consider the ways it will disrupt mathematical…
The recent discovery of a family of aperiodic monotiles, which includes David Smith's famous Hat, has shaken the field of plane tessellations. Music composers have already utilised the visual representation of plane tilings in their artwork…
One of the oldest and most enduring myths in human history is the belief that the Parthenon was cleverly designed with various curved structures and sizes in order to correct optical illusions, and therefore appear straight and regular. The…
We propose an interpretation of, and approach to, Helly's theorem that can be included quite early in the undergraduate curriculum. At the same time, the approach connects with contemporary models of data privacy and with sampling methods…
This paper asks what Brouwer might have replied to Dummett's interpretation of intuitionism. Complementing earlier literature, it treats Dummett's rejection of the ontological approach; the charge of psychologism and solipsism; indefinite…
What happens when a food product contains a version of itself? The Oreo Loaded -- a cookie whose filling contains real Oreo cookie crumbs -- can be viewed as the result of mixing a Mega Stuf Oreo into a Mega Stuf Oreo. Iterating this…
Mathematics is a mountain, but students need more than descriptions of the view: they need a trail they can actually walk. This paper presents the Math Teaching Atlas, a framework for mathematical exposition built around route units (single…
Inspired by the recent 90th anniversary of the Scottish Book we present some reflections about its impact. First we discuss new areas of mathematics it helped launch. Then we argue that it was actively used in stimulating the interests and…
This is an English translation and digitisation of Frobenius' and Stickelberger's "On the theory of elliptic functions" first published in Journal fur die reine und angewandte Mathematik (Crelle's journal), 83, 175-179 (1877) with the title…