历史与综述
Any act of problem-solving combines prior knowledge, local search, and a third element that is less often discussed: the extraction of information from search to update understanding. I propose a model of mathematical problem-solving as a…
Hyperbolic elliptic parabolic disks can be described by the inequality $\frac{x^2}{C^2}+2y^2-2y\leq0$ ($0<C<1$) in the unit disk based Beltrami--Cayley--Klein model of the hyperbolic geometry, up to hyperbolic congruences. The hyperbolic…
This is an English translation and digitisation of Sophie Kowalevski's (also know as Sofya Kovalevskaya) paper on what is now known as the Kovalevskaya Top. The original paper was written in French and published in Vol 12 of Acta…
We study triangles and quadrilaterals which are inscribed in a circle and circumscribed about a parabola. Although these are particular cases of the celebrated Poncelet's Theorem, in this paper we {\it do not assume} the theorem but prove…
Maxwells Demon is a mythical being, first described by the physicist James Clerk Maxwell (although named Maxwells Demon by Lord Kelvin). Maxwell used it in a thought experiment to potentially violate the Second Law of Thermodynamics by…
Vop\v{e}nka's Alternative Set Theory has been considered as a framework for modelling vague notions. This paper takes feasibility, pertaining to numbers as per some of Yessenin-Volpin's work, and tries to assess how this notion could be…
This is a study of the construction of particular regular sub-n-gons T in regular n-gons P using a special system of chords of P. In particular, some of these sub-n-gons have areas which are integer divisors of the area of the given n-gon…
The population and capital dynamics of three stylized investment systems are mathematically described using discrete-time difference equations with closed-form solutions. The models share a common capital budget equation but differ in their…
The effective teaching and learning of statistics persist as a challenge in K-12 education and has clear impacts in developing competence and confidence of students in entering STEM fields especially in today's digital age of data science.…
It is shown that Pythagorean triples can be used to generate matrices that have integer eigenvalues for all permutations of their coefficients, via simple formulas. For example, each and every permutation of the $2\times2$ matrix…
It is a well-known fact -- which can be shown by elementary calculus -- that the volume of the unit ball in $\mathbb{R}^n$ decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as $n \nearrow…
Graph colouring is a combinatorial optimisation problem with applications in several important domains, including sports scheduling, cartography, street map navigation, and timetabling. It is also of significant theoretical interest and a…
Imagine a student -- let's call him `E', and make him a "he" -- that is enrolled in Calculus 2, and who believes that to pass in maths courses he only needs to memorize methods and apply them quickly and without errors. Let's imagine that…
In this paper we sketch how some fundamental concepts of modern topology (as well as logic and category theory) can be understood philosophically in the light of Hegel's Science Logic as well how modern topological concepts can provide…
The $\pi$-series is attributed to Madhava, Gregory and Leibniz based on the chronology of its discovery. While this acknowledges the fact that Madhava of Sangamagrama an Indian mathematician who lived in Kerala in the 14th century along the…
This article provides information on the life and work of the number theorist Arnold Scholz. It is an English translation with modifications of an introduction to the correspondence of Hasse, Scholz and Taussky published in 2016.
We study Cantor's powerset theorem from a graph-theoretic perspective, consider some alternative proofs to Cantor's original, and provide a new proof.
We argue that it is neither necessary nor sufficient for a mathematical proof to have epistemic value that it be "correct", in the sense of formalizable in a formal proof system. We then present a view on the relationship between…
The widespread availability of generative artificial intelligence tools poses new challenges for school mathematics education, particularly regarding the formative role of traditional mathematical tasks. In post-AI educational contexts,…
The gap between high school and university level mathematics has long been deemed problematic. Felix Klein referred to this gap as the ''double discontinuity'' meaning that students come to university unprepared for university courses and…