历史与综述
Wittgenstein's paradoxical theses that unproved propositions are meaningless, proofs form new concepts and rules, and contradictions are of limited concern, led to a variety of interpretations, most of them centered on the rule-following…
In this text I present some problems which led to the introduction of special kinds of graphs as tools for studying singular points of algebraic surfaces. I explain how such graphs were first described using words, and how several…
Analogy has received attention as a form of inductive reasoning in the empirical sciences. However, its role in pure mathematics has received less consideration. This paper provides an account of how an analogy with a more familiar…
The aim of this paper is to prove wordlessly the sum formula of $1^{k}+2^{k}+\ldots +n^{k}$, $k\in\{1,2,3\}$.
This is a collection of teaching materials used in several Russian universities, schools, and mathematical circles. Most problems are chosen in such a way that in the course of the solution and discussion a reader learns important…
The classification of algebraic surfaces by the Italian School of algebraic geometry is universally recognized as a breakthrough in 20th-century mathematics. The methods by which it was achieved do not, however, meet the modern standard of…
We approach Guido Castelnuovo's intellectual world by focusing on a trilogy of papers published in 1889 and by drawing a few remarks about Castelnuovo's scientific interests and attitudes.
Fourier Series is the second of monographs we present on harmonic analysis. Harmonic analysis is one of the most fascinating areas of research in mathematics. Its centrality in the development of many areas of mathematics such as partial…
The Department of Mathematics & Statistics at Pomona College has long worked to create an inclusive and welcoming space for all individuals to study mathematics. Many years ago, our approach to the lack of diversity we saw in our majors was…
We give a short proof of the quadratic reciprocity law using Gauss's Lemma and Hermite's identity.
In this article we describe special type of mathematical problems that may help develop teaching methods that motivate students to explore patterns, formulate conjectures and find solutions without only memorizing formulas and procedures.…
In 1924, S. Banach and A. Tarski proved an astonishing, yet rather counterintuitive paradox: given a solid ball in $\mathbb{R}^3$, it is possible to partition it into finitely many pieces and reassemble them to form two solid balls, each…
This paper, which is mainly based on unpublished material, focuses on the scientific influence that Felice Casorati exerted on Salvatore Pincherle. This influence can be traced, in particular, in Casorati's work on the finite-difference…
In connection to the two fascinating constants $e$ and $\pi$, there are many beautiful visual proofs to the inequality $\pi^{e}<e^{\pi}$. The aim of this classroom capsule is to give three visual proofs to the more general inequality…
In this article, the notion of a mathematical model in science is attempted to be enlightened from several points of view. In particular, it is shown that mathematical models are introduced differently and used differently in different…
We report on a first experiment about an email based course that connects learning Python basics and introductory probability theory. In the experiment 7 short sequences of homework were sent out to prospective mathematics teachers who did…
We show the relevance of the logarithmic integral function in the development of mathematics in the first half of the 19th century. Its importance involved first level mathematicians such as Euler, Gauss, Bessel, Riemann. Our perspective is…
Mathematical objects are generally abstract and not very approachable. Illustrations and interactive visualizations help both students and professionals to comprehend mathematical material and to work with it. This approach lends itself…
This is an exposition of Gauss's proof of Descartes's rule of signs.
This paper states and proves a generalization of the well-known Desargues involution theorem from plane projective geometry.