Mathematics

Oftentimes, Stokes' theorem is derived by using, more or less explicitly, the invariance of the curl of the vector field with respect to translations and rotations. However, this invariance -- which is oftentimes described as the curl being a "physical" vector -- does not seem quite easy to verify, especially for undergraduate students. An even bigger problem with Stokes' theorem is to rigorously define such notions as ``the boundary curve remains to the left of the surface''. Here an apparently simpler and more general approach is suggested.

We present a simple but efficient concept for the realization of blended teaching of mathematics and its applications in theoretical mechanics that was conceived, tested and implemented at the University of Trento, Italy, during the COVID-19 pandemic. The concept foresees traditional blackboard lectures with a reduced number of students present in the lecture hall, while the same lectures are simultaneously made available to the remaining students via high quality low-bandwidth online streaming. Based on our first assumption that traditional blackboard lectures, including the gestures and the facial expressions of the professor, are still a very efficient and highly appreciated means of teaching mathematics, this paper deliberately does not want to propose a novel pedagogical concept of how to teach mathematics, but rather presents a technical concept how to preserve the quality of traditional blackboard lectures even during the pandemic and how to make them available to the students at home via online streaming with adequate audio and video quality at low internet bandwidth. The second assumption is that the teaching of mathematics is a dynamic creative process that requires the physical presence of students in the lecture hall as audience so that the professor can instantaneously fine-tune the evolution of the lecture according to his/her perception of the level of attention and the facial expressions of the students. The third assumption of this paper is that students need to have the possibility to interact with each other personally. We report on the necessary hardware, software and logistics, and on the perception of the proposed blended lectures by students from civil and environmental engineering at the University of Trento, compared to traditional lectures and also compared to the pure online lectures that were needed as emergency measure at the beginning of the pandemic.

In this article, we present a short, non-exhaustive study of an important and well-known property of combinatorial sequences - unimodality. We shall have a look at a sample of classical results on unimodality and related properties, and then proceed to understand the unimodality of the Gaussian polynomial in more detail. We will look at an outline of O'Hara's proof of the unimodality of the Gaussian polynomial. In order to grasp the challenge of the problem of obtaining an injective proof of the unimodality of the Gaussian polynomial (which is still an open question), we make several attempts and understand where these attempts fail.

This is an English translation of Henri Joris' article "Le chasseur perdu dans la forêt (Un problème de géométrie plane)" that appeared in Elemente der Mathematik v. 35 (1980) n. 1, 1--14. Given a point P and a line L in the plane, what is the shortest search path to find L , given its distance but not its direction from P ? The shortest search path was described by Isbell (1957), but a complete and detailed proof was not published until Joris (1980). I am thankful to Natalya Pluzhnikov for her dedicated work and to the Swiss Mathematical Society for permission to post this translation on the arXiv.

This is an English translation of V. A. Zalgaller's article "On a problem of the shortest space curve of unit width" that appeared in Matematicheskaya Fizika, Analiz, Geometriya v. 1 (1994) n. 3--4, 454--461. We refer interested readers to Ghomi (2018) for up-to-date discussion; the curve L 3 of length 3.9215... in Zalgaller (1994) still appears to be shortest, whereas the closed curve L 5 is provably not of unit width. I am thankful to Natalya Pluzhnikov for her dedicated work and to the B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine for permission to post this translation on the arXiv.

In this article, using only elementary knowledge of complex numbers, we sketch a proof of the celebrated Abel--Ruffini theorem, which states that the general solution to an algebraic equation of degree five or more cannot be written using radicals, that is, using its coefficients and arithmetic operations +,−,×,÷, and √ . The present article is written purposely with concise and pedagogical terms and dedicated to students and researchers not familiar with Galois theory, or even group theory in general, which are the usual tools used to prove this remarkable theorem. In particular, the proof is self-contained and gives some insight as to why formulae exist for equations of degree four or less (and how they are constructed), and why they do not for degree five or more.

The description of the image of cubic function f(x)= x 3 +x over finite field F 2 n was stated as a problem in the NSUCrypto olympiad in 2017. This problem was marked by organizers as <<unsolved>>. In this work we propose the full solution of this problem.

Herman Melville's novel Moby-Dick contains a surprising number of mathematical allusions. In this article we explore some of these, as well as discussing the questions that naturally follow: why did Melville choose to use so much mathematical imagery? How did Melville come to acquire the level of mathematical knowledge shown in the novel? And is it commensurate with the general level of mathematical literacy at that time?

Alan Baker, Fields Medallist, died on 4th February 2018 in Cambridge England after a severe stroke a few days earlier. In 1970 he was awarded the Fields Medal at the International Congress in Nice on the basis of his outstanding work on linear forms in logarithms and its consequences. Since then he received many honours including the prestigious Adams Prize of Cambridge University, the election to the Royal Society (1973) and the Academia Europaea; and he was made an honorary fellow of University College London, a foreign fellow of the Indian Academy of Science, a foreign fellow of the National Academy of Sciences India, an honorary member of the Hungarian Academy of Sciences, and a fellow of the American Mathematical Society. In this article we survey Alan Baker's achievements.

The book "A Course in Constructive Algebra" (1988) shows the way of understanding classical basic algebra in a constructive style similar to Bishop's Constructive Mathematics. Classical theorems are revisited, with a new flavour, and become much more precise. We are often surprised to find proofs that are simpler and more elegant than the usual ones. In fact, when one cannot use magic tools as the law of excluded middle, it is necessary to understand what is the true content of a classical proof. Also, usual shortcuts allowed in classical proofs introduce sometimes useless detours. In order to understand clearly a problem, prescience may be a handicap.