Mathematics

Some time ago Wastlund reformulated the Basel problem in terms of a physical system using the proportionality of the apparent brightness of a star to the inverse square of its distance. Inspired by this approach, we give another physical interpretation which, in our opinion, is simpler, natural enough, and very Eulerian in its spirit.

Writing assignments in any mathematics course always present several challenges, particularly in lower-level classes where the students are not expecting to write more than a few words at a time. Developed based on strategies from several sources, the two small writing assignments included in this paper represent a gentle introduction to the writing of mathematics and can be utilized in a variety of low-to-middle level courses in a mathematics major.

We introduce a methodology to analyze citation metrics across fields of Mathematics. We use this methodology to collect and analyze the MathSciNet profiles of Full Professors of Mathematics at all 131 R1, research oriented US universities. The data recorded was citations, field, and time since first publication. We perform basic analysis and provide a ranking of US math departments, based on age corrected and field adjusted citations.

Given a triangle, what is the equation of the line which bisects its area and has a given slope? The set of all lines bisecting the area of a triangle has been elegantly determined as a certain 'deltoid' envelope and this gives an indirect method of solution. We find that vector algebra allows the equation to be written down rather directly and neatly.

In ancient times, China made great contributions to world civilization and in particular to mathematics. However, modern sciences including mathematics came to China rather too late. The first Chinese university was founded in 1895. The first mathematics department in China was formally opened at the university only in 1913. At the beginning of the twentieth century, some Chinese went to Europe, the United States of America and Japan for higher education in modern mathematics and returned to China as the pioneer generation. They created mathematics departments at the Chinese universities and sowed the seeds of modern mathematics in China. In 1930s, when a dozen of Chinese universities already had mathematics departments, several leading mathematicians from Europe and USA visited China, including Wilhelm Blaschke, George D. Birkhoff, William F. Osgood, Norbert Wiener and Jacques Hadamard. Their visits not only had profound impact on the mathematical development in China, but also became social events sometimes. This paper tells the history of their visits.

This is a summary and verification of an elementary note written by John Blissard in 1862 for the Messenger of Mathematics. A general method of discovering trigonometric series having a closed-form sum is explained and illustrated with examples.

During the early 1830's Bernard Bolzano, working in Prague, wrote a manuscript giving a foundational account of numbers and their properties. In the final section of his work he described what he called `infinite number expressions' and `measurable numbers'. This work was evidently an attempt to provide an improved proof of the sufficiency of the criterion usually known as the `Cauchy criterion' for the convergence of an infinite sequence. Bolzano had in fact published this criterion four years earlier than Cauchy who, in his work of 1821, made no attempt at a proof. Any such proof required the construction or definition of real numbers and this, in essence, was what Bolzano achieved in his work on measurable numbers. It therefore pre-dates the well-known constructions of Dedekind, Cantor and many others by several decades. Bolzano's manuscript was partially published in 1962 and more fully published in 1976. We give an account of measurable numbers, the properties Bolzano proved about them, and the controversial reception they have prompted since their publication.

Boris R. Vainberg was born on March 17, 1938, in Moscow. His father was a Lead Engineer in an aviation design institute. His mother was a homemaker. From early age, Boris was attracted to mathematics and spent much of his time at home and in school working through collections of practice problems for the Moscow Mathematical Olympiad. His first mathematical library consisted of the books he received as one of the prize-winners of these olympiads.

This paper focuses on two mathematical topics, namely continuous probability distributions (CPD) and integral calculus (IC). These two sectors that are linked by a formula are quite compartmented in teaching classes in France. The main objective is to study whether French students can mobilize the sector of IC to solve tasks in CPD and vice versa at the transition from high school to higher education. Applying the theoretical framework of the Anthropological Theory of the Didactic (ATD), we describe a reference epistemological model (REM) and use it to elaborate a questionnaire in order to test the capacity of students to bridge CPD and IC at the onset of university. The analysis of the data essentially confirms the compartmentalisation of CPD and IC.

How do we move a robot efficiently from one position to another? To answer this question, we need to understand its configuration space, a 'map' where we can find every possible position of the robot. Unfortunately, these maps are very large, they live in high dimensions, and they are very difficult to visualize. Fortunately, for some discrete robots they are CAT(0) cubical complexes, a family of spaces with favorable properties. In this case, using ideas from combinatorics and geometric group theory, we can construct a 'remote control' to navigate these complicated maps, and move the robots optimally. Along the way, we face larger ethical questions that we cannot ignore.