Career opportunities for PhDs in the mathematical sciences have never been better. Traditional faculty positions in mathematics departments in colleges and universities range from all teaching to combined teaching and research responsibilities. Beyon
d those, a wide array of careers has now opened up to freshly minted graduates, in academics, industry, business, and government. It is well-understood that these all require somewhat different preparations for PhDs to be competitive. This commentary compares and contrasts mathematics graduate programs with Ph.D. programs in the life and biomedical sciences, which are structured in a way that allows considerable customization around each students career goals. While these programs may not be appropriate templates for the mathematical sciences, they have some features that might be informative. This commentary is intended to add perspective to the ongoing discussion around PhD training in the mathematical sciences. It also provides some concrete proposals for changes.
In this paper will be proved the existence of a formula to reduce a tetration of base $2^{k}$ and $5^{k}$ $mod 10^{n}$. Indeed, last digits of a tetration are the same starting from a certain hyper-exponent; In order to compute the last digits of tho
se expressions we reduce them $mod 10^{n}$. Lots of different formulas will be derived, for different cases of $k$ (where $k$ is the exponent of the base of the tetration). This kind of operation is fascinating, because the tetration grows very fast. But using these formulas we can actually have informations about the last digits of those expressions. Its possible to use these results on a software in order to reduce tetrations $mod 10^{n}$ faster.
Crossword puzzles lend themselves to mathematical inquiry. Several authors have already described the arrangement of crossword grids and associated combinatorics of answer numbers. In this paper, we present a new graph-theoretic representation of cro
ssword puzzle grid designs and describe the mathematical conditions placed on these graphs by well-known crossword construction conventions.
Quantitative methods and mathematical modeling are playing an increasingly important role across disciplines. As a result, interdisciplinary mathematics courses are increasing in popularity. However, teaching such courses at an advanced level can be
challenging. Students often arrive with different mathematical backgrounds, different interests, and divergent reasons for wanting to learn the material. Here we describe a course on stochastic processes in biology, delivered between September and December 2020 to a mixed audience of mathematicians and biologists. In addition to traditional lectures and homeworks, we incorporated a series of weekly computational challenges into the course. These challenges served to familiarize students with the main modeling concepts, and provide them with an introduction on how to implement them in a research-like setting. In order to account for the different academic backgrounds of the students, they worked on the challenges in small groups, and presented their results and code in a dedicated discussion class each week. We discuss our experience designing and implementing an element of problem-based learning in an applied mathematics course through computational challenges. We also discuss feedback from students, and describe the content of the challenges presented in the course. We provide all materials, along with example code for a number of challenges.
The authors have been using a largely algebraic form of ``computational discovery in various undergraduate classes at their respective institutions for some decades now to teach pure mathematics, applied mathematics, and computational mathematics. Th
is paper describes what we mean by ``computational discovery, what good it does for the students, and some specific techniques that we used.
Jacobis results on the computation of the order and of the normal forms of a differential system are translated in the formalism of differential algebra. In the quasi-regular case, we give complete proofs according to Jacobis arguments. The main resu
lt is {it Jacobis bound}, still conjectural in the general case: the order of a differential system $P_{1}, ldots, P_{n}$ is not greater than the maximum $cO$ of the sums $sum_{i=1}^{n} a_{i,sigma(i)}$, for all permutations $sigma$ of the indices, where $a_{i,j}:={rm ord}_{x_{j}}P_{i}$, emph{viz.} the emph{tropical determinant of the matrix $(a_{i,j})$}. The order is precisely equal to $cO$ iff Jacobis emph{truncated determinant} does not vanish. Jacobi also gave a polynomial time algorithm to compute $cO$, similar to Kuhns index{Hungarian method}``Hungarian method and some variants of shortest path algorithms, related to the computation of integers $ell_{i}$ such that a normal form may be obtained, in the generic case, by differentiating $ell_{i}$ times equation $P_{i}$. Fundamental results about changes of orderings and the various normal forms a system may have, including differential resolvents, are also provided.
The purpose of this note is to honour memory of Ubiratan DAmbrosio, a Brazilian mathematics educator, who passed away on May 12, 2021.
New understandings of the functioning of human brains engaged in mathematics raise interesting questions for mathematics educators. Novel lines of research are suggested by neuroscientific findings, and new light is shed on some longstanding issues in mathematics education.
Do you want to know what an anti-chiece Latin square is? Or what a non-consecutive toroidal modular Latin square is? We invented a ton of new types of Latin squares, some inspired by existing Sudoku variations. We cant wait to introduce them to you a
nd answer important questions, such as: do they even exist? If so, under what conditions? What are some of their interesting properties? And how do we generate them?
Nina Uraltseva has made lasting contributions to mathematics with her pioneering work in various directions in analysis and PDEs and the development of elegant and sophisticated analytical techniques. She is most renowned for her early work on linear
and quasilinear equations of elliptic and parabolic type in collaboration with Olga Ladyzhenskaya, which is the category of classics, but her contributions to the other areas such as degenerate and geometric equations, variational inequalities, and free boundaries are equally deep and significant. In this article, we give an overview of Nina Uraltsevas work with some details on selected results.
