Mathematics

In this short note, we discuss how the optimality conditions for the problem of minimizing a multivariate function subject to equality constraints have been dealt with in undergraduate Calculus. We are particularly interested in the 2 or 3-dimensional cases, which are the most common cases in Calculus courses. Besides giving sufficient conditions to a critical point to be a local minimizer, we also present and discuss counterexamples to some statements encountered in the undergraduate literature on Lagrange Multipliers, such as `among the critical points, the ones which have the smallest image (under the function) are minimizers' or `a single critical point (which is a local minimizer) is a global minimizer'.

Service sports include two-player contests such as volleyball, badminton, and squash. We analyze four rules, including the Standard Rule (SR), in which a player continues to serve until he or she loses. The Catch-Up Rule (CR) gives the serve to the player who has lost the previous point - as opposed to the player who won the previous point, as under SR. We also consider two Trailing Rules that make the server the player who trails in total score. Surprisingly, compared with SR, only CR gives the players the same probability of winning a game while increasing its expected length, thereby making it more competitive and exciting to watch. Unlike one of the Trailing Rules, CR is strategy-proof. By contrast, the rules of tennis fix who serves and when; its tiebreaker, however, keeps play competitive by being fair - not favoring either the player who serves first or who serves second.

Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique, Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research and teaching. Cauchy applied infinitesimals in an 1826 work in differential geometry where infinitesimals are used neither as variable quantities nor as sequences but rather as numbers. He also applied infinitesimals in an 1832 article on integral geometry, similarly as numbers. We explore these and other applications of Cauchy's infinitesimals as used in his textbooks and research articles. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis and geometry. We demonstrate the viability of Cauchy's infinitesimal techniques in fields as diverse as geometric probability, differential geometry, elasticity, Dirac delta functions, continuity and convergence. Keywords: Cauchy--Crofton formula; center of curvature; continuity; infinitesimals; integral geometry; limite; standard part; de Prony; Poisson

This paper discusses our experiences and challenges in teaching advanced undergraduate Real Analysis classes for Mathematics Education students at the University of PGRI (Persatuan Guru Republik Indonesia, Indonesian Teachers Association) Palembang, South Sumatra, Indonesia. We observe that the syllabus contains topics with a high level of difficulty for the students who are specialized in education and intend to teach mathematics at the secondary level. The conventional lecturing method is mainly implemented during the class, with some possible variations of the method, including the Texas method (also known as Moore's method) and the small group guided discovery method. In particular, the latter method has been implemented successfully for a Real Analysis class at Dartmouth College, New Hampshire by Dumitraşcu in 2006. Although it is a real challenge to apply a specific teaching method that will be able to accommodate a large number of students, the existing teaching activities can still be improved and a more effective method could be implemented in the future. Furthermore, the curriculum contents should be adapted for an audience in Mathematics Education to equip them for their future career as mathematics teachers. Any constructive suggestions are welcome for the improvement of our mathematics education system at the university as well as on the national scale.

The chaos game representation (CGR) is an interesting method to visualize one-dimensional sequences. In this paper, we show how to construct a chaos game representation. The applications mentioned here are biological, in which CGR was able to uncover patterns in DNA or proteins that were previously unknown. We also show how CGR might be introduced in the classroom, either in a modelling course or in a dynamical systems course. Some sequences that are tested are taken from the Online Encyclopedia of Integer Sequences, and others are taken from sequences that arose mainly from a course in experimental mathematics.

Dessin d'enfants (French for children's drawings) serve as a unique standpoint of studying classical complex analysis under the lens of combinatorial constructs. A thorough development of the background of this theory is developed with an emphasis on the relationship of monodromy to Dessins, which serve as a pathway to the Riemann Hilbert problem. This paper investigates representations of Dessins by permutations, the connection of Dessins to a particular class of Riemann surfaces established by Belyi's theorem and how these combinatorial objects provide another perspective of solving the discrete Riemann-Hilbert problem.

This is an English translation of the monograph "Die Klassenkörper der komplexen Multiplikation" by Max Deuring, published in 1958 as Enzyklopädie der Mathematischen Wissenschaften, Band I-2, Heft 10, Teil II by B.G. Teuber Verlagsgesellschaft, Stuttgart. It gives a systematic exposition of the analytic method in the theory of complex multiplication -- the interrelations between elliptic functions, modular functions and algebraic numbers. This translation was done as an exercise for learning the German language. The mathematical notation and terminology are kept as closely as possible to those given in the German original work. No attempt has been made to modernize the exposition, other than rectifying a (very) few obviously-correctable typographical errors.

This is a historical introduction to the theory of Stirling numbers of the second kind S(n,k) from the point of view of analysis. We tell the story of their birth in the book of James Stirling (1730) and show how they mature in the works of Johann Grunert (1843). We demonstrate their usefulness in several differentiation formulas. The reader can also see the connection to Bernoulli numbers, to Euler polynomials and to power sums.

Suppose there are n harmonic pencils of lines given in the plane. We are interested in the question whether certain triples of these lines are concurrent or if triples of intersection points of these lines are collinear, provided that we impose suitable conditions on the initial harmonic pencils. Such conditions can be that certain of the given lines coincide, are concurrent or that certain intersection points are collinear. The study of these questions for n=2,3,4 sheds light on some well known affine configurations and provides new results in the projective setting. As applications, we will formulate generalizations or stronger versions of the theorems of Pappus, Desargues, Ceva and Menelaos. Notably, the generalized theorems of Ceva and Menelaos suggest a new way to generalize the terms collinearity and concurrency.

These notes describe some results on dice comparisons when changing the numbers on the faces while the sum of all the face stay the same.