Mathematics

The use of Cauchy's method to prove Euler's well-known formula is an object of many controversies. The purpose of this paper is to prove that Cauchy's method applies for convex polyhedra and not only for them, but also for surfaces such as the torus, the projective plane, the Klein bottle and the pinched torus.

This evocative essay focuses on some landmarks that led the author to the study of principal curvature configurations on surfaces in R 3 , their structural stability and generic properties. The starting point was an encounter with the book of D. Struik and the reading of the references to the works of Euler, Monge and Darboux found there. The concatenation of these references with the work of Peixoto, 1962, on differential equations on surfaces, was a crucial second step. The circumstances of the convergence toward the theorems of Gutiérrez and Sotomayor, 1982 - 1983, are recounted here. The above 1982 - 1983 theorems are pointed out as the first encounter between the line of thought disclosed from the works of Monge, 1796, Dupin, 1815, and Darboux, 1896, with that transpiring from the achievements of Poincaré, 1881, Andronov - Pontrjagin, 1937, and Peixoto, 1962. Some mathematical developments sprouting from the 1982 - 1983 works are mentioned on the final section of this essay.

We looked at a method for estimating the complexity measure of game tree size (the number of legal games). It seems effective for a number of children's games such as Tic-Tac-Toe, Connect Four and Othello.

A perpetual calendar, a calendar designed to find out the day of the week for a given date, employs a rich arithmetical calculation using congruence. Zeller's congruence is a well-known algorithm to calculate the day of the week for any Julian or Gregorian calendar date. Another rather infamous perpetual calendar has been used for nearly four centuries among Javanese people in Indonesia. This Javanese calendar combines the Saka Hindu, lunar Islamic, and western Gregorian calendars. In addition to the regular seven-day, lunar month, and lunar year cycles, it also contains five-day pasaran, 35-day wetonan, 210-day pawukon, octo-year windu, and 120-year kurup cycles. The Javanese calendar is used for cultural and spiritual purposes, including a decision to tie the knot among couples. In this chapter, we will explore the relationship between mathematics and the culture of Javanese people and how they use their calendar and the arithmetic aspect of it in their daily lives. We also propose an unprecedented congruence formula to compute the pasaran day. We hope that this excursion provides an insightful idea that can be adopted for teaching and learning of congruence in number theory.

The Bernoulli function B(s,v)=−sζ(1−s,v) interpolates the Bernoulli numbers but can be introduced independently of the zeta function. The point of departure is a modification of the Stieltjes constants based on an integral representation given by J. Jansen. The functional equation of B(s,v) and its relation to the Riemann ζ and ξ function is explored. Classical results of Hadamard, Worpitzky, and Hasse are recast in terms of B(s,v). The extended Bernoulli function defines the Bernoulli numbers for odd indices harmonizing with rational numbers studied by Euler in 1735 and which are the bridge to the Euler and André numbers. Interpolating functions for both the signed and the unsigned case are given. The Swiss knife polynomials let the integer sequences of the Euler-Bernoulli family calculate easily.

An overview of the history of projective representations (= spin representations) of groups, preceded by the prehistory of studies on the theory of quaternion due to Rodrigues and Hamilton. Beginning with Schur, we cover many mathematicians until today, and also physicists Pauli and Dirac. This is a self translation of Appendix A of my book "Introduction to the theory of projective representations of groups" in Japanese, 2018, Sugakushobo, and may serve as an introduction to our paper arXiv: 1804.06063 [math.RT] which will appear in Kyoto J. Math.

In this paper we initiate some investigations on MathSciNet database. For many mathematicians this website is used on a regular basis, but surprisingly except for the information provided by MathSciNet itself, there exist almost no independent investigations or independent statistics on this database. This current research has been triggered by a rumor: do international collaborations increase the number of citations of an academic work in mathematics? We use MathSciNet for providing some information about this rumor, and more generally pave the way for further investigations on or with MathSciNet. Keywords: MathSciNet, tree-based methods, international collaborations

The theory uses methods and language of linear algebra to study nonlinear spaces. These techniques can be used particularly to describe analytic geometry of non-linear elliptic, hyperbolic, De Sitter and Anti de Sitter spaces. The main innovation of elaborated theory is space parameterization by introduction of space signature. This parameterization allows studying of different homogeneous spaces in one global framework. When the parameters are used as variables in definitions, axioms, equations, theorems, proofs, all these have exactly the same form that describes the reality of all homogeneous spaces simultaneously. When it is necessary to describe some space particularities or to see the difference between two concrete spaces, the concrete values can be put in parameters of each definition, axiom, equation, theorem and proof.

A simple way is shown to construct the length π from the unit length with 4 digits accuracy.

Robust preparation of future secondary mathematics teachers requires attention to the acquisition of mathematical knowledge for teaching. Many future teachers learn mathematics content primarily through mathematics major courses that are taught by mathematicians who do not specialize in teacher preparation. How can mathematics education researchers assist mathematicians in making explicit connections between the content of undergraduate mathematics courses and the content of secondary mathematics? We present an articulation of five types of connections that can be used in secondary mathematics teacher preparation and give examples of question prompts that mathematicians can use as applications of teaching secondary mathematics in undergraduate mathematics courses.