历史与综述
This Master's Thesis analyzes thoroughly the topic of the Mathematics Education in Russia and Mathematical Circles. It deals with the historical context of the Mathematics Education in Russia in the XVIIIth, XIXth and XXth centuries; the…
Supplements to Mehta & Normand (1997) are given, with regard to integrals involving Euclidean distances between n+1 random points in d-dimensional space, each visited once.
We will verify that the fair copy "Ueber die Anzahl der Primzahlen unter einer gegebenen Gr\"osse" found in Riemann's Nachlass is not in Riemann's hand. Further, we will show that this paper was written by Alfred Clebsch and that it was…
The conditions determining that two triangles are congruent play a basic role in planimetry. By comparing not congruent triangles with respect to given sets of corresponding elements it is important to discover if they have any common…
Dice odds in the board game RISK were first investigated by Tan, fixed by Osbourne, and extended by Blatt. We generalized dice odds further, varying the number of sides and the number of dice used in a single battle. We show that the…
A three-polar, cf. T. Gregor, J. Halu\v{s}ka, Lexicographical ordering and field operations in the complex plane. Stud. Mat. 41(2014), 123--133., $HSV-RGB$ Colour space $\triangle$ was introduced and studied. It was equipped with operations…
We present English translation of the classical article of Hermann Amadeus Schwarz (1843--1921) "Proof of the theorem that a surface area of a ball is smaller than of any other body of the same volume" which was published in 1884, in…
We point out that the proof of irrationality of $\pi$ by Niven can be modified to a proof by contraposition. As a warm-up, we also give a proof of irrationality of $\sqrt{2}$ and $\sqrt{3}$ in a similar way.
When studying the history of mathematical symbols, one finds that the development of mathematical symbols in China is a significant piece of Chinese history; however, between the beginning of mathematics and modern day mathematics in China,…
We will study the historical pathway of the emergence of Tessarines or Bicomplex numbers, from their origin as "imaginary" solutions of irrational equations, to their insertion in the context of study of the algebras of hypercomplex…
Explicit area expressions are known for a special case, due to Tao & Wu (1987), and lead to calculation of integrals in applied probability.
It is well known that $\sum_{p\le n} 1/p =\ln(\ln(n)) + O(1)$ where $p$ goes over the primes. We give several known proofs of this. We first present a a proof that $\ge \ln(\ln(n)) + O(1)$. This is based on Euler's proof that $\sum_p 1/p$…
Theorem. An irreducible cubic polynomial with rational coefficients has a root in a one step radical extension of Q if and only if the discriminate is a square of a rational number. Theorem. An irreducible polynomial x^4+px^2+qx+s with…
We present a detailed and elementary construction of the real numbers from the rational numbers a la Bourbaki. The real numbers are defined to be the set of all minimal Cauchy filters in $\mathbb{Q}$ (where the Cauchy condition is defined…
Reproducing kernel Hilbert spaces are elucidated without assuming prior familiarity with Hilbert spaces. Compared with extant pedagogic material, greater care is placed on motivating the definition of reproducing kernel Hilbert spaces and…
We present a new English translation of L.E.J. Brouwer's paper `De onbetrouwbaarheid der logische principes' (The unreliability of the logical principles) of 1908, together with a philosophical and historical introduction. In this paper…
Pierre van Hiele (1909-2010) suggested, both in 1957 and later repeatedly, wide application for the Van Hiele levels in insight, both for more disciplines and for different subjects in mathematics. David Tall (2013) suggests that Van Hiele…
Numbers are often used to define more complicated numbers. For example, two integers are used to define a rational number and two reals are used to define a complex number. It might be expected that an irrational power of an irrational…
Most democratic countries use election methods to transform election results into whole numbers which usually give the number of seats in a legislative body the parties obtained. Which election method does this best can be specified by…
We review the role of mathematics from a historical and a conceptual perspective in the light of modern data science.