历史与综述
Liouville's 1853 paper, in which he derived in closed form the general local solution of equation $u_{z\bar z}=\exp(u)$, is one of the few papers from the 19th century that 21st century mathematicians routinely quote as motivation for their…
In this paper, we first give a simple combinatorial proof of Tepper's identity. Then, as a by product of this interesting identity we present another proof of the well-known Wilson's identity in number theory. Finally, we obtain a…
Chapters 1 to 4 are the lecture notes of my course "Real Algebraic Geometry I" from the winter term 2020/2021. Chapters 5 to 8 are the lecture notes of its continuation "Real Algebraic Geometry II" from the summer term 2021. Chapters 9 and…
Legendre published the first attempted proof of the law of Quadratic Reciprocity. But, in its final form (1797), it had a gap. Some 125 years later Herman Teege published the first rigorous proof of the unproven hypothesis which formed that…
The diagonals of a quadrilateral form four component triangles (in two ways). For each of various shaped quadrilaterals, we examine 1000 triangle centers located in these four component triangles. Using a computer, we determine when the…
Pranata mangsa is a local knowledge based on a calendrical system utilized historically by Javanese peasant farmers in conducting their agricultural activities in cultivating plants and vegetation. It is also used by fishermen to guide them…
In this article, using only elementary knowledge of complex numbers, we sketch a proof of the celebrated Abel--Ruffini theorem, which states that the general solution to an algebraic equation of degree five or more cannot be written using…
We show that the four best-known substitutions used to integrate secant rely on different rational parametrizations of conic sections, coming from stereographic projections. We also investigate other possible explanations for the four…
In this missive, the author goes about ruining an online word guessing game called Wordle, by generally trying to mathify the whole thing. On the other hand, one might consider that the author is trying to spice up mathematics by applying…
In this didactic note, we describe a procedure to derive successive approximations of $\pi$ using Euler Beta functions. It is an interesting exercise for undergraduate students, since it involves polynomial roots, integral calculations,…
From antiquity the conceptual perception of space changed painfully and at a relatively slow pace. It went through mythological descriptions, religious beliefs, metaphysical worldviews and cosmological models with a mechanistic structure,…
This article proposes an artistic approach to increase and enrich the understanding of Julia Sets. This approach includes the mathematical, the playful, the artistic and the computational dimensions. It is argued that these four dimensions…
We discuss the idea that computers might soon help mathematicians to prove theorems in areas where they have not previously been useful. Furthermore we argue that these same computer tools will also help us in the communication and teaching…
We provide a short proof, not utilizing complex numbers, for the solution set of homogeneous second order linear differential equations with constant coefficients.
In this article, we reveal how Benjamin Franklin constructed his second $8 \times 8$ magic square. We also construct two new $8 \times 8$ Franklin squares.
We discuss some seemingly unrelated observations on integers, whose close or farther away neighbors show a complex of combinatorial, ordering, arithmetical or probabilistic properties, emphasizing puzzlement in more common expectations.
This paper presents a mathematical method of playing the puzzle game Wordle. In Wordle, the player has six tries to guess a secret word. After each guess the player is told how their guess compares to the secret word. With the available…
As educational systems move from onsite to online due to the COVID-19 pandemic, teachers face the difficulty of designing online examination formats which minimise opportunities for dishonesty. In this paper, we expose our design of such a…
Both metamathematics and physics are posited to emerge from samplings by observers of the unique ruliad structure that corresponds to the entangled limit of all possible computations. The possibility of higher-level mathematics accessible…
Cem Tezer was a fastidious, meticulous, highly idiosyncratic and versatile scientist. Without him Turkish community of mathematics would be incomplete. Our sense of gratitude for his work in various areas of mathematics, history of…