历史与综述
Given fixed distinct points $A, B, C, D$, we examine properties of the locus of points $X$ for which $(XA, XC)$, $(XB, XD)$ are isogonal. This locus is a cubic curve circumscribing $ABCD$. We characterize all possible such cubics $\mathcal…
The multi-layered nature of societal developments poses a challenge in the teaching and learning of history of mathematics. As an attempt to tackle this challenge we experimented with the use of Knowledge-Maps. The focus of our interest…
The main focus of this paper is on models of quartic surfaces, especially so-called complex surfaces. These are special fourth-degree surfaces that Julius Pl\"ucker introduced in the 1860s for visualizing the local structure of a quadratic…
This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the…
A summary of an experimental course on algebraic curves is given that was held for young learners at age 11. The course was a part of Epsilon camp, a program designed for very gifted students who have already demonstrated high interest in…
The description of the image of cubic function $f(x) = x^3+x$ over finite field $F_{2^n}$ was stated as a problem in the NSUCrypto olympiad in 2017. This problem was marked by organizers as <<unsolved>>. In this work we propose the full…
The Dedekind tessellation -- the regular tessellation of the upper half-plane by the Mobius action of the modular group -- is usually viewed as a system of ideal triangles. We change the focus from triangles to circles and give their…
We discuss fun problems, vaguely related to notions and theorems of a course in differential geometry. This paper can be regarded as a weekend "treasure chest" supplementing the course weekday lecture notes. The problems and solutions are…
In this article we generalize the classic "farm pen" optimization problem from a first course in calculus in a handful of different ways. We describe the solution to an $n$-dimensional rectangular variant, and then study the situation when…
In order to explore tonality outside of the `Pythagorean' paradigm of integer ratios, Robert Schneider introduced a musical scale based on the logarithm function. We seek to refine Schneider's scale so that the difference tones generated by…
Tilings are around us everywhere, and our curiosity draws us to study their properties. A tiling is a way of arranging pieces on a board, such that there is no space left uncovered, nor any space covered by more than one tile. In…
This article is an introduction to Mordell-Weil theorem. In this article, I introduced some basic properties about ellptic curves and proved the theorem in two different ways.
Leonhard Euler likely developed his summation formula in 1732, and soon used it to estimate the sum of the reciprocal squares to 14 digits --- a value mathematicians had been competing to determine since Leibniz's astonishing discovery that…
In this note, we use the concept of a polynomial ring to give an elementary proof to Cayley-Hamilton Theorem. We also give an elementary proof to Birkhoff theorem on Bi-stochastic matrices.
This chapter proposes a view from inside the DAD, starting from determining some essential resources missing of DAD, to proposing 10 programs of research/development for developing it. It could be considered as a follow-up of Chapter 1,…
We prove the twin prime conjecture and the generalized conjectures of Kronecker and Polignac. Key to the proofs is a new theoretical sieve that combines two concepts that go back to Eratosthenes: the 'sieve' filtering a finite set of…
Mar\'ia Andresa Casamayor de la Coma, born in Zaragoza, is known as the first woman who published a scientific book in Spain. In this paper we provide answers to several of the most important questions about her unknown biography such as…
A recurrence relations for sums of powers of complex functions can be written as a system of linear equation AX=B. Using properties of determinant and Cramer's rule for solving systems of linear equation, this paper presents an absolutely…
Configuration spaces form a rich class of topological objects which are not usually presented to an undergraduate audience. Our aim is to present configuration spaces in a manner accessible to the advanced undergraduate. We begin with a…
Remarks on the life and work of Paul Erdos.