相关论文: Euler and magic squares (De quadratis magicis)
Mathematical billiards is much like the real game: a point mass, representing the ball, rolls in a straight line on a (perfectly friction-less) table, striking the sides according to the law of reflection. A billiard trajectory is then…
This article, showing that almost all objects in the title are asymmetric, is re-typed from a manuscript I wrote somewhere around 1980 (after the papers of Bang and Friedland on the permanent conjecture but before those of Egorychev and…
The smallest Euler brick, discovered by Paul Halcke, has edges $(177, 44, 240) $ and face diagonals $(125, 267, 244 ) $, generated by the primitive Pythagorean triple $ (3, 4, 5) $. Let $ (u,v,w) $ primitive Pythagorean triple, Sounderson…
When people mention the number theoretical achievements in Ancient China, the famous Chinese Remainder Theorem always springs to mind. But, two more of them--the concept of primes and the algorithm for counting the greatest common divisor,…
The purpose of the work is to furnish a complete study of a discrete and special function, discovered by the author and named with the Arabian letter "SHIN" {The letter SHIN is the thirteenth letter of the Arabian alphabet}. It includes…
Hourglass is an equal-area pseudocylindrical map projection developed by John P. Snyder in mid 1940s. It was never published in a detailed way by its author, and only a couple of references exist in literature since 1991, both of them…
This article -- written in honour of Juergen Ehlers -- consists of two different, though interlocking parts: Section 1 describes my 54 years of perpetual experiences and exchanges with him, both science and episodes, whilst section 2…
We propose ten puzzles in particle theory: quark, asymmetries of quantum numbers in the Standard Model, point particle and interaction-distance, mystic Higgs mechanism, possible violation of the Pauli Exclusion Principle and some basic…
The geometry of three-dimensional space guides the search for a better model than the blackhole with its unwelcome singularity. An elementary construction produces on the 4-manifold of 2-spheres in a Riemannian 3-space a space-time metric…
The classical Euler's problem on stationary configurations of elastic rod with fixed endpoints and tangents at the endpoints is considered as a left-invariant optimal control problem on the group of motions of a two-dimensional plane…
In this short note we shall give connection between the most perfect "Khajuraho" magic square of order 4x4 discovered in 10th century and the "Lo-Shu" magic square of order 3x3 with the day October 1, 2010, i.e., 01.10.2010. The day has…
Extra dimensions can be utilized to simplify problems in classical mechanics, offering new insights. Here we show a simple example of how the motion of a test particle under the influence of an inverse-quadratic potential in 1D is…
Diaconis and Gamburd computed moments of secular coefficients in the CUE ensemble. We use the characteristic map to give a new combinatorial proof of their result. We also extend their computation to moments of traces of symmetric powers,…
We introduce Magic Gems, a geometric representation of magic squares as three-dimensional polyhedra. By mapping an n times n magic square onto a centered coordinate grid with cell values as vertical displacements, we construct a point cloud…
The purpose of this essay is to trace the historical development of geometry while focusing on how we acquired mathematical tools for describing the "shape of the universe." More specifically, our aim is to consider, without a claim to…
From the Rhind Papyrus and other extant sources, we know that the ancient Egyptians were very iterested in expressing a given fraction into a sum of unit fractions, that is fractions whose numerators are equal to 1. One of the problems that…
The global existence of weak solutions for the three-dimensional axisymmetric Euler-$\alpha$ (also known as Lagrangian-averaged Euler-$\alpha$) equations, without swirl, is established, whenever the initial unfiltered velocity $v_0$…
For over 25 years, a solution has existed to Einstein's vacuum equation that describes a space-time with two Kerr black holes. First formulated by Kramer and Neugebauer (KN) in 1980 [1], this solution has been extensively researched by…
Can you stretch and reform a curve such that it fills a square completely? This question dates back to 18th century, the origin of space-filling curves. It was proved affirmatively by many great mathematicians. In this document, we…
Prime Numbers clearly accumulate on defined spiral graphs,which run through the Square Root Spiral. These spiral graphs can be assigned to different spiral-systems, in which all spiral-graphs have the same direction of rotation and the same…