历史与综述

From a census of forty copies, we can distinguish three different editions of von Staudt's Geometrie der Lage: the first of 1847 and two undated ones from the 1870's.

How far can a stack of $n$ identical blocks be made to hang over the edge of a table? The question dates back to at least the middle of the 19th century and the answer to it was widely believed to be of order $\log n$. Recently, Paterson…

We consider an elementary discrete process which starts from purely random configuration and leads to well-ordered and stable state. Complete analytical solution to this problem is presented.

We observe that the elementary logistic differential equation dP/dt=(1-P/M)kP may be solved by first changing the variable to R=(M-P)/P. This reduces the logistic differential equation to the simple linear differential equation dR/dt=-kR,…

Translation of the Latin original, "Methodus generalis investigandi radices omnium aequationum per approximationem" (1776). E643 in the Enestrom index. Euler gives a series to find powers of roots of polynomials.

The divergence of the harmonic series is proved by direct comparison with a series whose nth partial sum telescopes to the natural logarithm of n. The key idea is to apply the classical inequality x>=log(1+x) (valid for x>-1) with x=1/k and…

By means of a graphical journey across the Mandelbrot set for the classic quadratic iterator $f(z):z^2+q$, we illustrate how connectivity breaks as the seed $z_0$ is no longer at the critical point of $f(z)$. Finally we suggest an attack to…

In this paper we provide a straightforward proof that if a pair of amicable numbers with different parity exists (one number odd and the other one even), then the odd amicable number must be a perfect square, while the even amicable number…

A review of Jaynes' posthumous book "Probability Theory--The Logic of Science." I use scientific and personality elements gathered from other papers by Jaynes to help throw light on the origins of Jaynes' life quest.

A formula for the radii and positions of four circles in the plane for an arbitrary linearly independent circle configuration is found. Among special cases is the recent extended Descartes Theorem on the Descartes configuration and an…

In his treatise on floating bodies Archimedes determines the equilibrium positions of a floating paraboloid segment, but only in the case when the basis of the segment is either completely outside of the fluid or completely submerged. Here…

Translation of the Latin original "Speculationes circa quasdam insignes proprietates numerorum" (1784). E564 in the Enestrom index. In this paper Euler talks about Farey sequences and proves some results about the phi function, the number…

This is an essay that considering the knowledge structure and language of a different nature, attempts to build on an explanation of the object of study and characteristics of the mathematical science. We end up with a learning cycle of…

In this paper we present three simple applications of probability and highlight and discuss their paradoxical flavour.

We show that the use of the main characteristics of the circle map leads naturally to establish a few statements on primes and pseudoprimes. In this way a Fermat's theorem on primes and some interesting properties of pseudoprimes are…

Though the truths of logic and pure mathematics are objective and independent of any contingent facts or laws of nature, our knowledge of these truths depends entirely on our knowledge of the laws of physics. Recent progress in the quantum…

This short note is devoted to the representative dynamics, which realizes a link between the theory of controlled systems and representation theory. Dynamical inverse problem of representation theory for controlled systems is considered: to…

This short article is devoted to the dynamics of controlled (and, therefore, open) systems. The internal forces, which appear only in the presence of external free controls and depend explicitely on them, are considered. Such interactive…

A methodical analysis of the research related to the article, ``Sur les groupes continus'', of Henri Poincar\'{e} reveals many historical misconceptions and inaccuracies regarding his contribution to Lie theory. A thorough reading of this…

We discuss the babylonian method of extracting the root square of a number, from the point of view of modern mathematics. We also speculate that the babylonian mathematics was rich enough for a generalization of this method, despite the…