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Euler calculus is based on integrating simple functions with respect to the Euler characteristic. This paper makes the case for extending Euler calculus to continuous integrands by integrating with respect to (Gaussian) curvature. This…

Starting from the well-known and elementary problem of inscribing the rectangle of the greatest area in an ellipse, we look at possible, gradually more and more complicated variants of this problem. Our goal is to demonstrate to an average…

I discuss some problems related to extreme mathematical realism, focusing on a recently proposed "shut-up-and-calculate" approach to physics (arXiv:0704.0646, arXiv:0709.4024). I offer arguments for a moderate alternative, the essence of…

This paper concerns the explicit treatment of substitutions in the lambda calculus. One of its contributions is the simplification and rationalization of the suspension calculus that embodies such a treatment. The earlier version of this…

The goal of this paper is to experiment new math concepts and theories, especially if they run counter to the classical ones. To prove that contradiction is not a catastrophe, and to learn to handle it in an (un)usual way. To transform the…

We present a replacement for traditional Riemann integrals in undergraduate calculus, which supplements naive precalculus and at the same time opens a way to more sophisticated theories such as Lebesgue integration.

The work presents two examples of simple mathematical formulas which are natural nonlinear modifications (one being a generalization) of Gielis' formula. These formulas involve a comparable number of parameters and provide non-Platonic…

This translation has been withdrawn due to certain imperfections and mistakes, which are corrected in the version uploaded at The Euler Archive (see E65 at http://www.eulerarchive.org/)

The Compositional Integral is defined, formally constructed, and discussed. A direct generalization of Riemann's construction of the integral; it is intended as an alternative way of looking at First Order Differential Equations. This brief…

This article provides a simple geometric interpretation of the quadratic formula. The geometry helps to demystify the formula's complex appearance and casts it into a much simpler existence, thus potentially benefits early algebra students.

We give a simple rigourous treatment of the classical results of the abelian sandpile model. Although we treat results which are well-known in the physics literature, in many cases we did not find complete proofs in the literature. The…

We derive an efficient algorithm to find solutions to Euler's concordant form problem and rational points on elliptic curves associated with this problem.

The goal of this notice is to present a proof of Bachet's conjecture based exclusively on the fundamental theorem of arithmetic. The novelty of this proof consists in its introduction of a partial order on rational integers through the…

This is a purely pedagogical paper with no new results. The goal of the paper is to give a fairly self-contained introduction to Judea Pearl's do-calculus, including proofs of his 3 rules.

A very simple closed-form formula for Sheppard's corrections is recovered by means of the classical umbral calculus. By means of this symbolic method, a more general closed-form formula for discrete parent distributions is provided and the…

Some thoughts are presented on the inter-relation between beauty and truth in science in general and theoretical physics in particular. Some conjectural procedures that can be used to create new ideas, concepts and results are illustrated…

In this paper we propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in $\mathbb{R}^n$ with $n\geq 2$. We give a reformulation of the Euler equations as a differential…

The classical quadratic formula and some of its lesser known variants for solving the quadratic equation are reviewed. Then, a new formula for the roots of a quadratic polynomial is presented.

We consider the problem of computing the Euler characteristic of an abstract simplicial complex given by its vertices and facets. We show that this problem is #P-complete and present two new practical algorithms for computing Euler…

In this note some philosophical thoughts and observations about mathematics are expressed, arranged as challenges to some common claims.