历史与综述

We give a brief review of the work of Ramanujan on cranks that is found in the Lost Notebook. Recent work by Bruce Berndt and his coauthors have brought to light many interesting results of Ramanujan on cranks, which we highlight in this…

We briefly review the distinction between abstract groups and symmetry groups of objects, and discuss the question of which groups have appeared as the symmetry groups of physical objects. To our knowledge, the quaternion group (a beautiful…

The paper suggests a short survey of integration algorithms which evolved since 1982. These theorems and algorithms form discrete versions of the calculus theorems.

The paper suggests a slightly more rigorous justification to Wang et al.'s work from 2007, and introduces the Slanted Line Integral.

This Note describe my own recollection of the first 30 years of Category Theory, it is not the result of any historical investigation. The choice of concepts and its evaluation is my own, necessarily subjective. It follows a chronological…

This paper serves as the announcement of my program---a joke version of the Langlands Program. In connection with this program, I discuss an old hat puzzle, introduce a new hat puzzle, and offer a puzzle for the reader.

In his list of open problems, Martin Erickson described a certain game: "Two players alternately put queens on an n x n chess board so that each new queen is not in range of any queen already on the board (the color of the queens is…

In this paper, we propose a new algorithm of calculating the day of the week for any given century, year, month and day in Gregorian calendar. We provide two simple formulas to convert the century and the year into two integers. Then we…

Developing a 21st Century Global Library for Mathematics Research discusses how information about what the mathematical literature contains can be formalized and made easier to express, encode, and explore. Many of the tools necessary to…

A common format for sports contests involves pairwise matches between two teams, with the #1 player of team A matched against the #1 player of team B, the #2 player of team A against the #2 player of team B, and so on. This paper addresses…

Egyptian decompositions of 2/D as a sum of two unit fractions are studied by means of certain divisors of D, namely r and s. Our analysis does not concern the method to find r and s, but just why the scribes have chosen a solution instead…

This paper introduces DD calculus and describes the basic calculus concepts of derivative and integral in a direct and non-traditional way, without limit definition: Derivative is computed from the point-slope equation of a tangent line and…

Gabriel's horn is the famous mathematical object that has finite volume and infinite surface area. This article gives a template for making Gabriel's horn out of paper cones. It also describes the mathematics behind the construction of the…

Telescoping sums very naturally lead to probability distributions on ${\mathbb Z}^+$. But are these distributions typically cosmetic and devoid of motivation? In this paper we give three examples of "first occurrence" distributions, each…

A Vuza canon is a non periodic factorization of the finite cyclic $ \mathbb{Z}_{N}$. The aim of this paper is to present some new results for generating Vuza canons, and to give new minoration of their number for some values of N.

If the denominator of a rational function of several variables is sum of even powers and the numerator is a monomial, then we give a numerical criterion, using the exponents involved in the expression of the rational function, to decide if…

The learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which…

This paper collects some reflections about an apparent incongruity between the usual (third-person) understanding of the probability of an event calculated for an extended period of time in the future (e.g., the expected probability of a…

These are notes and slides from a Pecha-Kucha talk given on March 6, 2013. The presentation tinkered with the question whether calculus on graphs could have emerged by the time of Archimedes, if the concept of a function would have been…

For h=3 or 4, Egyptian decompositions into h unit fractions, like 2/D = 1/D1 + ... +1/Dh, were given by using (h-1) divisors (di) of D1. This ancient modus operandi, well recognized today, provides Di=DD1/di for i greater than 1.…