## Splines and Bézier Curves and their application in Video Games

The geometry of curves and surfaces is of fundamental importance in computer graphics and in video game programming. In this article we will describe the mathematics of interpolation curves, in particular splines and Bézier curves, with examples of use in Read more…

## Ordinary Generating Functions and Recurrence Equations

1) Ordinary generating functions of a variable Generating functions are an important tool for solving combinatorial problems of various types. A typical problem is the counting of the number of objects as a function of the size $$n$$, which Read more…

## Lambert Series, the Arithmetic Function $$r(n)$$ and Gauss’s Probability Integral

In this article we will study some properties of Lambert series. Then, using Lambert series relative to the representation of integers as sum of two squares, we will compute the value of Gauss’s classical probability integral. 1) Dirichlet generating functions Let’s Read more…

## Iterated Function Systems, Fractals and Sierpinski Triangle

In a previous article we have introduced some examples of fractals, illustrating their main characteristics, both qualitative and quantitative: self-similarity, geometric irregularity, fractional dimension. To continue the study of fractal science, it’s first necessary to give a more rigorous definition Read more…

## Dirichlet’s Box Principle and Ramsey Numbers

The box principle is attributed to the German mathematician Dirichlet (1805-1859). It is also called the pigeonhole principle. This article illustrates Dirichlet’s principle and gives a brief introduction to Ramsey’s theory, with some examples of computation of Ramsey numbers. For Read more…

## Exercises in Elementary Number Theory (III)

Exercise 1 The Fermat numbers are defined as follows: $F_{n} = 2^{2^{n}} + 1 \quad n =0,1,2,\cdots$ Prove that all Fermat numbers with $$n \gt 1$$ have the last digit equal to $$7$$. HintThe number \(2^{2^{2}} = Read more…