Animation has an essential role in video games. Most video games try to simulate real world physics; in order to achieve this objective it is essential to be able to represent, in the most realistic way, the movement of people, Read more…
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…
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…
This article gives a brief illustration of Newton’s laws of motion in the plane and analyzes the tools that the Unity 2D engine makes available to programmers to simulate the movement of bodies in the 2D environment.For the study of Read more…
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…
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…
This article illustrates the properties of the Euler and Möbius functions, which have great importance in Number Theory and in other fields. As an example of application we describe the RSA algorithm for public key cryptography. 1) Arithmetic functions An Read more…
The birth of probability theory is usually set in the mid-seventeenth century. At that time the two great mathematicians Blaise Pascal (1623–1662) and Pierre de Fermat (1601– 1665) discussed together some gambling problems and defined the theoretical basis of the Read more…
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…
Exercise 1 Let \(x_{n} \) be a sequence of positive integers so defined: \[ \begin{array}{l} x_{1}= 2 \\ x_{n + 1} = x_{n}^{2} – x_{n} +1 \quad n \gt 1 \\ \end{array} \] Prove that the numbers \(x_{n} \) are Read more…