1) Algebra of matrices A matrix \(A(m,n)\) defined on the field of real numbers \(\mathbb{R}\) is a collection of real numbers \((a_{ij})\), indexed by natural numbers \(i, j\), with \(1\le i\le m\) and \( 1\le j\le n\). We can represent a matrix with a rectangular array of numbers arranged in Read more…
Exercise 1 A square and a triangle have the same area. Which shape has the greatest perimeter? HintIt may be useful to remember the following formula: Arithmetic Mean-Geometric Mean InequalityLet \(x_{1},x_{2}, \dots x_{n}\) be non-negative real numbers; then: \[ \dfrac {x_{1}+ x_{2} + \cdots x_{n}}{n} \ge \sqrt[n]{ x_{1}x_{2} \cdots x_{n}} Read more…
Problem Let \((a_{1},a_{2}, \cdots ,a_{n})\) be a permutation of the set \(\{1,2, \cdots,n \}\). Compute the average value, indicated with \(M_{n}\) , of the following sum: \[ (a_{1}- a_{2})^2 + (a_{2}-a_{3})^2 + \cdots + (a_{n-1} – a_{n})^2 \] taken on all permutations. If \(n=2\) the set of permutations is \(\{12,21\}\) Read more…
Euclidean geometry studies geometric objects such as lines, triangles, rectangles, circles, etc. Fractals are also geometric objects; however, they have specific properties that distinguish them and cannot be classified as objects of classical geometry. Although Mandelbrot (1924-2010) is generally considered the father of the scientific theory of fractals, in reality the ideas underlying Read more…
Problem There are four people A, B, C, D. A box with a red ball inside it is given to A, who can leave things unchanged with probability \(p\) or replace the red ball with a white one with probability \(q=1-p\). The box is passed to B who can leave Read more…