Fri, Jan 15, 2016

A *polyhedral set* in \(\mathbb{R}^d\) is the intersection of a finite
number of closed halfspaces, and a *(convex) polytope* is a bounded polyhedral
set. This definition of a polytope is called a *halfspace representation*
(*H-representation* or *H-description*).

Sun, Nov 8, 2015

\[ \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \]

Read more about SymbolsFri, Jul 31, 2015

Mon, Jun 29, 2015

The Fibonacci numbers are defined as \(f_0 = 0,\ f_1 = 1\) and, for \(i \ge 2,\ f_i = f_{i-1} + f_{i-2}\). Here is the beginning of the Fibonacci sequence:

\[0, 1, 1, 2, 3, 5, 8, 13, 21, \ldots\]

We generalize the definition above by changing the two initial values, for example with \(f_0 = 4,\ f_1 = 6\) we obtain

\[4, 6, 10, 16, 26, 42, \ldots\]

Read more about Fibonacci numbersMon, Jun 29, 2015

We are given \(k > 0 \in \mathbb{R}\) and \(a_1, a_2, \ldots, a_n \ge 1 \in \mathbb{N}\) such that

\[ a_1 < a_2 < \cdots < a_n. \]

We want to solve the following equation

\[ \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n} = k. \]

Read more about Inverse sum equationsWed, Jun 24, 2015

\[ \binom{n}{k} = \binom{n}{n-k} \]

holds because keeping \(k\) elements from \(n\) elements is equivalent to discarding \(n-k\) elements from \(n\) elements.

Read more about Binomial coefficient tricksTue, Jun 23, 2015

\( \frac{a+bi}{c+di} = \frac{a+bi}{c+di} \frac{c-di}{c-di} = \frac{ac+bd}{c^2+d^2}+\frac{bc-ad}{c^2+d^2}i \)

Read more about Complex numbers divisionFri, Jun 19, 2015