Fri, Jan 15, 2016

Polyhedral sets

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

Symbols

$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$

Fri, Jul 31, 2015

Probabilistic primality testing

Mon, Jun 29, 2015

Fibonacci numbers

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

$4, 6, 10, 16, 26, 42, \ldots$

Mon, Jun 29, 2015

Inverse sum equations

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

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

Wed, Jun 24, 2015

Binomial coefficient tricks

$\binom{n}{k} = \binom{n}{n-k}$

holds because keeping $$k$$ elements from $$n$$ elements is equivalent to discarding $$n-k$$ elements from $$n$$ elements.

Tue, Jun 23, 2015

Complex numbers division

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

Fri, Jun 19, 2015