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).

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Sun, Nov 8, 2015

Symbols

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

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Fri, Jul 31, 2015
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

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

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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

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

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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.

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Tue, Jun 23, 2015

Complex numbers division

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

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Fri, Jun 19, 2015