Aurélien's Math Notebook

Modular Arithmetic

Mon, 11 May 2020 00:00:00 +0000

Positional notation. An arbitrary precision integer on a computer is represented as a list of digits (limbs). They are just bigger. Today’s computers can hold 64 bits per limb. Modular arithmetic consists in applying addition and multiplication to integers modulo a certain number $n$. Given an implementation of addition, multiplication, and division over the integers, we can emulate addition or multiplication modulo $n$ by performing the corresponding operation on integers then taking the result modulo $n$.

The Drunkard's Walk

Wed, 17 Oct 2018 00:00:00 +0000

A drunkard is zigzagging home. At every steps forward (or backward) he is making, he also moves $1$ step to the left with probability $p$ and $1$ step to the right otherwise. He starts $i$ steps to the right of a river. What is the expected number of steps forward before he falls into the river?

Sums of geometric series

Sun, 23 Jul 2017 00:00:00 +0000

Among the identities that are useful in the analysis of algorithms, this one shows how sums of geometric series converge when the ratio is smaller than one (in absolute value). It occurs in divide an conquer schemes. For example, it allows to show that hierarchical cuttings for $n$ hyperplanes in $d$ dimensions only need space proportional to $n^d$.

\[ \sum_{i=0}^{\infty} \binom{i+j}{j} x^i = \frac{1}{ {(1-x)}^{j+1} }, \forall j \in \mathbb{N}, \forall x \in (-1,1). \]

Infinite Number of Primes

Sat, 22 Jul 2017 00:00:00 +0000

The erroneous proof I hear most often is: Suppose $P$ is a finite set that contains all the primes, then $p^* = 1 + \prod_{p \in P} p$ is prime. Indeed, the flaw is that $p^*$ is not necessarily prime but rather must be a multiple of some prime not in $P$.

Quantum Computer Algorithms

Fri, 21 Jul 2017 00:00:00 +0000

Notes from June 2015 containing the following: Phase kick-back, Deutsch’s algorithm, Fourier sampling, Deutsch-Jozsa algorithm, Bernstein-Vazirani algorithm, Preimage of a function, Simon’s algorithm, Grover’s algorithm, Amplitude amplification, Quantum Fourier Transform, and Shor’s algorithm.

Recurrences

Sat, 22 Apr 2017 00:00:00 +0000

Theorem

Let $a \in \mathbb{N} $, $t < a \in \mathbb{N} $, and $b \in \mathbb{R}$. Defining $A_t$to be some real number and

$$ A_N = (1 - \frac{a}{N}) A_{N-1} + b, N > t \in \mathbb{N}, $$

then

$$ A_N = \frac{b}{1+a} (N+1), N \ge a \in \mathbb{N}. $$

Converging series

Fri, 21 Apr 2017 00:00:00 +0000

Let $-1 < x < 1$,

$$ 1 + x + x^2 + x^3 + x^4 + \cdots = \frac{1}{1-x}. $$

Polynomial-time approximation schemes

Wed, 04 May 2016 00:00:00 +0000

PTAS

$(1\pm\varepsilon)$-approximation with complexity $O(n^{f(\varepsilon)})$.

Equivalence of 3SUM problems

Wed, 27 Apr 2016 00:00:00 +0000

Are different versions of the 3SUM problem equivalent?

Log Log Log

Thu, 14 Apr 2016 00:00:00 +0000

Theorem

$$ \frac{\log n}{\log \log n}! = \Theta(n) $$

VC-dimension

Tue, 12 Apr 2016 00:00:00 +0000

Definitions of VC-dimesion and $\varepsilon$-nets.

$d$ hyperplanes intersection bounds

Sun, 17 Jan 2016 00:00:00 +0100

We bound the position of the $0$-cells of an arrangement of hyperplanes in $\mathbb{R}^d$. This allows, for example, to build an hypercube that intersects all cells of the arrangement. Such an hypercube must contain at least one point of each cell of the arrangement. When $q > 0$, in order to fix which point of a $q$-cell we want to include in the hypercube, it suffices to add the $n$ hyperplanes of equation $x_i = 0$ to the arrangement. With those additional hyperplanes, the arrangement is such that each $q$-cell of the arrangement with $q > 0$ contains a $0$-cell of the arrangement, hence, we only need the hypercube to intersect, for each $0$-cell $\nu$ of the arrangement, an hypersphere of center $\nu$ and arbitrarily small radius. The inequalities of the polyhedral set defining our hypercube will thus only depend on the position of the $0$-cells of our arrangement.

Polyhedral sets

Fri, 15 Jan 2016 00:00:00 +0000

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).

Symbols

Sun, 08 Nov 2015 00:00:00 +0000

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

Probabilistic primality testing

Fri, 31 Jul 2015 00:00:00 +0000

Sage implementation.

Fibonacci numbers

Mon, 29 Jun 2015 00:00:00 +0000

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

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

Inverse sum equations

Mon, 29 Jun 2015 00:00:00 +0000

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

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

Binomial coefficient tricks

Wed, 24 Jun 2015 00:00:00 +0200

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

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

Complex numbers division

Tue, 23 Jun 2015 00:00:00 +0000

$$ \frac{a+bi}{c+di} = \ldots $$

Gaussian elimination

Fri, 19 Jun 2015 00:00:00 +0000

Python implementation.

Aurélien's Math Notebook

Modular Arithmetic

The Drunkard's Walk

Sums of geometric series

Infinite Number of Primes

Quantum Computer Algorithms

Recurrences

Theorem

Converging series

Polynomial-time approximation schemes

PTAS

Equivalence of 3SUM problems

Log Log Log

Theorem

VC-dimension

\(d\) hyperplanes intersection bounds

Polyhedral sets

Symbols

Probabilistic primality testing

Fibonacci numbers

Inverse sum equations

Binomial coefficient tricks

Complex numbers division

Gaussian elimination