Math | Mauricio Poppe

Queuing Theory for Systems Engineers

A practical, example-first guide to queuing theory in systems engineering: single-server M/M/1 response time and the 50% load paradox, the hockey stick latency curve, Pollaczek-Khinchine service variance, multi-server resource pooling (M/M/c), and capacity planning rules of thumb with interactive visualizers.

Published on Mon, Aug 24, 2026 • 15 min read

#system design #performance #queuing theory #distributed systems

Bayesian Networks

A Bayesian network is a directed graph in which each node is annotated with quantitative probability information. This article covers the definition of a Bayesian network with a graphical representation, the determination of independence between variables, and the problem of finding the probability distribution of a set of query values given some observed events.

Published on Thu, Mar 5, 2020 • 5 min read

#math #probability #bayesian networks #exact inference

Divisibility

Let $a,b \in \mathbb{Z}$. We say that $a$ divides $b$, written $a \given b$, if there’s an integer $n$ such that $b = na$. If $a$ divides $b$, then $b$ is divisible by $a$, and $a$ is a divisor or factor of $b$. Also, $b$ is called a multiple of $a$.

This article covers the greatest common divisor and how to find it using the Euclidean Algorithm, the Extended Euclidean Algorithm to find solutions to the equation $ax + by = gcd(a, b)$ where $x, y$ are unknowns.

Published on Sun, May 21, 2017 • 5 min read

#math #number theory #divisibility #modulo

Affine spaces

✦ Interactive

An affine space is a generalization of the notion of a vector space, but without the requirement of a fixed origin or a notion of “zero”.

Published on Tue, Mar 15, 2016 • 5 min read

#math #geometry #affine geometry #affine spaces

Geometric tests

Different algorithms to test geometric properties, such as finding the intersection of two lines.

Published on Wed, Mar 9, 2016 • 3 min read

#math #geometry #euclidean geometry #line line intersection

Eigenvalues and eigenvectors

An eigenvalue represents how an object scales (or stretches/compresses) a particular direction (or eigenvector) when acted upon by the object. This article covers how to find these values in a square matrix, as well as their applicability in computer graphics.

Published on Mon, Mar 7, 2016 • 2 min read

#math #linear algebra #eigenvalue #eigenvector

Projective space

In projective geometry, unlike Euclidean geometry, two parallel lines meet at a point. Desargues introduced the concept of a line at infinity where a point at infinity can be defined. This article covers the need for a point at infinity in projective space, the line at infinity, and the projective plane.

Published on Fri, Mar 4, 2016 • 4 min read

#math #geometry #projective geometry #projective space

Quaternions

Quaternions are a set of 4-dimensional vectors that are used to represent rotations in computer graphics. They were discovered by William Hamilton as an extension of 2D complex numbers to a 3D equivalent.

This article covers the definition of a quaternion, its notation, and operations.

Published on Tue, Sep 8, 2015 • 7 min read

#math #numeral systems #complex numbers #imaginary numbers

Complex Numbers

✦ Interactive

Imaginary numbers were invented to solve problems for equations with no real roots. Complex numbers extend imaginary numbers by adding a real number.

This article covers the definition of complex numbers and operations such as addition, product, norm, conjugate, inverse, and square root. Finally, this article covers the geometric and polar representations of complex numbers.

Published on Tue, Sep 8, 2015 • 8 min read

#math #numeral systems #complex numbers #imaginary numbers

Integer Factorization

Integer factorization is the process of decomposing a composite number into a product of smaller integers, if these integers are restricted to be prime numbers then the process is called prime factorization.

This article covers factorization using trial division and fermat factorization through Pollard's Rho algorithm and using the sieve of eratosthenes.

Published on Sun, Jun 14, 2015 • 8 min read

#math #number theory #integer factorization #divisibility

Divisor Function

The divisor function returns the number of divisors of an integer. This article covers important relations of the divisor function and prime numbers.

Published on Sat, Jun 13, 2015 • 4 min read

#math #number theory #divisor function #prime numbers

Primality Test

A prime number is a natural number greater than $1$ which has no positive divisors other than $1$ and itself.

This article covers different algorithms for checking if a number is prime or not, including a naive test, the Eratosthenes Sieve, the Euler Primality Test, and the Miller-Rabin Primality Test.

Published on Thu, Jun 11, 2015 • 7 min read

#math #number theory #prime numbers #erathostenes sieve

Special factorial modulo p

Let $n!_{\%p}$ be a special factorial where $n!$ is divided by the maximum exponent of $p$ that divides $n!$. This article describes this problem and its solution, with an implementation in C++.

Published on Tue, Jun 9, 2015 • 5 min read

#math #number theory #prime numbers #modular arithmetic

Prime factors of a factorial

This article describes and implements a solution for the following problem: given two numbers $n$ and $k$ find the greatest power of $k$ that divides $n!$

Published on Tue, Jun 9, 2015 • 2 min read

#math #number theory #prime numbers #integer factorization

Discrete Logarithm

The discrete logarithm finds a solution for $x$ in the congruence $a^x \equiv b \pmod{n}$ where $a$, $b$, and $n$ are integers, $a$ and $n$ are coprime. This article covers two algorithms to solve this problem: by trial multiplication and using Baby Step Giant Step.

Published on Mon, Jun 8, 2015 • 3 min read

#math #number theory #discrete logarithm

Chinese Remainder Theorem

The Chinese Remainder Theorem (CRT) is a theorem that deals with finding a solution to a system of congruences.

This article covers the definition of the CRT and an example implementation in C++.

Published on Fri, Jun 5, 2015 • 3 min read

#math #number theory #chinese remainder theorem #congruences

Modular Arithmetic

Modular arithmetic is a type of arithmetic that deals with integers and remains within a fixed range of values. It involves performing arithmetic operations such as addition, subtraction, multiplication, and division, but with the added concept of a “modulus” or a “mod” value.

This article covers the definition of a congruence relation, and some of its properties like addition, multiplication, exponentiation, and inverse. Next, I show how we can use the Extended Euclidean Algorithm to find the modular multiplicative inverse in a general case and in the case of coprime numbers.

Published on Thu, Jun 4, 2015 • 5 min read

#math #number theory #divisibility #modulo

Extended Euclidean Algorithm

The Extended Euclidean Algorithm finds solutions to the equation $ax + by = gcd(a, b)$ where $x, y$ are unknowns. This article covers a few applications of the Extended Euclidean Algorithm like finding the modular multiplicative inverse of a number and finding solutions for linear congruence equations.

Published on Tue, Jun 2, 2015 • 4 min read

#math #number theory #divisibility #modulo

Euclidean Algorithm

The Euclidean Algorithm finds the greatest common divisor of two numbers. In this article I implement the algorithm from scratch in C++.

Published on Mon, Jun 1, 2015 • 2 min read

#math #number theory #divisibility #modulo

Eratosthenes Sieve

The Eratosthenes Sieve is an algorithm to find prime numbers up to a positive number $n$ using $O(n)$ space.

Published on Mon, Jun 1, 2015 • 1 min read

#math #number theory #prime numbers #erathostenes sieve

Binary Exponentiation

Given two numbers $a$ and $n$, finding $a^n$ involves performing $n$ multiplications of $a$. However, it’s possible to do this in $log(n)$ operations using binary exponentiation.

Published on Mon, Jun 1, 2015 • 1 min read

#math #number theory #binary exponentiation

Euler's phi function

Euler’s phi function represented as $\phi(n)$ gives, for a number $n$, the number of coprimes in the range $[1..n]$; in other words, the quantity of numbers in the range $[1..n]$ whose greatest common divisor with $n$ is the unity. In this article, I try to explain how it works and implement it in C++.

Published on Mon, Jun 1, 2015 • 3 min read

#math #number theory #divisibility #euler's phi function

Taylor's Theorem and Infinite Series

Taylor series help approximate the value of a definite integral for a function whose antiderivative is hard to find. This article explains the key ideas behind Taylor’s Theorem and an example of approximating its value with a polynomial function.

Published on Thu, Apr 2, 2015 • 4 min read

#math #calculus #taylor series #function

Integral

✦ Interactive

An integral is a mathematical concept that represents the accumulation or summing up of quantities over a certain interval or region. In this article, we’ll discuss the properties of the integral by looking at an example of antidifferentiation and some examples of evaluating definite integrals.

Published on Thu, Apr 2, 2015 • 14 min read

#math #calculus #integral #maxima minima

Derivative

✦ Interactive

The derivative is a concept that represents the rate of change or the slope of a function at a particular point. It is a fundamental concept in calculus and is used to analyze how a function changes with respect to its input as the input changes very slightly.

This article covers the physical and geometric interpretation of the derivative, as well as some applications like finding maxima and minima in a function and the Newton-Raphson method.

Published on Thu, Apr 2, 2015 • 17 min read

#math #calculus #derivative #maxima minima

Introduction to Calculus

✦ Interactive

This article gives an introduction to calculus, starting with the concept of a function and how calculus helps us solve problems related to determining tangents to curves (expressed as functions), finding the minima/maxima like determining the maximum range of a projectile, and finding the length of curves, areas, and volumes.

Published on Tue, Mar 31, 2015 • 2 min read

#math #calculus #function