What is an equation, and why do we solve them that way?

In words, an equation is a technique to figure out the value of a quantity when the only thing we have is a convoluted and self-referential definition. Enough talking, now…

Why does Finite Elements Methods (FEM) tile domains with triangles?

Why is it the case that the Finite Elements Method (FEM) tiles domains with triangles? With so many geometrical shapes, is there anything special triangles have to offer? For starters,…

Overdetermined and underdetermined systems of equations put simply

Systems of equations are an evolution of equations, but they are often misunderstood. This article aims at providing real world examples and intuitions for systems of equations, and in particular…

Projection methods in linear algebra numerics

Linear algebra classes often jump straight to the definition of a projector (as a matrix) when talking about orthogonal projections in linear spaces. As often as it happens, it is…

Reproducing a transport instability in convection-diffusion equation

Drawing from Larson-Bengzon FEM book, I wanted to experiment with transport instabilities. It looks there might be an instability in my ocean-ice model but before being able to address that,…

What is the Rossby number?

The Rossby number is used to describe whether a phenomenon is large-scale, i.e. if it is affected by earth’s rotation. But do we actually quantify if a fluid flow is…

How do Dirichlet and Neumann boundary conditions affect Finite Element Methods variational formulations?

To solve a classical second-order differential problem     with FEM, we first need to derive its weak formulation. This is achieved by multiplying the equation by a test function…

A gentle (and short) introduction to Gröbner Bases

Taken from my report for a Computer Algebra course. Motivation We know there are plenty of methods to solve a system of linear equations (to name a few: Gauss elimination,…

What is the difference between Finite Differences and Finite Element Methods?

With Finite Differences, we discretize space (i.e. we put  a grid on it) and we seek the values of the solution function at the mesh points. We still solve a…

A note on the hopes for Fully Homomorphic Signatures

This is taken from my Master Thesis on Homomorphic Signatures over Lattices. See also But WHY is the Lattices Bounded Distance Decoding Problem difficult?. What are homomorphic signatures Imagine that…

But WHY is the Lattices Bounded Distance Decoding Problem difficult?

This is taken from my Master Thesis on Homomorphic Signatures over Lattices. Introduction to lattices and the Bounded Distance Decoding Problem A lattice is a discrete subgroup , where the…

The role of intuitions in mathematics

Some thoughts and questions about the role of intuition in mathematics: Is intuition needed to really understand a topic? I would say yes, since in the end we reason through…

Probability as a measure of ignorance

One of the most beautiful intuitions about probability measures came from Rovelli’s book, that took it in turn from Bruno de Finetti. What does a probability measure measure? Sure, the…

Conditional probability: why is it defined like that?

So, you want to calculate the probability of an event knowing that another has happened. There is a formula for that, it is called conditional probability, but why is it…

Diagonalizing a matrix NOT having full rank: what does it mean?

This is going to be a quick intuition about what it means to diagonalize a matrix that does not have full rank (i.e. has null determinant). Every matrix can be…

Finding paths of length n in a graph

Suppose you have a non-directed graph, represented through its adjacency matrix. How would you discover how many paths of length link any two nodes? For example, in the graph aside…

On the relationship between L^p spaces and C_c functions for p = infinity

Very quick post on the relationship between , and . I will assume you already know what I am talking about, I’ll just be sharing some intuition on what those mean,…

The meaning of F Value in the Analysis of Variance for Linear regression

This is a sample output for linear regression: The F Value is computed by dividing the value in the Mean Square column for Model with the value in the Mean…

On the meaning of hypothesis and p-value in statistical hypothesis testing

Statistical hypothesis testing is really an interesting topic. I’ll just briefly sum up what statistical hypothesis testing is about, and what you do to test an hypothesis, but will assume…

Why hash tables should use a prime-number size

I read in several books and online pages that hash tables should use a prime number for the size. Nobody really justified this statement properly. Here’s my attempt! I believe…