#mathematics

Power Series in Practice

2025-05-18
We saw in previous blogs (here and here) about infinite series of just numbers. In this blog we will look at a similar idea, but now the series will be of a variable. Here, the variable will be \(x\) or \(x - a\), which is called a power series. Power series are actually even more useful in physics than general infinit
#mathematics

Grad, Div, Curl

2025-05-10
In this article, our goal shall be to study three mathematical operations in vector calculusWe shall be studying these in a very formal mathematical way, breaking beyond just mere computation to understand why they exist, and what they really are. If you're only looking for a quick reference for grad, div, curl, this p
#mathematics

Testing Infinite Series for Convergence

2025-05-04
It is necessary when confronted with an infinite series to ask whether it converges or diverges. We've already worked with some convergent series in a previous article. Specifically geometric series where \(\vert r \vert < 1\). If \(\vert r \vert > 1\) then the series has no defined sum as it grows forever.Strang
#mathematics

Introduction to Sentential Logic Part 3 Solutions

2025-04-28
These are the solutions for this blog. You should try all the problems yourself first!\[P \leftrightarrow Q = P \rightarrow Q \wedge Q \rightarrow P\] \[P \rightarrow Q \wedge Q \rightarrow P = (\neg P \vee Q) \wedge (\neg Q \vee P)\] For this next step we make use of the distributive laws! \[(\neg P \vee Q) \wedge (\n
#logic #mathematics

Introduction to Sentential Logic Part 3

2025-04-28
This is part 3 of a series on sentential logic. You should read part 1 and part 2 firstA tautology is a logical statement which is always true. A contradiction is a logical statement that is always false. Consider \(P \vee \neg P\). Let's make a truth table for itIn other words, it does not matter what value \(P\) has,
#logic #mathematics

Introduction to Sentential Logic Part 2 Solutions

2025-04-19
You can find the relevant blog here
#logic #mathematics

Introduction to Sentential Logic Part 2

2025-04-19
This is a part 2 of a blog series, you can read the first part here.In this blog we're going to work more with variables, and find ways to manipulate logical expressions in pure mathematics.In the last blog we introduced some variables very briefly. We saw something likeThis is okay as we started to see the overall str
#logic #mathematics

Infinite Series and the Zeno Paradox

2025-04-12
Zeno of Elea was a pre-Socratic philosopher (whom I wrote about in a much much earlier blog) of which little is known. Nevertheless, his paradoxes have lived on in history. Zeno was a student of Parmenides, and they held that all experiences in reality are mere illusions. The basic idea was that there is no possibility
#mathematics #philosophy

Solving Polynomials by Hand Solutions

2025-04-02
These are the solutions to the Solving Polynomials by Hand blog. Don't look at these till you try all the problems!
#mathematics

Solving Polynomials by Hand

2025-04-02
Something we need to do all the time in mathematics is solve polynomials. There's many methods for doing this thankfully! In this blog we'll go over methods of solving polynomials by hand, and leave numerical methods for another day.If we have a quadratic formula, we can usually factorise by inspection. This is basical
#mathematics

Introduction to Complex Numbers

2025-03-30
In this blog we'll begin to understand imaginary and complex numbers: why they exist and their properties. We'll finish up with a few real world examples from physics to see that imaginary numbers are just a really bad name: they do actually exist!Consider the following quadratic equation\[x^2 - 4 = 0\]We can easily so
#mathematics #physics

Introduction to Sentential Logic Part 1 Solutions

2025-03-22
These are the solutions to the exercises given in this article, but make sure you try them all yourself first!Yes. If only one of two people can win, and one can't even make it, then there's only one possible winnerNo. There's no reason to believe that John will win the fishing contest again just because he won it befo
#logic #mathematics

Introduction to Sentential Logic Part 1

2025-03-22
This article will be part of a series, so make sure you check back for the next part!The study of logic is an ancient part of the liberal arts tradition. Unfortunately, this tradition has waned in the last century and now very few people study them. As part of my interest in polymath, I wish to contribute to a revival
#logic #mathematics

Kalman Filter Notation

2025-03-08
This article is a WIP - I'll add more notations as I find themKalman filtering notation is terrible. Every author uses a different, confusing, notation. I am not able to standardise it, but at least I can provide some of the key references and explain what notation they use.This article assumes some familiarity with th
#kalman-filtering #mathematics #signal-processing #science

How to Differentiate Solutions

2025-03-01
\[y = 3x\] \[\frac{dy}{dx} = \lim_{\Delta x \rightarrow 0} \frac{3(x + \Delta x) - 3x}{\Delta x}\] \[\frac{dy}{dx} = \lim_{\Delta x \rightarrow 0} \frac{3x + 3\Delta x - 3x}{\Delta x}\] \[\frac{dy}{dx} = \lim_{\Delta x \rightarrow 0} \frac{3 \Delta x}{\Delta x}\] \[\frac{dy}{dx} = 3\]\[y = 2x^2 + 5x\] \[\frac{dy}{dx} =
#mathematics #science

How to Differentiate

2025-03-01
Differentiation is an essential skill for anyone who wants to get into physics, mathematics, computer science, engineering or any technical field.Why do we want to differentiate? What purpose can it serve? The reason for differentiation is that we want to know the gradient, or steepness, of a function at any given poin
#mathematics #science

Kalman Filter: Filtering a Sin Wave

2025-02-23
\[ \newcommand{\Lagr}{\mathscr{L}} \]The sin wave tracking problem presents us with a unique opportunity. There are many possible solutions to this problem, some linear and some non-linear. This allows us to see the effect of different kinds of Kalman filters in actionFirst, let's just quickly write a simple method whi
#kalman-filtering #mathematics #signal-processing #science

Kalman Filter: Tracking a Cannon Ball

2025-02-15
So far, we have been using linear Kalman filters to track reasonably simple problems. However, more complex problems require more complex Kalman filters. One of the primary limitations of the Kalman filter is that it is not capable of handling non-linear situations very well. In this example, we will be tracking a cann
#kalman-filtering #mathematics #signal-processing #science

Kalman Filter: Falling with Control Term

2025-02-08
In previous blogs, we looked at objects falling under gravity. In those blogs we asked the Kalman filter to try and estimate the position, velocity and acceleration from the readings. However, we know in advance what gravity is doing. So, why don't we instead tell the Kalman filter that the object is accelerating? That
#kalman-filtering #mathematics #signal-processing #science

Kalman Filter: How to Solve the Riccati Equations

2025-01-25
I've already published two blogs on practical applications of the Kalman filter. In both of those blogs, I use the somewhat throw away line "simply solve the Riccati equations". If you are not familiar with the Riccati equations then this line could be very confusing indeed. So I hope to clear up any confusio
#kalman-filtering #mathematics #signal-processing #science

Kalman Filtering Object with Constant Velocity

2025-01-04
In a previous blog entry we looked at tracking a falling object. That was focused on the practical implementation. This time we will cover tracking an object moving at constant velocity but focus more on the actual derivation of the filter. Let's start with the differential state-space equations.\[ \begin{bmatrix} \dot
#programming #science #mathematics #kalman-filtering #signal-processing

Kalman Filtering Falling Objects

2025-01-04
In this blog we will track a falling object using a Kalman filter. Our goal will be to create a simulation of a falling object, and filter that with the Kalman filter. We'll plot the residuals and see how well we can reconstruct the original signal from the noisey signal. We'll code this in Rust, which is a great choic
#programming #science #mathematics #kalman-filtering #signal-processing

Introduction to Variables and Sets

2022-05-15
In the last article, we assigned truth values to statements. However, if these statements depend on variables, it can be less obvious what the truth value is. For example, it is very easy to assign a truth value to the statement "3 is a prime number", but what about the statement "x is a prime number&quo
#mathematics

Introduction to Truth Tables

2022-04-24
The truth table is a useful introductory tool in the field of sentinel logic. It allows us to compute the status of a logical statement by simply considering each of the statements that comprise the logical statement. It is obvious how "and" and "not" impact statements. Let's explore this while cons
#mathematics