What is the factorial of 1/2?

Fredrik alleged / Wikimedia Commons

What is your favorite function?

If your answer is not the gamma function, then I’ll ask you again after you have read this article.

Your answer might have changed…

Introduction

In the late 1720s, Leonhard Euler was thinking about how to extend the factorial to non-integer values.

This was the start of a rich theory used all over the scientific world. A theory of one of the most important functions in mathematics.

Leonhard Euler is, without doubt, one of the greatest mathematicians in history. To give you an idea of Euler’s powers here follows some examples that show his brilliance.

First…


The heart of Python

Image by Kevin Ku from Pexels

Python is an object-oriented programming language. In fact, it is often said that everything in Python is an object.

So in order to fully understand Python, you really need to understand the underlying structures like objects and classes to be able to understand what is really happening under the hood.

This article is about the concept of metaprogramming in Python and we will see how to manipulate classes and the instantiation of them.

Metaprogramming can be loosely defined as code that manipulates code.

An object is of course an instantiation of a class and as the curious beings that we…


And a differentiable function encoding them

Image from Wikimedia Commons — Bernhard Riemann in 1863.

In 1859 Bernhard Riemann published one of the most influential papers in analytic number theory concerning the distribution of the prime numbers by use of analytic methods.

It was titled Eber die Anzahl der Primzahlen unter einer gegebenen Grösse. It translates to something like On the Number of Primes Less than a Given Magnitude.

In particular, he investigated the zeta function (now called the Riemann zeta function in his honor) and proved that the zeta function encoded information about the distribution of the prime numbers through its zeros.

In this rather short article, I will give you some of my…


By taking a look under the hood

Image by Cecbur on Wikimedia Commons

In this article, I will train a deep neural network to classify images, but I will also give you an understanding of what is happening inside the neural network itself and how convolutions really work.

We will explore the following sections.

· Convolutions
Filters
Pooling
· The Network
Preprocessing
The model
Training the network
· Predicting
· What does the neural network “see”?

I will show you the network in action by displaying to you what it is actually “seeing” when it makes the decisions by diving into its “brain” — the layers of the…


The simplest neural network with TensorFlow 2

Image by mikemacmarketing on Wikimedia

There are tons of great theoretical articles out there on the anatomy and mathematics of artificial neural networks, so I am going to take another approach to writing about and teaching this subject. In this article, we are going to get our hands dirty right away!

Let’s jump right in. You can choose a section below if you want to skip the installation section.

· Installation
Option 1
Option 2
· The data
· The model
· Why TensorFlow?

Installation

If you already have the requirements installed, then you can skip this section.

Option 1

The first thing we need to…


Real-life applications of topology

You are sitting in a bar with drinks on the table. Unfortunately, the table has four legs making it susceptible to wobbling.

People are folding napkins and paper plates to make them fit under the supposedly too-short leg.

But none of these attempts will really work and you have to perform a small circus act of carrying all your stuff while dancing around with the table. Oh, and the whole bar is watching you perform this act…

Allow me to let you in on a little secret.
Mathematicians don’t have wobbly tables. Why? …


How I found a useful formula

Image by Amédée Félix Barthélemy Geille

Have you ever wondered how Fourier series can be used in number theory? Or if there is a formula for the difference between a sum and an integral and what that is?

Not to mention, if there is some tool that makes us capable of summing over all the prime numbers?

In this article, I will answer those questions and more. Hang tight.

I have always been fascinated by doing my own research and discovering new results in mathematics.

Usually, I rediscover something that I wasn’t aware was already discovered.

Normally, my “discovery" has been discovered by someone in the…


From scratch

Image by author

The world is a changing place. If we want to understand nature and the laws it needs to follow, we need a language and a tool well-suited for that. Calculus is such a tool and the first really useful tool you learn about in calculus is differentiation and derivatives.

In this article we will cover the following:

· My First Encounter with Derivatives
· Slope
· Examples and properties from first principles
f(x) = x^a
Notation
Linearity
· Differentiation Rules
The product rule
The quotient rule
The chain rule
· L’Hôpital’s Rule
· Taylor Series
Applications of Taylor…


With some interesting applications

Image by Claudio Rocchini on Wikimedia Commons

About a year ago I was trying to come up with a formula for the nth Fibonacci number which wasn’t Binet’s well-known formula. Fiddling around with the generating function, I was thinking about how to extract the coefficients without having to differentiate the beast n times.

This is a story about the result of this pursuit. We can prove it and use it to solve interesting problems and maybe even prove other theorems.

In this article, I will show you this beautiful formula for calculating derivatives without having to differentiate a single time. …


The beginning of calculus

Табуретка / Wikimedia Commons

This is part two in a series of articles where my goal is to teach calculus from scratch.

You can find the first one here.

The most fundamental and important concept to understand in calculus is that of a limit.

As with many other objects in mathematics, limits can be understood from many different levels. In this article, I will try to provide you with a couple of different levels of understanding so that you are able to understand the inner workings of calculus as we move on in the subject.

As you will see in the coming articles, limits…

Kasper Müller

Mathematician and Data Scientist interested in the mysteries of the Universe, fascinated by the human mind, music and things that I don’t understand.

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