Thinking back to my high-school years, linear algebra was a topic I was particularly fascinated with. It gave me the skill to solve large systems of linear equations and a geometric perspective on the problem making the whole process intuitive.
However, regarding matrix determinants, I was taught that they are numbers for matrices, how to compute them, and not much more. It took until my university courses that I learned the beauty behind determinants.
As soon as I learned about determinants’ geometric meaning, I was wondering why this wasn’t already taught in high school as it is very easy to…
Neural networks are riddled with uncertainty. After feeding an input through a neural network to retrieve an output, we cannot be sure that the output we got is a correct description of reality. Moreover, the uncertainty of a neural network needs to be separated into two different kinds:
1. Aleatoric Uncertainty: Does not resolve with more data.
2. Epistemic Uncertainty: Gets better with more data.
Separating uncertainty into these two independent components provides a better understanding of how and what neural nets actually learn. …
In our daily life, we constantly deal with uncertain situations. So — at least subconsciously — we are confronted with probabilities all the time. Moreover, it seems we have a good intuitive feeling for probabilities. Or do we really?
We know that flipping a coin 100 times will result in about 50 heads and 50 tails. Also, it’s clear to us that the probability of bumping into a friend on holidays abroad is very low, though not zero. However, there are situations when correct predictions following from probability theory are completely counter-intuitive.
Let’s dive into five paradoxes in probability theory…
Quantum computers can generate truly random numbers. That’s something a traditional computer cannot do without relying on external sources. Traditional computers can only generate numbers that appear to be random but in reality, are computed according to fixed rules. These numbers are called pseudo-random. If we generate enough of these pseudo-random numbers we will notice that they eventually repeat.
As a PhD researcher, it is crucial for my job to quickly code up an idea to see if it works or not. Python is an excellent tool enabling just that. It allows for focusing on the idea itself and not be bothered with boilerplate code and other tedious things.
However, Python comes with a major drawback: It is much slower than compiled languages like C or C++. So, what do we do after we tested an idea by building a Python prototype and now we want to turn it into a fast and performant tool? Most often than not…
Monads are programmable semicolons. That’s it. For a programmer, a monad provides functions that allow for sequencing actions. Moreover, between every two following actions, a specific code snippet is executed.
So, a monad is a semicolon — but one whose exact behavior you can configure.
In imperative languages like C or Java, semicolons are used to express a sequence of operations. The code in front of the semicolon is executed before the code after the semicolon. …
Fermat’s theorem on sums of two squares had famously been proven in just one single sentence.
Can prime numbers be written as the sum of two squares? For 13 this is possible (2²+3²), but for example, 3 cannot be written in this way. Pierre de Fermat came up with a theorem on this topic, for which Don Zagier, an American mathematician gave a proof, which astonishingly is just one sentence long.
Game theory is a branch of mathematics with a lot of applications. Especially in economics, game theory plays a very important role. That’s the reason why a lot of mathematicians are presented with the Nobel prize in economics. A widely known example is John Forbes Nash, who’s life is illustrated in the movie “A beautiful mind”.
As the name “game theory” suggests, there are some kinds of games involved. In fact “game” is very broadly defined. It means a scenario where two (or more) perfectly rational “players” have to make some kind of decision. The decisions they make determine whether…