Ingenium Blog

Nowadays, Ai is all over the place. People make use of Large Language models on a daily basis. Programmers do not need to go through the full internet to get working code. It has even the promise of making businesses more efficient. Currently there is also a heated debate going on whether it will ever culminate in the invention of artificial general intelligence. All these aspects aside, if Ai ever becomes something of significant value in a business, there is an important question to be made. Suppose an ai-model is on the verge of being shut down and the model has all tools in its hands to resist, then will it actually resist? Will it take matters in its own hands and act immorally to avoid being shut down? The complete working of Ai is something I want to talk about in another post. Hence, it might be a bit early to consider this as a blog post. However, I really want to write this post to put the recent video of Species (Documenting AGI) in a more nuanced perspective. Essential...

Last week we dipped our toes into the strange world of limits. We have seen in last post that many discret particles interacting with each other lead to a continuous phenomenon like diffusion! This week, we show how you can actually compute pi using calculus. Before doing so, we learn about one of the crown jewels of Calculus: Taylor series. It is the mathematical equivalent of taking apart a function’s DNA and rebuilding it out of simple pieces. This is a natural follow-up on the series of last week. First of all, we need to ask ourselves one big question. Today, we use calculus to find the value of pi. What is the Taylor series? In calculus, Taylor’s theorem gives an approximation of a $n$-times differentiable function around a given point $a$ by a polynomial of degree $n$. This is the n th order Taylor polynomial. For a smooth function, we can consider the polynomial for an arbitrary degree. In that case, the Taylor polynomial is the truncation at t...

After reading this blog a couple of times, you may ask: Well, Jasper. You say you like math and science, then what do you do for a living? Well, I am a PhD-student and work in the field of control theory and dynamical systems. What I exactly do is surely something I want to talk about in the future, but it might lead us too far for now. Since I am a PhD-student, I also have some teaching duties. Hence, I am giving the exercise sessions of the calculus course in the engineering science program. These exercise sessions are really nice because it provides some variation during the week. I can dive in the math I also had to study and it's nice to go back to the roots. In addition, these exercise sessions provide good suggestions for blog posts. When teaching calculus, one of the most common, and honest, student questions goes something like this: Why do we need to learn about limits and all these messy rules like L'Hôpital's and Taylor series? How will I ever decide if ...

Friday last week, I was talking with a friend on the train back home. The train just stopped in the middle of nowhere without any station in sight and we were thinking about what would cause this. I said it probably had to do with scheduling. Sometimes trains have to stop so that other trains can pass or clear a section of track. It’s actually a matter of optimization. I argued that you can model the entire rail network as a mathematical problem where you try to minimize the total delay time, while respecting all the constraints of routes, stations, and train frequencies. Maybe this is something for another blogpost because optimization theory is another huge interest of me. My friend laughed when I said I might work on this in my spare time. Then I noted that knowing a bit of math can help you in a quiz, like the one I once did about how to see easily if a number is divisible by three . And funnily enough, the following week I was actually preparing to host a quiz myself. I had ...

Some of you may have heard about The Monty Hall problem. It originates from the American television game show Let's Make a Deal and named after its original host, Monty Hall. It goes as follows: Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a sportscar. Behind the others there are only goats. You pick a door, say No. $1$, and the host, who knows what's behind the doors, opens another door, say No. $3$, which has a goat. He then says to you, "Do you want to pick door No. $2$?" Is it to your advantage to switch your choice? At first glance, this problem seems simple. But it’s one of those puzzles where our intuition completely misleads us. It’s a perfect example of how probability defies common sense. The original question became famous as a question from reader Craig F. Whitaker's letter quoted in, and solved by, Marilyn vos Savant in the Ask Marilyn column in Parade magazine in $1990$. Before we...