Déjà Vu 2022

S Association magazine of W.S.G. Abacus

Interview

Yanna Kraakman Anouk Beursgens

Review D.A.V Kronos Sem Geerts

Pattern formation in biology

Déjà Vu 2022

INDEX 4 7 8, 11

Columns Treasurer's Thoughts The Pen Your Story Mathematical Project Module 2 6 Pattern formation in biology 16

4 5 9 10, 14 11 22

2

Fun Review On the couch Meme Page Interview Puzzles Solutions Puzzles

Editorial Dear reader, this is the first Ideaal! of the year, and also the first one since nearly all Covid restrictions have been lifted. So in a sense this edition’s title, déjà vu, refers to us finally returning to normal life again. For me that means looking forward to some very nice activities, a Kick-In with everything that we want out of it.

However, some of you have not had this real experience yet: most second years have not studied before Covid, and most third years have only had half a year of Covid-free studying. So, for most of you reading, this is a completely new experience: drinks, activities and being at the university without restrictions nearly every day. So you might wonder, “why still the title déjà vu?”. The real reason I will keep from you: let’s just say we got a little confused about the day the Ideaal! would be released.

Anyways, in this Ideaal! you can find some nice pieces about the Bachelor assignment some students have already done in the first semester: if you’re interested how this was experienced read the interview with Lars! We also have some nice mathematical pieces: one from Sem about pattern formation in biology. And of course, everybody’s favourite: some more puzzles!

have some ideas for pieces to write for the Ideaal!. Even better, perhaps you would like to write a piece for the Ideaal! yourself. We are always looking for new pieces! You can reach us any time at ideaal@abacus.utwente.nl.

Will I see you around at some of the great upcoming activities, finally without any real Covid restrictions?

Jorg

AGENDA • 9 March 16:00 Drink lecture Ortec • 9 March 20:00 WeAD • 10 March 12:45 EEMCS-trip location reveal lunch • 14 March 12:30 Pi(e)-Day • 15 March 12:45 Cantus Game • 15 March 20:00 Shogi • 16 March 20:00 4th Abacantus

I hope you will have fun reading the Ideaal! and perhaps you will Colophon Editorial address: W.S.G. Abacus Postbus 217, 7500 AE Enschede Phone: 053 - 489 3435 ideaal@abacus.utwente.nl www.abacus.utwente.nl Editors: Jorg Gortemaker Tim Hut Daan van Kats Lavinia Lanting Fia Nase Timon Veurink

Guest writers: Eline Peeters (Treasurer's Thoughts), Anouk Beursgens (Review: D.A.V. Kronos), Anna Gesink (Project Module 2), Sam Baak (Your Story), Emma Donkers (Your Story), Jorn de Jong (Puzzle Solution: Zig Zag, Puzzle: Cutting Pi(e)), Diana Dalenoord (Interview: Yanna Kraakman), Sem Geerts (Pattern formation in biology), Thomas Kanger (Puzzle: Sudoku).

Ideaal! is the magazine issue of Wiskundig Studiegenootschap Abacus and the department of Applied Mathematics.

The responsibility for the articles published in this magazine lies with the original author(s).

The period of notice of Abacus is four weeks before the end of the association year (August 1st).

That means that you must cancel your membership before July 4th. If you have forgotten to cancel your membership before that time, you will pay contribution for another college year.

Would you like to cancel your membership? Send an email to board@abacus.utwente.nl.

3

Treasurer's Thoughts Text: Eline Peeters Welcome back! The title of this edition of the Ideaal! Is “Deja vu”, which is French for “I have already seen”. Well, let me tell you I have seen myself in my current situation quite some times already. You’re probably thinking “what situation are you talking about?”, so let me give you a little impression: someone from the Ideaal! (somehow it’s usually Jorg) asks me to write a specific piece for their new edition. I agree, and to make sure I don’t forget it, I add it to my long term to-do list and I put the deadline in my calendar. Usually this deadline is quite far in the future, which is nice because then I still have enough time to work on the piece and not stress about it too much. Or at least, that’s always my initial thought… Because

the situation that has happened all three times now, is that I postpone writing the piece until the day of the deadline itself and have no clue what to write about. Last two times it was a little easier, because I was asked to write about my holidays and my sports association, which are two quite clear topics. However, for this edition I had complete freedom. This freedom resulted in me writing about writing this piece: seems super normal right?!

Something else I have already seen before, is the COVID-19 situation.

Currently, we’re still restricted by some rules of the government, but it seems like clubs are going to open again! Those who know me well (or have experienced past Kick-In with me) know that I’m quite a fan of dancing to way too loud music with (a few) beer(s) in my hand. You probably can imagine that I’m very hyped about the very

likely new relaxations :D This is exactly the same situation as in July 2021: clubs were opening again, I got very excited, went to a party in the VB, and guess what I got as a reward: a COVID-19 infection!

Let’s hope the déjà vu stops at the third step this time, and I won’t get infected with corona again…

However, looking at the current infection rates, it seems like almost everyone will get infected in the coming two weeks. So by the time you’re reading this, I want to either congratulate you with having survived quarantine, or want to wish you a lot of luck in your student room for 7 days.

Kind regards, Eline Peeters Treasurer of the 54th board of WSG Abacus

Review: D.A.V. Kronos Text: Anouk Beursgens When I started studying, I wanted to learn a new sport, but which one? Most people that know me well, know that I am really bad at making choices. So, to solve the issue of having to decide which sport skill I wanted to learn, I just chose them all! How? Well, I joined D.A.V. Kronos, the athletics association! Most of you might immediately think of running, and yes that is offered in a variety of ways (sprinting, long sprint and endurance), but there are also SIX technical

4

events which you can learn (long jump, high jump, javelin throwing, hurdles, discus throwing and shot put) and, since you need strong muscles for all of these, there is also a circuit training once a week.

With so many things to do, there is training offered every day from Monday till Thursday! Depending on how busy you are, you can join all four trainings in one week or do only one training in four weeks. Also during trainings there is a lot of freedom: there are two events planned and if you still feel like doing another event, you can just do that.

Besides the really ‘gezellige’ trainings, Kronos of course also organises other activities, like an association weekend, the FLAAT (Flat Athletics Tournament) and the yearly Kronolympics. Every year Kronos organises the ‘Campusloop’

too. As the name suggests, this is a running race over the Campus and takes place at the start of April (April 6 th this year), so it is a perfect training for the Batavierenrace!

As you might have guessed from this not-so-subtle promotion, I am in the committee, so to put it even less subtle: make sure to join! ;)

On the Couch...

Text: Lavinia Lanting and Timon Veurink

1. Introduce yourself!

A: Hiiiiiii, I am Amalie, for the sjaars MAMAlie. I'm 20 years old and finishing my bachelor. I am also wearing a yellow sweater.

J: Hi, I’m Jeroen. I'm 21, and I am doing my second year of the bachelor. I am wearing a gray sweater.

O: Huye, I’m Olaf, 19, I study maths, I'm a representative of Audentis.

W: Hi, I’m Wessel, 21 and I’m wearing a beautiful black sweater.

J: Olaf is wearing a purple sweater!

O: Is that important?

J: Yes. 2. How long have you been sitting on this couch?

O: Two minutes, since I’ve been forced to join this interview.

J: Since 11:45.

A: Zero minutes! We are not sitting on THE couch.

3. What is your favorite memory on this couch?

O: I haven’t done it yet on THIS couch, Jeroen.

J: I did not receive any rose from you! Also, my favourite memories are the hour long Taipan games.

A: I think it's the fact that I get sca-

red every time somebody walks on the tables to come sit next to me.

W: I’m actually only ever doing home work here…

4. Who do you prefer to sit on the couch with?

O: With Emma en Sam. A: Who are Emma and Sam? Anyway, I like to sit on the couch with whoever. I accept anyone.

J: With the Clicker Heroes group.

O: With my own thoughts.

W: I don’t care who I’m sitting on the couch with, so long as I don’t have to do any homework.

5. What is your favorite article in the Ideaal!?

J: On the Couch.

A: The Pen. W: I really liked the cake reviews from last time.

6. What has been the highlight of your day so far?

O: I was having breakfast and then my housemate joined me with a random girl. That was fun.

J: Turning off the Vector Calculus lecture.

A: When I woke up to see the sun

had risen once again.

W: The fact that I made it to my train this morning and did not forget my gloves as opposed to the previous two times this week.

7. What will be the highlight of your day?

O: House evening with the boys!

W: I have chess lessons today! I’m going to learn how to exterminate everyone at the chess tournament.

J: Ice skating training. Next week I have a tournament, so I need to be in tip top shape, which is probably not going to happen.

A: I’m going to get cake to celebrate passing the written exam for my driver's license!

8. If you were in a committee together, how would that committee have been called?

O: Depression! Or Incest, that’s also very possible.

A: Incest is wincest! But wait, why Incest?

O: We are as close as family and Jeroen definitely wants to get some.

A: I'm leaving the committee.

O: And we are back to Depression.

A: DepresCo.

5

9. What would your perfect Valentine’s Day gift be?

J: A rose from Olaf.

O: A note that says:"It's almost February 15".

A: Valentine's Day is a scam!

10. What is your favorite math theorem?

O: I love Euler's Formula.

A: No way! You’re lying! It’s just the first thing that came to your mind.

O: True.

A: Caley-Hamilton.

J: The Fundamental Theorem of Arithmetic. It sounds very fancy.

A; You mean of Calculus, right?

J: No, but that exists too.

A: Apparently I'm not that far yet with the Bachelor.

W: I just did a Computer Science module and I learned Morgan's Law and that one is very cool.

11. Can Mathematics be a love language?

O: No. 12. Are you a cat or a dog person?

O: I am an animal person.

A: I hate all animals, except for rabbits and fish.

13. What would your perfect date look like?

O: First of all it would be with Jeroen. A home made dinner, then cuddling together under a blanket while watching a movie, then falling asleep.

15. What are you going to do next?

W: Jorrit and I need to do PDE. A: Im going to TA in Spiegel.

J: I'm going to Carré.

O: I will do Vector Calculus on the other couch.

16. Is there anyone special you would like to dedicate this interview to?

A: To all the haters.

O: To Chendo.

J: I have to let you go.

O: Can you make that a promise?

J: Your mom.

W: That person knows who they are.

14. Cayley or Hamilton?

A: BOTH! O: Hamilton.

J: Hamilton.

W: Hamilton.

Project Module 2 Text: Anna Gesink With the name of the second module being ‘Mathematical Proof Techniques’ it came as no surprise that the project would have something to do with proofs. Rewriting an existing proof for a theorem of some subject, to be exact. We had to gain intuitive knowledge for the proof and use scientific sources. Furthermore, we were asked to write about the history and applications of the theorem. This module we could pick our groups ourself, which resulted in the usual ‘do you wanna be my partner’-awkwardness we seem to have never left behind in primary school. Luckily, I already got my group formed beforehand, so we could focus on picking our topic. We could choose from a long list, and we decided on the equality of second order partial derivatives.

Or, as all of you will probably know it:

6

So we started an Overleaf, stared at the calculus book and got to work, accompanied by the usual ‘how are we gonna structure this’, ‘don’t forget citations’ and ‘why is the figure outside the margin’. Thanks to the C-situation we did most of our meetings online, which resulted in some intercontinental house tours, pet reveals and the occasional mom reveal. But we also got work done, and finished our draft just before the end of the Christmas break. We took the waiting time after this to focus on the pile of work we like to call Module Two. When we got our feedback we discussed this with our supervisor, and we could continue working. After the exams we finished up our report and handed it in. But we weren’t done yet, as we still had a presentation to do. Fortunately, this wasn’t so much work.

Since the presentations were not graded, the occasional meme appeared in some of the presentations on presentation day. For us this meant we could finally tell the joke we didn’t find professional enough for the report. Something about a

neighborhood not being a community of houses… The best presentations got prices, but those presentations were held in the morning, which unfortunately meant I could not attend those. We ended the presentation day with pizza, which was obviously very welcome.

The Pen Text: Lavinia Lanting Groundhog Day As you may or may not know, February 2 is known in North America as Groundhog Day. This observed tradition derives from the Pennsylvania Dutch superstition that if a groundhog emerges from its burrow on this day and sees its shadow due to clear weather, it will retreat to its den, and winter will go on for six more weeks, but if it does not see its shadow because of cloudiness, spring will arrive early.

The Groundhog Day ceremony held at Punxsutawney in western Pennsylvania, centring on a semimythical groundhog named Punxsutawney Phil, has become the most frequently attended and wellknown ceremony.

Now you may be wondering, what does this have to do with anything?

Why should I care for this or how this have to do with ‘déjà vu’, the theme is this edition of the Ideaal!?

Well, those of you who know me well or, like me, are movie enthusiasts, surely will have heard of ‘Groundhog Day’, the 1993 film starring Bill Murray in the role of Phil (yes, like the groundhog), a self-centred weatherman sent to Punxsutawney on assignment to cover the ceremony centring on Punxsutawney Phil. In the movie,

once arrived at Punxsutawney, Phil starts reliving Groundhog Day over and over again without any apparent reason or explanation as to why. Initially, Phil sees this as a disgrace and approaches each new iteration with anger to the point of almost madness. It is only later he finally starts seeing this time loop he is stuck in as an opportunity for self-improvement: he starts learning new languages and skills and takes the time to really get to know the inhabitants of Punxsutawney as well as his colleague Rita, their likes and dislikes and starts wanting to become a better person for all of them. Eventually, after achieving the perfect day (one centred around others), Phil breaks free from the loop.

Something that personally strikes me about this movie, is the intense change Phil undergoes. The film greatly downplays the amount of time passed during the loop: according to the writers, Phil spends about 30 years stuck in the loop.

Hearing this, terrorised me: the sole idea of being stuck in a time loop for 30 years gave me the chills. Just think about it: 30 years long spent reliving the same exact day over and over again, getting to know every single detail of it by heart, no new experiences, no new meetings with new people, just the same every single day. Right? Well, no and that is what makes Phil, in my eyes, a smart character: realising the helplessness of his situation, Phil finds his own way of making his day new every single day and he does this by learning. He learns about the people around him, but, as previously mentioned, he also learns new skills: passed the loop, Phil is capable of playing the piano and speaking French to perfection and has cultivated an interest in literature and poetry. Of the movie, this is what, later on, really left an impres-

sion on me: the idea that, in spite of the helplessness and repetition of the time loop, curiosity and eagerness to learn and improve oneself can overcome the feeling of hopelessness and madness brought upon you. I myself know that, given the time and money, there are a billion more things I would like to pursue: more books to read, more movies to watch, more languages and instruments to learn, more knowledge to be obtained. In a sense, ‘Groundhog Day’ gives me hope that it’s never too late to stop time and pursue self-improvement as well as own interests, so be sure to think over what you would like to change about yourself and make sure to also take the time to give yourself the attention you deserve.

On a side note, ‘Groundhog Day’ is an amazing movie, so make sure to give it a watch at some point as well.

7

Your Story Text: Sam Baak Hi guys, for the people who don’t know me, my name is Sam Baak.

I’m a third year student here in Applied Mathematics and that feels actually pretty weird. Because of Covid, the last past two years, if felt like the world stopped the time. But still, we had to study and make sure we pass our exams. Sadly enough, that didn’t go very well for me. But because of that, I met great people from second year, with who I became very good friends.

There is also another good thing about Covid. I know, there are not much, but during the lockdown it

was also not allowed to play volleyball. But my team came up with a great idea. We arranged a team evening every week. Hereby we saw each other in a different way, because now we had dinner together and chatted the whole evening, instead of that we played volleyball and that we really had to focus on the exercise we were doing. Actually, that was really nice, because we got to know each other way better and in a different way. Now while we are playing again, I can really feel the bond we build during that time.

better, by great activities at Abacus!

Make sure that you see people and that you don’t feel alone, because that time has been long enough.

I really liked it to write this for you, I hope you liked reading a little bit about me. I hope you all start enjoying your student life (again)!

Love, Sam

For now, I’m really happy that everything is starting again, and that we (as third year students) can get to know the second and first years

Text: Emma Donkers When I was told what the theme for this Ideaal! was, I had to laugh. The first thing Déjà Vu reminds me of is my depression.

Not many people at Abacus know that I struggle with mental illness, so maybe I’m writing my story here to break the stigma. If you struggle with fear or anxiety, self-hate or intense sadness, know that you are not alone.

The words déjà vu make me think of a circle: seeing something over and over again without being able to stop it. Sometimes it takes so long to get back exactly where you started that you don’t quite remember if it is really the same. That’s what my depression has felt like. Every few weeks I think I’m getting better and then I fall into a hole, again. I always forget how bad it was until I am in it once again.

Of course that circle also means that I get out of the hole a few days later. And I am surprised every time when I get out why I couldn’t get up earlier. Let me attempt to explain.

8

On good days I would see some bad news about a national or global crisis (you take your pick) and I am able to tell myself it will be fine, I am healthy and I have a roof over my head, I have amazing friends and a happy family. I would go on with my day. Maybe I would make a mistake or maybe my bike would get a flat tire. I would be able to say “Mistakes happen, this does not mean I am a bad person”. I would go on with my day.

My good days are when I forget I can’t talk myself out of the hole on bad days.

After months and months of therapy, several types of medication and taking a lot of time to rest and heal, these good days are getting better, and there are fewer bad days.

And now that my friends and family know what is going with me, they are often able to talk me out of the hole when I can’t do it myself. But I am always afraid that maybe this is just the “good” part of the cycle, and tomorrow the fall starts again.

I guess that is a type of bravery I still have to learn: knowing you

might fall tomorrow, but walking on anyway.

This is the story of my life the past year, and I am not the only one.

In 2007, about 7% of all people between the ages 18 and 24 struggled with depression (NEMESIS-2, Trimbos Institute). This was more than any other age category.

In 2020 the Dutch Central Bureau of Statistics measured that 9% of all people between the ages 18 and 35 struggled with depression.

If you are one of those people: please talk to someone. You are not alone. You are not a burden. If you don’t feel comfortable talking with family or friends, there are Friendly Faces at Abacus who will of course respect your privacy.

Meme page

9

Interview: Lars van der Velde

This time we took the time to look into what a student doing their Bachelor assignment goes through. For this reason we sat down to talk with Lars van der Velde.

Text: Jorg Gortemaker and Tim Hut

Could you please introduce yourself?

I’m Lars, I’m a fourth year student who is still trying to achieve their Bachelor's degree. For the rest...I box a little and I have just finished my Bachelor assignment. Not yet my Bachelor though. It will take me a bit more time for that.

What have you been working on for your bachelor assignment?

My research concerned mathematical modeling of the transport of medicines in the alveoli. In particular, it looks at how the size and diameter various medicines influence the speed of the transport.

Which type of mathematics did it involve?

What did you like most about your bachelor assignment?

In the end I had a lot of fun, especially when I started getting a lot of breakthroughs and when the model finally began working and something useful followed from it that I could use for my results.

What did you dislike about your bachelor assignment?

What I disliked and what I had most difficulty with was starting on the assignment, thinking about what needed to be done. Basically, it was hard to get that ball rolling.

Does working on your bachelor assignment give you insight on how to apply mathematics into real life?

Looking back, is there anything you would have done differently?

I should have given myself a clearer goal earlier and I should have made a better picture of what the endresult should look like. This would have given me more peace and clarity and, as a result, I would have had a lot more fun.

Would you choose a different subject if you could?

I was pleasantly surprised by this subject and I liked it better than I thought. I had given a different top three, but it was very nice to receive this subject. I think I would choose the same subject again if I could.

Do you have tips for students Not necessarily. When I looked at who are starting their bachelor I would say it mostly involved tech- the other presentation though I assignment this module? niques from Numerical Mathematics, since I had to make use of PDEs. I’ve mainly solved a partial differential equation and that was probably the hardest mathematics I've had to do for my assignment.

10

thought:'It's nice that you can apply mathematics in that field.' In relation to my own Bachelor assignment I mostly tried thinking about how you could combine it with physics, but I mostly learned the mindset on how to tackle such problems.

Make sure you have a clear goal in mind, try not to stress too much and just keep going. In the end, you'll see that you will get there and if I can do it, you can certainly do it too.

You still have some work left, but what are you going to do after you finish your Bachelor?

I think I’m going to do the Master direction of Operation Research here at the UT and after that we’ll see.

Small baby steps.

Do you have anything else you would like to say?

I wish you a great deal of strength and success. Don't drive yourself crazy with everything.

Your Story Text: Fia Nase The one thing I can say for almost all the people of Georgia is that we are absolutely terrified of snow.

Once in 2014, Atlanta received about 5 centimeters of snow, and the whole state went into a panic. I was 10 at the time, and the elementary schools quickly evacuated at the sight of the first snowflake. Lucky for me, I got off at the first bus stop. Unfortunately for my brother, he went to a private school 45 minutes away from our house that didn’t have buses, so our mom had to pick him up. All that to say, I was left to my own devices for at least 6 hours.

I turned our house into the pit stop for hot chocolate and consumed all the pork buns that my mom spent hours making the day before. I made a snowman, went sledding down the street, completely drenched the bottom floor with melted snow, and lived my best life completely oblivious to the fact that at the same time, miles away, my mom’s sports car was being pulled up the hill by some random man who must have been so happy to finally make use of his overpriced jeep.

Georgia went into a state of emergency for at least 3 days. Children who didn’t take the bus had to sleep in their schools, mattress stores opened up their doors to people who got stuck on the roads. And you might be looking down at the state for there being so much chaos and panic over a small amount of snow. But in its defense, Georgia is over three times the size of the Netherlands, and it only had one snowplow. No one had winter tires, and there wasn’t a sprinkle of salt on the ground, because why would we ever need it in the south?

I was reminded of this story because it snowed when I went back to Georgia over the Christmas break.

Days leading up to the “winter storm”, the grocery stores were raided. I went to go buy chicken nuggets and wonder bread, but all that was left was a random selection of gluten-free bread and “vegan nugget bites” shaped like dinosaurs.

School was canceled of course, but only for a day, instead of the week and a half we got off during Snowmageddon, it’s historical name.

This is because Georgia impulsively bought 39 more snowplows and enough salt to season every single potato in Idaho. How much

snow did we get that day? Half a centimeter. It melted in an hour.

And the kids still got to get the day off from school to play in the sun.

The next day, the temperature had a high of 23 degrees Celsius.

A little tip for travelers. If you are wondering how to gauge the intensity of an emergency in the south, look to the Waffle Houses. Waffle House is a popular chain of all-day breakfast places where the compilations of “crazy American” videos are made. The Waffle House Index is an actual recognized measurement for severe weather in the States. If you see a closed Waffle House, panic. Then run. Then hide.

Puzzle Solution: Zig Zag Text: Jorn de Jong The idea was to count the difference between all consecutive points.

Up and down was alternating. If this difference was 0, you get a space, if it’s 1, you get a A, etc.

The title spelled ‘GOOD LUCK’ and the text is ‘HAVE A HAPPY HOLIDAYS’ (woops). Also of course to all of you who didn’t get the message. ;)

11

Just Abacus

12

Things...

13

Interview: Yanna Kraakman Text: Diana Dalenoord Studying after the bachelor's degree in Applied Mathematics After my bachelor’s graduation, I decided not to say goodbye to mathematics just yet. I get very happy with mathematics, so it was logical to make the step to the Applied Mathematics Master. I decided to continue studying within Prof. Marc Uetz's research group, Discrete Mathematics and Mathematical Programming.

Mathematics is a hobby and a to model the movements of a skatway to solve (social) problems er. To do this, I set up differential Mathematics was for Yanna a way of relaxation. So it is not a surprise that she chose a study with a lot of mathematics. She ended up, after a great search, with the Bachelor (and Master) in Applied Mathematics and has applied her mathematical knowledge to several projects.

Some of those projects are explained below.

Skating motion modelling One of the fun module assignments I did was about motion modelling.

Together with my peers, we decided

14

equations for someone who was making skating movements. This was quite difficult and I watched many videos of how skaters move.

In this assignment, we used differential equations, programmed an animated movie, looked at what gravitational forces were involved and at what speeds they were moving, for example, at the drop-off at the start.

The Solar System Modelling is an important part of mathematics. This came up again during the assignment to model the solar system. We modelled several planets orbiting around our sun and a moon around one of those planets and looked at how the system behaved. In doing so, it is important to pick the right speed for each planet/moon. Otherwise, they might fly out of orbit or burn because they get too close to the sun. We orbited a moon around our sun and looked at how the planets behaved in relation to each other.

In addition to modelling and physics, programming was also involved during this assignment, which I enjoyed.

Internship at UT-Start-up QbayLogic QBayLogic is a digital design and consultancy company. Their focus is on leveraging FPGA technology in domains with difficult mathematical problems and solutions that need low latency and high performance. FPGA (field-programmable gate array) is an integrated circuit consisting of programmable logic components. They can be programmed as logic functions such as AND and XOR. These functions can be, for example, decoders or simple mathematical functions.

My assignment was to write a program that checks if a function termi-

nates, i.e. generates an answer for any input variable, or may get stuck in a loop. It was an exciting assignment because I was working out of my comfort zone: in a field linking mathematics with theoretical computer science. I learned a new programming language (Haskell) that I had not learned at UT and I learned what a formal language is (basically my own simple programming language) and how to program one.

The program that I created was to be used within the compiler that the company worked with. Such a compiler translates source code to computer code and simplifies the code when possible. The compiler cannot handle code that gets stuck in infinite loops, since this ultimately would result in an infinitely large FPGA chip, so this is where my program would come in handy.

the optimal outcome reached by working together. An example of a game is the Prisoners Dilemma, in which the overall outcome is best if the prisoners work together, even though this does not make sense from a selfish perspective. You may not always achieve the greatest benefit yourself, but you do ensure that the group achieves the best result.

My assignment was about the Price of Anarchy of a specific, abstract type of game. This game can be interpreted as a situation in which there are several computers and many employees who at the same time all need multiple computers to run computations. The more employees use the same computer, the slower it becomes. Selfishly, everyone wants to use the fastest computers. However, by working together and optimizing the distribution of computers, the employees may be done earlier on average. I have proved that in such games, the average computation time if players play selfishly may be up to twice as long as if they work together. It is thus really valuable to hire a supervisor who can optimize the distribution of the computers.

Graduating project in Berlin Through Marc Uetz, I received an interesting and challenging graduation assignment at the Technical University of Berlin. In Corona time, this meant that I mostly did my research alone in a room with occasional help from my German supervisor Prof. dr. Max Klimm and remotely, via online video calls, with Marc. My assignment was about the Price of Anarchy, which took me into the world of game theory.

The Price of Anarchy gives information about the possible outcomes of a game with multiple players. In particular, it compares the outcome when the players play selfishly with

An experience abroad It is great fun to do a part of your study abroad. I did my research on my own in the office (due to Covid), but during lunchtime, we went for a nice lunch together, with all the department employees. Berlin is a hip and modern city with lots of opportunities to soak up culture and party in Techno Clubs or by going to a Rave.

And my German has improved as well.

Student Life As a student, your life doesn't just consist of math and studying. I was a member of WSG Abacus, attended many activities and joined committees, so I got to know many students. It gave me a good feeling to be a member of this cool study association.

I was also a member of the Green Team Twente for a year. We investigated how we could drive a race track with our hydrogen car in the most economical way. As a strategy engineer, I calculated how you should drive at certain points (e.g. a corner) to be as economical as possible. I was also the driver and we had a test day on the Zandvoort circuit. It is very special to drive on such a circuit with a hydrogen car that drives 15km per hour, while Max Verstappen drove laps of over 300km/h. In the 2018/19 academic year we not only became European Champions, but in June 2019 we also became World Champions in London. It was great to be part of such an enthusiastic team.

What’s next I recently completed my master's degree in Applied Mathematics cum laude. And now, I am looking for a (research) job where my expertise in network/graph theory, game theory, optimization, automata and programming come in handy. Looking forward to my next challenge.

15

Pattern formation in biology Text: Sem Geerts Introduction Mathematics shows up everywhere in nature and biology, there are simply Wikipedia pages just talking about that (Wikipedia, 2021)1.

Just talking about patterns alone, we observe symmetry in many orFigure 1: Pattern formation on a Giant pufferfish (Tetraodon mbu) ganisms, say flowers or starfish, but also snowflakes; fractals appear lot about these kind of systems and d the diffusivity matrix, indicating in leaves and Romanesco broc- what mathematical tools can be how much each of the variables coli; spirals can be found in shells, used in order to examine this more wants to spread over space. Now sheep horns, and cacti; crystals closely. In this article, we will go over steady states u of course simply and generally grow in the shape of tessellations. what Turing instability means in the satisfy Maybe even your house cat dis- study of reaction-diffusion systems homogeneous steady states can plays very nice patterns like a tiger and look at a specific example of be found by solving the roots of the To deor a zebra. These patterns do not how you would apply this to a sys- reaction equation only end up in the skin of animals, tem describing two types of pig- termine the stability of this steady but also plants will grow in similar ments that can occur in the skin of state, we consider any inhomogetypes of pattern formation. Have an animal. Finally, we will also look neous perturbation v to the steady we obtain the you ever wondered how these ty- at a larger system where plants state by also grow in vegetation patterns.

In linearisation pes of patterns show up? Yeah of course: "Genetics!" you might say. the end I hope to highlight again the But how does every individual cell beauty of mathematics that shows knows how to turn black or white in that these patterns have to show a zebra? Well as it turns out, they up and not appear randomly or is where Df denotes the Jacobian don't. The natural interaction of two chosen by genetics. Of course, this matrix of the reaction equation. We competing types of pigments re- is also a recommendation to follow can now apply separation of variasults in the inescapable formation the course Mathematical Biology, bles by the form but sadly you will have for 2023 to of these patterns. make that happen. for some constant vector v . Inserting this in the linearisation results The process behind this pattern Reaction diffusion equations in formation is discovered by Alan In these systems, there might Turing. Yes, the guy that solved the Enigma code was also a ma- be multiple variables u that both thematical biologist. In fact it was spread over space in some diffu- after division by up Note now that even his last published paper that sive manner, but also interact with the left-hand side is time depenprovided one of the founding theo- one another. In biology this often dent only, and the right hand side is ries of mathematical biology. These works well, as many things are in- space-dependent only (as the funcpatterns are therefore also called teracting by either symbiosis or tion f does not operate on space or Turing Patterns and were disco- competition. Think of species com- time itself, so neither can the Jacovered in the study of reaction-dif- peting for space, plants competing bian). This can only be true if both fusion systems and are formed by for water or two type of pigments. sides are constant over space and a process called Turing instability. The way these interactions are lin- time, which means we can define ked are described by the reaction the constants During the Mastermath course Ma- equation thematical Biology I followed in the In general form, these equations which are the temporal and spatial first semester this year, we talked a read eigenvalues respectively. This defi1 Wikipedia. (2021). Patterns in nature. httnition results from the general seps://en.wikipedia.org/wiki/Patternsinnature- with D the Laplacian operator and paration of variables form Page 7 of 7.

16

Overall, we obtain the eigenvalue problem and the so-called dispersion relation Since we are interested in whether other spatial perturbations are resulting in instability of the homogeneous steady state, we consider the dispersion relation as a polynomial in temporal eigenvalue li for each spatial perturbation characterised by m. Whenever a given spatial mode (for a given m) gives rise to a positive eigenvalue li, we know that the homogeneous steady sta- Figure 2: Cheetah (Acinonyx jubatus) showing different patterns on its body te is unstable to these sinusoidal than on the end of its tail perturbations. The spatial perturand bation that gives rise to the largest positive eigenvalue is known as the most unstable mode.

It can be shown that for instability So for these kind of systems, where we are dealing with multiple variables u has dimension larger than 1), it is not sufficient to only look at the eigenvalues of the Jacobian matrix $Df(\vu) It was Alan Turing's idea that diffusion driven instability is possible in these kind of systems when the diffusion constants of the various components are sufficiently different. Pattern formation can take place when reaction-stable uniform (homogeneous) steady states turn unstable due to these different diffusivity parameters, meaning that the diffusion drives this instability of the steady state.

Patterns on Cheetahs Maybe the above was quite abstract and not easy to follow and does not immediately show you why there are different types of patterns. Therefore, we will now consider a specific example, known as the Gierer-Meinhardt model (Gierer and Meinhardt, 1972)2 2 Gierer, A., & Meinhardt, H. (1972). A theory of biological pattern formation.Kybernetik,12(1), 30–39.

on a one-dimensional domain of length L with no-flux (homogeneous Neumann) boundary conditions. All parameters are strictly positive.

This model describes the interaction between a short-range autocatalyic substance A (meaning that the presence of this reaction product will increase the production of this product even more, i.e. an activator) and the long-range inhibitor H of this reaction, which limits the formation of activator A but also itself.

Note that this is of course a highly idealized model and should be used with caution. However, it does actually produce quite significant results, especially when extended to a two-dimensional domain. With homogeneous Neumann boundary conditions on a domain of length L we of course solve resulting in The dispersion relation in this case satisfies where

of these spatial modes, we need a sufficient amount of diffusion P in order to achieve so. Steady states are namely stable w.r.t. homogeneous perturbations (the spatial mode with k) whenever meaning we need On the other hand, the dispersion relation shows that for instability we need We can, therefore, conclude that we need more diffusion of the inhibitor than the activator (P>1), otherwise this Turing instability cannot begin in the first place.

Now I have to agree with you, this seems all quite technical so what do these kind of things mean in practice. For that, we will look at a cheetah. We can see that on the body of the cheetah there is a clear spotted pattern visible, whereas the tail shows stripes. This happens for the same type of pigments, so the governing equations are likely to be the same, so why the different types of patterns? This actually (likely) has to do with the fact that

17

Figure 3: Different solutions of the Gierer-Meinhardt model for various diffusion constants (Mai et al.,2012)

the spatial lengths (like the L in the above model) are different for the two directions of the now two-dimensional domain. Every direction will give rise to a different spatial mode phi that is the most stable in that direction. When the two lengths are of relatively the same order, these spatial modes will be the same and therefore give rise to a dotted pattern. However, when one is much larger than the other, spatial modes are different at this will give rise to a striped pattern as on the end of its tail.

These different kind of patterns also show up when analysing the Gierer-Meinhardt model. When varying the diffusion constant, we get different solutions as shown in Figure 33. Evidently, this is not the same as the cheetah example, but when one would change the domain sizes, analogous results would follow.

Vegetation Patterns In my essay for the course Mathematical Biology, I considered pattern forming vegetation in semiarid sloped ecosystems. The study of vegetation pattern and their formation has started to rise in the midst of the 20th century, when several observational studies were conducted to investigate different 3 Mai, F., Qin, L., & Zhang, G. (2012). Turing instability for a semi-discrete gierer–meinhardt system.PhysicaA: Statistical Mechanics and its Applications,391(5), 2014–2022. https://doi.org/https://doi.org/10.1016/j.physa.2011.11.034

18

kinds of patterns formed by different species. One of the first to study the mechanism itself, and not just observe and diagnose the phenomenon, was by Watt, 19474.

Also research from other fields could be used. For example, aerial photographs for geological and water supply investigations in 1947 were used by Macfadyen, 19505 to generalise the concept without looking at species specifically, but by using geological factors and water flow. So far, mathematical models were left out and is a study topic of much more recent years. Much more detailed photographs have been taken in the following years after the initial discovery, some of which can be seen in Figure 46.

Here we note that these systems occur all over the world, but mostly in areas with a limited water supply.

Moreover, we see that they actually occur on several scales, with wavelengths ranging from centimetres to kilometres.

Semi-arid environments are ecosystems where water supply and -consumption are carefully balanced, and vegetation does not have an unlimited supply of water to grow everywhere. Due to the com-

4 W att, A. S. (1947). Pattern and process in the plant community.Journal of Ecology,35(1/2), 1–22. http://www.jstor.org/stable/2256497 5 Macfadyen, W. (1950). Vegetation patterns in the semi-desert plains of british somaliland.The GeographicalJournal,116(4/6), 199–211.

6 Meron, E. (2019). Vegetation pattern formation: The mechanisms behind the forms.

petition of water between plants, this can eventually lead to pattern forming vegetation, where vegetation is alternated with bare ground in 2D patterns. It turns out that in systems where a slight surface slope is present, these patterns run along the contours and are even more pronounced. Surface water runs off to lower ground, dispersing the water supply to a favourable side for the uphill vegetation. In nature, this happens for a wide range of grasses, shrubs and sometimes even trees.

We can even postulate that the patterns will migrate uphill. When water is sparse, vegetation will be mostly be spotted and not be present everywhere. Rainfall will run downstream and pile up at the uphill side of these vegetation spots. The downstream part of each vegetated area will therefore have less water available than the uphill part, and these densely vegetated areas will move uphill.

Klausmeier, 19997 was one of the first to investigate these kind of systems from a mathematical perspective. He described a 2D system on an infinitely large domain with a slope present. Here he assumed there is only one type of plant species present, or species acting in a similar fashion with the available water. Water is assumed to be pre7 Klausmeier, C. A. (1999). Regular and irregular patterns in semiarid vegetation.Science,284(5421), 1826–1828.

Figure 4: Areal photos of several vegetation patterns in semi-arid systems (Meron, 2019)

sent in surface water only. The relations in the reaction equation are all found from biological observations, leading to the set of equations

This system can be nondimensionalised to reduce the amount of free parameters, leading to

In this new system we interpret A and B as the amount of rainfall and plant decay respectively, similarly n now represents the influence of the slope on the system.

This system has three homogeneous steady states, one of which is just a trivial bare system with no plants being present (u=0) and water precisely balanced by the amount of rainfall (w=A). The others are non-trivial, but can easily be found from the reaction equation,

but only one of them is stable w.r.t. homogenous perturbations. However, non-uniform perturbations might still lead to divergence away from the steady state and possibly to some other solution! However, as it turns out, there are no other stationary solutions, but periodic solutions do exist!

The existence of migrating periodic solutions can be shown by inserting z=x-ct and rewrite the above equations using u(x,t)=U(z) and w(x,t)=W(z). When I performed a bifurcation analysis by varying the new parameter c for the wave speed in the new system, we indeed find a Hopf bifurcation. Some of you might know that this implies that a periodic solution comes into existence. Now a periodic solution in z actually implies that we also have a periodic solution in x, but that migrates to the right with velocity c.

However, this does not show that these solutions are stable. Sher-

ratt and Lord, 20078 are able to show this stability of several spatial modes. How these spatial modes look like is shown in Figure 5a and a bifurcation diagram in A showing the stability of several spatial modes can be seen in Figure 5b, where the dotted black line shows instability but a full line stability.

We therefore see that there can be multiple stable spatial solutions present in the system. One of the most intriguing properties in this model is that the theory of most unstable mode does not hold anymore in this system like in the example of the Gierer-Meinhardt model. It was previously always assumed using this Turing Instability that the system will always converge to the most unstable mode, namely the one with the largest negative eigenvalue. But here this turns out not the be the case.

8 Sherratt, J. A., & Lord, G. J. (2007). Nonlinear dynamics and pattern bifurcations in a model for vegetationstripes in semi-arid environments.Theoretical population biology,71(1), 1–11.

19

(a) Bifurcation diagrams

(b) Rep resentation of several modes for A=1.5

Figure 5: Results from the bifurcation analysis in parameterA(Sherratt and Lord, 2007)

2

We can show this by making numerical simulations. I have simu-

lated this system using a discri3 tization of 400 points, where we

vary the amount of rainfall A over time, which can be seen in Figure

4

Figure 6: Results from numerical simulation using ode45 in MATLAB

20

6. What can be seen is that for a certain constant value of A=1.5, which occurs twice, we do not have the same solution in both time instances. The history of the system also influences how the system is currently behaving, a phenomenon known as hysteresis. Simply adding the small advection term introducing the slope in the sytem in the usual reaction diffusion equations changes the system significantly enough to not be able to use regular Turing instability anymore. Apparantly, there is still a lot to study and to learn in these kind of systems.

To conclude One of the intriguing things I found when following this course, is again the power of mathematics that shows up. When simply going out in nature, we observe all these kinds of patterns of which we do not necessarily know why and how they show up. With mathematics, we do not try to describe equations for these patterns themselves, but use underlying studies from other fields (biology, geology) to describe fundamental equations. When analysing these systems equations, we see that these patterns have to show up and they cannot even be avoided. Therefore, the genetics don't determine which cell should be coloured which pigment, but the pigments have this self-ordering mechanism due to the diffusiondriven instability.

Mai, F., Qin, L., & Zhang, G. (2012).

Turing instability for a semi-discrete gierer–meinhardt system.

PhysicaA: Statistical Mechanics and its Applications,391(5), 2014– 2022. https://doi.org/https://doi. org/10.1016/j.physa.2011.11.034 Meron, E. (2019). Vegetation pattern formation: The mechanisms behind the forms.

Sherratt, J. A., & Lord, G. J. (2007).

Nonlinear dynamics and pattern bifurcations in a model for vegetationstripes in semi-arid environments.Theoretical population biology,71(1), 1–11.

Watt, A. S. (1947). Pattern and process in the plant community.Journal of Ecology,35(1/2), 1–22. http:// www.jstor.org/stable/2256497 Wikipedia. (2021). Patterns in nature. https://en.wikipedia.org/ wiki/PatternsinnaturePage 7 of 7

References Gierer, A., & Meinhardt, H. (1972).

A theory of biological pattern formation.Kybernetik,12(1), 30–39.

Klausmeier, C. A. (1999). Regular and irregular patterns in semiarid vegetation.Science,284(5421), 1826–1828.

Macfadyen, W. (1950). Vegetation patterns in the semi-desert plains of british somaliland.The GeographicalJournal,116(4/6), 199–211.

21

Puzzle: Sudoku Text: Thomas Kanger 13

13

14

14

2

13

14

13

2

14

14

Puzzle: Cutting Pi(e) Text: Jorn de Jong It’s almost Pi-day. Therefore, I have a puzzle related to +-8.5397. As we all know, with one straight cut you can slice a pie into 2 pieces. With another cut, that crosses the first one, you can have 4 pieces of pie.

How many pieces can you get with 5 cuts? You can of course try it out during Pi-day ;)

22

54

14

13

54

1

13

Rules: normal sudoku rules apply. Cells in cages, indicated by a purple or yellow color, must sum to the clue given in the top left corner of the cage.

ORTEC, WCPN and Leiden European City of Science 2022 present:

The Dutch

Puzzle and Sudoku Championships

2022

Do you enjoy logic puzzles?

Do you solve the daily sudoku in the newspaper?

Do you want to compete with the best puzzlers of The Netherlands?

If so, please register for the Dutch Puzzle & Sudoku Championships!

The Dutch Championships will be held in the summer.

You can qualify online at a preferred time between Friday, March 25th and Monday, March 28th.

Scan the QR-code or go to nk.wcpn.nl to register for the championships!

The website also has all information, of puzzles examples and tips from Dutch champions.

23

24