\end{Summer} 2021 by Ideaal! (Abacus)

S Association magazine of W.S.G. Abacus

Interview

Dies Committee A Day At...

the Subway Eline Peeters

Strategic location models

\end{Summer} 2021

INDEX 4 4 6 10

Columns Internal Thoughts Your Story The Pen Staff Says Mathematical Continued Fractions 14 Strategic location models 19

5 7 8 9, 21, 23 16 18 22 24

Fun On the couch Meme Page Interview: Dies committee Puzzles A Day At... Review: Harambee Interview: Linda ten Klooster Review: The Cookie Corner

Editorial Hi everyone! Great to see you all on the university again. I hope the first module went well for you all. This ideaal! is going to be a special one, since it is not only the first ideaal! of the new year, but also the first one to be printed in a really long time. And I get to write the redaction… Pressure is on. As you probably have read, the title of this ideaal! is \ end{summer}. For all the new members, this is a reference to an ideaal! we published in July 2020 called \begin{summer} and as true mathematicians, we always finish up our statement. But as any LaTeX writer knows, the end of one statement (almost) always signifies the possibility to start a new one. So what will it be for you? \begin{student life}, \begin{party like there is no tomorrow}, \begin{board year} or maybe even \begin{make new friends}. It is all up to you.

able fashion is probably my biggest \begin{} for this academic year. I usually never share my goals, because I believe it means bad luck in actually achieving them. But reaching goals like this is simply not something that you do alone. And the same is true for anyone who has set the goal to finish a math bachelor or master. There will always be people around willing to help you, if not your friends, then your teachers. I have been asking and accepting quite a lot of help recently and it should have been something I started doing way earlier. Sadly to only figure this out after my fifth year of studying. But your \begin{} can of course be completely independent of your study, there are many things that are more important that might need attention first. Whatever it is I wish you the best of luck with it.

Personally, have started my Timon Veurink board year as the Secretary of the University gymnastics association: Linea Recta this year, which is challenging, but a lot of fun. I realise that I still have a lot to learn about both being board and studying well, but trying to do this in a reliable and sustain-

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: Thomas Kanger (Internal Thoughts) Sytze Horjus (Your Story) Martijn ter Steege (Puzzle) Johannes Schmidt-Hieber (Staff Says) Nick Muntendam (Review: Harambee) Eline Peeters (Strategic Location Models) Diana Dalenoord (Interview: Linda ten Klooster) Jorn de Jong (Puzzle)

AGENDA • 19 November 12:45 Dies Cake Lunch • 23 November 12:45 Lunch lecture Nobleo • 30 November 12:45 Lunch lecture BouWatch • 1 December 12:45 Homemade Wednesday • 8 December 16:00 Sinterklaas • 8 December 18:00 Kantelpiet

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.

Internal Thoughts Text: Thomas Kanger The title of this Ideaal! is \ end{Summer}, which takes me back to the Ideaal! of 10 July 2020, called \ begin{Summer}. In that Ideaal! there was an interview with the five student candidate board members at that time, right now they have already completed their board year. During this very long period, a lot of things have changed. I would like to take some time to look at the differences and similarities. The association has gone from the 52nd to the 53rd to the 54th board and the Ideaal! will have its first physically interviewed 'On the Couch’ in a very long time. During the Kick-In of this year, a lot more things were possible than initially

thought and a lot more things were possible than last years Kick-In, there were amazing parties and there was a physical study-related programme. We also went from a cancelled symposium during 20192020 to an epic online NWS (National Mathematics Symposium) last year. During this long period, I have written a couple of pieces for the Ideaal!, been on the first online 'On the Couch' and made some chess puzzles for the wonderful Ideaal! Puzzle booklet, which you should have a look at. While there has not been a Kick-In camp during this period, the preparations for the special First and Second years Camp have already started. While lots of committees had to organise online activities, physical activities are becoming the norm again, which I am excited about. With all these differences, I become

Your Story Text: Sytze Horjus The Kick-In was something to look out for during my summer holiday. Especially because last year’s Kick-In was partly cancelled due to corona. This year however I would be an official dogroup parent together with Niels Berg leading the Abstract Moose pack. However things turned out a little bit different. Namely, our dogroup got cancelled and Niels and I were separated. That’s how I ended up in the second best dogroup: the Amazing Mammoths together with Jelle, Leanne and Quinten, who is better known as Taipanmaster. This title of his unfortunately never got the respect he wanted. Although he was already explaining the kiddo’s the basics of Taipan the very first

moment we got to see them, he would soon be beaten in every Taipan game he played that Kick-in. This famous card game was played a lot that Kick-In. And when I say a lot I mean a LOT. However I never doubted my raising skills as a parent, I would have never suspected our kiddo’s to be addicted to Taipan that fast. It was therefore not really surprising that after the foxhunt, which was our first activity in Enschede, a brand-new pack of Taipan cards was nowhere to be found anymore in Enschede. I can still remember the face of the shopkeeper of Comicasa when our dogroup was the third group buying only taipan that day. It is therefore that, besides the fact that there were also amazing activities and parties, I could summarize this Kick-In using only one word: Taipan.

appreciative of the current situation. While some things are still limited by government measures, life is becoming a lot more stimulating. Just like Luuk van der Werf did in his `Internal Thoughts’, I would like to encourage everyone to live life to the fullest. I would like to end my piece like I end the weekly mail, with a nice rhyme. Summer has ended, now Autumn is here, We have all had a strange year, The puzzles in the Ideaal! are always amazing, They are die-hard puzzles and deserve praising, I wish the reader a very nice day, The game of Jass is a great game to play.

On the Couch... Text: Jorg Gortemaker and Timon Veurink

P: Daan Pluister Mts: Martijn ter Steege T: Thomas Middelkamp V: Marnix Vos Ideaal!: GOOD MORNING EVERYONE! 1. Introduce yourself! T: I'm Martijn Mts: I'm Thomas T: Martijn is a super sporty boy and played football on Saturday and is quietly enjoying a cup of coffee. (Pluister comes in) Mts: For the people who don't know Daan Pluister yet, this is Jochem Pluister. P: No, my twin is Frits. Mts: This is Daan, in his spare time he likes to do his homework on the train. P: So this is Marnix, he had a really good day until "GOEDEMORGEN!" happened. V: Let me start by saying it's a bit of a weird task, I have to introduce the man who needed no introduction. 2. How long have you been sitting on the couch? T: I think 10 minutes. P: I think 5 minutes V: I think 3 minutes. Mts: Me a little longer. 3. What is your favorite memory on this couch? Mts: Gossiping about the "GOEDEMORGEN" sir.

V: Is this 18+? P: No active memories. T: I like the conversation I just had with Martijn, my best active memory 4. Who do you prefer to sit on the couch with? T: This group is really magical and beautiful, and Paul and Bozho. V: And Kenneth. T: I think that's a nice composition, those are people you can mess around with. Mts: I do want to sit on the couch with a few models. T: The question is, of course, who do you prefer to lie on the couch with? Mts: Then with Paul of course. 5. What is your favorite article in the Ideaal!? Mts: Poldertalk V: Daan has made a very nice puzzle. P: I always really like Stefan's puzzles. P: I like Marnix's piece the most, who might not have written it. Mts: I like Thomas' piece about his bachelor assignment 6. What was the highlight of your day? V: Everything before "GOEDEMORGEN!" happened. T: That I found out that I had to hand in an assignment before 12 at noon.

P: Last night’s party, it was my girlfriend's birthday. Mts: When I had to go to the bathroom last night. 7. What will be the highlight of your day? P: It doesn't change. Mts: The liana of Marnix. V: Tonight we are going to play volleyball again with NSE. T: The TWE, we’ll start preparing this evening. 8. What would your committee be called if you were in a committee together Mts: The CoBoCo, the cool boys committee. T: MNSCDCo, the Monday Night Sports Center Drink Committee, but then we miss Bozho, Lars, Erik, Pelle, Hugo and other sportsmen V: We’ll change it to MTS. P: But the admins are Lars, Pelle, Lucas and SCOTT E JONKER. T: Yeah, we still missed Scott. 9. What would your perfect lunch be? V: Not my salad. I do like the croquette sandwiches, as a realistic lunch. T: But on brown bread. Mts: I’m always in for a bowl of noodles. P: The leftover free lunch if there is any left.

10. What is your favorite math theorem? P: I think it was the theorem I said last time Mts: The Cramer-Ráo Bound T: Let's have a look (grabs book and laptop).. V: n is not np theorem, but that is not theorem, so gotcha T: Just do the Radon-Nikodym theorem Mts: Shout out to the Bolzano Weierstrass theorem 11. Are you a cat or a dog person? V: Dog, I have a dog, cats suck. Mts: Do a chihuahua P: I'm not really an animal person.

V: Daan's point of view is actually: 13. What are you going to do next? if I can't throw coal in it then I don't V: Discrete Optimization with Daan. need it. P: Yes, me too with Daan. And Marnix. 12. Cayley or Hamilton? T: I'm going to study in the Waaier, T: They of course have a common with friends. And I'm going to study theorem. my friends at Waaier. And Maarten V: Hamilton, good guy, but I do hate too, because that is Marnix's real him again after yesterday. Some- name and probability and integrals times he just needs to keep his with Lebesque. mouth shut. V: and probability! T: Yeah, he just needs to shut up. That was what happened on Saturday, Hamiltonian cycles. P: What I replied last time. T: Mts goes for Cayley of course.

The Pen Text: Fia Nase Then: Turducken It’s almost the third Thursday of November which means a very special holiday season is coming for Americans. It’s time to put the butter mold in the freezer and take the canned cranberry sauce out from the back of your pantry. It is officially turducken time. You might find yourself wondering, what in the world could this turducken thing be? Well, to answer this question you need to have some basic understanding of word math. By definition: Turkey + Duck + Chicken = Turducken Now, there isn’t a lot of real information about this delectable bird dish available on the internet, so I am asking for you to trust me on this one. I also ask that you do not search for the recipe of a vegetarian or vegan option of this meal, the results may be offensive. With that being said, I will explain the importance of the turducken

on Thanksgiving day. There isn’t any. Most people go for the traditional bread stuffed turkeys that are more likely to explode in the fryer than end up on your dinner plate.

But you have to admit, the word is fun to say and there will always be a grandma somewhere out there that insists on putting three birds in one and then forces their grandchildren to eat three different textures of meat to break their strict diet of dino-nuggets. In order to make one, you buy each bird and then put the chicken in the duck in the turkey. Turduckens have no bones in them and are very easy to carve. They also happen to be massive, on average each tribird will weigh 14kg. I have never seen a turducken being completely finished, I don’t think anyone has.

There will always be leftovers and there will most definitely be turducken sandwiches being served for lunch each day until Christmas. The purpose of this bird is to celebrate all that we are thankful for. Turduckens solve most of the world's problems. If you suddenly find yourself stranded, but with money, in a country town somewhere in Minnesota, then I advise you to make your way to the closest supermarket and pick up one of the ready-made turducken meals. It will probably taste gross and warming it up in the microwave will overcook the meat, but at least you aren’t starving. If your angry in-laws were acting up at the last family function, send them a turducken. The dominance asserted by this action will make them sit quietly at the Thanksgiving table. In conclusion, solve all problems with turduckens: If: (insert problem) Then: Turducken

Meme page

Interview: Dies Committee

This year Abacus celebrates its 52th birthday with a full week of activities. This could not have been organised without the Dies committee: Justus, Lieke, Linda, Marjolein, Sanne

Text: Jorg Gortemaker and Daan van Kats

Can you introduce yourself to our youngest readers? (Lin: Are you calling us old?) Lie: My name is Lieke, I am 18 years old and I am a freshman, I come from Friesland and I am the chairman of the Dies committee. Lin: I'm Linda, I'm a fifth year and Commissaris commissieuitje, right? Sa: I'm Sanne, I'm also a fifth year student just like Linda and I'm the secretary J: I'm Justus and I'm an even older prick than the others here and I'm Commissaris ‘Ik ben niet de penningmeester’ and I'm tipsy Se: I'm Sem, 23, male, sixth year, 1.86m, and I'm Commissaris vreetzak en geld. M: I'm Marjolein, I'm 22, I don't do math anymore and I'm the Feestactieve commissaris.

Why do you think it is important to celebrate Abacus's birthday?

Lin, Sa, M: 52 'vo J: And now a normal answer, from Sem? Lin: Maybe we can't give a normal answer indeed. Se: Anyway, why is a birthday fun? Actually, I don't like a birthday at all Sa: Birthdays are just really shitty, why are we doing this anyway? We’ll cancel everything! Lin: But throwing parties is fun! Se: It's just a reason to party, a reason to eat more, and one to spend money. So we're just going to do that. Sa: Every reason for celebration is a good reason Lin: And that's how we get these soft sweaters. Se: We're just paying for these though!

Ju: Okay, there were some nice people on the 52nd board, but that's really not the reason to celebrate it Lin: Sure it is Se: My mother is turning 52 Sa: Congratulations! Se: Not until next year Lin: All connected to 52! Se: See? So many reasons why the 52nd is important!

What do you think is special about Abacus' 52nd birthday?

How did you come up with this theme for the Dies?

Lin, Sa, M: 52 'vo Sa: Because a deck of cards has 52 cards Se: And 52 weeks in a year

What do you think will be the best activity of the Diesweek? J, Sa: Everything! M, Lie, Lin: The party! Se: I actually just feel like eating. J: The red thread game. We're going to make sure Hugo doesn't win this time, that's the nice thing about it. Lin: Hugo can't even win this year.

Lin: Marjolein came up with the link of 52 with the cards. Se: The title wasn't the first idea actually, but then all of a sudden we

had a logo and it was pretty good. J: Actually, we all had to think about the theme, and only Marjolein did that. Sa: And that became the theme.

You all came up with a character that fits the theme, what is your link with the character?

Se: I don't want to link myself to…. J: I really want to link Sem to Kevin Spacey Se: I didn't mind delving into my character a lot, by watching a lot of series Lin: It was very hard for everyone. Sa: For me as well, but in a different way. What has been your highlight at Abacus so far? J: The old-board day. Lin: My first active members weekend. Sa: Being board. M: Being 52nd board! Se: I have no idea, Abacus isn't that great. Sa: Um, sorry? Se: Oh Abacus is great! But I can't call it a highlight, it's all that great. J: That the idea box has become online. M, Lin, Lie, Se, Sa: That's the low point!

What do you hope will be the highlight of Abacus' coming year?

Se: The moment when I get the high score in “Regenwormen” J: The moment we can finally get out of here because we graduated. Se: I'm just putting that off. Sa: The dies! M: More parties! Se: A birthday catch-up drink! Lin: I'm still in for that. Se: With a hundred mugs, the hundred boots drink on the hundredth day of the following year. Lin: I already feel sorry for the bartenders. J: And I feel sorry for the emergency response officers.

Why did you join the Dies committee, and why should people join it next year?

Lin, M, Sa: 52 'vo J: I give the floor to our chairman. (next dies though) Sa: 53? ew Se: Look this is all a bad reason, why did Lieke join the Dies? Lie: I was forc…no… M: I asked very nicely! But you did feel intimidated because I knew your name Lie: A little bit, because I didn't know you but... M: But I knew your name was Lieke. Sa: We were sitting here: “Lieke do you want to join the Dies committee!?” Lie: And I didn't know who they were, but they asked me… very nicely. Sa: Very calm and civilized.

Lie: And then I wanted to think about it for a moment and then I wasn't allowed to and then I said yes. Sa: Yes, you were allowed to think about it, you even walked away and came back three minutes later: "Okay." Lin: What do you think of your decision? Lie: Yeah, no regrets.

Do you have a hint for the Red thread game? J: Do your best. Sa: Be creative. Se: Make sure you're not on a team with Hugo. M: Oh yeah then you can't win. Lin: Then you're going to lose. Sa: Bring poker chips from home. M: We can send something. Would you like to give something else to the readers of the Ideaal!? Sa: Yes, cards. Lie: Enrol! Lin: Come to the party. M: Eat cake. Sa: Come to the party and bring all your friends… We’ll come alone too. J: Become an old prick and then do the Dies committee, twice preferably. Within the field of cards, there is a specific way the cards are grouped that will help you find the order and end solution.

Puzzle: What's the PIN? Text: Martijn ter Steege Susan has a PIN code that does not contain multiple numbers. She tells Mimi the first 2 digits of the code and Lana the last 2 digits of the code. Mimi says: "My numbers are an even number".

Lana says: "My digits are prime and the 2 digits separately are both prime." Mimi says: "I know the code", to which Lana says: "Then I also know the code". What is the code?

Staff Says Text: Johannes Schmidt-Hieber When I got the opportunity to write for Ideaal!, I thought it would be nice to write about the new statistics group. First of all, statistics is extremely important. The impact on our everyday life is overwhelming. For instance, in the past one and a half year, the imposed corona measures were suggested by experts based on sophisticated statistical models from epidemiology that predict the spread of the virus and the hospitalisation rate. Another example: A few weeks ago, a new robot vacuum cleaner was launched that uses artificial intelligence methods to detect dog poop. That robot cleaners drive over dog poop and distribute it over whole apartments was named the poopocalypse in a Guardian article in 2016. If you are interested in the details and some stories read the full article (https://bit.ly/2WjdvCg). Based on a large image dataset of dog poop on floors, finally these robot vacuum cleaners could be trained to circumvent the poopocalypse. Overcoming the poopocalypse was not the last unsolved statistical problem of mankind. There are so many left. For some problems, state of the art methods still perform terribly bad and we need further research from statisticians on how to overcome this. We read about the steady stream of game changing machine learning applications in the news every day. At the same time the mathematical foundations are developed to understand when AI methods work well and when they fail. The new statistics group at the UT is highly visible and active in these developments. Together we develop mathematical tools to analyse data. Currently the group consists of five full time faculty members, a postdocs and a number of PhD students. Annika Betken studies data

with an underlying time structure; Cristóbal Guzman works at the interface between optimization and data privacy; Hans Hang worked earlier for Samsung's AI lab and has a research focus on statistical learning; Katharina Proksch is an expert for multiple statistical testing procedures and applies this currently to sports data; and I am working on the foundations of machine learning. We have a clear focus on mathematical theory but also do applied projects. For instance, when I was still in Leiden, I once supervised a master student who used a large database of cooking recipes to program a generator for new ones. Clearly, new cooking recipes should be new, but they should also still lead to edible food and should not be completely random. His program suggested pepper cake. A cake with pepper in it. It was definitely new and after a round of tasting we also agreed that it was edible. In another applied project, I recently supervised a master student who wanted to use machine learning tools to predict the 'mol' in the Dutch television program 'Wie is de Mol?'. The idea here is that if one gathers data from the previous seasons (this is very tedious and time consuming work), then, one can learn patterns that can help to identify the mol. For instance it seems that in the past years the mol got slightly less screen time in the individual episodes. Such characteristics can be automatically learned from data and are used to predict the mol in a new season. Next to our research and supervision of students, the statistics group takes an active role in the transformation of the statistics curriculum. There is a lot going on, regarding both the creation and the change of statistics courses. In the bachelor program, Katharina Proksch added Multiple Hypotheses Testing as a new choice in Module 11. In the applied mathematics master, we teach now for the second time the statistical learning course. We

also added a Capita Selecta for statistics. This is a seminar, where new developments within statistics are discussed. There is moreover a new track called AI4Health within the master mathematics. While the new track has some overlap with the specialisation datascience, the focus is more applied. For this track, we have created a new course on causality. While in traditional statistical inference, one can only detect correlations without saying something about the underlying causal relationships, causal models impose more structure such that also 'A causes B' conclusions can be made from data. For the AI4Health track we also provide two so called chair cases. These cases apply the theoretical knowledge of the statistical learning course and the causality course to real data. For the statistical learning case, we plan to work as a group on an open data challenge or a statistical consulting case. Data challenges are organised for instance on Kaggle. com. Next to the math line, we are also active in the service teaching. Last year, our former postdoc Alexis Derumigny (now an assistant professor in Delft) created a new statistics course for ATLAS. The statistics of today is not comparable to the statistics of twenty years ago. The fundamental changes in the field make it necessary to constantly update the existing statistics curriculum. While computations by hand were a main focus in statistical education until recently, the availability of excellent statistical software make computations by hand superfluous. Secondly, there are endless numbers of different models and methods in statistics. Every day new ones are proposed and even experts do not know all of them. Currently the hot stuff is everything related to neural networks and deep learning. But the past tells us that trends can change quickly. Over ten years, there could be something else. Support vector machines were state of the art before the deep learning took over. It

could happen that suddenly support vector machines have a comeback. Whether a student who takes a statistics class in 2021 will stay in academia or apply mathematics in a company, they will definitely have to adapt to many changes during their career. An ideal statistics program should therefore not only teach the statistics of today but also prepare them for these changes. Let's make students ready to adapt to new developments. Here is the idea: Behind all these countless methods there are a few basic principles. Neural networks for instance heavily rely on the maximum likelihood principle and extend linear models, one of the most basic objects in statistics. Understanding of the basic principles paves the way through the statistics jungle and should be the main focus of the mathematical statistics education. This will teach course participants to understand the motivation behind statistical methods, be able to adapt to changes, and allow a master student to propose variations of existing methods. Nothing has only advantages. Whatever choice we make for the teaching, this will come with some

drawbacks. Statistics relies for instance on many approximations. To teach the basic principles of statistics in Module 5, it is nearly irrelevant how good these approximations are. Instead of taking a lot of time to derive some approximation bounds, we just skip over it. The mathematics becomes then as sloppy as I remember it from the physics courses that I took as an undergraduate student. Working constantly with unexplained approximations leaves class participants with a feeling of arbitrariness and uncertainty. Moreover, we have to rely on our own lecture notes as all the available introductory statistics textbooks appear to me unsuitable. Often they fall within one of these categories: outdated, too much material, heavy use of measure theory, complete lack of mathematical rigour, unusual selection of topics, no connection between theory and applications. To keep the course always up to date, we therefore use our own lecture notes. Wei-Ting Sun helped us a lot to update the notes and I have invested a considerable amount of time myself on it. Still they need to be polished and improved. It will take some more time, but we will do our best.

If you are interested in statistics and plan to write a thesis on a statistics related subject, you should first take as many statistics and probability courses as possible. For the master, you should either choose the specialisation datascience or the specialisation AI4Health. As mentioned before, datascience is more theoretical and successful candidates have definitely the background to apply for a PhD position within our group. AI4Health is more application oriented. If you are interested in data, you have next to the statistics group a lot of other options at the UT. The Stochastics Operations Research (SOR) group works, among other topics, on scheduling problems based on data; the Applied Analysis (AA) group connects deep learning with imaging; the Optimization group works on clever algorithms for combinatorial optimisation tasks. I do not know of any other math department in the Netherlands with such a strong focus on data.

Just Abacus

Things...

Continued fractions: The golden ratio and Fibonacci Text: Jorg Gortemaker A well-known fact of the Fibonacci sequence is that when one takes the ratios of two consecutive entries, the ratio tends to the Golden ratio. In this piece I would like to take you through a proof using continued fractions. Perhaps you’ll learn some new things along the way.

continued fraction prematurely, you can write the nth convergent as follows:

To start, what is a continued fraction? A continued fraction is of the following form:

Theorem 1

Definition 1

Proof: Proof by induction, basis step:

our equation for Ak following from our assumption:

Definition 2

With Pn being the numerator and Qn the denominator when written as a fraction. We have the following theorem: This theorem will be the key in proving that the Fibonacci sequence ratios do indeed tend to the golden ratio. The golden ratio is a well known constant, being given by: Definition 3.1

We can use this representation for rational numbers with a finite number of entries, and for irrational numbers with an infinite number of entries. The proof of this is rather trivial, if it were finite for an irrational number we’d be able to write it as a simple fraction, making it rational which would be a contradiction. If it were infinite for a rational number, we’d never be able to simplify it to a single fraction, making it irrational, which is a contradiction. This is a nice property of continued fraction, making it quite easy to see if a number is rational when the representation is finite, whilst for decimal representation it could continue on forever, even if it’s rational. Another nice property is that we can approximate irrational numbers with it, we’ll show this with the golden ratio in a bit. First I would like to show an interesting proof regarding convergents of continued fractions. A convergent is when you cut off a

Induction step: We assume it holds for k, prove it’s true for k+1:

This equation follows from its definition, described in Euclid’s elements for the first known time: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.” To write this in another way:

Notice that when we have the kth convergent we can find the (k+1)th convergent by substituting qk with qk+1/qk+1: Definition 3.2

We can also do this substitution in

Using this we can find its continued fraction representation by substituting the equation into itself: This continues on forever, it’s ones

all the way down. Then let’s also introduce the Fibonacci sequence and some additional notation. The Fibonacci sequence is defined as:

Induction step, assume it’s true for Ak-1 and Ak:

Definition 4

Fn is the nth Fibonacci number. Now let’s look at the convergents of the continued fraction of the golden ratio. Example 1

Since we have qk=1 for all k for the golden ratio we have:

What we can show next is that the sequence of P is the same as the Fibonacci sequence. We can do this by induction once again. Theorem 3 For convergents of the continued fraction of the golden ratio we have:

Notice Q1=P0 and Q2=P1. With using this as the basis step and theorem 1 we can show by induction the following: Theorem 2 For convergents of the golden ratio we have:

Proof: Basis step, see example 1:

Proof: Basis step:

Induction step: Assume Pk-1=Fk and Pk=Fk+1: By theorem 1 Since qk=1 for all k:

By definition 4: What this shows is that not only do the ratios of the Fibonacci numbers approach the golden ratio, they are even the convergents of the continued fractions. Substituting the result of theorem 3 in theorem 2 we have for convergents of the golden ratio:

The fact that this is true means some nice properties of continued fractions are carried over to the ratios of the Fibonacci numbers, the main one I would like to talk about here is that the convergents of a continued fraction are “optimal”. Optimal here talks about how when we have a convergent of a continued fraction with denominator Q it has the smallest error with the number we try to approximate in comparison to every convergent with a smaller denominator. So in this sense the ratios of the Fibonacci sequence also convert optimally. The only issue that remains is the fact that the golden ratio is very difficult to rationally approximate, this is also evident when looking at the continued fraction, every step we nudge the fraction by the smallest possible value, which makes the error shrink ever so slightly between convergents. To end I would like to leave you with the fact that it is possible to show similar results for every metallic ratios, all with their own corresponding sequence. The first step is to show for the nth metallic ratio that the continued fraction has all entries equal to n i.e. qk=n for all k for the nth metallic ratio. I leave the proof as an exercise to the reader. Good luck if you’ll be attempting to prove it, and if not then I hope you learned something new about continued fractions, the golden ratio and the Fibonacci sequence.

A Day At: the Subway

We had the pleasure to spend some time and speak with Carlos, the person who effectively runs Subway, ensures all of our lunch lectures are catered an who had Subway be placed in th e building Bastille. Text: Lavinia Lanting

As many of you may know, strolling at the back of the campus may lead you to a certain building called Bastille. Here one may find many different things such as the Drakenkelder, where you can go to play a game or read a good book, or the Student Union shop, where many of us get the readers needed for some of the courses we follow. In this particular building, one may also find the shop that has been providing for many of the lunch events we have had at Abacus as of recently: the Subway.

ted and then I set tent here. The location is really good and so is the target audience.” Carlos has thus been working at the Subway in the Bastille since the very beginning, even before its opening in July of 2017. He was the person who came with the idea of opening a Subway there and had the big open space that it was be rebuilt and shaped into what it is today.

How and why did you start?

Can you give an introduction of “I had been looking for a suitable yourself? location for a while. I had contact “My name is Carlos and I run the Subway in the Bastille. We opened in July of 2017. When I first saw the location that we now occupy, I really thought it could use some work: the back, where we now store the needed ingredients, was used as a storage for all sorts of stuff. I had the place be renova-

with many people and places and, eventually, I got in contact with the Student Union. I was the one to initiate the contacts and discussions between us and soon we found out that the Student Union definitely had an interest in granting me some space to set up the Subway. This is because the Student Union had done some research on what

students were still missing at the university and what they felt was still needed and it came out that a place like Subway was one such thing.” Carlos further remarks that the Student Union has been the best business partner he could have wished for himself, already since day one. “Everything is done in consultation with each other and we have accomplished many things together.”

Could you explain what your job consists of? “This is really my own business and I do loads of things here. You could say I’m a sort of factotum. I busy myself with about everything that has to do with Subway, from making the sandwiches to taking orders and ensuring they are ready to be picked up on time to standing here with you. I’m really an all around employee.”

How does your typical working day look like? “Well, I like to begin quite early in the morning. I live in Laren, near Lochem and I usually start my day here at around 7:00-8:00 a.m. As you might know, we also do catering, among other things, and catering costs quite some time and needs to be picked up early sometimes. Baking the bread can take quite a bit of time, so sometimes I need to get up really early. Because baking the bread costs so much time, that’s how my day starts. While the bread is baking, I then begin prepping the shop. This includes making a inventory of what I think will be needed for that day. We want our ingredients to be fresh, so in the morning we also cut everything that we think we will be needing aside from the salad, since that we buy already cut. After prepping, we open up shop and begin preparing sandwiches for lunch lectures. Then lunch time comes around and the shop gets quite busy, which is why it is important that the catering part is about done by that time. Around 2:00 p.m. the business fades and we can start cleaning and prepping for dinner. It is mostly international students who come around for dinner, while the Dutch one mostly come around for lunch. We have though quite a good deal for dinner, between 5:00 and 7:00 pm, with the Student Union. After dinner time we clean, reorganise things, I do some administration work and we set things ready for the day after. The only thing that has to already have happened before is placing the orders for ingredients, as that has to be done before 3:00 p.m. We only order twice per week since we want our ingredients to be fresh and we try to minimise spillage.”

What’s it like to work at Subway at a university? Do students make for nice clients? “Well, I like the fact that no two clients are the same and it is quite fun to work with young people. It is really different than when working in a city, where mostly adults and elderly come along. In addition, usually the shop would be only busy during lunch and dinner, but I must say that the business here is more equally spread throughout the day. Currently we are open 7 days a week due to Corona and I have worked on all 7 days for quite a while, but I am trying to go back to working 5 days per week. This is because I also need to take care of all of the administration, which takes a lot of time, especially when it comes to requesting working licenses for students that come from outside of Europe, but even though this costs time and effort, I believe these students should also be given the chance to find a parttime job.”

Do you have any interesting stories about peculiar clients you’ve had? “Chinese students have sometimes difficulty in making themselves be understood in English. One time a very much drunk Chinese student came along. He was so drunk that he could barely find the words to speak, let alone make clear to me what it was exactly that he wanted, but I also did not want to send him away. Eventually he managed to order something. It was not the best experience, but it is also very rare that something like this happens. Students here are usually all well-educated and well-kept.”

What’s your favourite Subway sandwich to eat? “My favourite one would be a wrap with just vegetables and a strip of sauce. Simple, but very tasty and it fits well with my philosophy that meat is not always needed. In that same trend, we are going to participate in the Sustainability Week, during which we will suggest and sell products that are better for both health as well as the environment. Although I must say that students here already make quite good choices for both their health and the environment. They eat whole grain bread and many don’t eat a lot of meat and I like that a lot. In addition, these days vegetarian options are many more than they used to and are quite tasty, although one must also consider that soja is also not that good for the environment, so even then you should always try to find a balance.” What’s your favourite Subway sandwich to prepare? Why? “My favourite would probably be the Spicy Stake Bacon. It’s a big sandwich which requires some work, so it is quite satisfying to make.”

Since last year, Abacus has been quite often in contact with you to order Subway lunch for various events that take place during a break. Since Abacus is the study association of Applied Mathematics, what do you think of mathematicians based on your experience with them? “I’d say you have two types of mathematicians, if I may rely on my experiences thus far. Some mathematicians are just not that good

at communicating, which I think is also very fitting of a technical study to some extent. I think it must be very different in a city such as Leiden.”

Is there anything you want to share with the readers of the Ideaal? “I really like that you’re here and that you are showing interest 2 in what other people at the university do. I can also appreciate the fact that you guys are thinking out of

the box. Keep up the good work!” It was quite pleasant to be able to spend an afternoon with Carlos, who was very friendly and welcoming and had a lot to share with us. So if by any chance you find yourself in the Bastille (or even just nearby it) and are feeling like eating something, be sure to swing by the Subway and to say hello.

Review: Harambee Text: Nick Muntendam The place to be if you want to play vol2 leyball among students in Enschede: Harambee. Harambee is one of the biggest student volleyball associations with currently 10 men’s teams, 11 women’s teams, 12 mix team’s and 4 Nevobo mix teams. Because of this vast number of teams, there is something to be found for every level 4 of volleyball, whether you are a complete beginner or have been playing for several years already. Most teams at Harambee train two times per week for 1.5 hours in the evening. Besides that, you also play a match on Friday or Saturday once per one or two weeks. Harambee usually tries to schedule all the home matches on the same weekend such that you can cheer the other Harambee teams to victory before or after your own match singing one of the many Harambee songs. These home weekends are always a big event with a Toto where you can bet on the results of the matches or a bingo card with relatable things that can happen during a match. After the matches the Sports Canteen

is open to share drinks with your own team or other teams. The highlights of the home weekends are definitely the matches of Gents 1 and Ladies 1. They both play in the third division which is the highest regional competition in volleyball. Next to the volleyball activities at Harambee, there are also a lot of parties and fun activities to attend, usually accompanied with a good amount of beer. If this is nothing for you, that’s perfectly fine, since none of these activities are mandatory. But if you do like to party, there is the monthly night out, where you can go dancing with your fellow volleyball players in the city center or join people to raid the Vesting Bar after the Sports Canteen has closed. All these activities are of course organized by a big number of nice committees who are always

4 looking for new members to help them. Besides party committees there are also committees for more serious things if that has your fancy.

Strategic location models Text: Eline Peeters Have you ever noticed that most fast food restaurants are located very close to each other? Or that there are usually two supermarkets exactly facing each other? This is not only convenient for us, the customers, but it also gives the highest profit to the companies. The placement of those companies and the fraction of customers they attract can be seen as a strategic location model. The mathematics behind this is part of Operations Research. A famous example of such a strategic location model is the so-called ice cream vendor problem. In this problem, there are two different ice cream vendors, selling ice cream on the beach. This beach is represented by a line, with locatoin values 0 to 1. The vendors can place their ice cream carts anywhere they want on the line: x ∈ [0,1]. Let’s denote the location of vendor

A with a and the location of vendor B with b, where a, b ∈ [0,1] and a ≤ b. The quality of the two different ice creams can be assumed to be exactly the same, as all clients will always visit the cart that is located closest to them. As a consequence, vendor A will attract all clients on the interval [0, a], and vendor B will attract all clients located on [b, 1]. Additionally, the set of clients in between a and b will distribute evenly over the two vendors: vendor A will attract [a, a+(a+b)/2] and vendor B attracts [a + (a+b)/2, b]. The goal of both vendors is to maximize the fraction of customers to their ice cream cart. Therefore they continuously move their cart a little more towards the middle to steal some customers from the other vendor. Eventually, both ice cream vendors end up in the middle, a = b = 0.5. In this state, both vendors will have lower profit if they move to another location, causing both carts to stay in their position. This is called the

equilibrium state. Harold Hotelling, an American mathematician from the 20th century, analysed the exact same case with even more companies in the game. He proved that there is always stability in competition, which means an equilibrium state can always be found. These models can of course also be extended to represent the reality more accurately. For example, the locations can be represented by graphs, instead of a one dimensional line. The vertices of the graph represent possible locations for the companies to locate their stores. Directed edges are created between these vertices if customers are able to travel from one location to the other. At every vertex, there is also one (set of) client(s). They are given a vertex weight to represent the spending capacity of the customers, which can be different for every client. The companies can select a location themselves, where it is also possible to have multiple different companies located on one vertex.

In this game, there are two different parties that each have different objectives. All companies strive for maximum profit, while all customers will go to the cheapest company in their neighbourhood. Therefore, this game is called a strategic two-sided facility location game. The goal is to find an equilibrium in the division of the company locations, and their corresponding profit. All companies start at a random vertex each, and for every company it is checked whether it could benefit from a reposition to another vertex. If this is the case, the company will make that move, and the profits are analysed again for this new situation. This process of calculating and comparing profits continues until there is no single company that could benefit from a location change anymore: the equilibrium! A location game can have multiple equilibria, which can all differ in their efficiency. The efficiency of an equilibrium indicates how well the total profit relates to the maximum possible total profit. Algorithms exist to speed up the process of finding an equilibrium

location distribution and the corresponding efficiency. Of course, graphs can be very different from each other and can vary in a lot of properties, such as the edge density and number of vertices. These have an impact on the computation time of the algorithm and the efficiency of the found equilibrium.

We see that the algorithm needs more steps to find an equilibrium if the ratio between facilities and vertices increases. Also, the graph type that generally gives the lowest computation time is the Power law graph. In such a graph, the degree distribution follows a power law. This means that there are many

nodes with low degree, and only a few nodes with a high degree. This causes clustering in the graph. There are way more graph properties to change, such as the input location distribution, and the distribution of the vertex weights. If you want to know more about the subject, you can check my Bachelor’s assignment :).

Puzzle: Nonogram Text: Jorg Gortemaker & Lavinia Lanting Here as a nice puzzle a full page of Nonogram. As a joke I wanted to put the lyrics of "There's no limit" here, if you don't get that joke, too bad. If you do get it here is some blank space to still put the lyrics:

Interview: Linda ten Klooster Text: Diana Dalenoord For as long as Linda can remember, she has been good at maths and solving all kinds of puzzles. So it is no surprise that she chose to study mathematics. After visiting Open Days and student for a days at various universities, she chose the University of Twente. Linda: "In the end it was the atmosphere at UT that really appealed to me, and the campus had something personal and friendly about it. It was immediately clear to me that I would feel at home at the UT.”

Is the programme what you had imagined?

"Actually, not at all. I think the problem with maths courses is that most of it is stuff you have never seen before. The courses in calculus are most similar to Maths B, and from Maths D I knew something about algebra, imaginary numbers and probability, but the subject Linear Structures came completely out of the blue for me. It was a shock at first, but then I tried to see these new subjects of mathematics as puzzles to be solved, and that helped me. The project work was a bit like what I had expected, because that was clearly explained during the Open Day. I also got the hang of the pace after a few months."

Which topics do you find most interesting and why?

"I liked Module 6, Dynamical Systems because this was a practical project. This increased my interest in Differential Equations. I saw what you could do with it. I especially like seeing that mathematics can be applied to areas that nonmathematicians benefit from. Like the field of Operations Research.

In module 8 we made timetables for a hospital. Because I would like to be meaningful to others after my graduation, I find it interesting to take up applications in healthcare, education or climate."

Can you give an example of a module assignment?

"During Module 8, we worked together with students from Civil Engineering and Technical Business Administration. We made a timetable for a fictitious hospital in which patients came in at certain times and were sent on to different departments without too many queues. So you had to make assumptions to come up with a solution. First, formulate the problem, where are the bottlenecks, and what is an acceptable solution? It was difficult to determine what everyone's role was, this took a lot of communication. It was challenging to come up with a solution in a short period."

Why did you do the educational minor?

"During my entire secondary schooling, I was already giving tutoring and helped with homework. I have always been good at explaining things and I was curious whether I would be able to handle being a teacher. I also thought it would be useful to do an internship during the bachelor, where you don't gain any real practical experience. I am very happy that I did it, because I learned a lot that you can't learn during

normal courses. Not only the ability to explain problems, strategies and solutions in different ways, but also patience, self-confidence and a combination of perseverance and flexibility. I was also keen to tell girls that a technical education is doable for them. I used to have few female examples in engineering, and I didn't know then that there are quite a few girls studying mathematics."

Can you tell us something about your bachelor's assignment?

"I did my bachelor’s assignment at the Department of Applied Analysis, under the supervision of PhD student Len Spek and associate professor Christoph Brune. My assignment was a combination of Differential Equations, which I liked but sometimes found complex, and Deep Learning, an area we learn nothing about during the bachelor’s. During module 11, I spent quite some time learning the mathematical structure behind Deep Learning. This was tough, but I also enjoyed learning the material of a new subject in my way. I researched how neural ODEs, a fairly recent and interesting development within deep learning, can be used to approximate differential equations. The bachelor’s assignment required a lot of discipline from me, especially because it was completely online. Motivation was a difficulty, but good for my personal development in terms of self-confi-

cooperation, communication, presenting, etc. I got to know myself better, both the good and bad sides, and it gave me some insight into the obstacles of a real working life."

dence and perseverance. My thesis is called "Approximating differential equations using neural ODEs", and I am proud of the final result."

Why did you become a board member of Abacus?

"In my first year I did committee work at Abacus. I noticed that I learned useful things that were not covered during my studies. You do learn to work together, but at Abacus I also developed a sense of organisation. I considered whether I wanted to join the board, because I did not want to delay my studies too much. But I decided to go for it because it is very instructive, but

also a lot of fun! I like organising, leading and overseeing things. That is why I became chairman. In addition, I became commissioner for educational affairs because of my interest in the organisation of our study programme and what could be improved. As Commissioner for Educational Affairs, you are a bridge between students and staff. Because of these two functions, I had many meetings with students, other associations and employees of UT. Part of my duties also consisted of supervising committees and organising activities, both education-related and recreational. In that year, I learned about leadership,

As Linda mentioned above, Applied Mathematics is a great programme that gives you many options. And besides her presidency at Abacus she has been an enthusiastic Open Days team leader for three years. After obtaining her Bachelor's degree, she decided to do a double Master's in Applied Mathematics and Education & Communication in Science, so she can also obtain her first grade teaching qualification. Linda: "During the minor 'Leren Lesgeven’ (teaching minor) I noticed that I like to pass on the love for mathematics to young people. That is why I decided to do the master in becoming a teacher in mathematics alongside the Applied Mathematics Master's degree."

Puzzle: Rectangles in a square Text: Jorn de Jong We start with a square, and we are going to fill the square with rectangles. This is of course easy if there is no constrain on the ratio of the rectangle, so they must have a ratio of 2:1. This means the length has to be 2 times the width. So this would be a fine way to fill the square with two rectangles. Not all rectangles you use need to have the same size, but it is allowed to use multiple rectangles of the same size.

Now the question, for what numbers of rectangles is it possible to fill the square? I already showed that 2 works, so only infinitely more for you to check ;). If you don’t understand the question or want a hint, feel free to text me at +31 6 18614483. (This question was originally asked during Oxford University admissions interview)

Review: The Cookie Corner Text: Lavinia Lanting The Ideaal! was curious which cookies and snacks are most preferred by the Abacus members. Therefore we thought it was time for a special review of The Cookie Corner. The following snacks came out the survey as the Top 3: 1. Peppermints with 46% 2. Haribo Candy with 35% Chips with 35% 3. Waffles with 30% Lollies with 30% Oreos with 30% M&M's with 30% Some special/weird/interesting suggestions: ● Fruit, Just keep the fruit. If there is only fruit it is even better ● Rice cakes ● "Ontbijtkoeken" ● Peppermints xxl for when someone really has a bad breath ● "Pepernoten" ● Instead of Unox noodles, please buy yumyums ● "Roze koeken" sometimes? ● Hamburgers ● Fryer ● Snack tomatoes ● Snack cucumbers ● Vegetarian bapao