Take it or Leaf it 2021 by Ideaal! (Abacus)

S Association magazine of W.S.G. Abacus

Staff Says

Ruben Hoeksma A Day With...

Attentive Listeners Martijn ter Steege

Statistical Disclosure Control

Take it or Leaf it 2021

INDEX 4 4 6 10 11 11 25

Columns Chairman's Thoughts The Pen Staff Says Your Story General Adjunct's Thoughts National Mathematics Symposium Congratulations, Tracy! Mathematical Statistical Disclosure Control when publishing on Thematic Maps 18 The Tinetti Test 26

5 12 16 16 17 22

Fun On the Couch A Day With... Review Puzzle Solution Memes Interview

Editorial Dear reader,

AGENDA

In the last four years, I have been a proud member of the committee that makes the Ideaal! possible. It was a great time: meetings that sometimes were more nonsense than sense, lots of contact with Abacus members and getting to know our lovely staff. I had hoped to keep helping out for the rest of my student time, but it turns out life is not always peaches and sunshine. Sometimes you have to make tough decisions. That’s why this will be the last Ideaal! that I’m a part of.

• 22 April Candidate Board Announcement • 23 April Casual Friday • 24 April Batavierenrace

And what an edition! Some great mathematical additions have been provided by peers, two interviews as always and some wonderful columns. The committee has been working like crazy to get it together throughout exam stress and online meetings, mostly without me being able to help.

• 28 April Coffee morning Pizza Feud

In my personal opinion, life is a string of choices: do or do not. Take it or leaf it. For me, the time has come to leave the Ideaal! behind, but I hope with all my heart that other creative people will keep taking part in this great magazine.

• 29 April Procam Pitch Workshop

Now to end this appreciation party: enjoy this edition of the Ideaal!. Emma Donkers

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: Emma Donkers Jorg Gortemaker Tim Hut Daan van Kats Lavinia Lanting Timon Veurink

Guest writers: Daan Velthuis (Chairman's Thoughts), Ruben Hoeksma (Staff Says), Jeroen Fränzel (Your Story), Clara Stegehuis (General Adjunct's Thoughts), Thomas Kanger (Review: D.T.T.V. Thibats), Martijn ter Steege (Statistical Disclosure Control when publishing on Thematic Maps).

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.

Chairman's Thoughts Text: Daan Velthuis The title of the Ideaal! “Take it or leaf it” suits this time really well in my opinion. Of course spring is coming along and everywhere you look the nature turns green again. It is getting warmer by the day, except for the few days that it suddenly starts snowing. This describes the “leaf it” part, but the “take it” fits the association. Every committee is taking the current situation and is making the best of it. They are finding alternatives no one even saw as a possibility before this year. Look at the

Christmas dinner committee for example. The “Christmas” dinner is coming up, not in the usual way all together in a big room, but all together in a big virtual room. The Ab-Actie has to stretch in their brainstorms to keep on coming with new fresh ideas, but I am honestly impressed with the number of fun activities they are organising each module. With a little bias, I especially like the new weekly activity “Smarter by the Second”, which I have the honour of hosting.

entific lounge, which I found really interesting. All in all, the committee has shown this year that they do not need to use a free lunch to lure people to their activities (although you can get free lunch again!). These are only a few examples of members of Abacus really taking it. I wanted to say that I am feeling lucky to be the chairman of an association with such active committees. Thank you all!

The education committee have provided us with an excellent education week with a brand-new sci-

The Pen Text: Jorg Gortemaker Take it or leaf it, immediately reminds me of a song by the Cage the Elephant. The song is about either going all in into a relationship or just simply ending it all; “Stay with me, or cut me free”. Lately I’ve been finding a lot of parallels with this sentiment and trying to follow your courses as a student, will you be able to stick with it or will you let go, only to have to try and catch up again. Will you take it or leave it. Before the whole horrible pandemic thing this definitely has been a part of the student culture in my experience. For example some students have a hard time trying to keep up in the first year. You’ll see them for the first few weeks after which they seem to disappear, either they have trouble keeping up or they found out the study isn’t something for them. I really don’t want this to be underestimated however, keeping up can be quite hard. Missing a single lecture, for whatever reason, means you have to catch up. This is hard. And as I have tried to point out already, it doesn’t take much

to miss out on more lectures. It’s take it or leave it, either you’re all in or you’re out. Why would you go to the next lecture if you’ve missed the previous one, you won’t get it anyways since you’ve missed part of the course, and then the same is the case for the next lecture and the next lecture, so on and so on. Catching up is quite troublesome, nobody wants to have to do it. In my opinion, trying to keep people hanging on is a big task for the student organisation. Having contacts within the study is a big motivation to keep trying and for me it also makes it easier to keep trying, studying means hanging out with the friends you made at the study, it makes it a lot easier to stick around. I’m guessing you can figure out where the pandemic comes in here. The pandemic has brought in several issues for students, all with their own negative effects. Not being able to physically attend lectures makes it rather hard to pay attention to them. But what’s important here, trying to keep up is extremely hard without the environment of other students. What I miss most is the Educafé, the Abacus room.

This gave me the environment to study well in, it gave room for focus and also room for something relaxing to do when that focus was gone for a bit. Was it just some talking, grabbing some coffee or tea from the Abacus room or -my favourite option- grabbing a game from the Abacus room and playing for a bit. It’s a great environment and it kept you on track, even if you only studied for half an hour at least you did something and you weren’t instantly tired from having to try and focus that hard. Although currently people try to recreate this environment by studying together online, I have tried it myself, it is still quite easy to lose grip. When somebody doesn’t show up, perhaps they missed a lecture and don’t feel like studying or are watching the recording of the lecture. Do this for every student in the group and after just a few lectures no one’s left. Also during these study sessions trying to recreate the environment is hardly possible, you can’t expect to have the same amount of focus during these online study sessions, you’re still stuck staring at a computer screen. And forget about the quick talk, the quick coffee and tea break and playing games in between. Al

though all possible online, in my experience they require more energy/focus than you get in return. The situation for the first years is even worse I think, how would one find such a group, the ways to get to know people is quite limited. As a member of the Ab-Actie I’d say, we’re trying our best, but in the end they’re still online activities and trying to pull people towards them is quite hard, especially first years who are quite hard to reach online. Abacus comes in here as they have the task to try and keep people connected within the study, among

other things of course. There have been several activities in this regard, the most obvious one being the members meet members, a great way to get to know other members of the association. Also several activities by the Ab-Actie have been organised to try and keep members in touch with each other, but as stated before we have had a hard time attracting newer members to activities as they are hard to reach and aren’t as connected to the association it seems. So hey, if you haven’t joined an Abactie activity yet, there are several

great activities coming up this last module so go check them out. For my final words I would like to end a little more positive, the end of the pandemic is coming up, we hope, and hopefully this means that we can go back to the normal way of studying, including all the other things in case we lose our focus. For the time being, hang in there, try to wrestle through these final weeks in the online environment and try to make the best of it, perhaps by joining a few of the upcoming activities.

On the Couch with The Prom committee Text: Jorg Gortemaker patterns. L: That’s also possible on a bow tie however. L: Oh wait, I just found a tie with a duckie pattern, I think I’ll go for tie.

1. Could you introduce yourself and tell us what you’re doing in the committee? E: I am Eline, third year student Applied Mathematics and I am the chairwoman of the Prom Committee. L: Hey, I am Leanne, a first year student and I do the decorations for the Prom Committee. 2. This is of course the on the couch, so, is anyone seated on a couch? Everyone: No! E: I am sitting on my desk chair. L: Yep, same for me.

L: Yeah, a lot of what we’re doing has several alternatives. 5. How has organizing the prom been going so far? E: Mwoah. E: You’re living with a lot of uncertainty, for example looking for a venue is troublesome, since they don’t know what is possible and we don’t know what is possible. 6. Is there something you’re looking forward to? L: I’m very enthusiastic to talk about the theme. E: Yes, same for me.

3. How long have you been sitting on your chair? E: 10 minutes. L: 5 minutes, I sat down perfectly in time for this interview.

7. Could you perhaps already reveal something about the prom? E: After some discussion we decided to reveal a small hint towards the theme: Brabant.

4. You are of course members of the Prom Committee, could you tell us shortly what you do? E: At the moment it’s just a lot of working out scenarios.

8. Usually I would ask about Cayley or Hamilton or if you like cats or dogs better, in this case I would like to know, bow tie or tie? E: I’d say tie, because of the fun

9. What is the best drink to get at a prom? E: I would say, try out every single drink possible. L: I haven’t been to a prom while being at the legal drinking age yet, so I wouldn’t really know, I like mixed drinks though. 10. What are you doing afterwards? L: I am going to work on my project. E: I’ll have lunch outside because it’s nice weather out. I also have a Twick-in meeting to go to. 11. Is there something you would like to share with the readers of the Ideaal!? E: Be sure to get yourself a nice dress or a nice suit. L: Stay hydrated.

Staff Says Text: Ruben Hoeksma A very mathematical card game: SET (and SUPERSET) I am aware that this is one of the most corny phrases of the last year, but we live in strange times. Most of us long for social activities that do not involve a screen, but we settle for replica’s of those activities with a screen. Personally, I like to play board games and, while playing games over the table is almost always preferable to playing them behind a computer screen, you don’t get to play over the table with friends that do not live close to you. Here, I want to talk to you about the game SET and one of its variants, SUPERSET, which I got to play with some of my friends again during this period. The game SET Some of you, or maybe even most of you, will know the card game SET. For those that do not, let me introduce you. SET is a game played with a special deck of cards. Each card has four attributes: the number, shading, color1 and type of the displayed shape(s). Each of these attributes can take three values (see Table 1), and each of the possible 34=81 combinations of these values is contained exactly once as a card in the deck (see Figure 12).

As such you have a card three open blue diamonds, a card two striped purple squiggles and a card one solid orange oval (see Figure 2).

1. In the original game red, green and purple. 2. The SET symbols and cards are copyrights of the publisher, therefore I have used my self-created versions of them for the figures.

A SET consists of three cards that, in each of the attributes, are all equal or all different. Thus, one solid blue squiggle, two solid blue squiggles and three solid blue squiggles form a SET, since all three cards are solid blue squiggles and each card has a different number of shapes (see Figure 3).

The game starts with twelve cards

face-open in the middle of a table of (any number of) players. The players look for a SET within the twelve cards. If a player finds one, they shout “SET!” and point out the SET to the other players. If it is indeed a SET, the player receives the three cards and they are replaced by three fresh cards. If at any point the cards on the table do not contain a SET (or the players cannot find it) three more cards are added. When all cards have been turned open and no SET is left in the cards on the table, the game ends and the player with the most cards wins. This game with very simple rules is a lot of fun, can be played with almost anyone and is very addictive. If you want to try it out right now,

you can do this on the open source website https://setwithfriends.com/. A variant of SET: SUPERSET When players get the hang of the game, they get faster in recognizing SETs, which can result in hectic game play (more so in real life than online). To slow things down again, you can make the game more difficult. The most straightforward way to do so is by adding attributes to the cards. However, this has the disadvantage that one has to create their own deck of 243 cards as well as come up with these other attribute (attributes like smell do not sound that appealing). Alternatively, you can play the following, arguably more fun, variant that we call SUPERSET, but is known under other names as well (I have heard INTERSET and ULTRASET). A big advantage

is that it is played with the original deck of cards. First, notice that for each pair of cards, there is a unique other card that completes the pair to a SET. Conversely, every card completes forty different pairs to a SET. Both these properties are easily proven when we have discussed the mathematics behind SET and I will leave it to you to find those proofs. Now, start with a number of cards open on the table. We suggest nine, but more on that later. Instead of looking for a SET, the players look for a SUPERSET, which contains four cards with the following property. The four cards can be split into two pairs such that each pair is completed to a SET with the same card (which does not need to be on the table). An example of a SUPERSET is given in Figure 4. For the experienced SET player, SUPERSET is definitely worthwhile to try and even for less experienced players it can be more enjoyable to play

against an experienced SET player, since it levels the playing field a bit. Mathematics surrounding SET Now, of course, as mathematicians many questions arise in your brains. “Is it possible for twelve cards to not contain a SET?”. “What is the maximum number of cards that may not contain a SET?”. “How likely is it to have a SET on the table?”. “Why do you call that a squiggle?”. Somewhat surprisingly, the second question was answered even before the game SET was invented (1971 vs. 1974). The answer comes from the following connection with vector spaces. Consider the (finite) field F3 with three elements3. The vector space F34, consists of vectors (x1, x2, x3, x4), each consisting of four 3. F3 can be represented to have the element set {0, 1, 2}, with addition and multiplication under modulo 3 arithmetic.

coordinates with one of three possible values each. Through the natural bijection arising from mapping the four SET attributes to the four coordinates, we can map SET cards to F34 and visa versa. For example, the vector (1, 1, 0, 2) maps to “one solid orange oval”, the third card in Figure 2. Given this representation of SETcards as vectors in F34, we can investigate how we recognize a SET. One way to do so, is by first considering a smaller set of cards. The set of all orange cards can be represented with vectors in F33, the set of all orange diamond cards as vectors in F32 and the set of all solid orange diamond cards as vectors in F31 (see Figure 5). A natural way to represent these vector spaces is by grids, F3d is then represented by a 3d grid. We can represent a subset of cards with marks in the grid. As such we have represented SETs in F31, F32 and F33 in Figure 6. Of course, the only way to form a SET in F31 is to take all three cards and we gain little information from this. However, when we observe F32, we can see a pattern emerging.

Lemma 1. Three vectors x, y, z in F3d are a SET if and only if x+y+z=0. The proof of Lemma 1 follows from the fact that 0+1+2≡0 mod3 and for all i in {0, 1, 2}, we have i+i+i=3i≡0 mod3. Another way of viewing a set is by realizing that they lie on a line. We say that they are collinear. You realize this when you see that the difference between any coordinate of two vectors is either 0, 1 or 2 (mod3), and for a SET, x, y, z in F3d, those differences, x−y,

y−z and z−x are all equal (see the orange lines in Figure 6b), since the values of the i-th coordinates of x, y and z are either all equal or one of 6 possibilities

per and even a computer drawing in four dimensions is not one of the options of our era or my drawing skills. Therefore we flatten our four dimensional grid into Figure 7. If we let the cells of the grid in Figure 7 correspond one-to-one (by there relative ordering) to the cards in Figure 1, the set in Figure 7 consists of three blue striped ovals, one blue striped squiggle and two blue striped diamonds (see Figure 8).

Now, it is easy to check that indeed the differences are equal and, moreover, if the any coordinate does not satisfy Lemma 1, the vectors are not collinear. So, when we look for a SET in twelve cards, we look for three collinear points in a grid like Figure 7. However, we cannot just draw some lines into the grid of Figure 7. The grid should be viewed as a four dimensional one. Sadly, we lack four dimensional pa-

The maximum number of cards without a SET Now, let us get back to one of the questions that we posed earlier: “What is the maximum number of cards that may not contain a SET?”. We can use our new-found correspondence to pose the same question more generally:“What is the maximum size of a set of points in F3d that may not contain a line?”. We call such a set of points in F3d a d-cap. Finding the maximum size 1-cap is a trivial exercise. It turns out that for 2 and 3-caps we can use the same technique to compute their maximum size. First consider the 2, 3 and 4-caps in Figure 9 and convince yourself that these in fact do not contain any line. From the following theorem, I will provide only a proof for the 2-cap. However, the technique can be used to prove the maximum-size 3-cap as well (the proof for the maximum-size 4-cap needs a more involved argument).

Theorem 2. A maximum-size

2-cap has 4 elements, a maximumsize 3-cap has 9 elements, and a maximum-size 4-cap has 20 elements. Proof. Figure 9a provides proof that a 2-cap of size 4 exists.

We prove that this is the maximum size of any 2-cap by contradiction. Assume that a 2-cap of size 5 exists. Let its elements be x1, . . . , x5. We can partition the three-by-three grid that represents F32 with three horizontal lines. Since a cap does not contain a line, at most two of the elements x1, . . . , x5 are contained in each of the three horizontal lines. This means that two lines contain two elements and one line contains a single element. W.l.o.g., let the single element be x1 and the line containing it be L. It is not hard to see, e.g., by enumeration, that each point in F32 is contained in exactly four lines. In the case of x1, let these lines be L, L1, L2 and L3. Since L contains only x1 and no other elements, the remaining elements of the 2-cap must be contained in the other three lines, L1, L2 and L3. Then, by the pigeon hole principle, one of L1, L2 and L3 must contain two of the other elements. But then there is a line that contains x1 and two other elements of the 2-cap and this contradicts that the 2-cap does not contain a line.

Conclusion There is much more that can be said about the mathematics of SET. We have only scratched the surface. The maximum size of a 4-cap was first proven by Giuseppe Pellegrino in 1971. This was 3 years before SET was even invented. So even without the joy of playing the game, mathematicians were already proving the important theorems for it. The maximum size of a 4-cap can be proven by counting the same thing in different ways and concluding that it does not exist [2]. In a paper by myself and my friends and coauthors [1], we have used similar techniques to prove the maximum size of supercaps. A supercap is, obviously, a set of cards that does not contain a SUPERSET. It turns out that the maximum size of a 4-supercap is 9. Which is why I would suggest nine to be the number of cards to start with when playing SUPERSET. You need the tension of the possibility that no SUPERSET is present.

If this short introduction has spiked your interest, I definitely recommend the 2003 paper by Davis and Maclagan [2]. Most of what I told you here, is based on that article, but there is a lot more literature available (among which is the book “The Joy of SET” [3] not to be confused with “The Joy of Sets”, a book on set-theory). If you are interested in the mathematics of SUPERSET, a lot less literature is available that actually speaks of the game (again, there are mathematicians who study this without the joy of playing the game), so in this case you are condemned to read our paper [1]. Now that you have read me going on about the mathematics of the game, you should treat yourself and go play. That is truth-be-told, probably the best way to enjoy SET (or SUPERSET!). As the url says, get some friends to plays with you: https://setwithfriends.com/. Even SUPERSET is implemented, although they got the name wrong and call it ultraset. Perhaps, when

theorems about SET or even new variants. Finally, I will leave you with a puzzle. Figure 10 contains exactly six SETs. Can you find all of them? Have fun with SET!

References [1] Fábio Botler, Andrés Cristi, Ruben Hoeksma, Kevin Schewior, and Andreas Tönnis. SUPERSET: A (Super) Natural Variant of the Card Game SET. In FUN 2018, volume 100 of LIPIcs, pages 12:1–12:17. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2018. doi:10.4230/LIPIcs.FUN.2018.12. [2] Benjamin Lent Davis and Diane Maclagan. The card game set. The Mathematical Intelligencer, 25(3):33–40, 2003. doi:10.1007/BF02984846. [3] L. McMahon, G. Gordon, H. Gordon, and R. Gordon. The Joy of SET: The Many Mathematical Dimensions of a Seemingly Simple Card Game. Princeton University Press, 2016. URL: https://books.google.cl/ books?id=8JojDQAAQBAJ.

Your Story Text: Jeroen Fränzel The life behind the screens. This would be my way to describe my experience so far of the university. All day sitting behind the desk, reading the books, listening to online lectures, trying to keep up with all and try to enjoy yourself with the little things we have left in our daily routine. I must also say, I do not feel so lonely anymore. I have found my place. But is this place, behind the computer, the place I want to be? The last three quarters of a year have been rough, and we had to adjust. Some periods were better than others, but I got through it. Now I am here, sitting behind that screen and writing about my experience of this school year.

Let me recap, the first module. What an exciting time, starting at a new place to learn what I wanted for a long time. Luckily, I was handed some days at the university and took them with pleasure. The second module was less on campus and more online. The dreaded time has come, as higher year students had warned me about. And it was rough: too much work, too little enjoyments. But I made friends, talked a lot with them online, played games and tried to stay positive. The third module was really something different. Again, others had warned me, but you never fully know what they are talking about. This whole new way of thinking was a lot harder than anticipated. Still, with help of friends, family (of course also from my amazing girlfriend), organized activities and the

study advisor I can get through this period. I still enjoy the study, not to worry, and I learnt to enjoy it behind a screen.

General Adjunct's Thoughts Text: Clara Stegehuis The previous year was not great in many aspects, but luckily 2020 did not only have bad things in store for me. We welcomed our first child in January 2020, I obtained a research grant, and I also started some new research projects that I find very interesting, for example on calculating the effectiveness of contact tracing. Another highlight is that I got the opportunity to be part of a great Abacus board, and observe how Abacus tried to make the most of this academic year. While online activities are not optimal, we were pleasantly surprised by how many students and employees showed up at the different activities, like the scientific lounge and Sinterklaas. So apparently, even at home, students and employees like to meet. I think this shows great promise for the next board, who hopefully can translate this enthusiasm to meet into great live activities and meetups. Pandemic survival schedule Still, the previous year was probably not the best year ever for most

people (except for my son, who has lived almost his entire life in lockdown, and does not even seem to notice). The same holds for us: without the possibility of much social interactions, the evenings were becoming a bit boring. So, as two applied mathematicians, we decided to overcome this in the most structured way possible: by making a weekly activity schedule. Now our curfew-evenings are filled with activities we enjoy: • Monday: cooking something good for breakfast/lunch tomorrow • Tuesday: follow online course • Wednesday: free • Thursday: sports • Friday: free • Saturday: movies or board ames • Sunday: sports

the evenings, but at least it helps to overcome the pandemic boreout for a bit! Furthermore, following the online course on neural biology gave me even more respect for all students who have been motivating themselves to do all coursework for the past year by themselves. Keeping concentrated while watching all those lecture videos is sometimes difficult for me as well, and I am only doing it one evening a week, voluntarily! So my deepest respect to all of this year’s students, and I hope that we will be able to interact in classes on campus soon!

So on Tuesdays I learn about neurons in our brain, last Monday we prepared pancakes for breakfast, while on Thursday and Sunday evenings I make sure to get some exercise as well. On Saturday, we solved some murder mysteries from the Belgium Crimibox puzzle games. Of course, we still hope that soon we can also invite people over again in

National Mathematics Symposium Text: Lavinia Lanting Annually, the National Mathematics Symposium (NWS) takes place and this year, W.S.G. Abacus from the University of Twente has the honour of organising this amazing symposium! This symposium will take place on June 4th 2021, and has the theme “Find your limit’’. During this day, several interesting lectures and workshops will be held in a beautiful location. In addition, lunch will be provided and all of this for just €2! Throughout the day many interesting topics will be touched upon related to our theme, such as how

social media algorithms may cause misinformations or what the limits are to the human life span based on Extreme Value Theory, subjects treated respectively by Jerry Spanakis from Maastricht University and John Einmahl from Tilburg University. In addition an orator from Optiver, Dennis Fleurbaaij, will be giving a talk about trading and two cases will be provided by the companies Da Vinci and Nobleo.

On our website you will also find links to all of our social media accounts where we post updates about the symposium, so be sure to follow us on there as well! We hope to see you all on June 4th 2021, NWS committee 2021

For more information about our theme, orators and topics and to enroll for the symposium, you may go to our website: https://abacus. utwente.nl/nws

A Day With...

Since this year, Abacus has two attentive listeners for the members: Mariya Karlashchuk and Raymon de Lange. We asked them to give more insight into who they are, what they do as an attentive listener and what being one means in their perspective.

Text: Daan van Kats

How did you become an attentive listener, and what did motivate you for that? R: I thought about it and believed that I’m in a lucky position in the whole lockdown-scenario: I have contact with friends of mine, and am still having activities which keep me in contact with other people online. But I do realize that a lot of people are stuck on their own in their student room or otherwise and do not get to meet a lot of people very regularly. M: I saw the announcement some time ago and thought to myself: I also do know quite some people who are having a really hard time during the pandemic-thing. I can imagine that some people have more difficult times than I do, and it’s nice to help these people out. Why I actually decided to do it is also because in my student house, I’m usually the person who people go to if they want to talk or something. Especially in this pandemic, some people started to open up when they had difficult times and it’s like I had crying people in my arms already four times in the past couple of months. It doesn’t sound very positive, but it kind of gives you the confidence that you are good enough to be able to listen to other people’s stories and you can help them out by listening. R: I suppose also the experience of

knowing what works when listening to people and what doesn’t helps a lot. M: Maybe we are like the mum and dad of Abacus, that might be possible. R: I do love dad jokes. With what types of things, questions or problems can members come to you? R: The general idea is that you can come with literally anything you feel that has any form of (emotional) baggage. You can basically just drop it off while talking with one of the two of us. M: If you have something on your plate then you can talk to us, always, anyways. Just contact us, then we figure something out. If the problem is rather large, for example if we suspect someone is dealing with more serious issues, like severe anxiety or depression, then we will likely at some point suggest help from appropriate professionals. R: It’s really supposed to be the lowest barrier to entry: even just having a bit of a bad day is a reason. Whatever happened or how you feel like, you can complain or just talk it out with someone. It might also be a bit more serious than that, and in that case we are not the professionals and then we’re just there to help out with smaller stuff and

maybe we can recommend ways to deal with whatever problem there is or with who you can go to for serious help. How can members contact you, and what would an appointment with you be like? R: There is an official mail that we have, but if you can find us on Discord or WhatsApp, we wouldn’t mind messaging through that as well. The point is to meet up, so whatever ways of contacting me would work for me in that case. When we meet, it will probably look like that person beginning to talk about whatever they want to talk about. Of course the point is to be an attentive listener, so it is for us mostly keeping the conversation going and listening to the other. M: It is actually kind of like when you talk to your friends about a problem, we listen and we might give some advice. It’s just like a normal conversation that you have, except that it is about a personal issue and not about something random. What would you say is the most rewarding part of being an attentive listener? R: Of course helping people out is always nice, but especially the part where there is a certain part of trust involved is rewarding. It is difficult for people to open up and talk

about whatever bothers you. I myself always like to do my best to be the person that’s easy to open up to, it is very rewarding and also nice to see people do that, since people tend to very often bubble things up more than they should. M: What is rewarding is the fact that people trust you with these problems and that you can really help them out. Usually, the thing that makes me happy the most is when I see the other person becoming more relieved of their problems or issues or whatsoever. Or at least that they are also happy again. R: It is very nice if you can almost physically see the baggage being lifted from someone's shoulders. Do you have any general tips regarding mental health that you would like to share or that you think are important to go by? R: Of course, we are not professionals, so our advices are solely based on personal experiences and should not be taken as highly as from someone who actually studied for it. But what I believe helps a lot is, maybe a little bit cliche, just talking with someone about whatever is on your mind. I found out that for most of my problems, talking with my housemates or friends helps the most, especially when I have something weighing me down. That's a tip for the smaller stuff, and for the bigger things: don’t be afraid to look for help. I think that many people have the feeling that if you go to therapy, it must be really bad. But it is perfectly normal, even when you think you have nothing really going on, you can go to a specialist to check up. In the same sense that it is okay to go to the doctor even if you don’t really feel strange, just to see if everything is still alright. M: If you feel a day less, some advices which seems really stupid really also may help. Usually exercising and moving around helps out these days when we’re stuck at home. It is always good to go outside, even if it’s raining or the wind blows you away. Just being outside already

helps, just like being in nature and R: Going to the gym was also for me walking. the thing that used to give me the structure and discipline in my life. What was your own experience You’re not looking forward to going when you went studying online there, but it’s good to have done it. etc. last year? And also, getting in that habit geneR: I would say that the first half year rally makes it so that the rest of the went really well for me. I always day also has some more structure. had a pretty strong discipline that I could keep up throughout my life, What would you say to people who so as soon as the online schooling are doubting about contacting started I just made myself always one of you? get up at the same time to be sure M: Don’t doubt! We’re very friendly that I didn’t sleep in, even though I people and would like to help you didn’t have to be in a lecture. I kept out. a certain list of things that I wanted R: The barrier to enter should be bato get done every day, which went sically zero, there is no prerequisite really well for the first half year. and no reason not to do it if you feel I must say that the second half like you need it. We are not profesyear, especially since the lockdown sionals, and in that sense we can’t became again more serious, it has offer serious help, but therefore, become difficult for myself as well. there is no reason not to contact It definitely has been difficult to us, as there is no level of seriousremain productive every day, and ness you need to have. I’ve found out that I have had to M: Don't hesitate to send us take much longer to achieve smal- an e-mail at our mail-adresses: ler things, so for now I’ve mostly mariya@abacus.utwente.nl been “trying to get by”. and raymon@abacus.utwente.nl. Fortunately, I have been able to manage the change to online teaching well enough. I can work with most of the things that are online but I feel like that is different for everybody, some people really struggle with that. M: The first couple of months were actually a relief, because before this I had a very busy lifestyle, which some of my friends will probably recognize when they read this. Every evening was full and if I wanted to have an evening with my boyfriend together, I had to plan it two weeks in advance. Only like after half a year or so, I started feeling more shitty and especially in the second lockdown the corona-vibes hit me very big. I don’t know, but it was not nice and I felt like shit the whole time. With the studying I didn’t have problems, only the fact that I miss real life conversations with my project mates instead of via Discord. But for the rest it was actually quite okay, I don’t feel terrible about this situation, I just do miss the gym for example.

Just Abacus things...

Review: D.T.T.V. Thibats Text: Thomas Kanger Drienerlose tafeltennisvereniging Thibats is the table tennis association for students in Twente. Thibats has approximately 50 members. When the Sports Centre reopens we will have training in SC 6. The beginner group trains every Monday for 2 hours and the more advanced players train for 2 hours every Tuesday. During our training, we focus on one playing technique, like playing forehand or giving topspin to the ball. We generally play matches with only that

technique KOTH style and to end the training we usually play some normal matches. After practice, we are generally getting a drink in the cantine and we regularly finish the night in the Vestingbar. Besides the training sessions, you can also play freely for an extra hour after the training sessions. And additionally, you can join a Thibats team to play competition against other table tennis teams. Thibats’ TEK committee organizes fun activities many times a year. Some of these activities have something to do with table tennis like

Puzzle Solution: Knight Sudoku Puzzle Text: Timon Veurink To the right of this text you find the solution of the Knight Sudoku puzzle. A full explanation is available on YouTube in a video called: ”Chess Sudoku: A Knight Puzzle!” by the Channel: Cracking The Cryptic. But if you still find yourself having much difficulty solving this puzzle, don’t be too harsh on yourself. Even the best horse trips sometimes.

the yearly Sinterklaas tournament, other activities have nothing to do with table tennis, like going karting or going bowling. Due to Covid they now organize online activities, like a chess tournament, which are fun. Thibats also organises the International Moekotte-tournament every year, table tennis players from the region and some students from other universities join to play in this table tennis tournament. Unfortunately, the 2020 Moekottetournament got cancelled due to Covid but hopefully, there will be a 2021 edition of the tournament.ersity back then.

Meme Page

Statistical disclosure control when publishing on Thematic Maps Text: Martijn ter Steege Statistical disclose control is essential when publishing data. Public authori-ties, such as Statistics Netherlands (Dutch: CBS), do research about various topics. Some of these topics are more sensible than others: for instance publishing the average age of a group of people is not sensible, but the average income is. Therefore, the privacy of each individual within a group should be guaranteed. In short, statistical disclose control makes sure that everyone’s privacy is ensured. The next question is, how can we make sure that each individuals privacy is ensured? In a previous paper by Douwe Hut [1], this was done by adding noise using the (p,q)-rule. This rule makes sure that it is impossible to estimate some individual data within p%, if all other contributions’ lower bound are known within q%. I won’t go into too much details, so if you want to read about this, I’d recommend you to read [1]. In my Bachelor assignment, I’ve looked into another mechanism that protects group data: the Pufferfish framework. Don’t ask me or my supervisor why this method is called Pufferfish, because we do not know. What we do know is that this method was explained in [2], but that not a lot of further research was done. Therefore, I chose to investigate into this method and t check if this method had potential to be used. The Pufferfish framework is based on the following equations:

where T denotes the data, function

M denotes a mechanism to protect the data, si and sj are mutually exclusive statements about the data (for instance: si: ‘individual h is not in the data set’, sj: ‘individual h is in the data set’), θ denotes all information that is known about the data. Both equations are sufficient to have Pufferfish privacy, where ε determines the strength of the privacy. The reader can easily verify that1 these equations can be rewritten into

using some basic probability theory. Don’t look this up in my report, because I didn’t even compute this. It is good to understand the interpretation of this computation. We can interpret

as being the initial probability that si is more likely to be true than sj. This can be seen as pre-knowledge, that is not influenced by the published data.

can be interpreted as the postknowledge, the knowledge you have after observing the data. Due to the boundaries of e±ε, and assuming that ε should be small for large privacy, we get that both boundaries are close to 1. Hence, both probability fractions should be nearly equal, so it can be assumed that no new information can be obtained by evaluating the data. That means that a mechanism M ensures privacy, when this equation holds. 1

myself!

I always wanted to use this

For my research, I proved that the addition of noise according to the Laplace distribution satisfies the aforementioned property. This Laplace noise is dependent on the amount of contributions in the group and the privacy parameter ε: a larger group or larger ε means less noise. I won’t go into much detail here, but if you want to see the proof, then you should read my paper. Another mechanism I managed to create was the relative error mechanism. This mechanism does not add noise, but it has multiplicative noise. The reason why this is useful is that we can protect data relatively now: we can say that our mechanism protects the data within k%. This multiplicative noise also has Laplace distribution, but also involves a logarithm. Now we have two mechanisms that can be applied for group privacy. Looking at my research title, I haven’t talked about the publication of thematic maps. Thematic maps can be visualized in several ways, but in my research I focussed on publishing grid maps. In Figure 1 you can see such a grid map that is used by Statistics Netherlands.

Now we will focus on publishing any data on thematic maps. We can use all data of one grid cell as group, and use the mentioned pri-

vacy mechanisms to protect them. The results of the absolute error protection maps are visible in Figures 2 and 3 and the relative error

protection maps are visible in Figures 4 and 5

In the Figures, we see that different cell sizes produce different scales of noise. This is an immediate consequence of the group size, and therefore logical behaviour. We see that the absolute error protection method works really well, especially with the larger cell sizes. Comparing these unprotected and protected grid maps show little differences, which is desirable for useful data publication.The relative error protection case shows less appealing grid maps. This follows since the multiplicating noise ((b) plots) of one cell is significantly larger than the others, which results in a useless protected map. This problem is only resolved when taking appropriately sized cell sizes, but that is not always possible in data publication. In all c) plots, we see the grid maps that could be published online, because this is proved to be safe. However, we notice that some cells produce negative averages, while all individual contributions of

a group are non-negative. This is unrealistic behaviour and should be tackled. In my paper, I’ve also introduced a privacy mechanism with bounded output, such that negative outputs can be prevented. If you are interested in this, please check my paper via https://essay. utwente.nl/85628/. That concludes my Bachelor assignment. I have to say that my research topic was really interesting, since it was different than the typical suggested Bachelor assignments. It was really fun to do, since I had the feeling that I really contributed into new research. If you still have any questions about my Bachelor assignment, feel free to ask me, because I’m more than willing to explain everything in more detail!

dx.doi.org/10.1007/978-3-03057521-2_14. [2] Daniel Kifer and Ashwin Machanavajjhala. “Pufferfish: A framework for mathematical privacy definitions.” In: ACM Transactions on Database Systems 39.1 (2014), pp. 1–36. DOI: http://dx.doi. org/10.1145/2514689.

References [1] D. A. Hut. “Statistical disclosure control when publishing on thematic maps.” In: (2020). DOI: http://

Interview: Active Members Text: Lavinia Lanting

Justus Sleurink bination of organizing something together with people and working towards something, which in most cases is something nice.

1. Introduce yourself. My name is Justus, I am a 5th year in the meantime. I have almost finished my bachelor and I have actually been active at Abacus all throughout my studies and I also did a board year 2 years ago. I have been a member of 10 committees in total, but that will only decrease from now on. 2. How long have you been an active member of Abacus? Way too long. No, just kidding, I can still see myself remaining active in some capacity as long as I'm still studying here. 3. Why did you decide to become an active member? Because I liked the Kick-In and liked the people and then the committee market came by and then I ended up in the Freshmen Committee. I liked it and I started doing more and more and then I did a board year. The rest will go without saying if you have nice people around you. 4. What is your favourite part about being an active member at Abacus? It's a tough question. Well…it is mainly the part of organizing nice things with nice people. The com-

5. Which committees are you currently in and in which role? Oh help. I'll just grab the list on the Abacus website. I am currently the chairman of the Advisory Council. I am still the Study Trip Committee's chairman, which is also not a committee of Abacus of course, and also of Stichting RSA. In addition, I am also still in the CoCo. I don't really have a position there at the moment and besides that I am currently still in the (H) ero committee and in the BataCo and I have no real position there either. I am an emergency response team member, so I am on the (H) ero committee and I organize the Batavieren race. 6. Which of these is your favourite and why? Of those that I'm in now? Good question. The most fun thing to do is no longer RSA at the moment, unfortunately, as we decided to cancel it last week. It takes a lot of time, but in the end there is not really a pay off at the moment, despite the fact that it was a lot of fun to work on. In the end I think I would choose the BataCo. It's fun, not too strict, not too small, not too planned. 7. Which other committees have you been in and in which role? I started with the Freshman Committee for which I was treasurer. Not much of that has remained for the rest of my career. Then I joined Ab-Actie for a year in my second year and I chaired the committee as well. I have been an Officer of Logistics for the Twick-In. I also spent a while in the Translacie: that was mainly to translate documents

and then the committee was disbanded during my board year. Last year I was on the Sports Day Committee, but that was more as part of the BataCo. And I have also been on the Dies committee this year as Chairman. 8. You’ve also been a board member. Why did you choose to do a board year for Abacus? Initially because I liked it so much at Abacus and I wanted to spend more time being active. I also really learned things from it on top of the fact that it was really so much fun. My main reason was that I had had such a great time at Abacus myself by being active and I wanted to give back to the other people, I wanted to make them enthusiastic about being active and I wanted to show them how much fun it is at Abacus. 9. Are you planning on being active next year as well? If yes, which committees are you planning on being a member of next year? If it were up to me, I'd do another year in the Advisory Council. I will then see if there is something small to do next to it and if nice people want to participate, such as with the Dies this year. At the moment I am mainly thinking about another year in the Advisory Council and perhaps at Stichting RSA in the Supervisory Council after we have finished handling and found a new committee. Most of what I wanted to do I have already done. 10. Which committee that you have not joined (nor will be joining) do you think does fantastic work? In any case the Christmas dinner committee. They always give me a fantastic and pleasant evening, so they always do well as far as I am concerned, but I would never want to be in it myself, because then I have to get involved and work in-

stead of sitting and enjoying and having a good time and drinking. The (A)bac(ch)us I think does amazing work for similar reasons, but in general, all committees do just fine. 11. How would you convince non active members of Abacus to become active? There is a whole strategic approach to it. You have to get them drunk during the Kick-In, befriend them

and then have them join all kinds of committees. But seriously, if you like Abacus or people from Abacus or an activity and you ever want to give something back to the association, no matter how large the committee, it is certainly worthwhile to become active. Oh yes and otherwise you have to convince people with the Active Members Weekend because that is the highlight of the year.

12. Do you have anything left to say? That is a good question. Sign up for all future activities that I organize, I would say. Just promotion for things that don't even exist yet. There you go!

Study Advice first. But I really liked the activities and I really enjoy organizing things, so that's how I ended up at the Ab-Actie. As for the Christmas Dinner committee, I only helped with cooking last year and it seemed fun to organise it too. These are all things that I went to myself and that I really enjoy going to and I just really like organizing things and that's how I got active.

7. Are you planning on being active next year as well? If yes, which committees are you planning on being a member of next year? I do want to stay in the Ab-Actie. I think they also will be very happy with that. I really enjoyed the Christmas Dinner Committee this year, but it is a pity that it is not allowed to take place physically, so maybe I will stay in it for another year so that I can also organize it physically, but maybe there is something else I could do.

Alex Schopbarteld

1. Introduce yourself. My name is Alex, I am a 2nd year student in Applied Mathematics and Applied Physics. In my spare time I try to play the guitar. I always play D&D with my friends on weekends. I really enjoy cooking, that's a really big hobby of mine, and I'm also an active member. I don't have a lot of time left next to all of this. 2. How long have you been an active member of Abacus? From the beginning of this year, so that's approximately ¾ of a year 3. Why did you decide to become an active member? I already wanted to in my first year, but I wanted to get my Binding

4. What is your favourite part about being an active member at Abacus? I just really like the people. You are in a meeting and you are working on something, but I really enjoy being together with the group and I enjoy devising and organizing activities and that it is appreciated. 5. Which committees are you currently in and in which role? I am the secretary of the Christmas Dinner committee, I am the treasurer of the Twick-In and I am the poster boy of the Ab-Actie. I also like it so much that I am listed on the website as poster boy. I think that's an honor actually. 6. Which of these is your favourite and why? I think I like making the poster is the most when I have the time. When I have little time for it, I don't like it very much because I feel like I have to rush it. I can just spend too much time on it because I like it so much.

8. Which committee that you have not joined (nor will be joining) do you think does fantastic work? I like the PixCie and the Ideaal! very nice. I don't see myself walking around with a camera though. I also haven't been on a trip yet, so I don't know for certain if they do a good job, but I'm sure they do. 9. How would you convince non active members of Abacus to become active? I just really like it myself and I think everyone who likes the activities will also enjoy organizing them. I think you should just try it out and if it is something for you, you will notice right away. 10. Do you have anything left to say? No not really. When will this come out? Do I still have to promote the Christmas Dinner?

Wessel Heerink red with the Freshmen Committee and then I thought: sure, I can do that too. 4. What is your favourite part about being an active member at Abacus? It's just fun. You have, especially in this time, people to speak to every week instead of just sitting alone in your room. I also like organizing things, so that's why I'm also part of the Twick-In and Christmas Dinner Committee this year.

1. Introduce yourself. My name is Wessel and I am 20 years old. I am a 2nd year Applied Mathematics student and I live in Nijverdaal.

5. Which committees are you currently in and in which role? I am in the www.com, but we have no real roles there. Everyone just works on the website. I have no real role within the Christmas Dinner Committee and I am secretary of the Twick-In.

2. How long have you been an active member of Abacus? Ever since my first year as member of the Freshmen Committee and of the www.com.

6. Which of these is your favourite and why? I think that would be the Twick-In. Surely it is special that I can organize something so big. A lot has to be done, but with our efforts it still all works out fine.

3. Why did you decide to become an active member? At first I joined the www.com because I love programming myself. Then only one person had registe-

7. Which other committees have you been in and in which role? I was chairman of the Freshmen Committee.

8. Are you planning on being active next year as well? If yes, which committees are you planning on being a member of next year? I think I will only be member of the www.com next year. The Christmas Dinner committee is a bit less fun than I expected, but maybe that's also because of Covid. 9. Which committee that you have not joined (nor will be joining) do you think does fantastic work? I always like the activities of the AbActie. It doesn't really matter what it is, but I really enjoy being with others. 10. How would you convince non active members of Abacus to become active? You get to know other people from other years. You build up contacts with people who also study mathematics and it is simply recommended. 11. Do you have anything left to say? Not really. The Ideaal! comes out too late to promote the Christmas Dinner.

Hugo Hof 1. Introduce yourself. My name is Hugo, I am 6th year and I am now in the first year of my master. I am doing the master DMMP and I live in the center of Enschede. I am vegan and I really love to make music. 2. How long have you been an active member of Abacus? From my second year, I think. 3. Why did you decide to become an active member? At first for the fun and I thought the

Twick-In would be great because I was allowed to participate in the Kick-In one more time. It felt worth it, also because it was so wonderful for everyone at Abacus. I also joined the (A)bac(ch)us for the same reason: because I liked the drinks so much and I wanted to offer the same experience to others. 4. What is your favourite part about being an active member at Abacus? I like committees in which you have to communicate a lot with people. I

just think that's super fun. At some committees the meetings or the committee outings and things like that are really great fun. 5. Which committees are you currently in and in which role? Currently I am the chairman of the NWS committee. I am the secretary of the Advisory council. I am the secretary of the (A)bac(ch)us.

8. You’ve also been a board member. Why did you choose to do a board year for Abacus? It felt worthwhile because I wanted to dedicate myself to giving to other people as well, given how beautiful it is at Abacus. It was of course very nice, but also very useful. I got a lot out of it for myself.

6. Which of these is your favourite and why? Depends in terms of what. In terms of fun, perhaps the (A)bac(ch)us, but we don't do anything now, but that is normally the best in terms of fun. The Advisory Council is also really fun, but the NWS even more if we can really organise it. If we can continue organising it, then it will probably really be super rewarding. It depends on how you look at it.

9. Are you planning on being active next year as well? If yes, which committees are you planning on being a member of next year? I don't think I'm getting out of the (A)bac(ch)us yet. I will also check whether there are enough people for the Advisory Council next year and whether I will still be asked. In any case, I will not be picking up new committees as I will be doing an internship and graduation next year.

7. Which other committees have you been in and in which role? At the beginning of this year I was treasurer of the Dies 2020 and for the Twick-In 2017 I was known as War: we were all named after sauces for chips.

10. Which committee that you have not joined (nor will be joining) do you think does fantastic work? I'll just take the committee list. I think the Christmas Dinner Committee does a fantastic job. I often

like to help cook on that day. And I think the Ab-Actie also works very well with all those activities that many people enjoy. But basically everyone does a good job. 10. How would you convince non active members of Abacus to become active? I would convince them by talking to them about how nice it is at Abacus and how useful it is for your student days. I convinced Gavin in this way during the Christmas Dinner to join the EEMCS Trip Committee and with freshmen if the first one wants, the rest will too. 11. Do you have anything left to say? No, but thanks for the interview. The Ideaal! is also doing a really good job.

Congratulations, Tracy! Text: Lavinia Lanting Although this academic year has been a tough one, some have managed to shine through it all. One such person is the lecturer Tracy Craig, who you may know from the Mathematics Line, International Collaborators or as a colleague if you are a Teacher Assistant of a member of the staff. This year Tracy has been nominated by the students of Applied Mathematics for the Applied Mathematics Educational Prize (AMEP), during Abacus' Education Week. Tracy gave then a beautiful speech over the theme 'Online Mathematics', which crowned her winner of the AMEP and awarded the title of 'Teacher of the Year'

from Abacus. This is, though, not all, as Tracy has actually won 3 prizes this year. Astatine also awarded her with the title of 'Teacher of the Year' and Scintilla with that of 'Best Digital Teacher'. Through this little column we would like to congratulate Tracy Craig on being awarded these three pizes and titles this year. We also would like to truly thank her for the amazing job she has been doing and for the effort she puts into teaching her students. Thank you for your hard work, Tracy, and congratulations!

The Tinetti Test Text: Tim Hut In the second part of the project of module 6 my group (consisting of me, Marion Fonseca Hoeltgebaum and Ties Martens) tackled a balancing problem regarding the Tinetti Test. The Tinetti Test [1] is used to assess the balance and stability of people in order to determine their fall risk. It consists of two parts, one related to the patient’s gait, and the other related to their balance while standing. We looked at the pushing section of the latter and in short it is: a patient is pushed lightly, if they can remain standing on their feet they get 2 point; if they need to take a step backwards they get 1 point and if they can do neither (and need to be stabilized by the physiotherapist) they get no point. This test has been determined to have quite a reasonable reliability [2], but The test can sometimes be impractical to apply, despite its low amount of necessary equipment and training, due to the potential discomfort of the patient, who might not enjoy being pushed, as well as the dangers of falling. Therefore, it would be beneficial to be able to simulate this test through modelling and predict the outcome using information about the patient which could be collected in some other way. Hence, our research question is: can we predict the outcome of the Tinetti Test based on the measured muscle strength of the patient? Method First we start by creating a system of differential equations using the Euler-Lagrange equation. To represent a body, we used a three-link and a four-link, visible in figure 1a and 2a. The x and y coordinates of the endpoints of the kth segment of the n-link are rewritten in the form xk=xm+lksin(φk) and yk=ym+cos(φk), where m=k−1 if k≠n and m=2 if k=n. All variables related to the nth segment are often denoted with υ to signify the upper body. Next,

the kinetic energy T, the potential energy U and the Lagrangian L are calculated.

Here mk is the mass at (xk, yk) and g is the gravitational acceleration. Next we use the Euler-Lagrange equation: We rewrite it in the form: Here the M represents the mass matrix depending on elements of φ and C is a column vector depending on elements of, among others, φ, where φ=[φ1 · · · φn]T. Extra torque Q is put into the system, therefore it changes into: Let since matrix M is invertible, it follows that the system is:

If we would plot and animate the nlink for Q=0 it will collapse and fall through "the ground". Therefore the torque Q is made a feedback controller and is created such that the model convergences to a reference point. Since we have a nonlinear multiple input multiple output system, we used a method called feedback linearization. For this method it is required that the system is in companion form, that is for 1≤k≤n−1, and which is clearly the case for our system, if you let x=φ and u=Q. First the input state Q=h(φ,v,p) is found such that the system is transformed in an equivalent linear time invariant system:

Q is chosen to be in the form b(φ)−1(p−f(φ,v)), thus Q=M(φ)+C(φ,v). Therefore, when this controller is applied, the obtained system is linear:

Now we can create another controller for p such that the poles of the characteristic polynomial are negative. That way, the system will become asymptotically stable. The controller is chosen to be in the form p=−α1φ−α2v for α1,α2≥ 0. And the characteristic equation becomes (s2+α2s+α1)n=0, which has negative poles. Lastly, to steer the system to the reference point the controller is adapted to include the values of the desired position, such that the desired position becomes the equilibrium. The new controller is: Using this we write everything in one equation for Q, Simulation To describe the entire movement, starting from an initial position we consider three phases after the push: returning to the start position, the swing phase for the step and the impact from the step with the ground. Each phase is adapted from the first system we introduced. Extra torque is added to represent the push and to allow proper knee bending since the human body has is restrictions how much it can bend and we made it proportional to the push. Let the total torque Q=Q1+Q2, then Q1 is the equation we last discussed and Q2 is in the form βH(−(t−tβ))+γH(−(t−tγ)), where H(t) is the step function and β and γ are constants representing the energy by the push and the extra torque, which are active on the intervals [0, tβ] and [0, tγ] of time t. The person starts in an initial position standing straight for t<0 and is stabilized. At t=0 the push becomes active. We consider three possible events following from the phases coinciding with the scoring

of the Tinetti Test. The first is, a person is pushed and is able to return to the initial position modeled as a three-link. While the push is active the controller can’t stabilize to the equilibrium and upper body and the upper leg start rotating counterclockwise. At t=tβ the push becomes inactive and the person returns to the initial position. This is the first phase and succeeds if the torque from equation 2 doesn’t exceed the maximum torque. Then the person gains two points. If the first phase doesn’t succeed, the person needs to take a step backwards. Starting in the first phase it goes to the swing phase for the step as soon as the torque for the first time exceeds the maximum torque. The model changes to a four-link and the controller moves to a new equilibrium where it has one leg behind. Once again equation 2 is checked if the torques don’t go over the maximum torques. If the second phase succeeds it goes to take the impact with the ground. The last phase starts when the second leg for the first time hits the ground. The knee of the person is considered a spring, it stores energy as the knee is bent making the leg shorter, and it uses this energy to make the knee straight again. We use Hooke’s law, the spring force which acts on the movable side of the spring pushes the hip forward to a stable position: F=k(x−x0), to calculate an extra torque that acts on φ2. Here k is the spring constant and (x−x0) is the deviation from

the length x0 of the spring and x is the length of the spring in meters. Qspring=Fspring×l2=kl2(x−l3). For the last time the torques are checked, if the last phase succeeds the person gets two points. If the second or the last phase fails, the person is unable to take a step and needs to be caught by the physiotherapist. Now comes the bad news, our model doesn’t predict the actual outcome of the Tinetti test. The main problem is that during our trials the torques used in the second and last phase would always be higher then the torques in the first phase. This resulted that a person would always ether get 2 points or 0 points. It is unclear if that is due to overly high torque caused by our methods during swing and impact phase or due to overly low torque during stabilization, or both. But comparing with literature suggests the former.

References [1] Seifollah Jahantabi-Nejad and Akram Azad. Predictive accuracy of performance oriented mobility assessment for falls in older adults: A systematic review. Medical Journal of the Islamic Republic of Iran, 2019. 10.34171/mjiri.33.38. [2] M J Faber, R J Bosscher, and P C W van Wieringen. Clinimetric properties of the performance-oriented mobility assessment. Physical Therapy and Rehabilitation Journal, 2006. https://doi.org/10.1093/ ptj/86.7.944.

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