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# Playing with L-Systems

For today’s session of the math circle I jointly run for 5-7 year-olds, we got the kids to play with Lindenmayer Systems (L-Systems for short). L-Systems can be used as compact representations of complex geometric shapes, including fractals. The aim of the session was for children to understand that simple formulae can describe complex geometric objects, building on the intuition that properties of shapes can be described algebraically that we got through a previous session on symmetry and algebra.

I stumbled across this excellent L-System generator on the web, which was perfect for our needs as we didn’t need to install any software on the school laptops. After illustrating how the Koch Snowflake could be generated, we simply let them loose to experiment, suggesting that each time they set the number of iterations to 1 before exploring a greater depth of iteration. They seemed to really enjoy it. On a one-to-one basis, we discussed the reason that various formulae generated their corresponding shapes, trying to embed the link between the equations and the graphical representation, but the main emphasis was generating visually pleasing images.

Here are some of the curves they produced. In each case, the caption is of the form: number of iterations, angle, axiom, production rule.

I would have liked to have the time to discuss in more depth why the curve that appeared to fill the triangle had no white space visible.

Once we had finished, we finally drew together where I presented a simple L-System for the Sierpinski Triangle, an object they’d seen before in a previous session. There were several exclamations of awe, which are always great to hear!

# Algebra and symmetry for 5-7 year-olds

At the encouragement of one of the teachers, I have resumed the math circle I was running a few years ago, jointly with her. Previous posts can be found here. This time we are targeting some of the youngest primary school children. So far, I’ve attended two sessions. In the first, we played with Polydron and I mainly used the session to observe the children and see what they’re capable of. In the second – described here – we experimented with symmetry.

Before the session, we created a few cardboard templates of three well-defined shapes, numbered 1, 2, and 3 in the picture below.

We told the children that we would be exploring a way of describing shapes as codes, and first had them cut out several copies of the shapes from coloured paper. This was not as easy as I had hoped – the children varied considerably in speed and accuracy of cutting.

We then told them that there were two operations that could be written down in our code, “next to”, written “;” meaning “the thing on the left of the semicolon is placed immediately to the left of the thing on the right” and “F” meaning “flip the shape in a vertical axis”. They seemed to understand this quite easily, and had no problems coming up and seeking clarification of any points. I hadn’t considered that they had not encountered the semicolon before, though, but this was not a problem, even if it was rendered more like a “j” in most of their written formulae.

We asked them to construct a couple of shapes, such as 2;F2 and F1;2, and then experiment with their own. They seemed to have a lot of fun, and the lower part of the photograph shows some that they were keen to show us when we reconvened in a circle on the carpet.

The children noticed that some shapes such as 2;F2 and F2;2 were symmetric, whereas some were not. I pointed out that nobody had tried to construct a shape such as FF1, and asked what that might look like. Several children correctly identified that FF1 would be indistinguishable from 1, so I wrote “FF1 = 1”, and I think I saw a flicker of an “ah ha!” moment in at least one child who seemed excited by the use of equations here.

At this point we ran out of time in our 1hr session. I would like to take this simple algebra further. In particular, I would like to get them to explore:

• the generalisation from the observed FF1 = 1 to ∀x.FFx=x.
• the observation that symmetry in a vertical mirror line corresponds to the statement Fx = x
• the observation that all shapes of the form x;Fx are symmetric, and relate this to the algebraic definition of symmetry
• the property F(x;y) = Fy;Fx

For those of a mathematical bent, these are the properties of a semigroup with involution.

I enjoyed myself, and I think the children did too!

# Math Circle Session 13: Infinity

Given the number of questions we’ve been asked by the children about infinity in recent weeks, I decided that today we’d have a session just on infinity.

I opened the session by asking the children whether they had any questions about infinity they’d like to try to answer in this session. I received the following suggestions:

• Why does infinity have the symbol $\infty$?
• What is infinity plus infinity?
• Why do we sometimes say infinity is a number and sometimes not?
• What are the rules for adding and subtracting with Aleph numbers? (From a child who had discussed infinite cardinals at home in the past.)

We began the session by one child answering the first question: “it represents going round and round and never stopping, which would also be a circle, but that’s zero.” I hoped the others would crop up during the discussion that followed.

We began by discussing prime numbers. All children had heard of primes, but only the older children could give a clear unambiguous definition. I had them list out some prime numbers, and asked how many there were. Opinion was divided: one child thought there would be an infinite number (after all, the session was on infinity!), while another child thought that this can’t be true “because most numbers are not prime, and there are infinite counting numbers, so there can’t be infinite primes too.” I then showed them Euclid’s proof that there are infinite primes.

Starting first with a hotel with a finite number of rooms, I then explained Hilbert’s  Hotel. For blog readers, there is an excellent animation of these ideas on TED. This seemed to get them quite excited, which is what I was hoping to see! We covered one new arrival and an infinitely large coach of new arrivals, each time stopping to see whether they could contrive of a construction that would allow the new arrivals to fit in the full hotel. The child who had previously seen infinite cardinals suggested a solution to one new arrival, but nobody came up with a solution to the infinitely large coach of new arrivals. However, it was great to see the thought going into this (most children wanted to move each person from their room to one infinitely further down the corridor.)

At this point, there were two directions I had thought we might go, given enough time: one on an infinite number of infinitely large coaches using a prime construction, given our proof of Euclid’s Theorem, and the other to look at a diagonal argument proof of the uncountability of the continuum, as a route into the final question the children asked. However, given we only had five minutes left, I decided it was better to just let the children’s discussion free-range, since they were still talkative after the hotel story. A variety of discussion points emerged involving infinity, and I just let them talk to each other: is space infinite, how can space not be infinite, are there an infinite number of life-supporting planets out there, what happened “before” the big bang, etc.! It was great to see this discussion, even if we hadn’t manage to answer all the questions posed at the beginning.

# Math Circle Sesion 12: Eulerian Circuits

Yesterday we turned our attention to Eulerian circuits in math circle. This was a completely “homebrew” session.

We started by asking the children to solve the classical puzzle of drawing this “house” without taking pen off paper, and without retracing your steps.

Surprisingly, they hadn’t seen it before, but it didn’t take long for them to solve, and also to discover that the path was not unique. We then asked whether it is possible to draw the same shape, following the same rules, but ending up at the end of the drawing back where you started. The children made many attempts at this. Nobody could find a way, but nobody suggested a reason why it wasn’t possible. One child noticed that if you remove the bottom most edge, the problem has a solution.

As a result, we began to enumerate all the connected graphs with one vertex, with two vertices, with three vertices, and with four vertices. For each graph, children quickly solved the problem where it had a solution. For small enough graphs, where there wasn’t a solution, they could see this through complete enumeration of paths.

One child suggested that there may be a link between the parity of the number of vertices in the graph, and whether or not an Eulerian circuit exists, but I demonstrated this to be false by reference to the two houses – the complete one and the one with the missing floor.

I suggested we look at the act of entering and exiting a vertex, using two edges each time. That, combined with drawing the order on each vertex of each of the graphs we’d enumerated, resulted in the conjecture from one of the kids that Eulerian circuits can only exist when all vertices have even order, and we together constructed an argument why this should be the case.

I then posed the (much harder) converse question: does an Eulerian circuit exist for every connected graph with even order vertices. I am not sure the children clearly understood the difference between this and the previous question; since they had shown that every graph with an Eulerian circuit has even order vertices, I think this seemed “obvious” to them (we must look at logic during one session – any reader with a child-friendly introduction to logic, please let me know!) I asked them to try to produce a graph with even order vertices that does not admit an Eulerian circuit, and they tried for some time before suggesting it is not possible, though none were clear why.

In a step towards trying to answer this question, I wanted to demonstrate that every graph with only even order vertices contains a circuit. I had them first try to construct graphs where each vertex has even order but the graph contains no circuit. Some elaborate failed attempts ensued, but then one child made the following remarkable claim: “it’s possible if the number of vertices is infinite”. I queried this child, and he drew the following “graph”:

His argument was that all these vertices have order two, yet the graph clearly contains no circuit. I thought this a fairly remarkable construction for a 7 or 8 year old! After restricting ourselves to finite graphs (infinity crops up so often, and absorbs so much interest, we should have a session on infinity also!) we all agreed that every graph with even order vertices contains a cycle.

From that point it was a small step to prove by induction that every graph with even order vertices admits and Eulerian circuit. I think they understood this, but I don’t think I had time to properly emphasise the inductive nature of the proof, and again proof by induction is something we must revisit in the future.

As a final exercise, I had them find an Eulerian circuit in what I had imagined to be a complex case: a pentagram inscribed inside a pentagon, with vertices at all points of intersection. They solved this easily, after which my infinity-inspired child noticed that there is a small pentagon formed by the pentagram, and asked what would happen if we put a pentagram inside this pentagon, and so ad infinitum, forming a fractal graph: would the property still hold?

Lots of things to pick up in future sessions, I think! If I were to change anything when redoing this session, it would be to make the problems more concrete, e.g. using towns as vertices and roads as edges, as my wife had suggested before the session! I think the children “got” it without this, but it might have captured the imagination slightly more.

An Interlude: Is Infinity Odd or Even?

During the session, when infinity came up, I was asked a direct question by one of the younger children: “is infinity odd or even?” At a lull in the main session, I decided we should use this question as a way to talk about the importance of definitions once more. All children had presupposed an arithmetic in which infinity plays the same role as any “other number”, and therefore already had in their minds that, for example, $2 \times \infty = \infty$ and $\infty + 1 = \infty$. I didn’t question these suppositions (again – a possible topic for another session!) but instead decided to work with them. I asked for a definition of even, and got:

Definition: A number is even if it is two times a whole number.

I asked for a definition of odd, and got:

Definition: A whole number is odd if it is not even.

Definition: A whole number is odd if it is one more than an even number.

Children then quickly worked out for themselves that by their definition (and under their arithmetic) infinity is even. By the first definition of odd, infinity is not odd, but by the second definition, infinity is both odd and even! Thus the two definition of odd coincide for integers but not for their version of arithmetic with infinities. We left that as a talking and discussion point about the importance of definitions, rather than trying to draw any strong conclusions.

# Math Circle Session 11: More Platonic Solids

Tuesday saw the next installment of our math circle. In the previous session, we had been investigating Platonic solids, and the aim of this session was to build up to a proof that there are exactly five such solids.

Since in the previous session the children had produced a lot of their own definitions, I produced a sheet summarising their definitions as well as standard mathematical definitions of some of the concepts we had been dealing with during the previous session. Below you can see the summary sheet of the children’s definitions, which was laid alongside the standard definitions. Names have been changed for privacy reasons.

Definition: Line-Shape

A line-shape is any shape made by a finite number of straight lines.

(Note the Thomas-Adrian Condition: The number of lines used by the shape must be finite, because an infinite number could lead to “curvy shapes”.)

Definition: Eve-Regular

An Eve-Regular line-shape is one whose sides all have the same length.

Definition: Line-Shape-Shape

A line-shape-shape is any 3D shape made by a finite number of line-shapes.

(Note the Christopher Convexity Test: Place a pencil on the object so it rests on at least two points; if you cannot create a space between the pencil and the object between these two points, the object is convex.)

The first part of the session was then taken up comparing these definitions to more standard ones. In particular, I pointed out that the standard name for Eve-Regular polygons is equilateral, and that polygons that are both equilateral and equiangular are referred to as regular. This approach seemed to work very well, especially with the younger children, who were very excited to see their names in print as part of a definition! Fortunately, I had been able to include the names of all children who had been present in the previous session, as all had been able to contribute.

After reminding the children of the {n,m} notation for the Schläfli symbol of a Platonic solid, I led them towards trying to systematically enumerate the solids. They were able to quickly tell me that the minimum values of n and m must be 3. We then worked our way through all possible Schläfli symbols until they found that m is bounded from above for each n, due to the need for the internal angles to add to a value strictly less than 360 degrees. By physically building each polyhedron as they went, they were able to show that every allowable m does indeed lead to a polyhedron, and we were able to construct all 5, showing that there were indeed only 5 of them in the process: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron.

In case they finished early, my plan was to have them compute the Euler characteristic for each Platonic solid, and then try to build other polyhedra to see whether the value held, however the proof of five solids took up the whole session. After two sessions building polyhedra with Polydron, some of the children may have reached the limits of their interest in the topic for now, so if we do explore Euler characteristics for polyhedra, it will be in a later session.

# Math Circle Session 10: Platonic Solids

Today was the 10th session of our math circle, and the first session I’ve led without the support of my wife. I had asked the school headteacher whether he could attend, so I was not alone faced with five children, and he thankfully agreed!

Today we explored three dimensional geometry. Over the Easter break, I’ve been reading a book by Kaplan and Kaplan on their experiences with their math circles (review to come in a future blog post), and this inspired me to try something different. We would spend a considerable time on definitions (which are the heart of mathematics, and often lost in trying to make mathematics appeal to primary school pupils) but these would – often – be definitions that the children themselves came up with through discussion and refinement. I was pleasantly surprised that even the “jumpiest” math circle members seemed to be OK with this approach, and contributed to the process of forming definitions.

We started by looking at 2D shapes. I drew some polygons on the board and one curvy shape and asked what the general name for the polygons was, expecting one of the older children to suggest the name “polygon”. However, this was not forthcoming. So we began discussing the differences between the shapes. Here we hit the second unexpected perspective: one of the children – rather creatively – suggested that “every” 2D shape is a polygon, because if you zoom in enough then the boundary becomes straight. We thus took a brief informal detour into limits, and ended up by concluding that this child had made a great contribution by ensuring “straight” and “finite” ended up in any definition they came up with. Since they were unsure what these shapes were called, following Kaplan and Kaplan, I suggested they find a name for them, and they settled on line-shape. Their final definition was: a line-shape is any shape made by a finite number of straight lines.

During the discussion on naming, two of the younger children suggested the word regular for those 2D shapes made from straight line segments. The older children disagreed, because they had seen regular polygons before, and instead insisted that a regular shape is a shape where the sides have the same length. (Nobody mentioned angles.)

Extending to 3D shapes, the children also did not naturally suggest the word “polyhedron“, however I was very impressed by the reasoning of one child, who suggested the term line-shape-shape “because if line-shapes are 2D shapes made of lines, then line-shape-shapes should be 3D shapes made of line-shapes!” After exploring these definitions a little more, I revealed the names polygon and polyhedron, and suggested we use them to avoid my own confusion!

I then had children make polyhedra using Polydron. As expected, they made a wide variety of weird and wonderful polyhedra. The oldest child had seen a (regular) icosahedron before and wanted to build one, but was trying to do so by constructing its net blind, unsuccessfully. All the polyhedra they constructed were convex, so I built a non-convex polyhedron and asked the children what made it different. They all agreed that it was because it “went in”, but I challenged them to figure out how they would convince someone else that this polyhedron “went in”. Eventually, I took a pencil and placed it on the polyhedron, at which one of the children noticed that there is a part of the pencil between the two touching points which is not itself part of the polyhedron. Thus we came to an agreed understanding of convexity.

These definitions were sufficient for me to (incorrectly at first!) define a Platonic solid as a convex polyhedron whose faces are regular polygons of the same shape and size.

I asked children to construct as many Platonic solids as possible, and they quickly came up with a tetrahedron and a cube. One child also came up with a decahedron consisting of two pentagonal-base pyramids joined, at which point I realised I had forgotten to include “and with the same number of faces meeting at each vertex” in my definition of a Platonic solid!

I introduced the children to Schläfli symbols {n,m} for a Platonic solid with m n-gon faces meeting at each vertex, and asked them to explore other Platonic solids. One child quickly discovered that hexagons cannot form the faces of a Platonic solid (because they tessellate three around a vertex, tiling the plane – in the words of the child “because of the angles”), but just moved onto another search rather than highlighting this important negative result; luckily I spotted it and queried him!

The school headteacher found {3,4}, the octahedron (identified immediately as ‘the shape of a d8’, i.e. an 8-sided die, by one of the children). The child who had previously tried to construct the icosahedron found that {3,6} does not exist, and therefore conjectured that {3,5} would be the missing icosahedron, which it was, much to her delight. The Schläfli symbol provided the extra information to enable her to make it, and she asked for permission to take a photo of it at the end “to show my dad!”

I was hoping to get to a proof that there are exactly five Platonic solids, but I will postpone this for next time, I think. And in the mean time, I will produce a worksheet summarising the definitions and results from this session.

# Math Circle Session 9: Parity

Today’s session was inspired by a different math circle book: that by Anna Burago, which is targeted at slightly older children.

We started with 7 coins, and asked the children to place them so that 4 were heads up and three tails up. The rules of the game are as follows:

• in each turn, you can pick any two coins of your choice
• you must turn both those coins over

The task is impossible, but the children didn’t see so immediately. One child even thought he had found a solution, but was unable to repeat it. One child noticed that he could get them to all tails, but not to all heads. Very quickly, children of all ages noticed that “odds” and “evens” had something to do with this difficulty.

One of the older children offered a partial explanation: by turning over two heads, the number of heads-up coins decreases by 2; by turning over two tails, it increases by 2; by turning over one head and one tail, it stays the same. So the only possibilities after one move are to have 2, 4, or 6 heads face up, and all these are even.

I helped the children to generalise this to “an even change in the number of heads up” after any number of moves. Unexpectedly, one child objected to zero being considered an even number, which led me to take a detour. One serious concern I have with typical primary school mathematical education, which I share with Hung-Hsi Wu, is the absence of clear definitions. So I viewed this as an ideal opportunity to get the children to formulate the definition of an even number. I received these suggestions from the children:

• a number that you can divide by two
• a number that is a multiple of two
• a number that ends in 0, 2, 4, 6 or 8

One child objected to the first definition, “because I can divide 1 by 2, to get 0.5”. So after some prompting, the children changed the first definition to “a number that you can divide by two and the answer is a whole number”.

We then worked through each definition in turn, testing it for whether zero was an even number. The oldest child provided a clear verbal proof that the first two definitions were equivalent, and a proof that any number meeting the third criterion also met the other two (but not vice versa).

The child who had objected to zero being even seemed to accept that by these definitions zero was even, but suggested that this would make $\frac{0}{0} = 2$, “which is not the case because $\frac{0}{0} = 0$“. I paused this question, writing on the board to return to, because we were in danger of getting too far off track.

For the last part of the session, I tried to get the children to reason algebraically, suggesting that – from their own definition – any even number can be written as $2n$ and, therefore because $4 + 2n = 2 \times 2 + 2n = 2(2+n)$, we still have an even number after any number of moves. I think I overestimated the preparedness of the children for this form of reasoning. The older kids were fine with this, one of the middle kids who had obviously seen but not understood algebraic notation before, upon seeing the $n$ exclaimed “oh no, those numbers, I hate those numbers!” and the youngest had not met brackets in arithmetic expressions. Trying to cover all this at once was probably too much, but I hope we managed to get the basic ideas across.

Finally, I returned to the question, which I had rephrased, “what is zero divided by zero”. We looked at the equation $2 \times 0 = 0$, and how – if the child who suggested it were correct, we could equally conclude from $3 \times 0 = 0$  that the answer was “3” and from $4 \times 0 = 0$ that the answer was 4. I suggested that there is no well defined answer, so we refer to it as undefined. Seemingly unsatisfied with this answer, one child left to get a calculator. I was hoping this to be my moment of triumph where the calculator reported “-E-“. Stupid calculator reported “0” (!) However, at least one child obviously could see what was going on, and asked whether 1 divided by zero would therefore be infinity. In the last few moments, I managed to explore this question with a subset of the children by looking at the growth of $\frac{1}{1/n}$ as $n$ increases in size, while the others packed away.

I’d view this as a mixed session. Lots of very interesting questions raised, not all answered satisfactorily, but at least most of the children were thinking mathematically.

# Math Circle Session 8: The Circle Game & Puzzles

Today was the 8th session of our math circle for primary school children. We only had four of the original six children today, for a variety of reasons, but it seemed to work well.

We first ran through a puzzle we set the kids to think about over the last week. The idea of the puzzle, taken from Martin Gardner’s book, is as follows. Take a piece of paper shown below. Imagine this is a map, that you have to fold up (I find folding up maps really hard!)

You have to fold only along the lines shown, and must form a folded “map” with the 1 on top, facing up. The next number down after the 1 must be the 2, after the 2 must be the 3, and so on until the 8 lies on the bottom.

Some kids had tried this at home, some had not (as a policy, we never make “homework” compulsory) but none had found the solution, so we covered this today.

We then moved onto the main part of the session, “The Circle Game”, adapted from Rozhkovskaya’s book. The kids were each given a worksheet consisting of 16 points equally spaced around a circle, and asked to play a game in pairs. The rules of the game are as follows:

• take it in turns
• on each turn, connect one point to another point with a straight line; any pair of points can be chosen, but no line drawn can cross another line already drawn

The last player to be able to draw a line is the winner.

The kids enthusiastically played this game, trying to outsmart each other. However, at the end of play, we had three games in which the first player to move had always won (though the points selected were different). I also asked the kids to count the number of moves made, which was found to be 29 in each game. The question was posed: “why does the first player always win?”

One child quickly pointed out that if the number of moves is an odd number, it must always be the first player who wins. Another child suggested that we should try 17 points around the circle as then “it would be an even number of moves and Player 2 would win”. I suggested that instead of making the game more complicated, we simplify it to the smallest possible number of points on the circle, which the children identified as two points.

Children very quickly found that two points would always result in one move, three points in three moves, and four points in five moves. At this point the child who suggested that 17 points would result in a win for Player 2 withdrew his opinion, and all children were convinced that the number of moves would always be odd and therefore Player 1 would always win.

We then looked at generalising the number of moves required for $n$ points. Children very quickly noticed that the number of moves increased by two for each new point added. Due to the significant difference in ages in the group, I wasn’t sure how comfortable children would be with an algebraic generalisation of the result, but it turns out that even the youngest was comfortable with the formula $moves = 2 \times points - 3$, which they got to via my suggestion that if the numbers were going up by two each time, try looking at the relationship between $moves$ and $2 \times points$.

I was surprised by how quickly we had reached this point in the session, so I rounded off the exploration by demonstrating an inductive proof of this formula in graphical form on the whiteboard (see, for example, Theorem 1.8 at http://press.princeton.edu/chapters/s9489.pdf).

One child was keen to play this game with his dad, and it was suggested that the “youngest goes first” rule should be applied, possibly after agreeing a prize for winning!

With the remaining spare time for the session, we set the following puzzle – also from Gardner’s book – to begin during the session and to continue to think about over the week. Consider the grid below.

The puzzle is to fill each empty cell with a single digit, so that the bottom row forms a number between $0$ and $10^{10}-1$. The challenge is that the digit under the heading zero must equal the total number of zero digits in this number, the digit under the 1 must equal the total number of 1 digits in this number, and so on. I thought it might be hard for the kids to understand the puzzle, but actually they understood it quite easily. They were amazed when I told them that the answer is unique: there is only one of the 10 billion possible numbers that works! It was interesting to observe the different reactions to this: some kids decided they’d never be able to find the solution, and therefore decided not to put much effort into the task. Others decided this was a challenge, and kept plugging away. One child was determined to take it home so that his dad would help.

Overall, I think this session went very well.

Update: I received an email tonight from the school headteacher with the solution to this number puzzle. I love that he’s so into it!

# Math Circle Session 7: Combinatorics, Pascal’s Triangle, and the Sierpinski Triangle Again

Today saw Session 7 of our math circle. No blog on Session 6, as I was away for work – my wife explored The Towers of Hanoi in my absence.

Today’s session was based on a combination of two sessions in Rozhkovskaya’s book. The idea is as follows. Imagine a maze at a tourist attraction. The maze is such that there is one entrance, but seven exits labelled 0 to 6. Imagine that you are standing facing north at the entrance. At this entrance, and at each intersection, you can move ahead-left on a bearing of 315 degrees by one unit or ahead-right on a bearing of 45 degrees by one unit until you meet the next intersection or exit. Thus the maze decomposes into 7 levels (including the entrance and exits). Now further imagine that you must roll a die at each decision point, taking the left route if you roll even and the right route if you roll odd.

The first question: If I were an ice-cream vendor, where should I place my van to capture most people coming out? Children had an intuitive feeling that most people would come out in the middle. Only one child (the oldest) offered a reasoned explanation for this: “we expect as many odd throws as even throws, which means the most likely exit would be the middle one”.

We got the children to experiment by rolling routes through the maze, and collected a tally of how many trials ended up at each exit, confirming this suspicion. Children naturally noticed that exits zero and seven were unlikely, as they would require rolling “all odd” or “all even”.

This naturally led onto the next part of the exercise, to annotate each decision point with the number of ways in which it could be reached. Children found this hard, and made many mistakes, because they were trying to count all the ways to reach a given decision point from the maze entrance from scratch for each decision point. I explained that there are only at most two ways to reach any decision point, so it is sufficient to sum the possibilities found to these two preceding points.

Thus, Pascal’s triangle was revealed. We asked the kids to sum the rows of Pascal’s triangle, and they discovered they summed to successive powers of two.

Finally, we had the kids colour all the even numbers in a large version of Pascal’s triangle (pre-printed). One of the kids immediately recognised a fractal-like shape. Once complete, we pointed out the resemblance to Sierpinski’s triangle, created in Session 7. Oddly (to me) this wasn’t immediately apparent, even to the kids who were there.

Overall, I was a little disappointed with today’s session. I think the material was good, but it was a sunny day, and two of the kids showed no sign of wanting to be there, inside after school. This meant I had to spend a lot more time than I ever imagined trying to get these kids on task, meaning a lot less time to discuss the mathematics of what we were seeing, which was a shame. My one rule for Math Circle was that the kids should want to be there and, for whatever reason, I don’t think this was true for 1/3 of them this time. I hope it’s a temporary blip, but I need to ask for views from primary school teachers; I was again reminded today how different kids are to the adults I’m used to!

# Math Circle Session 5: Powers, Limits, and Fractals

Yesterday saw the next installment of our math circle with primary school children. Again, following but extending Rozhkovskaya’s book, we used fractals as an interesting way in to consolidate (and in the case of the younger pupils, introduce) powers of numbers, to explore the children’s intuition about infinite series, and to make some pretty pictures!

We started by producing a fractal tree. The rule is simple: after Year 1, the trunk has grown. At the end of Year 2, two branches have grown from that, after Year 3, two further branches from each existing branch, and so on. Children were quickly able to tell us that the number of branches doubled each year, some took great pleasure in calculating or reciting powers of two. The template for this exercise was taken from Rozhkovskaya’s book, and is quite clever: lines are drawn across the page heights of 8 squares, 12 squares, 14 squares, 15 squares, and 15.5 squares, and each successive year’s branches should be drawn to reach the corresponding height. A bird is drawn flying around 18 squares up. The question is posed: will the tree ever reach the bird?

The children had mixed views on whether the tree will reach the bird. The most common view was that it must. One of the older children was able to provide a line of reasoning: “Each time, the height of the tree increases. Since it is always increasing, even by small amounts, it must eventually reach any height.” We then looked at how much gap there is between the height of the tree and a line of height 16 as the years progress: 16 at the beginning, 8 at the end of Year 1, 4 at the end of Year 2, and so on. Children could see that there was always a positive gap between the height of the tree and the height 16, for any finite number of years. However, this clashed with several children’s intuition, and it took quite a lot of discussion before it was generally accepted that an infinite series can sum to a finite limit. One of the youngest children in the group asked some very probing questions, such as “is infinity a number”, which led to some useful side discussions. Random banter between children later in the session about “hacking” their siblings’ passwords led onto a consolidation of this discussion by asking the question “If you had to press an infinite number of keys on a keyboard, could you do it in a finite amount of time? No? What if the first key took 8 seconds to press, the next 4, the next 2, and so on…”

We then moved onto a practical activity of constructing a sequence of approximations to the Sierpinski triangle by sticking little white triangles on coloured paper (pictured). One of the children noticed quickly, and was able to clearly explain, that the number of white triangles was increasing in powers of three. At this point we ran out of time, and our session ended. I intend to pick up and use the Sierpinski triangle again soon in math circle, through Pascal’s triangle modulo 2.


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# Lesson 1.2 Lines and their equations

This is a free lesson. We trust you enjoy it!

## The equation of a curve

Watch this video, which explains the idea of the equation of a curve.

## The equation of a curve

1. The equation of a curve indicates the relationship between the  $$x$$- and  $$y$$-coordinates of all points on the curve. Each type of function/relation has a characteristic shape.
2. A curve (or line) is the set of all points whose coordinates satisfy the equation of the curve (or line).
3. If a point lies on a curve, its coordinates will satisfy the equation of the curve. If a point does not lie on the curve, its coordinates will not satisfy the equation of the curve.

## Straight lines

A straight line is a "curve" with a constant gradient — think about why this is the case! The following video explains how we derive two equations for a straight line. You don't have to know how to derive these formulae, but you must know them well, and be able to use them. You will do so hundreds of times before completing this course, and you will definitely have to use at least one of them in the exam!

What about horizontal and vertical lines? Can you suggest what their equations will look like?

## The equation of a straight line

The equation of a straight line with gradient $$m$$ and passing through point $$A$$ with coordinates $$x_A$$ and $$y_A$$ is given by

$$y-y_A = m(x-x_A)$$

The equation of a straight line can be written in the form

$$y=mx+c$$

where  $$m$$ is the gradient of the line and  $$c$$ is the  $$y$$-intercept.

The equation of a horizontal line has the form  $$y=k$$,

and the equation of a vertical line has the form  $$x=k$$,

where  $$k$$ is a constant.

The following videos show you how to find the equation of a straight line when you are given either the gradient and one point on the line, or two points on the line.

Read sections 1.7 and 1.8 in your text book and make sure you understand them. The following video shows you how to find the point of intersection of

$$x+5y=22$$  and  $$3x+2y=14$$.

(See Exercise 1A no. 11(e).)

## Weight lifting

Complete the following exercises from your text book:

Exercise 1B

• Question 1 b, d, f, h.
• Question 2

Hints:

• For questions 5 - 10, note that parallel lines have the same gradient. In questions 7 & 8, you must first work out the gradient of the line through the last two points given; this allows you to work out the equation of the line through the first point.
• For question 11 you must solve simultaneous equations (see section 1.8 in the text book).

format: plain


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# Compute thickness¶

This tutorial shows how to use the main functionality of STRATAGem, to compute the mass thickness of a layer from experimentally measured k-ratios. For this example, we used the calculated k-ratios from the previous tutorial (Custom standard) as “experimental k-ratios”. We should therefore compute a Al2O3 layer with a thickness of 30 nm.

As usual, let’s start by importing the important classes and constants.

import math
from stratagemtools.sample import Sample, composition_from_formula
from stratagemtools.experiment import Experiment, LINE_KA
from stratagemtools.stratagem import Stratagem, PRZMODE_XPP, FLUORESCENCE_LINE_CONT

The next step is to define our Sample. Note here that we do not specify any thickness for the Al2O3 layer. No argument is given, which is equivalent to setting the thickness to None.

unknown = Sample({14: 1.0})

From the previous tutorial (Custom standard), the following Al and O $$\text{K}\alpha$$ k-ratios were respectively calculated 0.034 and 0.058 using the Al2O3 standard. We now use these values to define the experiments. We do not need to know the k-ratio for the Si $$\text{K}\alpha$$, but we need to specify this element as not analyzed.

energy_eV = 15e3
exp_si = Experiment(14, LINE_KA, energy_eV, analyzed=False)
exp_al = Experiment(13, LINE_KA, energy_eV, kratio=0.034, standard=std_al2o3)
exp_o = Experiment(8, LINE_KA, energy_eV, kratio=0.058, standard=std_al2o3)

We then execute the method compute from the Stratagem interface to compute the unknown thickness. The method returns a new Sample object with the calculated thickness. Note that the new Sample object could also be retrieved from the method get_sample or the property sample after executing the compute method.

with Stratagem() as strata:


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# Research Grants

## Externally funded research projects commencing in 2022

ARC Discovery Projects

— at the University of Sydney

• Associate Professor Florica-Corina Cîrstea and Professor Yihong Du (chief investigators): Singular solutions for nonlinear elliptic and parabolic equations. ($427,000) [DP220101816] Summary The analysis of many models fundamental to physical and biological sciences is obstructed by singularities. This project aims to discover and classify the singular solutions for two important types of nonlinear equations: elliptic and parabolic. The project expects to generate novel methods to decipher singularities by using innovative approaches from geometric analysis and dynamical systems. Expected outcomes of this project include new and powerful tools to advance a more general theory of singularities. This should provide significant benefits, such as new mathematical knowledge on key issues on singularities lying at the forefront of international research and enhanced expertise in an area of worldwide recognition for Australia. • Professor Georg Gottwald (chief investigator): A dynamical systems theory approach to machine learning. ($356,000) [DP220100931]
Summary

Forecasting the future state of a high-dimensional complex multi-scale system is a challenge we face in areas ranging from climate science to epidemiology. Even when basic physical mechanisms have been identified, the actual evolution equations are often unknown. This project will develop a computationally cheap machine learning framework for forecasting. The proposed mathematical framework provides a forecast together with a quantification of its uncertainty. We will develop sophisticated mathematical theory underpinning the novel methodology, as well as applying it to the perennial problem of subgrid-scale parametrisation of tropical convection, a missing key element in current climate models.

• Professor Mary Myerscough (chief investigator): Space, time and boundary conditions: Mathematics for evolving plaques. ($421,000) [DP220101454] Summary This project aims to create new mathematical theory to model the morphology of atherosclerotic plaques, which cause heart attacks and strokes, as plaques grow or regress. The project expects to devise new mathematical tools for formulating novel spatial models for cellular processes inside the plaque. These should give a new window into plaque growth and spatial structures . Expected outcomes include powerful and reliable mathematical models, new tools to understand plaque evolution, and national and international collaborations with scientists and mathematicians. This should provide significant benefits including increased capacity to use mathematical models in vascular biology and training young researchers in interdisciplinary methods. • Dr Jonathan Spreer and Professor Francisco Santos (chief investigators): Triangulations: linking geometry and topology with combinatorics. ($429,000) [DP220102588]
Summary

Triangulations are the method of choice to represent geometric objects given by a finite sample of points. Prominent examples include the pictures produced by the finite element method, polytopes in optimisation, or surfaces in computer graphics. Knowledge about the triangulations of an object and how they relate to each other is essential for these applications. Seemingly canonical and straightforward methods perform well – or not at all, depending on intricate and highly involved mathematical properties. In this project we combine geometric and topological viewpoints to tackle high-profile questions about triangulations. This will unlock the full potential of combinatorial methods and practical algorithms in applications.

• Associate Professor Leo Tzou and Dr Justin Tzou (chief investigators): Microlocal Analysis – A Unified Approach for Geometric Models in Biology. ($405,000) [DP220101808] Summary This project will use microlocal analysis to create a unified approach for predicting the outcome of a broad class of diffusion and reaction-diffusion models. This will replace the traditional theory which is no longer adequate for the level of geometric complexity demanded of current models arising in biology/ecology. This project will address the urgent need for a systematic theoretical underpinning of diffusion/reaction-diffusion in geometric settings whose scope of application is broader than the the existing patchwork of methods. • Professor Martin Wechselberger (chief investigator): A coordinate-independent theory for multi-time-scales dynamical systems. ($432,000) [DP220101817]
Summary

Biochemical reaction networks operate inherently on many disparate timescales, and identifying this temporal hierarchy is key to understanding biological behaviour. Currently, the existing dynamical systems theory is not able to rigorously analyse many important biological systems and networks due to this inherent non-standard multi-time-scale splitting. This project aims to remove these stumbling blocks and develop a coordinate-independent mathematical theory that weaves together results from geometric singular perturbation theory, differential and algebraic geometry and reaction network theory to decompose and explain the structure in the dynamic hierarchy of events in non-standard multi-time-scale systems and networks.

— at Monash University

• Dr Julie Clutterbuck, Dr Daniel Hauer, Dr Paul Bryan and Professor Guofang Wei (chief investigators): Optimal shapes in geometry and physics: Isoperimetry in modern analysis. ($295,000) [DP220100067] Summary This project will find the best isoperimetric shapes in curved spaces: shapes that optimise geometric or analytic quantities, such as the volume enclosed by a surface of a given area, or the resonant frequency of a drum of given area. The optimal shapes lead to tools that are widely used in differential equations, geometric analysis, statistical physics, probability theory, and quantum computing. Through this work, we will forge connections between the geometry of curved spaces, and the physics of operators therein. The significant benefits of this project include increasing fundamental mathematical knowledge, building capacity in Australia’s world-class geometric analysis community, and strong links with international partners. • Professor Fima Klebaner, Associate Professor Kais Hamza, Dr Jie Yen Fan, Professor Andrew Barbour and Professor Bohdan Maslowski (chief investigators): New universality in stochastic systems. ($410,000) [DP220100973]
Summary

This project aims to uncover new analyses and effects in the complex behaviour of non-linear systems with random noise. Many systems originate near an unstable equilibrium. This project will develop a new mathematical theory that establishes a universality in the way the long term effect of noise expresses itself as random initial conditions in the dynamics. It will fill gaps in Mathematics and make refinements to existing fundamental scientific laws by including random initial conditions as predicted by our theory. This will advance our understanding of complex systems subjected to noise and will provide significant benefits in the scientific discoveries in Biology, Ecology, Physics and other Sciences where such systems are frequently met.

• Dr Ivan Guo, Dr Zhou Zhou, Dr Anna Aksamit Dr Kihun Nam and Professor Marek Rutkowski (chief investigators): Can green investors drive the transition to a low emissions economy? ($390,000) [DP220103106] Summary The project aims to develop a game-theoretical approach to model the impact of climate change on financial markets by studying the interactions between the government, companies and investors. Expected outcomes include novel solution concepts for stochastic games with heterogeneous beliefs, asymmetric information, and model uncertainty, as well as optimal investment and production strategies under climate driven economic transitions. Results will be used to validate and improve the recently launched Australian based climate transition index. The project should yield significant benefits for the financial industry and investors by providing novel insights into financial risks during the transition to a low emissions economy. ARC Discovery Early Career Researchers • Dr Shila Ghazanfar: Statistical approaches for spatial genomics at single cell resolution. ($443,869)
Summary

Cells cooperate to form complex, dynamic and varied tissue structures. This project aims to develop statistical and computational approaches to analyse spatial genomics data, a novel technology that retains vital spatial information at single cell resolution while detecting RNA molecules for hundreds of genes. Observing the molecular activity of cells in their spatial context is critical for tackling key biological questions, such as how tumour cells behave during malignancy or how stem cells determine their fate. Expected outcomes also include techniques to fully harmonise spatial and non-spatial genomics datasets, and methods toward understanding the complex relationships among cells in their environment, revealing novel cell biology

• Dr Ashish Goyal: Multiscale mathematical modelling to gain insights into hepatitis viruses. ($444,068) Summary This project aims to use mathematical modelling to study hepatitis viruses at multiple levels. The project expects to develop complex yet analysable mathematical models to comprehend the fundamental biology of hepatitis viruses by elucidating longitudinal patterns in viral and immune markers at intracellular and cellular levels, and advance a new subfield in mathematical biology, i.e., modelling codependent human viruses. Expected outcomes of the project include new generalized mathematical tools, biological insights that may aid research beyond the scope of this project, and strong interdisciplinary collaborations. Expected benefits include an increased capacity of the research community in Australia to use mathematical models in virology. ## Externally funded research projects commencing in 2021 Australian Laureate Fellowship • Professor Helen Byrne: New approaches to understand how form and function shape complex. ($3,021,288.)
Summary

As biology and medicine transform into quantitative sciences, existing mathematical methods are often inadequate to explain the data they generate. This project aims to unlock the potential of such biomedical data through the development of new mathematical approaches that combine concepts from pure and applied mathematics, statistics and data science, and then to investigate their ability to generate mechanistic insight into fundamental biomedical processes. In this way, the project expects to affect a paradigm shift in mathematical biology while strengthening Australia’s reputation as a world-leader in mathematical biology. An outcome from this project could be new mathematical models that guide decision making in the clinic.

ARC Discovery Projects

— at the University of Sydney

• Associate Professor Damian Birney and Professor Sally Cripps (chief investigators), with Prof Dr Jens Beckmann and A/Prof Rui Nouchi (partner investigators): A paradigm shift in understanding cognitive flexibility. ($493,123.) Summary The project aims to model cognitive flexibility as a dynamic process within people that varies across situations and occasions using advanced data analytics. Significance: The project intends to generate new knowledge in intelligence theory using recent advances that overcome known theory-testing limitations that have historically been ignored. Expected Outcomes: An authentic account of cognitive flexibility and a new paradigm for developing and testing models of dynamic change within people. Benefits: Dynamic models are needed to understand authentic problem-solving and cognitive function. The advances benefit research and applied areas where dynamic processes are important, including education, work, and cognitive aging. • Dr Kevin Coulembier (chief investigator) with Dr Pavel Etingof (partner investigator): New constructions and techniques for tensor categories. ($422,887.)
Summary

The goal of this project is to make fundamental advances in the structure theory of tensor categories. Such categories play crucial roles in numerous fields of mathematics, physics and beyond. New methods, theory and examples will be developed, inspired by algebra, representation theory and geometry. These will then be applied in the foundational study of tensor categories for (dis)proving several of the most important open conjectures in the field. This will open new perspectives for applications in other areas, most notably in representation theory. Other benefits include enhanced international collaboration and scientific capacity in Australia.

• Professor Holger Dullin and Dr Robert Marangell (chief investigators), with Prof Dr Yuri Latushkin (partner investigator): Spectral Theory of Hamiltonian Dynamical Systems. ($310,000.) Summary Stability theory of steady states, travelling waves, periodic waves, and other coherent structures in nonlinear Hamiltonian partial differential equations is a cornerstone of modern dynamical systems. In particular it is of utmost importance to reliably compute eigenvalues, which determine the stability or instability of such structures. This project will develop methods to compute the spectrum of Hamiltonian operators in more than one spatial dimension. It will use the powerful geometric tools of the Maslov index and the Evans function. We will use these to simultaneously advance, and bring together the theories of the two dimensional Euler equations and Jacobi operators. • Dr Alexander Fish (chief investigator): Additive combinatorics of infinite sets via ergodic theoretic approach. ($340,000.)
Summary

The proposed project will utilise innovative ergodic theoretic approaches to enable us to address important questions in Additive Combinatorics (Number Theory) and Fractal Geometry. In particular, we will resolve long-standing inverse additive problems for infinite sets, discover sum-product phenomena in Number Theory, and find a plethora of finite configurations in fractal sets. We will also extend the structure theory of one of the most popular mathematical models of quasi-crystals to a more extensive class of groups. This project will make significant contributions to Additive Combinatorics and Ergodic Theory and will bring the Australian research in these fields to ever greater heights

• Professor Nalini Joshi (chief investigator): Dynamics on space-filling shapes. ($501,777.) Summary Modern science derives its power from mathematical models and tools that enable us to predict their behaviours. The project aims to construct new models given by dynamical systems that move consistently from one tile to another in a lattice of higher-dimensional shapes called polytopes. The construction is expected to lead to new functions with properties that will provide extensions of current models of growth processes. The intended outcomes of the projec include predictive tools that describe nonlinear special functions and information about their symmetry reductions. This should provide significant benefits, such as new mathematical knowledge innovative techniques, and enhanced scientific capacity in Australia. • Associate Professor John Ormerod, Dr Garth Tarr and Professor Samuel Muller (chief investigators): Fast flexible feature selection for high dimensional challenging data. ($390,000.)
Summary

The project aims to provide new frameworks for fast flexible feature selection and appropriate modelling of heterogeneous data through structural varying-coefficient regression models. The outcomes will be a series of new statistical methods and concepts enabling more powerful modelling of complex bioscience data. The project will create the science for building reliable statistical models taking model uncertainty into account, impacting how results will be interpreted, and with accompanying software. This will be a significant improvement in the assessment of model confidence in the food and health research priority areas including areas such as meat science, Huntington’s disease, and kidney transplantation.

— at La Trobe University

• Dr Yuri Nikolayevsky and Professor Holger Dullin (chief investigators) with Prof Dr Vladimir Matveev (partner investigator): Finite dimensional integrable systems and differential geometry. ($390,000.) Summary Mathematical models of many processes in science (physics, engineering) and in the real world (nature, economics) are governed by complicated systems of differential equations. An important, distinguished class of such models is described by integrable systems, the systems for which one can provide a comprehensive qualitative picture, and in many cases, a complete solution. Using recently developed, powerful methods of integrable systems and differential geometry, this project will focus on a range of important, interconnected theoretical problems in both disciplines. The expected outcomes will provide new, deep, mathematically and physically significant results which will lead to applications and developments across a range of fields. ## Externally funded research projects commencing in 2020 ARC Future Fellowship • A/Prof Peter Kim: Mathematical modelling unravels the impact of social dynamics on evolution. (Four years,$1,028,533.)
Summary

This project aims to mathematically model human evolution as a dynamical process. The anticipated goal is to quantitatively analyse theories of human origins. The project expects to develop innovative mathematical models, improve our understanding of the evolutionary process, and advance a unique area of interdisciplinary collaboration: applied mathematics and anthropology. Expected outcomes include refined methods for mathematical modelling of human evolution and improved techniques for analysing such models. It should provide benefits, such as increasing research in mathematical biology, an important growth area of science in Australia, and advancing mathematical approaches to engaging questions arising from anthropology.

ARC Discovery Projects

— at the University of Sydney

• Dr Lamiae Azizi, with Professor Margaret Barbour, Dr Daniel Tholen, Professor John Evans, Assistant Professor Craig Brodersen, Dr Andrew McElrone and Dr Thomas Buckley. (Four years, $500,000.) Summary This project aims to develop leaf anatomical ideotypes with improved photosynthesis and water-use efficiency for wheat, rice, chickpea and cotton using novel three dimensional imaging and modelling techniques. This project expects to generate new understanding of the role of leaf anatomy on leaf function. Expected outcomes of this project include the world's first 3D spatially-explicit, anatomically accurate model of leaves of crop plants to allow virtual experiments identifying optimized anatomy for improved photosynthetic performance. Benefits to the agricultural industry include increased crop productivity and water-use efficiency to meet future global food demand and to make the most of Australia's limited water resources. • Professor Beniamin Goldys, with Associate Professor Thanh Tran and Dr Kim Ngan Le: Mathematics for breaking limits of speed and density in magnetic memories. (Three years,$525,000.)
Summary

The aim of this project is to develop a mathematical theory and numerical models of stochastic partial differential equations for magnetic nano-structures. Such materials will yield next-generation magnetic memories with up to three orders of magnitude faster switching speeds and dramatically increased data storage density. New mathematical theories will help understand their sensitivity to small random fluctuations that can destroy stored information. This project aims to revolutionise mathematical modelling of magnetic memories and put Australia at the forefront of international research. Technological advances to create much smaller and faster memory devices are expected to enable groundbreaking ways of managing and mining big data.

• Professor Nalini Joshi and Dr Milena Radnović: Geometric analysis of nonlinear systems. (Three years, $426,000.) Summary Modern science derives its power from mathematics. The project aims to capture, identify and describe pivotal, transcendental solutions of nonlinear systems that are universal in science, in the sense that they always arise as mathematical models under certain physical limits. The project expects to produce new mathematical methods to describe such functions by using a newly discovered geometric framework. Expected outcomes include the description of elusive solutions of discrete and higher-dimensional nonlinear systems. This should provide significant benefits, such as new mathematical knowledge, innovative techniques, enhanced scientific capacity in Australia. • Professor Andrew Mathas: Graded semisimple deformations. (Three years,$474,000.)
Summary

Recent advances in representation theory have revealed beautiful new structures in the classical representation theory of the symmetric groups and Hecke algebras. These discoveries have provided us with new algebras, the cyclotomic KLR algebras, that encode deep properties of fundamental objects in algebraic combinatorics and geometric representation theory. The cyclotomic quiver Hecke algebras are central to several open problems in mathematics but they are still poorly understood, with even basic properties like their dimensions being unknown. This project will establish a new framework for studying these algebras that will remove the current obstacles in this field and alllow us to prove substantial new results that advance the theory.

• Professor Mary Myerscough, with Dr Christina Bursill and Professor Helen Byrne: New mathematics for lipids and cells: structured models for atherosclerosis. (Three years, $500,000.) Summary The project aims to create new mathematical theory for immune cell behaviour which leads to heart attacks and strokes. This includes formulation and analysis of new types of mathematical models for atherosclerotic plaque development, leading to the creation of new mathematical tools to investigate cell fate in plaques and to generate new hypotheses for experimental research. Expected outcomes of this project include powerful and reliable mathematical models ready for application, and national and international collaborations with scientists and mathematicians. This should provide significant benefits including increased capacity to use mathematical models in vascular biology and training young researchers in interdisciplinary methods. • Dr James Parkinson, with Dr Jérémie Guilhot and Professor Bernhard Mühlherr: Lusztig's conjectures for Hecke algebras with unequal parameters. (Three years,$453,000.)
Summary

The goal of this project is to make fundamental advances in representation theory, a powerful branch of mathematics focused on taking abstract mathematical structures and "representing" them in a concrete and useful way. In particular we aim to prove a series of long standing and influential conjectures by George Lusztig concerning the representation theory of Hecke algebras, objects which are ubiquitous in modern algebra. Our work will lead to new discoveries, a fundamentally deeper understanding of Kazhdan-Lusztig theory, and will drive future research. Benefits include enhanced international collaboration and increasing capacity in pure mathematics, especially in the cutting-edge area of representation theory.

• Professor Marek Rutkowski, with Associate Professor Shige Peng: Fair pricing of superannuation guaranteed benefits with downturn risk. (Three years, $390,000.) Summary Australia is the fourth largest holder of pension fund assets worldwide. Hence the impact of market fluctuations on financial well-being of retirees can be detrimental, especially during market downturns associated with economic crises. The finance industry addresses this issue by complementing variable annuities with riders designed to protect the income stream of retirees. This project aims to develop a novel approach to fair pricing and optimal withdrawals and surrender policies for superannuation guaranteed benefit products through a comprehensive analysis of complex optimisation problems in stochastic models of financial markets with downturn risk. • Professor Martin Wechselberger and Dr Robert Marangell, with Dr Bronwyn Hajek and Dr Petrus van Heijster: A novel geometry approach to shocks in reaction-nonlinear diffusion models. (Three years,$480,000.)
Summary

Reaction-nonlinear diffusion models play a vital role in the study of cell migration and population dynamics. However, the presence of aggregation, or backward diffusion, leads to the formation of shock waves - distinct, sharp interfaces between different populations of densities of cells - and the breakdown of the model. This project will develop new geometric methods to explain the formation and temporal evolution of these shock waves, while simultaneously unifying existing regularisation techniques under a single, geometric banner. It will devise innovative tools in singular perturbation theory and stability analysis that will identify key parameters in the creation of shock waves, as well as their dynamic behaviour.

— at Macquarie University

• Dr Daniel Hauer, with Associate Professor Adam Sikora, Associate Professor Zihua Guo and Dr Melissa Tacy: Nonlinear harmonic analysis and dispersive partial differential equations. (Three years, $420,000.) Summary This proposal is devoted to linear and nonlinear harmonic analysis. It aims to unify the most significant attributes of harmonic analysis such as restriction estimates, dispersive properties of differential operators, spectral multipliers, uniform Sobolev estimates and sharp Weyl formula. Such unification will strongly improve tools for mathematical modelling in all areas of technology and science. Notable applications include medical imaging, fluid dynamics and subatomic modelling using quantum interpretation. It will solve several important open problems in spectral analysis of partial differential operators and develop new cutting-edge techniques in harmonic analysis with application to nonlinear partial differential equations. — at the University of Wollongong • Professor Jacqui Ramagge and Dr Nathan Brownlowe, with Professor Aidan Sims, Associate Professor David Pask and Associate Professor Lisa Clark: There and back again: operator algebras, algebras and dynamical systems. (Three years,$461,000.)
Summary

The aim of this project is to develop mathematics that enables us to transfer information back and forth between dynamical systems and algebras, including operator algebras. Dynamical systems - systems that change over time - are ubiquitous, and central to modern mathematics and its applications. In mathematics, dualities allow us to translate questions from one context to another in which they are easier to solve and then translate the answer back again. Expected outcomes include increased understanding of the relationship between operator algebras and the dynamical systems that they represent. Benefits include enhanced international collaboration, and increased Australian capacity in pure mathematics, particularly operator algebras.

ARC Discovery Early Career Researchers

• Dr Anna Aksamit: How to beat model uncertainty with more information. (Three years, $427,008) Summary Experience of the 2008 financial crisis exposed a weakness in our over-reliance on mathematical models. The main aim of this project is to develop mathematical tools to investigate the role of information in reducing model uncertainty. The project will undertake pressing research in robust finance, which is now one of the most active and dynamic topics in financial mathematics. It expects to quantify the value of information under uncertainty in mathematical modelling. It will generate new knowledge in probability theory and stochastic processes providing a significant mathematical contribution in its own right. • Dr Ellis Patrick: Statistical frameworks for high-parameter imaging cytometry data. (Three years,$427,068)
Summary

The project aims to develop statistical and bioinformatics methodology for characterising the complex interactions between cells in their native environment. Recent advances in imaging cytometry technologies have made it possible to observe the behaviour of multiple cell-types in tissue concurrently. The intended outcome is a suite of statistical methodologies that are crucial for addressing a variety of biological problems with these state-of-the-art technologies. This work will advance knowledge in bioinformatics, statistics and image analysis, providing benefits to scientists studying the fundamental behaviour of cells and underlying disease mechanisms.

## Externally funded research projects commencing in 2019

ARC Discovery Projects

— at the University of Sydney

ARC Discovery Early Career Researchers

• Dr Wenshuai Jiang: Singularity analysis for manifolds with Ricci curvature bounds. ($345,000) • Dr Ulrich Thiel: Representation theory: studies of symmetry shadows. ($420,256)
• Dr Guangbo Xu: Gauged sigma model, mirror symmetry, and related topics. ($350,000) ## Externally funded research projects commencing in 2018 ARC Discovery Projects — at the University of Sydney • Dr Oded Yacobi, Dr Kevin Coulembier and Professor Julia Pevtsova: New Dualities in Modular Representation Theory. (2018–2020:$273,485.)
• Dr Anne Thomas, Dr Elizabeth Townsend Milicevic and Assistant Professor Petra Schwer: Affine flags, Euclidean reflection groups and folded galleries. (2018–2020: $317,288.) • Prof Georg Gottwald: Mathematical model reduction for complex networks. (2018–2020:$357,072.)
• Prof Alexander Molev: Quantum vertex algebras. (2018–2020: $411,584.) • A/Prof Peter Kim, Dr Federico Frascoli, A/Prof Adelle Coster and Prof Chae-Ok Yun: Dynamical systems theory and mathematical modelling of viral infections. (2018–2020:$401,706.)
• Prof Martin Wechselberger: A geometric theory for nonstandard relaxation oscillators. (2018–2020: $401,706.) — at Monash University — at the Australian National University • A/Prof Samuel Müller, Prof Alan Welsh, Dr Francis Hui and Prof Yanyuan Ma: Dimension reduction and model selection for statistically challenging data. (2018–2020:$359,083.)

ARC Discovery Early Career Researchers

• Dr Alistair Senior: Diet, Variance and Individual Variability in Life-History. (2018–2020: $365,058.) • Dr Haotian Wu: Singularity Analysis for Ricci Flow and Mean Curvature Flow. (2018–2020:$328,075.)
• Dr Rachel Wang: Statistical theory and algorithms for joint inference of complex networks. (2018–2020: $343,450.) ## Externally funded research projects commencing in 2017 NHMRC Project • Professor David James, Dr Jacky Stoeckli and Professor Jean Yang: Dissecting Rapamycin sensitive and insensitive effects of mTOR. (2017–2020:$1,100,00.)

ARC Discovery Projects

• Professor Anthony Henderson and Associate Professor Pramod Achar: Modular character sheaves. (2017–2019: $345,00.) Summary This project aims to complete the fundamental mathematical theory of modular group representations, the algebraic description of symmetry over finite number systems. Group representation theory can be applied to any linear problem involving symmetry. However, the modular case, where the characteristic of the underlying field is a prime number, is less understood than real or complex scalars, and this lack of understanding blocks potential applications. This project will use geometric methods to answer questions about modular representations of the finite groups of Lie type, the most important class of finite groups. This project could make modular representation theory essential for computations, enabling faster solutions to problems of linear algebra and allowing future applications in such areas as data transmission technology. • Professor Jacqui Ramagge, Dr Nathan Brownlowe, Professor Iain Raeburn and Professor Marcelo Laca: From actions to operator algebras and their equilibrium states. (2017–2019:$286,000.)
• Associate Professor Qiying Wang, Professor Shiqing Ling and Professor Weidong Liu: Non-linear cointegrating regression with endogeneity. (2017–2019: $288,471.) • Professor Jean Yang, Dr John Ormerod, Associate Professor Samuel Müller, Dr Pengyi Yang and Professor Graham Mann: Prognosis based network-type feature extraction for complex biological data. (2017–2019:$354,500.)
• Professor Ruibin Zhang: Geometric themes in the theory of Lie supergroups and their quantisations. (2017–2019: $416,500.) ARC Discovery Early Career Researchers • Dr Kevin Coulembier: Quasi-hereditary categories in Lie theory. (2017–2019:$360,000.)
• Dr Zsuzsanna Dancso: Homological methods in combinatorics, algebra and geometry. (2017–2019: $360,000.) • Dr Pengyi Yang: Trans-omic networks. (2017–2019:$372,000.)

## Externally funded research projects commencing in 2016

NHMRC Project

• A/Prof Jean Yang: Statistical bioinformatics for network based prognostic and precision therapy in complex disease. (2016–2019: $463,652.) ARC Future Fellowship • Dr Zhou Zhang: Comprehensive Study of Kahler-Ricci Flows. (2016–2019:$764,960.)

ARC Discovery Projects

• Prof John Cannon and Prof Derek Holt: Composition algorithms for large matrix groups. (2016–2018: $305,500.) • Prof Nalini Joshi and Prof Kenji Kajiwara: Reflection groups and discrete dynamical systems. (2016–2018:$495,700.)
• Dr Peter Kim and Prof Kristen Hawkes: Human longevity: Modelling social changes that propelled its evolution. (2016–2018: $396,338.) • Prof Gus Lehrer, Dr Anthony Henderson and Dr Geordie Williamson: Algebraic Schubert geometry and unitary reflection groups. (2016–2018:$519,300.)
• A/Prof Mary Myerscough, A/Prof Charlie Macaskill and Dr Christina Bursill: Dynamics of atherosclerotic plaque formation, growth and regression. (2016–2018: $342,200.) • Dr Stephan Tillmann, Prof Joachim Rubinstein and A/Prof Craig Hodgson: Invariants, geometric and discrete structures on manifolds. (2016–2018:$334,000.)
• Prof Beniamin Goldys, Prof Thanh Tran, Prof Zdzisław Brzeźniak, Prof Andreas Prohl, Prof Ernst Stephan and A/Prof Salim Meddahi: Novel Approaches for Problems with Uncertainties. (2016–2018: $329,377.) ARC Discovery Early Career Researchers • Dr Andrew Papanicolaou: Solving non-Markov optimistisation problems using forward-backward stochastic differential equations. (2016–2018:$295,020.)

## Externally funded research projects commencing in 2015

ARC Discovery Projects

• Prof Gus Lehrer and Prof Ruibin Zhang: Symmetry via braiding, diagrammatics and cellularity. (2015: $130,000; 2016:$124,700; 2017: $130,000. 2018:$130,000. 2019: $130,000.) • Prof Andrew Mathas: The dimension problem for Hecke algebras. (2015:$135,000; 2016: $129,500; 2017:$135,000.)
• Prof Alex Molev: Classical and affine $$W$$-algebras. (2015: $115,000; 2016:$114,000; 2017: $124,000.) ARC Discovery Early Career Researchers • Dr Peter McNamara: Higher Representation Theory. (2015:$120,000; 2016: $120,000; 2017:$120,000.)
• Dr Oded Yacobi: Quantum Groups and Categorification in Geometric Representation Theory. (2015: $110,000; 2016:$110,000; 2017: $110,000.) ## Externally funded research projects commencing in 2014 ARC Discovery Projects • Prof Alan H Welsh and Dr Samuel Müller: Prediction, inference and their application to modelling correlated data. (2014:$117,000; 2015: $117,000; 2016:$117,000.)
• Dr Stephan Tillmann: Moduli Spaces of Geometric Structures. (2014: $90,000; 2015:$110,000; 2016: $70,000.) • Ruibin Zhang: Super Duality and Deformations in the Representation Theory of Lie Superalgebras. (2014:$120,000; 2015: $120,000; 2016:$120,000.)

ARC Discovery Early Career Researchers

• Dr Geoff Vasil: Computational geophysical and astrophysical fluid dynamics at the petascale. (2014: $110,540; 2015:$111,140; 2016: $111,140.) ARC Future Fellowship • Leo Tzou, Inverse Problems for Partial Differential Equations—A Geometric Analysis Perspective. (2013:$85,240; 2014: $159,905; 2015:$148,085; 2016: $144,085; 2017:$70,665.)

## Externally funded research projects commencing in 2013

ARC Discovery Projects

• Prof JJ Cannon and Prof DF Holt: Constructive Representation Theory. (2013: $130,000; 2014:$137,000; 2015: $145,000.) • A/Prof S Yan, Prof EN Dancer and Prof Y Du: Singularity, degeneracy and related problems in nonlinear partial differential equations. (2013:$100,000; 2014: $100,000; 2015:$100,000.)
• Prof N Joshi: Critical solutions of nonlinear systems. (2013: $110,000; 2014:$110,000; 2015: $110,000.) • Prof JH Rubinstein, A/Prof CD Hodgson and Dr S Tillmann: Triangulations in dimensions 3 and 4: discrete and geometric structures. (2013:$120,000; 2014: $120,000; 2015:$120,000.)
• Prof A Ward, A/Prof MR Myerscough, Prof J Kruse and Dr J Buhl: Leadership matters: the emergence of informed leaders and their influence on group movement. (2013: $120,000; 2014:$130,000; 2015: $110,000.) • A/Prof Q Wang, Prof W Liu and Prof C Tudor: Asymptotics in non-linear cointegrating regression: theory and applications. (2013:$50,000; 2014: $60,000; 2015:$70,000.)
• A/Prof J Yang, A/Prof S Müller and Prof GJ Mann: Vertically integrated statistical modelling in multi-layered omics studies. (2013: $130,000; 2014:$130,000; 2015: $130,000.) ARC Discovery Early Career Researchers • Dr Sheehan OlverA new class of fast and reliable spectral methods for partial differential equations. (2013:$108,221; 2014: $106,655; 2015:$100,764.)
• Dr John OrmerodScalable Bayesian model selection for massive data sets. (2013: $124,969; 2014:$121,789; 2015: $123,652.) Australian Laureate Fellowship • Prof N Joshi: Geometric construction of critical solutions of nonlinear systems. (2012:$313,558; 2013: $632,847; 2014:$643,960; 2015: $653,644; 2016:$634,810; 2017: $305,837.) ARC Future Fellowship • Dr Martin WechselbergerGeometric methods in mathematical physiology. (2012:$83,957; 2013: $167,914; 2014:$167,914; 2015: $167,914; 2016:$83,957.)

## Externally funded research projects commencing in 2012

ARC Discovery Projects

• Prof Maria Byrne, Dr Jean Y Yang, Prof Gregory A Wray: Heads or tails – which did echinoderms lose in the evolution of radial symmetry? (2012: $120,000; 2013:$110,000; 2014: $110,000.) • Dr Florica C CîrsteaAnalysis of nonlinear partial differential equations describing singular phenomena. (2012:$30,000; 2013: $30,000; 2014:$30,000.)
• A/Prof Georg Gottwald, A/Prof Gary A Froyland: Extracting macroscopic variables and their dynamics in multiscale systems with metastable states. (2012: $90,000; 2013:$90,000; 2014: $90,000.) • Prof Gus Lehrer, Prof Ruibin Zhang: Quantised algebras, supersymmetry and invariant theory. (2012:$110,000; 2013: $110,000; 2014:$105,000.)
• Prof Marek Rutkowski: Multi-person stochastic games with idiosyncratic information flows. (2012: $108,000; 2013:$117,000; 2014: $125,000.) ARC Future Fellowship • Dr Anthony HendersonSpringer fibres, nilpotent cones and representation theory. (2011:$81,672; 2012: $166,281; 2013:$169,046; 2014: $164,106; 2015:$79,669.)

ARC Discovery Early Career Researcher

• Dr Peter S KimMathematical modelling of breast cancer immunity: guiding the development of preventative breast cancer vaccines. (2012: $125,000; 2013:$125,000; 2014: $125,000.) Go8 – Germany Joint Research Co-operation Scheme • Dr Benjamin Burton (Qld), Prof Michael Joswig (TU Darmstadt), Dr Andreas Paffenholz (TU Darmstadt), Dr Stephan TillmannAlgorithmic methods in combinatorial topology. (2012:$14,000; 2013: $14,000.) Marsden Fund (NZ) • V.Kirk, J. Sneyd, H. Osinga, Dr Martin WechselbergerApplications of multiscale excitable systems to calcium dynamics and neuroscience. (2012–2014:$605,000.)

WUN Research Development Fund

• B. Krauskopf, H. Osinga,V. Kirk, J. Sneyd, R. Bogacz, A. Randall, K. Tsaneva-Atanasova, Dr Martin WechselbergerMathematics of non-communicable diseases: understanding failure of cell signalling. (2012–2013: $50,000.) ## Externally funded research projects commencing in 2011 ARC Discovery Projects • Dr James Atkinson: Algebraic interpretations of discrete integrable equations. (2011:$82,000; 2012: $82,000; 2013:$82,000.)
• Prof E Norman Dancer: Stable and Finite Morse index solutions and peak solutions of nonlinear elliptic equations. (2011: $120,000; 2012:$120,000; 2013: $110,000.) • A/Prof Holger R Dullin, Prof Nalini Joshi: Geometry and analysis of discrete integrable systems. (2011:$100,000; 2012: $100,000; 2013:$100,000.)
• Prof Gus Lehrer: Flag varieties and configuration spaces in algebra. (2011: $120,000; 2012:$120,000; 2013: $120,000.) • A/Prof Andrew Mathas, Prof Jonathan Brundan: Graded representations of Hecke algebras. (2011:$149,000; 2012: $130,000; 2013:$130,000.)
• A/Prof Alexander I Molev: Vertex algebras and representations of quantum groups. (2011: $110,000; 2012:$110,000; 2013: $110,000.) • Dr James Parkinson, Asst Prof Joel Kamnitzer: The geometry and combinatorics of loop groups. (2011:$56,200.00 2012: $44,000; 2013:$43,000.)
• Dr Anne C Thomas, Dr Inna A Capdeboscq: Lattices in locally compact groups. (2011: $73,185; 2012:$67,885; 2013: $74,285; 2014:$61,385.)
Australian Postdoctoral Fellowship awarded to Dr Anne C Thomas.
• Dr Benjamin A Burton, Dr Murray J Elder, Dr Stephan Tillmann: Generic complexity in computational topology: breaking through the bottlenecks. (2011: $85,000; 2012:$85,000; 2013: $85,000.) • Dr Martin Wechselberger, Prof Graeme J Pettet, Prof Christopher K Jones: A geometric theory for travelling waves in advection-reaction-diffusion models. (2011:$85,000; 2012: $85,000; 2013:$85,000.)
• Dr Zhou Zhang: Topological and analytic aspects of the Kaehler-Ricci flow. (2011: $22,560.00; 2012:$22,560.00; 2013: $22,560.00.) • Prof Matt P Wand, Dr John T Ormerod, Prof Yongmei Michelle Wang: Fast approximate inference methods for flexible regression. (2011:$110,000; 2012: $110,000; 2013:$110,000.)
• Prof Alan H Welsh, Dr Samuel Müller Building models for complex data. (2011: $130,000; 2012:$110,000; 2013: $110,000.) Human Frontiers of Science Program • Dr Guy Lyons, Dr Silvio Gutkind, A/Prof Mary Myerscough: Cell cooperation in cancer. (2011: US$300,000; 2012: US$300,000; 2013: US$300,000.)

## Externally funded research projects commencing in 2010

ARC Discovery Projects

• Prof JJ Cannon, Dr DF Holt, Prof J Carlson: Constructive Module Theory for Algebras. (2010: $100,000; 2011:$100,000; 2012: $100,000.) • Dr C Ewald: Quantitative and qualitative aspects of Asian and Australian options. (2010:$50,000; 2011: $50,000; 2012:$50,000.)
• A/Prof G Gottwald: Stochastic methods in mathematical geophysical fluid dynamics. (2010: $90,000; 2011:$100,000; 2012: $100,000; 2013:$55,000; 2014: $55,000.) Australian Research Fellowship awarded to Georg Gottwald. • Dr V Jayaswal: Statistical methods for analysing multi-source microarray data and building gene regulatory networks. (2010:$85,000; 2011: $83,000; 2012:$83,000.)
Australian Postdoctoral Fellowship awarded to Dr V Jayaswal.
• Dr L Paunescu: The canonical stratification of jet spaces. (2010: $50,000; 2011:$50,000; 2012: $50,000.) • Prof JH Rubinstein, A/Prof CD Hodgson, Dr S Tillmann: Triangulations in dimension three: algorithms and geometric structures. (2010:$120,000; 2011: $125,000; 2012:$100,000.)

• A/Prof Holger Dullin, Damien Sinclair O'Meara and Peter Singh Surya:  Bodies in Space. (2010: $27,500.00; 2011:$55,000.00; 2012: $55,000.00; 2013:$27,500.00. New South Wales Institute of Sport.)

## Externally funded research projects commencing in 2009

ARC Discovery Projects

• Dr A Henderson: The geometry of exotic nilpotent cones. (2009: $20,000; 2010:$20,000.)
• Prof N Joshi: Integrable Lattice Equations. (2009: $108,000; 2010:$80,000; 2011: $90,000.) • A/Prof A Mathas; Dr A Henderson: Pyramids and decomposition numbers for the symmetric and general linear groups. (2009:$98,000; 2010: $85,000; 2011:$78,000; 2012: $80,000; 2013:$87,000.)
Australian Professorial Fellowship awarded to A/Prof Andrew Mathas.
• Dr S Santra: Blow-up phenomena in semilinear elliptic partial differential equations. (2009: $69,000; 2010:$67,000; 2011: $67,000; 2012:$67,000.)
Australian Postdoctoral Fellowship awarded to Sanjiban Santra.
• A/Prof R Zhang: Noncommutative geometry in representation theory and quantum physics. (2009: $90,000; 2010:$90,000; 2011: $80,000; 2012:$90,000; 2013: $100,000.) Australian Professorial Fellowship awarded to A/Prof Ruibin Zhang. ARC Future Fellowships • Dr Georg Gottwald: Stochastic Methods in Mathematical Geophysical Fluid Dynamics. (2009:$95,250; 2010: $185,650; 2011:$185,650; 2012: $185,650; 2013:$90,400.)
• Dr Jean Yang: New statistical methods for identifying micro-ribonucleic acid (miRNA) regulatory networks. (2009: $80,800; 2010:$151,600; 2011: $151,600; 2012:$151,600; 2013: $70,800.) ARC Linkage Project • Prof J George, Dr Jean Yang, Dr FC McKay, Dr V Suppiah; Dr DR Booth, Prof G Stewart, Dr M Bahlo: Functional Genomics to Predict and Enhance Response to Interferon. (2009:$145,000; 2010: $110,000; 2011:$115,000.)


text:
## 2016/11/28

### A note on estimating chaotic systems

This is a short note unlike other posts but it is necessary: Recently I came across with some complex systems presentations in Youtube. Most of the scientists, especially complex system scientists, treat chaotic dynamical systems as mysterious objects. The models are beautiful (so you should be careful around them)  and they are not that difficult to deal with in real world. The typical example you would hear that a chaotic system is extremely sensitive to the uncertainty in the initial condition. That's true: If I give you the initial condition of a deterministic chaotic system with a machine epsilon uncertainty along with the exact dynamics, the trajectory you would predict will differ from the real one vastly after a while. This, then they say, can be counted as the evidence of undecidability in chaotic systems.

## 2016/10/26

### A primer on filtering

Here, I discuss the core of the filtering idea in a relatively simple language. I will not introduce particle filters here but at the end you should have a really solid idea about what they are aiming at. In the following, I assume some familiarity with probability densities and fundamental rules (e.g. marginalisation or conditional independence or difference between a random variable and its realisation).

## 2016/09/23

### A simple bound for optimisation using a grid

If I give you a function on $[0,1]$ and a computer and want you to find the minimum, what would you do? Since you have the computer, you can be lazy: Just compute a grid on $[0,1]$, evaluate the grid points and take the minimum. This will give you something close to the true minimum. But how much?

## 2016/01/17

### An $L_2$ bound for Perfect Monte Carlo

Monte Carlo methods are widely used for estimating expectations of complicated probability distributions. Here I provide the well-known $L_2$ bound.

## 2015/09/07

### Matrix Factorisation with Linear Filters (and discussion)

I submitted a preprint on matrix factorisations and linear filters. I managed to derive some factorisation algorithms as linear filtering algorithms.

In the paper, I left a discussion to here, estimating parameters via maximising marginal likelihood. So here it is.

In the paper, I consider the following probabilistic model, \begin{align} p(c) &= \NPDF(c; c_0, V_0 \otimes I) \label{priorC} \\ p(y_k|c,x_k) &= \NPDF(y_k; (x_k \otimes I_m) c, \lambda \otimes I). \label{LikelihoodCform1} \end{align} where each $x_k$ is a static model parameter vector, and $C$ is the latent matrix. Note that $(x_k^\top \otimes I_m) c = C x_k$ (see the paper). We would like to solve the following problem to estimate $x_k$ for each $k$, \begin{align*} x_k^* = \argmax_{x_k} \log p(y_k | y_{1:k-1},x_k). \end{align*} So we would like to obtain the incremental marginal likelihood $p(y_k | y_{1:k-1},x_k)$. We define it as, \begin{align*} p(y_k | y_{1:k-1},x_k) = \int p(y_k | c, x_k) p(c | y_{1:k-1}) \mbox{d}c \end{align*} where $c = \vect(C)$. The important thing here is $p(c | y_{1:k-1})$ is not the exact distribution but it is an approximation to the true posterior as one has to use $x_{k-1}^*$ to obtain the distribution (If $X$ were known, it would be exact). So we know from the paper, \begin{align*} p(c | y_{1:k-1}) = \NPDF(c; c_{k-1}, V_{k-1} \otimes I_m), \end{align*} and, \begin{align*} p(y_k | c, x_k) = \NPDF(y_k; (x_k \otimes I_m) c, \lambda \otimes I_m). \end{align*} Using formulas in Bishop's PRML (2006, page 93), we can arrive the marginal $p(y_k|y_{1:k-1},x_k)$ as the following, \begin{align*} p(y_k | y_{1:k-1},x_k) = \NPDF(y_k; (x_k^\top \otimes I_m) c_k, (\lambda \otimes I_m) + (x_k^\top \otimes I_m) (V_{k-1} \otimes I_m) (x_k \otimes I_m)). \end{align*} Using properties given in the Section I.A of the paper, we compactly obtain, \begin{align}\label{MarginalLikelihoodOfMFRLF} p(y_k | y_{1:k-1},x_k) = \NPDF(y_k; C_{k-1} x_k, (\lambda + x_k^\top V_{k-1} x_k) \otimes I_m) \end{align} Although integration is analytic, maximising the logarithm of \eqref{MarginalLikelihoodOfMFRLF} is not possible analytically. However the likelihood is differentiable, so one can use nonlinear optimisation methods. But in terms of performance this does not bring any advantage: One gains a bit in the overall error part but loses too much in terms of computation time. I have tried quasi-Newton methods (BFGS), and conjugate gradients but no cure.

## 2015/06/19

### Online Matrix Factorization via Broyden Updates

I arXived a new preprint titled Online Matrix Factorization via Broyden Updates.

Around this April, I was reading quasi-Newton methods (from this very nice paper of Philipp Hennig) and when I saw the derivation of the Broyden update, I immediately realized that this idea may be used for computing factorizations. Furthermore, it will lead to an online scheme, more preferable!

The idea is to solve the following optimization problem at each iteration $k$:\begin{align*} \min_{x_k,C_k} \big\| y_k - C_k x_k \big\|_2^2 + \lambda \big\|C_k - C_{k-1}\big\|_F^2.\end{align*}
The motivation behind this cost is in the manuscript.

Although the basic idea was explicit, I set a few goals. First of all, I would like to develop a method that one can sample any column of the dataset and use it immediately. So I modified the notation a bit, as you can see from Eq. (2) in the manuscript. Secondly, I wanted that one must be able to use mini-batches as well, a group of columns at each time. Thirdly, it was obvious that a modern matrix factorization method must handle the missing data, so I had to extend the algorithm to handle the missing data. Consequently, I have sorted out all of this except a rule for missing data with mini-batches which turned out to be harder, so I left out that for this work.

## 2015/06/18

### MH sampler for changepoint inference

The following Metropolis-Hastings exercise was a homework of a course I assisted.

In this post, we will design an independent MH sampler to localise a changepoint in a relatively short time series for tutorial purposes.

## 2015/03/17

### Trapezoid rule as Bayesian inference

I learned a very interesting relation between numerical integration and Bayesian inference from Diaconis, which I shortly review in what follows.

## 2015/03/08

### Tinkering around logistic map

I was tinkering around logistic map today, I accidentally discovered a property of it. That is, if you look at histogram of the sequence generated by it, it will be same independent of the input. This immediately hit me that there may be an underlying probability density of this map. In fact, after Googled, I found out that Ulam and von Neumann studied this! So here we go.

## 2015/03/04

### Monte Carlo as Intuition

I am doing teaching assistantship for Monte Carlo methods course. I explained the following exercise as an application of the Monte Carlo principle. It is nice because it coincides with a very intuitive guess.


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# Ribbon Cable Noise Susceptibility

As part of a product upgrade we are looking at replacing looms of wires bound together by cable ties or spiral wrap with ribbon cable. We are doing this to try and cut cost and assembly times.

We are happy with the current noise performance of the system, but ribbon cable always seems to be referred to as "Bad" in terms of noise and interference.

The signals consist of a few medium speed digital signals (4 MHz SSI interface) and predominantly +-10V differential analogue signals with a 2 kHz filter on the receiving end. All up we are planning on using a 50 Core cable.

The environment is lab/light industrial, the cable will be approx 300mm long running parallel thin steel plate.

We are looking at using the standard 1.27mm Pitch cable (2 row 2.56mm plugs).

Ultimately, while rated as bad, can I expect at least similar performance from a ribbon cable as bundled wires.

• Why don't you try and build one - you'll get far better feedback by prototyping it. Dec 15, 2015 at 9:44
• I am working on the design, I just thought it was worthwhile asking first as it is much easier to design things in rather than hack them in later. Dec 15, 2015 at 10:29
• Design = come up with idea, try it out (test prototype), proper implementation (test again). I don't think getting SE feedback on something as hard to define as this is going to help much. Just my opinion based on countless years in the industry! Dec 15, 2015 at 10:54
• I unfortunately don't have countless years in the industry, yet. We all need to start somewhere. And yes lots of testing is planned. Dec 15, 2015 at 10:57

If you compare a ribbon cable with some random wires, laced into into a bundle (which I presume is being done at the moment) then the differences would be described as 'good'. What you get is a well defined impedance between adjacent lines. IIRC, a gnd-sig-gnd triple on 50thou pitch is around 100$\Omega$, but you had better check! You can use the well defined ordering to impose further ground lines between signals and pairs that you do not want to cross-talk.

• I went with this answer as while it is not as complete as KyranF's answer it more directly answers what relative difference can be expected, and why. Jan 28, 2016 at 0:56

If the ribbon itself is mostly shielded by steel plate from the major sources of industrial plant noise, this will help.

If you have enough spare wires in the ribbon, you should separate each differential pair on each side with ground wire signals, and keep one side dedicated to the serial signals, while the other side is all of your analog pairs (again, make sure to separate unrelated pairs with spare ground wires).

You can actually get a shielded ribbon cable with twisted pairs already organized for you, which would be great for differential pair serial comms and for the analog signal pairs. You can get some at Digikey for example: here and datasheet here which are "3M" brand and you can get 30 metres for $1050. I'm sure buying direct in larger amounts would make a huge saving, as Digikey only offers these as "value add" which can be grossly priced. Not including tax, this makes it only$1 of cable for your 300mm you want per unit, for the full service of shielded twisted pairs in an easy to assemble ribbon cable, which can be easily IDC terminated and grounded on your PCB/chassis ground.

The capacitance is just as bad as similar multi-core cables I've seen, between 15-30pf/foot, and I've run gigabit ethernet over similar capacitance in twisted pairs in custom cables over 5 metres so i'm sure this will do whatever communications you want in addition to your analog signals.

With your signals being +-10V and such a low frequency filter on the other end, i'm sure un-shielded will still be fine, provided you try doing the signal types as far away from each other as you can, using blank ground wires between each different pair.

• I have seen the availability of shielded cables, but ruled them out due to needing to have in the range of 3-6 connectors per cable. I forgot to mention this in the question, this also makes it pretty hard to use the twisted ribbon cables. Dec 15, 2015 at 6:12
• So along the 300mm you will be having 3-6 IDC connectors attached along it?? wow. Dec 15, 2015 at 6:15
• the twisted pairs are just a handy feature, but if it makes it too difficult then go for the normal ribbon and space signals accordingly Dec 15, 2015 at 6:16
• Yea, our board has to hook into an external product with a number of modules. The ribbon connects our main board and number of adapter boards that plug directly into their device. Dec 15, 2015 at 6:20
• Are the blank ground wires for the purpose of protecting against contamination between signals for from external influence. Dec 15, 2015 at 6:21