Jens von Bergmann has run the numbers on land use in various municipalities in Metro Vancouver. The City of Vancouver in particular has lot of land tied up in streets and detached housing.

Use	CoV land
single-family detached houses and duplexes	34.0%
roads and right-of-way	28.1%
recreation, open space, and natural areas	15.2%
commercial	3.9%
low-rise apartments (residential or mixed-use)	4.1%
high-rise apartments (residential or mixed-use)	1.9%

Because the City of Vancouver has so little area left undeveloped, any proposals for new housing, schools, parks, stores, and so forth will displace some existing use of the land.

Owl is a Beamer color theme for real-world conditions. Its dark and light themes and projector-optimized palette help you create slides you can count on to be readable in the presentation room.

Owl is available on CTAN and comes bundled with the latest TeX Live distribution.

Preparing a presentation in LaTeX? Metropolis provides a simple, modern Beamer theme suitable for anyone to use.

Metropolis is available on CTAN and comes bundled with the latest TeX Live distribution.

I was a major contributor to Metropolis from 2015 to 2016. If you want to help make the theme better, you can join the development efforts on Matthias Vogelgesang’s GitHub page for the project.

I have a new paper, coauthored with my supervisor Bojan Mohar and colleague Hehui Wu and presented at the SIAM Symposium on Discrete Algorithms! It is my first foray into graph immersions with parity restrictions.

I am grateful to NSERC for supporting this research through an Alexander Graham Bell Canada Graduate Scholarship.

The word pea was originally pease in the singular and peasen in the plural. Eventually, speakers understandably interpreted the -s in pease as the plural suffix rather than just a sound in the original Latin pisum/pisa and Greek πίσον, and the English singular pea was born.

For example, a 15th-century cookbook has the following recipe for what we would today call pea soup:

Take grene pesyn, an washe hem clene an caste hem on a potte, an boyle hem tyl þey breste, an þanne take hem vppe of þe potte, an put hem with brothe yn a-noþer potte, and lete hem kele; þan draw hem þorw a straynowre in-to a fayre potte, an þan take oynonys…

Harleian manuscript 279

Pease also functioned as a mass noun, like bread or oatmeal.

Yisterday I ete cale and pes, & to-day I eete pes & cale, & to-morn I mon eate pess with cale, & after to-morn I mon eate cale with pease.

Alphabet of Tales

Unfortunately, the latter quote is taken from a religious anecdote promoting a moderate and uniform diet, and not a hilariously sarcastic comment by a medieval peasant.

A disproportionate number of my tweets are exactly 140 characters. I don’t know whether that means I’m really good at Twitter or really bad. Sometimes it’s the result of a too-long idea being meticulously edited down to size; sometimes it’s purely chance. Either way, I find 140-character tweets oddly satisfying — and based on a large dataset of tweets, it looks like I’m not the only one.

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The dataset paints a fascinating picture of the distribution of tweet lengths. Extremely short tweets are understandably very rare, but it doesn’t take long for the distribution to reach its first mode at 35 characters. The curve gradually and smoothly trails off to a local minimum around 116 characters, before positively spiking after 135. The average length is a bit more than 68 characters and the median a bit lower at 62.

The American copyright status of the song “Happy Birthday to You” has finally been resolved in the case Rupa Marya v. Warner Chappell Music. (Here in Canada, the song has been in the public domain since 1997.)

At the time of lawsuit, Warner was collecting royalties — around $2 million a year — for “Happy Birthday to You” despite the fact that the melody was in the American public domain. They claimed that the lyrics were still under copyright and that they owned the rights to them.

Although Warner had acquired some “Happy Birthday”-related rights, it wasn’t clear what those rights covered since the original transfer agreements had been lost. The judge ruled that the secondary sources did not support Warner’s claim on the lyrics specifically, assuming they were still under copyright at all. Settlement terms following the summary judgement definitively assigned the song to the American public domain.

As far as I can tell, the European copyright to both the “Happy Birthday” lyrics and melody would have been still valid, albeit with disputed ownership, until it expired in 2017.

In collaboration with the SFU Library and my fellow grad students, I’ve written a LaTeX template from which graduate students at Simon Fraser University can start writing their thesis or dissertation.

The project offers a LaTeX class file called sfuthesis that automatically sets your thesis according to the SFU Library’s style requirements. With its help, you can focus on writing up your research instead of fiddling with formatting.

Get started now by downloading a copy from the SFU Library website!

Rubber duck problem solving describes the phenomenon where you realize the solution to a problem in the middle of explaining it to someone else. The name stems from apocryphal stories in which stumped engineers are advised to get help from inanimate objects, including a literal rubber duck.

The technique works because communication forces us to arrange our thoughts and prevents us from taking shortcuts that would leave our audience behind. As one developer explains:

When you force yourself to verbalize something, you take poorly formed mind-stuff and slot it into discretely packaged concepts (words) whose meanings are agreed upon by other humans. This alone adds an important layer of organization to your thinking by taking non-verbal soup and giving it shape.

Joseph Pacheco

I have a new paper published in Graphs and Combinatorics! It’s my favourite paper to come out of my research with Jing Huang at the University of Victoria — the third written chronologically, and the last to be published. The main result is that the structure of monopolar partitions in claw-free graphs can be fully understood by looking at small subgraphs and following their direct implications on vertex pairs.

One of the most recognizable features of Japanese architecture is the matted flooring. The individual mats, called tatami, are made from rice straw and have a standard size and $1 \times 2$ rectangular shape. Tatami flooring has been widespread in Japan since the 17th and 18th centuries, but it took three hundred years before mathematicians got their hands on it.

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According to the traditional rules for arranging tatami, grid patterns called bushūgishiki (不祝儀敷き) are used only for funerals.¹ In all other situations, tatami mats are arranged in shūgishiki (祝儀敷き), where no four mats meet at the same point. In other words, the junctions between mats are allowed to form ┬, ┤, ┴, and ├ shapes but not ┼ shapes.

Shūgishiki tatami arrangements were first considered as combinatorial objects by Kotani in 2001 and gained some attention after Knuth including them in The Art of Computer Programming.

Construction

Once you lay down the first couple tatami, you’ll find there aren’t many ways to extend them to a shūgishiki. For example, two side-by-side tatami force the position of all of the surrounding mats until you hit a wall.

This observation can be used to decompose rectangular shūgishiki into

• $(m-2)\times m$ blocks forced by vertical tatami,
• $m \times m$ blocks forced by horizontal tatami, and
• $1\times m$ strips of vertical tiles,

and to derive their generating function

$$T(x) = \frac{(1+x)(1+x^{m-2}+x^m)}{1-x^{m-1}-x^{m+1}}$$

Four-and-a-half tatami rooms can also be found in Japanese homes and tea houses, so naturally mathematicians have also looked into tatami tilings with half-tatami. Alejandro Erickson’s PhD thesis reviews and extends the research into this area. Alejandro has also published a book of puzzles about tatami layouts.

1. In reality, grid layouts are also used for practical reasons in inns, temples, and other large gathering halls.

When the Ontario cities of Fort William and Port Arthur amalgamated in 1970, residents voted for a new name for their new city.

Candidate name	Votes
Thunder Bay	15,870
Lakehead	15,302
The Lakehead	8,377

The result deserves a place of honour in voting theory textbooks.

I have a new paper with Jing Huang in Graphs and Combinatorics! This was the culmination of my undergraduate research, and shows that a single strategy can be used to solve the monopolar partition problem in all graph classes for which the problem was previously known to be tractable, including line graphs and claw-free graphs.

This research was completed in the summer of 2010, my last undergraduate research term. I am grateful to NSERC for funding my work with a Undergraduate Student Research Award, and to my supervisor and coauthor Jing Huang.

Image Evolution is a very interesting Javascript tool based on Roger Johansson’s Evo-Lisa idea. It uses a genetic algorithm to represent images as a collection of overlapping polygons.

We start from [a set of random] polygons that are invisible. In each optimization step we randomly modify one parameter (like color components or polygon vertices) and check whether such new variant looks more like the original image. If it is, we keep it, and continue to mutate this one instead.

Just feed it an image and hit start, and a random collection of coloured polygons will gradually evolve into a cool abstract rendition of your picture.

Ever wonder why LaTeX doesn’t provide a way for printing the title and author once \maketitle has been issued? I did. So I asked a question on the TeX StackExchange and received an interesting answer. Turns out it’s an artifact of the times when memory was in extremely short supply.

The main reason was “main-memory” back in those days. LaTeX was effectively eating up half of the available space just through macro definitions. So with complicated pages or with some picture environments etc you could hit the limit. So freeing up any bit was essential and you still see traces of this in the code.

Frank Mittelbach

Dark times!

I have successfully defended my master’s thesis on graph-transverse matching problems! It considers the computational complexity of deciding whether a given graph admits a matching which covers every copy of a fixed tree or cycle.

The thesis is related to my previous work on cycle-transverse matchings and $P_4$-transverse matchings and, roughly speaking, shows that H-transverse matchings are NP-hard to find when $H$ is a big cycle or tree, and tractable when $H$ is a triangle or a small tree.

I am grateful to NSERC for funding my degree with a Alexander Graham Bell Canada Graduate Scholarship, and to my supervisor Jing Huang.

I defend my thesis in two weeks, but I’ll be prepared for the snake fight portion thanks to McSweeney’s guide:

Do I have to kill the snake?

University guidelines state that you have to “defeat” the snake. There are many ways to accomplish this. Lots of students choose to wrestle the snake. Some construct decoys and elaborate traps to confuse and then ensnare the snake. One student brought a flute and played a song to lull the snake to sleep. Then he threw the snake out a window.

Are the snakes big?

We have lots of different snakes. The quality of your work determines which snake you will fight. The better your thesis is, the smaller the snake will be.

In 1852, then-student Francis Guthrie wondered any if possible map required more than four colours. By the end of the century, Guthrie and his fellow colonists had drawn a map on Africa that needed five.

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The Four-Colour Theorem says that, no matter what the borders on your map are, you only need four colours to make sure that neighbouring regions are coloured differently. The theorem doesn’t apply if you let some regions claim other disconnected regions as their own, and in fact the map of European claims on Africa required five colours by the end of the 19th century.

Francis Guthrie, who moved to the South African Cape Colony in 1861, could well have owned a map like the above. Five colours are necessary to properly colour the land that Britain (red), France (orange), Portugal (yellow), Germany (green), and Belgium’s King Leopold II (purple) decided should belong to them.

Five territorities in the center are key to the map colouring:

	Area	Colonizer
🟣	Congo basin	King Leopold II
🟠	north of the Congo	France
🟡	south of the Kwango	Portugal
🔴	upper Zambezi basin	Britain
🟢	African Great Lakes	Germany

The boundaries between these colonies separate seven different pairs of empires. Borders between other African colonies account for the other three possible sets of neighbours:

• French Congo was next to German Kamerun;
• German East Africa to Portuguese Mozambique; and
• French claims in the Ubangi-Chari area were next to Anglo-Egyption Sudan.

In short, the adjacency graph between these empires was the complete graph $K_5$.

A little while ago, I did some sleuthing to find out the Erdős number of Brian May, astrophysicist and guitarist from Queen. My travels led me to Timeblimp, who threw together three measures of professional collaboration to make a rather fun parlour game. Assuming that the people in your parlour are three kinds of nerds and enjoy long and complicated internet scavenger hunts. Which I am and I do.

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The game is to find a well-known person who has published academically, released a song, and been involved in a movie or TV show. Then, you play three versions of Six Degrees of Kevin Bacon: find a series of movies to connect them to prolific actor Kevin Bacon, a series of coauthored papers to connect them to the eccentric mathematician Paul Erdős, and a series of musical collaborations to get to Black Sabbath. Add up all the links and you get the Erdős-Bacon-Sabbath number.

Brian Cox has an Erdős-Bacon-Sabbath number

If anyone has an Erdős-Bacon-Sabbath number, Brian Cox is exactly the sort of person you might expect to have one. The keyboardist, particle physicist, and BBC science presenter is no more than 7+3+3 degrees of separation from the centers of the EBS graph.

Sean from Timeblimp first suggested the possibility of Brian Cox having a well-defined Erdős-Bacon-Sabbath number, but to my knowledge nobody had worked out his Erdős number until now. I managed to find a path of length seven.

Erdős number	Researcher	Citation
7	Brian Cox	Hard colour singlet exchange at the Tevatron
6	Leif Lönnblad	Small-x dipole evolution beyond the large Nc limit
5	Gösta Gustafson	The action-angle variables for the massless relativistic string in 1+1 dimensions
4	Bo Söderberg	Scaling laws for mode locking in circle maps
3	Boris Shraiman	Scaling theory for noisy period-doubling transitions to chaos
2	C Eugene Wayne	The Euler-Bernoulli beam equation with boundary energy dissipation
1	Steven George Krantz	Intersection graphs for families of balls in ℝn
	Paul Erdős

The above connections use only papers with three coauthors or fewer. Cox has worked in gigantic collaborations like ATLAS, so it’s quite possible that there might be a shorter path.

Brian Cox — not to be confused with the other Brian Cox — is three degrees of separation from Kevin Bacon through his many TV appearances, including cameos on Doctor Who.

Bacon number	Actor	Credit
3	Brian Cox	Doctor Who “The Power of Three”
2	Matt Smith	Charlie Says
1	Sosie Bacon	Story of a Girl
	Kevin Bacon

Brian Cox had a successful music career before finishing his PhD and also has a Sabbath number of three.

Sabbath number	Musician	Band/Album
3	Brian Cox	Dare
2	Darren Wharton	Thin Lizzy
1	Tommy Aldridge	Speak of the Devil
	Ozzy Osbourne	Black Sabbath

Aftermath

After I published this post, someone brought it to the attention to none other than Brian Cox himself!

The resulting hullabaloo led to the discovery of many other Erdős-Bacon-Sabbath numbers. Eventually, I retired from EBS research after realizing its flaws as a game and as a social construct.

Pokémon Gold and Silver introduced the roaming legendary beasts: three one-of-a-kind Pokémon that move from route to route instead of sticking to a fixed habitat. Catching a roaming Pokémon amounts to winning a graph pursuit game — so what can we learn about it from the latest mathematical results?

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To review the Pokémon mechanics, each species can normally be found in a handful of fixed habitats. If you want to catch Abra, you go to Route 24; if you’re looking for Jigglypuff, head to Route 46.

The legendary Entei, Raikou, and Suicune¹ are different. There’s only one of each species, each situated on a random route. Each time the player character moves to a new location, the roaming Pokémon each move to a randomly-selected route adjacent to the one they were just on. In graph theory terms, the player and Pokémon are engaged in a pursuit game where the Pokémon’s strategy follows a random walk.

The study of graph pursuit games is a fascinating and active area of research. Classically, researchers have asked how many “cops” it takes to guarantee the capture of an evasive “robber” travelling around a graph. Depending on the graph, many cops might be needed to catch a clever robber; there is a deep open problem about the worst-case cop numbers of large graphs.

Because the graph corresponding to the Pokémon region of Johto contains a long cycle as an isometric subgraph, its cop number is more than one — in other words, it’s possible that a roaming Pokémon could theoretically evade a lone Pokémon trainer forever! Fortunately, the legendary beasts play randomly, not perfectly, so the worst-case scenario doesn’t apply.

A random walk in an arbitrary $n$-vertex, $m$-edge graph can be expected to spend $\deg(v)/(2m)$ of its time at each location $v$, and to visit the whole graph after at most roughly $4n^3/27$ steps. So any trainer who isn’t actively trying to avoid Entei should end up bumping into it eventually — and an intelligent trainer should be able to do much better.

The first place to start is the “greedy” strategy I originally tried as a kid: every time Entei moves, check the map, and move to any route that gets me closer to them. After Entei makes its random move, the distance between us could be unchanged (with Entei’s move offsetting mine), or it could go down by one, or it could go down by two in the lucky $1/\Delta$ chance that Entei moves towards me. If I start at a distance of $\ell$ steps away from Entei and get lucky $\ell/2$ times, I’ll have caught up — so using a negative binomial distribution bound,

$$\text{E[capture time]} \leq \frac{\Delta \ell}{2}.$$

In the grand scheme of things, this isn’t too bad — especially if $\Delta$ is low. But it still takes a frustratingly large time for a 12-year-old, and in general it’s possible to do better.

Recently, Peter Winkler and Natasha Komarov found a strategy for general graphs which gives a better bound on the expected capture time. Somewhat counterintuitively, it involves aiming for where the robber was — rather than their current location — until the cop is very close to catching him. The Komarov-Winkler strategy has an expected capture time of $n+o(n)$, where $n$ is the number of locations on the map. This is essentially best possible on certain graphs, and is better than the above $\Delta\ell/2$ bound when the graph has vertices with large degree.

For graphs without high-degree vertices — like the Pokémon world map — it is possible that a simpler solution could beat the Komarov–Winkler strategy. The problem is: simpler strategies may not be simpler to analyse. In her PhD thesis, Natasha wondered whether a greedy algorithm with random tiebreakers could guarantee $n+o(n)$ expected capture time. It is an open question to find a general bound for the “randomly greedy” strategy’s expected performance that would prove her right.

1. I’m including Suicune in this list since it roamed in the original Gold and Silver, but its behaviour is different from the others in Pokémon Crystal, HeartGold, and SoulSilver.

Dr Robb Fry, one of my professors from my Thompson Rivers University days, passed away earlier this year at far too young an age. Robb was a real character, a great teacher, and a lot of fun to know.

I took my second course in linear algebra with Robb, and it was one of the most entertaining courses of my first two years. While visiting my parents over the holidays, I dug out my course notes — the only full set of notes I ever took in undergrad — so I could share some memorable episodes from my time with him.

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Introducing the notation ”∃!”

It means “there exists a unique…” but I always read it like it’s a William Shatner thing. THERE EXISTS!

On terminology

ROBB: An oval…

CLASS: Don’t you mean an ellipse?

ROBB: Yeah, whatever the real term for that is.

On yellow chalk

I’m going to avoid yellow chalk today, because I have a suspicion that one day they’ll find that the stuff that makes it yellow is toxic. That’s going to be someone’s Ph.D. thesis one day, The Toxic Effects of Yellow Chalk, and I don’t want to be part of the study group.

On the kernel (which gets “killed” by a map)

The best bumper sticker I’ve ever seen had a picture of the colonel from KFC with “I am dead.”

On writing

If you’re ever reading a paper, and they say they have to prove a technical lemma, brace yourself for some horrific math.

On stable/invariant sets

I use “invariant” instead of “stable.” A stable set sounds like something for horses. I like horses, mind you, but they shouldn’t be confused with mathematics.

You taught me, inspired me, and motivated me to continue on the path to becoming a mathematician. But more importantly, it was a lot of fun to know you. Thanks, Robb.