In case you’re new to this: every now and then I encounter a paper or a book or an article that grabs my interest but isn’t directly useful for anything. It might be about some niche sub-sub-subtopic I’ve never heard of, or it might talk about something old from a new angle, or it might just have a funny title. I put these things in my Interesting Esoterica collection on Mendeley. And then when I’ve gathered up enough, I collect them here.

In this post the titles are links to the original sources, and I try to add some interpretation or explanation of why I think each thing is interesting below the abstract.

Some things might not be freely available, or even available for a reasonable price. Sorry.

On sphere-filling ropes

What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope’s thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simplest ones are similar to the seamlines of a tennis ball, others exhibit a striking resemblance to Turing patterns in chemistry, or to ordered phases of long elastic rods stuffed into spherical shells.

Spherical geometry makes the world go round.

The Lost Squares of Dr. Franklin: Ben Franklin’s Missing Squares and the Secret of the Magic Circle

If I mention Benjamin Franklin and mathematics in the same breath, your reaction will likely be:
1. “I didn’t know Franklin did mathematics,” or:
2. “I already know all about that subject.”

My counter-reply, as appropriate:
1. Yes, he did.
2. No, you don’t.

A lengthy piece from the American Mathematical Monthly about Benjamin Franklin’s adventures with magic squares. Closed access on JSTOR, $12.

Papy’s minicomputer

Georges Papy was gloriously batty, in that inimitably high-faluting way accessible only to French-speakers. Before I talk about the minicomputer, I have to direct you to this review of his book Groups, which is my absolute least favourite maths book, or the book I most love to hate, in New Scientist magazine. My eternal thanks to the reviewer for stepping up to the plate and smacking it out of the park with this snippet:

A final 32 plates in colour have all the discontinuity of a death-bed repentance. They leave disappointment at an opportunity wasted. Many are trivial, all are insufficiently captioned and – quite inexcusably – some exhibit colours other than those named in the accompanying text.

My thoughts exactly. Anyway, the minicomputer. Georges collaborated closely with his wife Frédérique, and she wrote this piece for the Bulletin of the Association of Teachers of Mathematics in 1970. It’s about a square piece of card which was meant to help children learn about numbers. I’m non convainçu.

Conway’s Wizards

I present and discuss a puzzle about wizards invented by John H. Conway.

Tanya Khovanova is doing an admirable job of collecting trickier puzzles and providing lucid discussion of their solutions, all on the arXiv for everyone to see. This one’s a logic puzzle involving two wizards on a bus.

Picture-hanging puzzles

We show how to hang a picture by wrapping rope around n nails, making a polynomial number of twists, such that the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get removed. This construction makes for some fun mathematical magic performances. More generally, we characterize the possible Boolean functions characterizing when the picture falls in terms of which nails get removed as all monotone Boolean functions. This construction requires an exponential number of twists in the worst case, but exponential complexity is almost always necessary for general functions.

A wonderful puzzle. Good luck forgetting the definition of a commutator after working through this paper. This was so interesting that I made it the subject of our first recreational maths seminar.

A stratification of the space of all $k$-planes in $\mathbb{C}^n$

To each $k \times n$ matrix $M$ of rank $k$, we associate a juggling pattern of periodicity $n$ with $k$ balls. The juggling pattern actually only depends on the $k$-plane spanned by the rows, so gives a decomposition of the “Grassmannian” of all $k$-planes in $n$-space.
There are many connections between the geometry and the juggling. For example, the natural topology on the space of matrices induces a partial order on the space of juggling patterns, which indicates whether one pattern is “more excited” than another.
This same decomposition turns out to naturally arise from totally positive geometry [Lusztig 1994, Postnikov ∼2004], characteristic $p$ geometry [Knutson-Lam-Speyer 2011], and noncommutative geometry [Brown-Goodearl-Yakimov 2005]. It also arises by projection from the manifold of full flags in $n$-space, where there is no cyclic symmetry.

Allen Knutson has spent more time than probably anyone else thinking about the maths of juggling. These are some slides from a talk about the connection between juggling and some very abstract geometry.

How to eat 4/9 of a pizza

Given two players alternately picking pieces of a pizza sliced by radial cuts, in such a way that after the first piece is taken every subsequent chosen piece is adjacent to some previously taken piece, we provide a strategy for the starting player to get 4/9 of the pizza. This is best possible and settles a conjecture of Peter Winkler.

Biologically unavoidable sequences

A biologically unavoidable sequence is an infinite gender sequence which occurs in every gendered, infinite genealogical network satisfying certain tame conditions. We show that every eventually periodic sequence is biologically unavoidable (this generalizes Koenig’s Lemma), and we exhibit some biologically avoidable sequences. Finally we give an application of unavoidable sequences to cellular automata.

I saw the main result of this paper written very clearly somewhere, probably Google+, but I’ve forgotten it and I’m finding the paper quite hard to read.

Figures for “impossible fractals”

Take “impossible” shapes and turn them into fractals. Admire pretty pictures.

The paramagnetic and glass transitions in sudoku

We study the statistical mechanics of a model glassy system based on a familiar and popular mathematical puzzle. Sudoku puzzles provide a very rare example of a class of frustrated systems with a unique groundstate without symmetry. Here, the puzzle is recast as thermodynamic system where the number of violated rules defines the energy. We use Monte Carlo simulation to show that the “Sudoku Hamiltonian” exhibits two transitions as a function of temperature, a paramagnetic and a glass transition. Of these, the intermediate condensed phase is the only one which visits the ground state (i.e. it solves the puzzle, though this is not the purpose of the study). Both transitions are associated with an entropy change, paramagnetism measured from the dynamics of the Monte Carlo run, showing a peak in specific heat, while the residual glass entropy is determined by finding multiple instances of the glass by repeated annealing. There are relatively few such simple models for frustrated or glassy systems which exhibit both ordering and glass transitions, sudoku puzzles are unique for the ease with which they can be obtained with the proof of the existence of a unique ground state via the satisfiability of all constraints. Simulations suggest that in the glass phase there is an increase in information entropy with lowering temperature. In fact, we have shown that sudoku have the type of rugged energy landscape with multiple minima which typifies glasses in many physical systems, and this puzzling result is a manifestation of the paradox of the residual glass entropy. These readily-available puzzles can now be used as solvable model Hamiltonian systems for studying the glass transition.

Invited Commentary: The Perils of Birth Weight—A Lesson from Directed Acyclic Graphs

The strong association of birth weight with infant mortality is complicated by a paradoxical finding: Small babies in high-risk populations usually have lower risk than small babies in low-risk populations. In this issue of the Journal, Hernández-Díaz et al. (Am J Epidemiol 2006;164:1115–20) address this “birth weight paradox” using directed acyclic graphs (DAGs). They conclude that the paradox is the result of bias created by adjustment for a factor (birth weight) that is affected by the exposure of interest and at the same time shares causes with the outcome (mortality). While this bias has been discussed before, the DAGs presented by Hernández-Díaz et al. provide more firmly grounded criticism. The DAGs demonstrate (as do many other examples) that seemingly reasonable adjustments can distort epidemiologic results. In this commentary, the birth weight paradox is shown to be an illustration of Simpson’s Paradox. It is possible for a factor to be protective within every stratum of a variable and yet be damaging overall. Questions remain as to the causal role of birth weight.

Algorithmic Self-Assembly of DNA Sierpinski Triangles

Algorithms and information, fundamental to technological and biological organization, are also an essential aspect of many elementary physical phenomena, such as molecular self-assembly. Here we report the molecular realization, using two-dimensional self-assembly of DNA tiles, of a cellular automaton whose update rule computes the binary function XOR and thus fabricates a fractal pattern—a Sierpinski triangle—as it grows. To achieve this, abstract tiles were translated into DNA tiles based on double-crossover motifs. Serving as input for the computation, long single-stranded DNA molecules were used to nucleate growth of tiles into algorithmic crystals. For both of two independent molecular realizations, atomic force microscopy revealed recognizable Sierpinski triangles containing 100–200 correct tiles. Error rates during assembly appear to range from 1% to 10%. Although imperfect, the growth of Sierpinski triangles demonstrates all the necessary mechanisms for the molecular implementation of arbitrary cellular automata. This shows that engineered DNA self-assembly can be treated as a Turing-universal biomolecular system, capable of implementing any desired algorithm for computation or construction tasks.

Delay can stabilize: Love affairs dynamics

We discuss two models of interpersonal interactions with delay. The first model is linear, and allows the presentation of a rigorous mathematical analysis of stability, while the second is nonlinear and a typical local stability analysis is thus performed. The linear model is a direct extension of the classic Strogatz model. On the other hand, as interpersonal relations are nonlinear dynamical processes, the nonlinear model should better reflect real interactions. Both models involve immediate reaction on partner’s state and a correction of the reaction after some time.

The models we discuss belong to the class of two-variable systems with one delay for which appropriate delay stabilizes an unstable steady state. We formulate a theorem and prove that stabilization takes place in our case. We conclude that considerable (meaning large enough, but not too large) values of time delay involved in the model can stabilize love affairs dynamics.

Modelling romantic relationships as dynamical systems, with some resulting advice for couples going through a rough patch. Closed access, ScienceDirect, $27.95.

Incorporating Voice Permutations into the Theory of Neo-Riemannian Groups and Lewinian Duality

A familiar problem in neo-Riemannian theory is that the P, L, and R operations defined as contextual inversions on pitch-class segments do not produce parsimonious voice leading. We incorporate permutations into T/I-PLR-duality to resolve this issue and simultaneously broaden the applicability of this duality. More precisely, we construct the dual group to the permutation group acting on n-tuples with distinct entries, and prove that the dual group to permutations adjoined with a group G of invertible affine maps Z12 -> Z12 is the internal direct product of the dual to permutations and the dual to G. Musical examples include Liszt, R. W. Venezia, S. 201 and Schoenberg, String Quartet Number 1, Opus 7. We also prove that the Fiore–Noll construction of the dual group in the finite case works, and clarify the relationship of permutations with the RICH transformation.

I saw this on the arXiv’s math.GR new preprints feed and put it in my collection before I really understood what it was about. Apparently some music theorists get a bit too theoretical and start applying group theory. Richard Green wrote an explanation of what it’s all about on Google+.

A note on paradoxical metric spaces

Via a submission to the arXiv with an intriguing title – “Invariant means of the wobbling group” – I found this paper which gives a definition of a wonderful term: a wobbling bijection. According to the paper on the wobbling group, they’re used in the solution to Tarski’s circle-squaring problem (exposited accessibly in this article from Math Horizons). Furthermore, according to the abstract of the paper “Geometrical bijections in discrete lattices”, wobbling mappings occur during earthquakes! Bonanza!

Conway’s rational tangles

I learned about this “trick” in a lecture by John Conway a number of years ago. He calls it “Rational Tangles” and there is plenty of information about it on the internet. Since then I have used it myself in classrooms of students of middle school age and older. The underlying mathematics is very interesting, but it is not necessary that the students understand the mathematics to get a lot out of the trick. In fact, some of the mathematics I do not understand.

Algebra and ropes pair up for a second time today.

Markets are efficient if and only if P = NP

I prove that if markets are efficient, meaning current prices fully reflect all information available in past prices, then P = NP, meaning every computational problem whose solution can be verified in polynomial time can also be solved in polynomial time. I also prove the converse by showing how we can “program” the market to solve NP-complete problems. Since P probably does not equal NP, markets are probably not efficient. Specifically, markets become increasingly inefficient as the time series lengthens or becomes more frequent. An illustration by way of partitioning the excess returns to momentum strategies based on data availability confirms this prediction.

Economists love assuming that markets are efficient, because it lets them apply all sorts of maths to them. Well, this chap says in this maths-light, word-heavy paper that maths might prove markets to be inefficient. What nonsense.

What are some of the most ridiculous proofs in mathematics?

Respondents seem to be conflating “ridiculous” with “short” but the top answer, which uses Fermat’s last theorem to prove $\sqrt[3]{2}$ is irrational, is sublime.

Embedding countable groups in 2-generator groups

One of the postgrads I share an office with said they’d heard something about every countable group being generated by two elements. Here’s a proof. It’s such a pleasing fact about the universe.

The Muddy Children: a logic for public announcement

Public announcement logic is pretty cool. One of the very first entries in my collection is the paper “‘Knowable’ as ‘known after an announcement’” (closed access, Cambridge, £30). These slides present the idea through a very accessible problem – three kids each may or may not have mud on their faces, and a fourth kid (being wilfully obtuse) tells them that at least one of them is muddy. Is that information to tell if you have mud on your face? By repeatedly announcing whether they know if they have mud on their faces, all three kids can eventually work out if they need to make a trip to the bathroom.

Circular reasoning: who first proved that $C/d$ is a constant?

We answer the question: who first proved that $C/d$ is a constant? We argue that Archimedes proved that the ratio of the circumference of a circle to its diameter is a constant independent of the circle and that the circumference constant equals the area constant ($C/d=A/r^{2}$). He stated neither result explicitly, but both are implied by his work. His proof required the addition of two axioms beyond those in Euclid’s Elements; this was the first step toward a rigorous theory of arc length. We also discuss how Archimedes’s work coexisted with the 2000-year belief — championed by scholars from Aristotle to Descartes — that it is impossible to find the ratio of a curved line to a straight line.

Zeroless Arithmetic: representing integers ONLY using ONE

We use recurrence equations (alias difference equations) to enumerate the number of formula-representations of positive integers using only addition and multiplication, and using addition, multiplication, and exponentiation, where all the inputs are ones. We also describe efficient algorithms for the random generation of such representations, and use Dynamical Programming to find a shortest possible formula representing any given positive integer.

Doron Zeilberger and an acolyte cement their place in my bad books by denying the utility of zero. Along the way, they neglect the entirety of Peano arithmetic by coming up with some approximate solutions to problems which almost definitely have closed form answers. He’s got a nerve.

Way back when, before we even launched this site, Katie and I made a video about a paper I’d found called Cryptographic and Physical Zero-Knowledge Proof Systems for Solutions of Sudoku Puzzles. The paper described a scheme for proving you can solve a sudoku puzzle without revealing your solution, using only paper and scissors. We tried it out, and it worked!

I’ve been wanting to do more videos like that, but papers describing new mathematical things to do with household objects are quite rare. I also have the problem that most of my Interesting Esoterica collection goes unread, even though there are quite a few things I’d really like to look at in more detail. So I’ve connected those two thoughts and come up with the ‘recreational maths seminar’.

The idea is this: I pick a paper from my collection that looks interesting, and post it along with a date to get together and discuss it. We all pile in to a Google+ On Air hangout and work through it for an hour or two, and leave as better-educated people, with a YouTube recording of the session for people who couldn’t make it.

I had a small test-run a couple of days ago with Colin Beveridge, where we looked at a very interesting paper titled Twin Towers of Hanoi. Here’s Google+’s recording of the session (I wasn’t particularly paying attention to the fact it was being recorded, so it starts and ends quite abruptly):

I need to pick a time that will suit the most people, and that I can make, so if you like the sound of my idea please put a comment below saying which of the following are convenient for you. I expect each seminar will last between one and two hours.

• Weekdays between 1800 and 2300 GMT
  (1300-1800 EST, 1900-0000 CET, 0300-0800 EDT)
• Weekdays between 0700 and 0900 GMT
  (0200-0400 EST, 0800-1000 CET, 1800-2000 EDT)
• Weekends between 0700 and 1100 GMT
  (0200-0600 EST, 0800-1200 CET, 1800-2200 EDT)
• Weekends between 1600 and 2300 GMT
  (1100-1800 EST, 1700-0000 CET, 0100-0800 EDT)

Today is Halloween, the day when skeletons and spooks and statisticians¹ roam the earth to wreak their awful revenge on the innocent.

Maths has a habit of borrowing peculiar words from the vernacular, so I thought I’d go on a witch-hunt in the arXiv and see what bone-chilling titles I could find. Here’s what fell into my genuine recreation Ghostbusters Ghost Trap:

Limiting Risk by Turning Manifest Phantoms into Evil Zombies

On the effect of ghost force in the quasicontinuum method: dynamic problems in one dimension

Taming the Ghost in Pais-Uhlenbeck Oscillator

Recovering Missing Slices of the Discrete Fourier Transform using Ghosts

An algebraic approach to laying a ghost to rest

Conclusion: mathematicians ain’t afraid of no ghosts.

Have you got any blood-curdling paper titles to share around the campfire?

1. While The Aperiodical is an equal opportunity employer, the author maintains a legitimate fear of “approximate counting”

In case you’re new to this: every now and then I encounter a paper or a book or an article that grabs my interest but isn’t directly useful for anything. It might be about some niche sub-sub-subtopic I’ve never heard of, or it might talk about something old from a new angle, or it might just have a funny title. I put these things in my Interesting Esoterica collection on Mendeley. And then when I’ve gathered up enough, I collect them here.

In this post the titles are links to the original sources, and I try to add some interpretation or explanation of why I think each thing is interesting below the abstract.

Some things might not be freely available, or even available for a reasonable price. Sorry.

The usefulness of useless knowledge

A rather long essay from a 1939 issue of Harper’s Bazaar extolling the virtues of apparently pointless scientific inquiry. People still feel the need to make this kind of argument today; they needn’t bother – this one’s decent and most of the examples are the same ones you’d pick today.

A New Rose : The First Simple Symmetric 11-Venn Diagram

A symmetric Venn diagram is one that is invariant under rotation, up to a relabeling of curves. A simple Venn diagram is one in which at most two curves intersect at any point. In this paper we introduce a new property of Venn diagrams called crosscut symmetry, which is related to dihedral symmetry. Utilizing a computer search restricted to crosscut symmetry we found many simple symmetric Venn diagrams with 11 curves. This answers an existence question that has been open since the 1960′s. The first such diagram that was discovered is shown here.

I wrote about this in the News section a while ago, but it belongs here in the Interesting Esoterica collection.

The Fastest and Shortest Algorithm for All Well-Defined Problems

An algorithm $M$ is described that solves any well-defined problem $p$ as quickly as the fastest algorithm computing a solution to $p$, save for a factor of 5 and low-order additive terms. $M$ optimally distributes resources between the execution of provably correct $p$-solving programs and an enumeration of all proofs, including relevant proofs of program correctness and of time bounds on program runtimes. $M$ avoids Blum’s speed-up theorem by ignoring programs without correctness proof. $M$ has broader applicability and can be faster than Levin’s universal search, the fastest method for inverting functions save for a large multiplicative constant. An extension of Kolmogorov complexity and two novel natural measures of function complexity are used to show that the most efficient program computing some function $f$ is also among the shortest programs provably computing $f$.

Top titling here: some wonderfully sweep-of-the-arm terms, with a barely noticeable get-out adjective hidden in the middle. The author tackles the oldest problem in algorithm design — can I write an algorithm to do write all my other algorithms for me? — but amazingly answers in the positive. The trick is that he only looks at “well-defined” problems, “well-defined” meaning, of course, the kind of thing tractable by his approach. It’s still a pretty fantastic result, anyway, because the world of computable, well-defined problems is still pretty enormous. Just don’t expect it to have an application.

The Canonical Basis of $\dot{\mathbf{U}}$ for Type $A_{2}$

The modified quantized enveloping algebra has a remarkable basis, called the canonical basis, which was introduced by Lusztig. In this paper, all these monomial elements of the canonical basis for type $A_{2}$ are determined and we also give a conjecture about all polynomial elements of the canonical basis.

I’m not interested in the content of this paper as much as I am in its presentation. Richard Green pointed out that Theorem 2 starts on page 4 and ends on page 8, and consists solely of 66 lines of mind-numbingly similar-looking algebraic expressions which might as well be gibberish. Some times these kinds of papers have to be written, though I’d rather they weren’t.

Mastermind is NP-Complete

In this paper we show that the Mastermind Satisfiability Problem (MSP) is NP-complete. The Mastermind is a popular game which can be turned into a logical puzzle called Mastermind Satisfiability Problem in a similar spirit to the Minesweeper puzzle. By proving that MSP is NP-complete, we reveal its intrinsic computational property that makes it challenging and interesting. This serves as an addition to our knowledge about a host of other puzzles, such as Minesweeper, Mah-Jongg, and the 15-puzzle.

Yeah yeah, another problem is NP-Complete. Add it to the pile and move on.

VIP-club phenomenon: emergence of elites and masterminds in social networks

Hubs, or vertices with large degrees, play massive roles in, for example, epidemic dynamics, innovation diffusion, and synchronization on networks. However, costs of owning edges can motivate agents to decrease their degrees and avoid becoming hubs, whereas they would somehow like to keep access to a major part of the network. By analyzing a model and tennis players’ partnership networks, we show that combination of vertex fitness and homophily yields a VIP club made of elite vertices that are influential but not easily accessed from the majority. Intentionally formed VIP members can even serve as masterminds, which manipulate hubs to control the entire network without exposing themselves to a large mass. From conventional viewpoints based on network topology and edge direction, elites are not distinguished from many other vertices. Understanding network data is far from sufficient; individualistic factors greatly affect network structure and functions per se.

How far can Tarzan jump?

The tree-based rope swing is a popular recreation facility, often installed in outdoor areas, giving pleasure to thrill-seekers. In the setting, one drops down from a high platform, hanging from a rope, then swings at a great speed like “Tarzan”, and finally jumps ahead to land on the ground. The question now arises: How far can Tarzan jump by the swing? In this article, I present an introductory analysis of the Tarzan swing mechanics, a big pendulum-like swing with Tarzan himself attached as weight. The analysis enables determination of how farther forward Tarzan can jump using a given swing apparatus. The discussion is based on elementary mechanics and, therefore, expected to provide rich opportunities for investigations using analytic and numerical methods.

This was covered quite widely in the popular press. It’s in the arXiv’s Popular Physics section. A nice one to show to a keen child or under-educated adult.

A Hamiltonian circuit for Rubik’s Cube

At last, the Hamiltonian circuit problem for Rubik’s Cube has a solution! To be a little more mathematically precise, a Hamiltonian circuit of the quarter-turn metric Cayley graph for the Rubik’s Cube group has been found.

Basically it is a sequence of quarter-turn moves that would (in theory) put a Rubik’s cube through all of its 43,252,003,274,489,856,000 positions without repeating any of them, and then one more move restores the cube to the starting position. Note that if we have any legally scrambled Rubik’s Cube position as the starting point, then applying the sequence would result in the cube being solved at some point within the sequence.

I was playing with my cube one day, and I wondered if a Hamiltonian path exists. Turns out it does! If I was a well-regarded artist I would set up a robot programmed to follow this sequence of moves and install it amongst some ironic placards about the Rat Race and The Futility of Being and so on.

How Java’s Floating-Point Hurts Everyone Everywhere

Java’s floating-point arithmetic is blighted by five gratuitous mistakes:
1. Linguistically legislated exact reproducibility is at best mere wishful thinking.
2. Of two traditional policies for mixed precision evaluation, Java chose the worse.
3. Infinities and NaNs unleashed without the protection of floating-point traps and flags
mandated by IEEE Standards 754/854 belie Java’s claim to robustness.
4. Every programmer’s prospects for success are diminished by Java’s refusal to grant access
to capabilities built into over 95% of today’s floating-point hardware.
5. Java has rejected even mildly disciplined infix operator overloading, without which extensions
to arithmetic with everyday mathematical types like complex numbers, intervals, matrices,
geometrical objects and arbitrarily high precision become extremely inconvenient.
To leave these mistakes uncorrected would be a tragic sixth mistake.

The dude responsible for the international standard on floating point numbers explains at length, and with barely contained fury, why Java’s implementation is wrong even though it doesn’t need to be. In summary: dude shakes fist at world still using cheese and a spring even after he invented the perfect mousetrap. This is definitely computer science and not maths, but most people will encounter floating point weirdness if they ever do any numerical computing.

Similarly, another of Kahan’s papers jumped out at me because of its title: Beastly Numbers.

Modified Pascal Triangle and Pascal Surfaces

 We present a way of modifying the known Pascal Triangle and some regularities of the suggested model. We extend this model continuously to what we define as Pascal Surfaces and discuss some properties of the new object.

The thought occurred to me that, much like the $\Gamma$ function generalises factorials to a continuous domain, it should be possible to make a continuous version of Pascal’s triangle that fills in the gaps between the integer points. This paper was the only one I could find to do that. I’m a bit worried that I can’t find a peer-reviewed version, and that it was written by an engineering graduate student. I’d be very interested to hear from anyone who has a better reference.

Magic: The Gathering is Turing-complete

Yeah yeah, another system is Turing-complete. Add it to the pile and move on.

I’m only collecting these for completeness’s sake now. It’ll take a really mad application to rekindle my interest.

Online Dating Recommender Systems: The Split-complex Number Approach

A typical recommender setting is based on two kinds of relations: similarity between users (or between objects) and the taste of users towards certain objects. In environments such as online dating websites, these two relations are difficult to separate, as the users can be similar to each other, but also have preferences towards other users, i.e., rate other users. In this paper, we present a novel and unified way to model this duality of the relations by using split-complex numbers, a number system related to the complex numbers that is used in mathematics, physics and other fields. We show that this unified representation is capable of modeling both notions of relations between users in a joint expression and apply it for recommending potential partners. In experiments with the Czech dating website Libimseti.cz we show that our modeling approach leads to an improvement over baseline recommendation methods in this scenario.

Application of an esoteric bit of maths to an unexpected subject. Excellent!

Algebraic theory of Penrose’s non-periodic tilings of the plane

I had some fun making a bit of a Wieringa roof from bits of paper at the last Newcastle MathsJam. I tried to find some more information on the Wieringa roof but didn’t have much luck. Finally, I found this superb exposition by de Bruijn, which says on page 49 pretty much everything that needs to be said to convince you it works. I’m not sure if this paper is the origin of the phrase “Wieringa roof”. It looks like it might be.

Twin towers of Hanoi

In the Twin Towers of Hanoi version of the well known Towers of Hanoi Problem there are two coupled sets of pegs. In each move, one chooses a pair of pegs in one of the sets and performs the only possible legal transfer of a disk between the chosen pegs (the smallest disk from one of the pegs is moved to the other peg), but also, simultaneously, between the corresponding pair of pegs in the coupled set (thus the same sequence of moves is always used in both sets). We provide upper and lower bounds on the length of the optimal solutions to problems of the following type. Given an initial and a final position of N disks in each of the coupled sets, what is the smallest number of moves needed to simultaneously obtain the final position from the initial one in each set? Our analysis is based on the use of a group, called Hanoi Towers group, of rooted ternary tree automorphisms, which models the original problem in such a way that the configurations on N disks are the vertices at level N of the tree and the action of the generators of the group represents the three possible moves between the three pegs. The twin version of the problem is analyzed by considering the action of Hanoi Towers group on pairs of vertices.

I love the towers of Hanoi. It sits right at my IQ grade. This paper applies some group theory to a doubled-up version of the game. I want to have a proper read of this and try it out myself later.

The topology of the minimal regular cover of the Archimedean tessellations

In this article we determine, for an infinite family of maps on the plane, the topology of the surface on which the minimal regular covering occurs. This infinite family includes all Archimedean maps.

I think this is a pretty important result in geometric topology, but this paper was brought to my attention because on page 2 it has a picture of the Loch Ness monster.

The Loch Ness monster was named by honorary Aperiodicalist Étienne Ghys. Dude is too cool!

Earliest Uses of Symbols of Calculus

In the midst of a discussion/argument about the best notation to use for derivatives, I found this page. I’m not sure how scholarly rigorous it is, but he cites Cajori so it can’t be all lies.

In case you’re new to this: every now and then I encounter a paper or a book or an article that grabs my interest but isn’t directly useful for anything. It might be about some niche sub-sub-subtopic I’ve never heard of, or it might talk about something old from a new angle, or it might just have a funny title. I put these things in my Interesting Esoterica collection on Mendeley. And then when I’ve gathered up enough, I collect them here.

In this post the titles are links to the original sources, and I try to add some interpretation or explanation of why I think each thing is interesting below the abstract.

Some things might not be freely available, or even available for a reasonable price. Sorry.

Because there are so many entries this time round, I’ve picked a few particularly interesting or esoteric items that you should definitely have a look at. The rest might or might not interest you.

Definitely have a look

Light reflecting off Christmas-tree balls

‘Twas the night before Christmas and under the tree
Was a heap of new balls, stacked tight as can be.
The balls so gleaming, they reflect all light rays,
Which bounce in the stack every which way.
When, what to my wondering mind does occur:
A question of interest; I hope you concur!
From each point outside, I wondered if light
Could reach deep inside through gaps so tight?

Somebody asked a question on MathOverflow and, since it was about Christmas-tree baubles, posed it with a rhyme inspired by The Night Before Christmas.

And this is a good world we live in, because some wonderful person managed to give a beautifully concise, clear answer also in rhyme! (It doesn’t quite scan, but you can’t have everything)

A mathematician’s survival guide

A career’s-worth of advice from Peter Castazza for new mathematicians on the subject of being a mathematician and taking part in maths culture, from dealing with “the public” to having the confidence to continue in research to seating mathematicians at dinner. Lots of good quotes and anecdotes.

Topology Explains Why Automobile Sunshades Fold Oddly

We use braids and linking number to explain why automobile shades fold into an odd number of loops.

I like to think I’m the kind of man who understands things, but those foldy-outy windscreen shades always bamboozle me. The result in this paper might actually help me next time I need to fold one up. It’s a really interesting fact, and explained very well by the authors. I think this is in the same category of “maths you can explain to your parents while on a day trip” as the proof that use the intermediate value theorem to show that you can make a wobbly table stable just by rotating it.

The wobbly garden table

And here’s that proof. It’s presented as a satirical comparison of the different ways engineers, physicists and mathematicians approach problems. Of course, the mathematician comes out looking considerably more rational than the other two. If all you want is the proof without the snarky tone, just read the mathematician’s bit.

A cohomological viewpoint on elementary school arithmetic

From finite group theory to algebraic geometry to complex analysis, cohomological methods play a major role in modern mathematics. The subject has a long history throughout much of the twentieth century and strongly influenced the development of modern mathematics. Mathematicians view such techniques as powerful but sophisticated tools applicable to a remarkably wide field of study, but they usually react with surprise to learn that the ubiquity of cohomology in mathematics extends even to arithmetic at the primary school level.

It’s always fun when someone deliberately applies a sledgehammer to crack a very tiny nut. Somebody on Google+ mentioned that someone had told them that “carrying is a cocycle”, and wanted to know more. A commentor obliged with a link to this surprisingly not completely pointless article. Closed access on JSTOR, $12.

Train sets

Suppose we have a train set — large stocks of straight and curved track, bridge-building materials, different kinds of sets of points, and a single engine, say clockwork for the sake of nostalgia. What can we do?

I think this was the first bit of interesting maths esoterica I ever encountered. When I was in sixth form my maths teacher read about this in the paper and brought it in to show us. Adam Chalcraft and Michael Greene show how to construct logic gates, and hence a Turing machine, using just train tracks and lazy (unmanned, uncontrolled) points. For years I’d forgotten who the authors were and tried in vain to find it again, so I’m very happy to finally have it in my collection.

The top ten prime numbers

This isn’t a list of the top 10 prime numbers. It’s a list of lists of the top 10 prime numbers so far found of various types. For what is effectively just a big list of numbers, it’s a surprisingly gripping read. There are all sorts of kinds of prime numbers I’d never heard about, like Anti-Yarborough primes or strobogrammatic primes. I think I found this via the entry for the undulating palindromic primes on the OEIS.

Seven staggering sequences

Neil Sloane, the man in charge of the Online Encyclopedia of Integer Sequences, wrote this paper for G4G7 about seven sequences that he found especially interesting. If anyone knows about really interesting integer sequences, it’s Sloane!

The rest

The Euler spiral: a mathematical history

The beautiful Euler spiral, defined by the linear relationship between curvature and arclength, was first proposed as a problem of elasticity by James Bernoulli, then solved accurately by Leonhard Euler. Since then, it has been independently reinvented twice, first by Augustin Fresnel to compute diffraction of light through a slit, and again by Arthur Talbot to produce an ideal shape for a railway transition curve connecting a straight section with a section of given curvature. Though it has gathered many names throughout its history, the curve retains its aesthetic and mathematical beauty as Euler had clearly visualized. Its equation is related to the Gamma function, the Gauss error function (erf), and is a special case of the confluent hypergeometric function.

I love the Euler spiral. I don’t know why. Maybe it’s because I first learnt of it as the “clothoid”, which is an excellent name, or maybe it’s because it gives me something to think about when I’m driving.

This shortish essay by Raph Levien gives a readable potted history of the spiral’s multiple discoveries and applications, illustrated with some lovely sparse diagrams of the sort that maths-illiterate Etsy craftspeople love.

High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams

This volume consists of a selection of papers based on presentations made at the international conference on number theory held in honor of Hugh Williams’ sixtieth birthday. The papers address topics in the areas of computational and explicit number theory and its applications. The material is suitable for graduate students and researchers interested in number theory.

I think I just kept this because of the title. That’s a good reason because it’s a good title, but I can’t remember why I found it. Maybe it contains something interesting.

Statistical Modeling of Gang Violence in Los Angeles

Gang violence has plagued the Los Angeles policing district of Hollenbeck for over half a century. With sophisticated models, police may better understand and predict the region’s frequent gang crimes. The purpose of this paper is to model Hollenbeck’s gang rivalries. A self-exciting point process called a Hawkes process is used to model rivalries over time. While this is shown to fit the data well, an agent based model is presented which is able to accurately simulate gang crimes not only temporally but also spatially. Finally, we compare random graphs generated by the agent model to existing models developed to incorporate geography into random graphs.

This is really interesting. I’ve had it in my folder since April, but didn’t get round to writing a post about it here before Peter scooped me thanks to UCLA’s hard-working press officer.

Theory and History of Geometric Models

Irene Polo-Blanco’s PhD thesis about the history of mathematical models: physical sculptures of surfaces of the sort Felix Klein liked to make. It’s very long, but the first chapter gives a decent history. The rest is a pretty uninspiring exposition of the maths behind the models.

Robust Soldier Crab Ball Gate

Based on the field observation of soldier crabs, we previously proposed a model for a swarm of soldier crabs. Here, we describe the interaction of coherent swarms in the simulation model, which is implemented in a logical gate. Because a swarm is generated by inherent perturbation, a swarm can be generated and maintained under highly perturbed conditions. Thus, the model reveals a robust logical gate rather than stable one. In addition, we show that the logical gate of swarms is also implemented by real soldier crabs (Mictyris guinotae).

The title of this paper sounded very promising, and the content did not disappoint. It does exactly what it says on the tin: they’ve made a logic gate using balls of swarming soldier crabs, and it’s robust. Splendid. Not so splendidly, it’s closed access and a download from the AIP costs $28.

Computer analysis of Sprouts with nimbers

Sprouts is a two-player topological game, invented in 1967 in the University of Cambridge by John Conway and Michael Paterson. The game starts with $p$ spots, and ends in at most $3p-1$ moves. The first player who cannot play loses. The complexity of the $p$-spot game is very high, so that the best hand-checked proof only shows who the winner is for the $7$-spot game, and the best previous computer analysis reached $p=11$. We have written a computer program, using mainly two new ideas. The nimber (also known as Sprague-Grundy number) allows us to compute separately independent subgames; and when the exploration of a part of the game tree seems to be too difficult, we can manually force the program to search elsewhere. Thanks to these improvements, we reached up to $p=32$. The outcome of the $33$-spot game is still unknown, but the biggest computed value is the $47$-spot game ! All the computed values support the Sprouts conjecture: the first player has a winning strategy if and only if $p$ is $3$, $4$ or $5$ modulo $6$. We have also used a check algorithm to reduce the number of positions needed to prove which player is the winner. It is now possible to hand-check all the games until $p=11$ in a reasonable amount of time.

I had always thought that Sprouts had been completely solved. I tried to find a reference which would persuade someone else, but I found this instead. Nimbers are a fun topic; see also this typewritten paper on multiplying games of Nim.

Long finite sequences

Let $k$ be a positive integer. There is a longest finite sequence $x_1,\dots,x_n$ in $k$ letters in which no consecutive block $x_i,\dots,x_{2i}$ is a subsequence of any other consecutive block $x_j,\dots,x_{2j}$. Let $n(k)$ be this longest length. We prove that $n(1) = 3$, $n(2) = 11$, and $n(3)$ is incomprehensibly large. We give a lower bound for $n(3)$ in terms of the familiar Ackerman hierarchy. We also give asymptotic upper and lower bounds for $n(k)$. We view $n(3)$ as a particularly elemental description of an incomprehensibly large integer. Related problems involving binary sequences (two letters) are also addressed. We also report on some recent computer explorations of R. Dougherty which we use to raise the lower bound for $n(3)$.

First of all: this paper is set in fixed-space type and yet, inexplicably, it was published in 1998. The mind boggles.

Anyway, I like this sequence because it starts innocently enough and gets stupid very quickly.

Calculus Made Easy

Being a very simplest introduction to
those beautiful methods of reckoning
which are generally called by the
terrifying names of the
DIFFERENTIAL CALCULUS
and the
INTEGRAL CALCULUS

Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.

Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics — and they are mostly clever fools — seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.

Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the
parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.

I came across this when I noticed that Project Gutenberg were making TeX versions of old books. It’s supposed to be the one calculus book that gets it right. A hundred years on, every calculus book I’ve seen is still obtuse and long-winded and hard to understand, so maybe we should stick with this one.

Cardinal arithmetic for skeptics

When modern set theory is applied to conventional mathematical problems, it has a disconcerting tendency to produce independence results rather than theorems in the usual sense. The resulting preoccupation with “consistency” rather than “truth” may be felt to give the subject an air of unreality. Even elementary questions about the basic arithmetical operations of exponentiation in the context of infinite cardinalities, like the value of $2^{\aleph_0}$, cannot be settled on the basis of the usual axioms of set theory (ZFC).

Although much can be said in favor of such independence results, rather than undertaking to challenge such prejudices, we have a more modest goal; we wish to point out an area of contemporary set theory in which theorems are abundant, although the conventional wisdom views the subject as dominated by independence results, namely, cardinal arithmetic.

Pretty sure I kept this just for the title.

Survey on fusible numbers

We point out that the recursive formula that appears in Erickson’s presentation “Fusible Numbers” is incorrect, and pose an alternate conjecture about the structure of fusible numbers. Although we are unable to solve the conjecture, we succeed in establishing some basic properties of fusible numbers. We suggest some possible approaches to the conjecture, and list further problems in the final chapter.

Jeff Erickon’s fusible numbers are a fun class of fractions. However, Junyan Xu reckons he got his sums confused(!)

A categorical foundation for Bayesian probability

Given two measurable spaces $H$ and $D$ with countably generated $\sigma$-algebras, a perfect prior probability measure $P_H$ on $H$ and a sampling distribution $\mathcal{S}\colon H \rightarrow D$, there is a corresponding inference map $\mathcal{I}\colon D \rightarrow H$ which is unique up to a set of measure zero. Thus, given a data measurement $\mu\colon 1 \rightarrow D$, a posterior probability $\hat{P_H}=\mathcal{I} \circ \mu$ can be computed. This procedure is iterative: with each updated probability $P_H$, we obtain a new joint distribution which in turn yields a new inference map $\mathcal{I}$ and the process repeats with each additional measurement. The main result uses an existence theorem for regular conditional probabilities by Faden, which holds in more generality than the setting of Polish spaces. This less stringent setting then allows for non-trivial decision rules (Eilenberg–Moore algebras) on finite (as well as non finite) spaces, and also provides for a common framework for decision theory and Bayesian probability.

Category theory, yeah!

Estimating the Effect of the Red Card in Soccer

We study the effect of the red card in a soccer game. A red card is given by a referee to signify that a player has been sent off following a serious misconduct. The player who has been sent off must leave the game immediately and cannot be replaced during the game. His team must continue the game with one player fewer. We estimate the effect of the red card from betting data on the FIFA World Cup 2006 and Euro 2008, showing that the scoring intensity of the penalized team drops significantly, while the scoring intensity of the opposing team increases slightly. We show that a red card typically leads to a smaller number of goals scored during the game when a stronger team is penalized, but it can lead to an increased number of goals when a weaker team is punished. We also show when it is better to commit a red card offense in exchange for the prevention of a goal opportunity.

Proper real football statistics.

Lectures on lost mathematics

Notes from a series of lectures given in 1975 about “lost mathematics”, which might more properly be called “outsider maths” – discoveries by people in other disciplines, and which weren’t known by mathematicians at the time. Lots of nice, although poorly photocopied, pictures.

On an error in the star puzzle by Henry E. Dudeney

We found a solution of the star puzzle (a path on a chessboard from c5 to d4 in 14 straight strokes) in 14 queen moves, which has been claimed by the author as impossible.

The star puzzle is very old. This dude decided he wouldn’t trust the author that a short solution was impossible, so worked one out. That’s all there is to this paper.

To Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus with Animated Reduction

The lambda calculus, and the closely related theory of combinators, are important in the foundations of mathematics, logic and computer science. This paper provides an informal and entertaining introduction by means of an animated graphical notation.

I love lambda calculus, me. I couldn’t follow this, though. I think the author’s made the classic error of picking a really weird analogy to to try to make it more accessible, and it just hasn’t worked for me. Though it says the bird terminology comes from a book by Smullyan; maybe I should read that instead.

On distributions computable by random walks on graphs

We answer a question raised by Donald E. Knuth and Andrew C. Yao, concerning the class of polynomials on $[0,1]$ that can be realized as the distribution function of a random variable, whose binary expansion is the output of a finite state automaton driven by unbiased coin tosses. The polynomial distribution functions which can be obtained in this way are precisely those with rational coefficients, whose derivative has no irrational roots on $[0,1]$. We also show, strengthening a result of Knuth and Yao, that all smooth distribution functions which can be obtained by such automata are polynomials.

I was thinking about extending the princess on a graph puzzle to use finite state automata. This looks like a good place to start. Sadly, it’s closed access – a download costs $15 to the punter on the street.

Equilibrium Solution to the Lowest Unique Positive Integer Game

We address the equilibrium concept of a reverse auction game so that no one can enhance the individual payoff by a unilateral change when all the others follow a certain strategy. In this approach the combinatorial possibilities to consider become very much involved even for a small number of players, which has hindered a precise analysis in previous works. We here present a systematic way to reach the solution for a general number of players, and show that this game is an example of conflict between the group and the individual interests.

I looked this up when I read here that an author was giving away copies of his book to fans who picked the lowest unique prime numbers. It’s odd that this game has an equilibrium solution, though I think the authors have assumed the players’ strategies have a certain form.

A reminder: every now and then I encounter a paper or a book or an article that grabs my interest but isn’t directly useful for anything. It might be about some niche sub-sub-subtopic I’ve never heard of, or it might talk about something old from a new angle, or it might just have a funny title. I put these things in my Interesting Esoterica collection on Mendeley.

In this post the titles are links to the original sources, and I try to add some interpretation or explanation of why I think each thing is interesting below the abstract.

Continue reading “Interesting Esoterica Summation, volume 3″ on cp’s mathem-o-blog

A reminder of what it’s all about: every now and then I encounter a paper or a book or an article that grabs my interest but isn’t directly useful for anything. It might be about some niche sub-sub-subtopic I’ve never heard of, or it might talk about something old from a new angle, or it might just have a funny title. I put these things in my Interesting Esoterica collection on Mendeley.

In this post the titles are links to the original sources, and I try to add some interpretation or explanation of why I think each thing is interesting below the abstract.

Click here to continue reading Interesting Esoterica Summation on cp’s mathem-o-blog

As this is the first one, and I’ve added loads of stuff in January, for this first post I’m using everything  I’ve added since the New Year. Future posts shouldn’t be anywhere near as long.

I should explain what the Interesting Esoterica collection is about.

Click here to continue reading Interesting Esoterica Summation on cp’s mathem-o-blog