Archive for maths


Posted in academina, culture, maths with tags , on August 31, 2008 by Philonous
Paul Baum (USA) giving a seminar

For those of you that didn’t know, I’m a grad student in mathematics these days. My two advisors are from the Russian school of mathematics, specifically Moscow State University back in the Soviet days. It seems that the culture of mathematics in Russia and the Soviet Union is/was completely different to that in the west. There is a much greater emphasis placed on examples and simplicity of exposition as well as a much smaller divide between pure and applied mathematics.

Nowhere is this difference more apparent than in seminars in Russia and in the UK. British seminars normally last for an hour and consist of a speaker talking about some part of their current research to an audience of academics who are invited at the end, if there is some time left, to pose some questions relating to the talk. Because of the time constraints, it is rather difficult to ask questions throughout the talk for fear of putting the speaker under time pressure towards the end. Perhaps as a result, there is a terrible risk at any given seminar of being completely lost before the seminar has begun. For instance, suppose the seminar begins

‘Let S be a category fibred in groupoids over a topological space X…’

If you are not familiar with one or more of these words, there is a good chance that the remainder of the seminar will be spent counting water stains on the ceiling, doodling and trying not to fall asleep.

In contrast, my impression of Russian seminars is that they have no fixed end-time. This means that foolish questions from people not completely acquainted with the specifics of the topic at hand are welcome. It seems that these seminars may last up to four or five hours, with tea and snacks served throughout. There is therefore an expectation that attending a Russian seminar, one will understand something or other by the end.

Of course this leads to difficulties when Russian and western mathematicians meet at seminars. Russian mathematicians expect to have learned something by the end while western ones are content with the possibility of being bored witless by a string of incomprehensible phrases, knowing however that it will be over in an hour.

No seminar I have attended has resulted in the following chaos, taken I think from a (sociology?) seminar in the US.  I think sitting through this would be much more excruciating, if less soporific than an hour of incomprehensible maths.

Math Capella

Posted in humour, maths, music, nerd pride with tags , on August 15, 2008 by Philonous

I was first sent this back in 2005 at which point I didn’t really know what it was all about. Now that I’m finally in the position where I actually get the jokes, I figure I can share it with you without loss of face. Behold the Klein Four: an A Capella group composed of mathematics students at Northwestern University near Chicago. They’re clearly differential geometers/mathematical physicists. It’s what all the cool kids do.

This simply cannot be beaten for sheer density of maths puns. Buy T-shirts here.

Roots (disambiguation)

Posted in culture, film, Food, history, literature, maths, music, science with tags , , , , , on July 23, 2008 by Philonous

In Euclidean Space:

Let V be a finite-dimensional Euclidean space, with the standard Euclidean inner product denoted by . A root system in V is a finite set Φ of non-zero vectors (called roots) that satisfy the following properties:

  1. The roots span V.
  2. The only scalar multiples of a root α ∈ Φ that belong to Φ are α itself and −α.
  3. For every root α ∈ Φ, the set Φ is closed under reflection through the hyperplane perpendicular to α. That is, for any two roots α and β, the set Φ contains the reflection of β in the plane perpendicular to α.
  4. (Integrality condition) If α and β are roots in Φ, then the projection of β onto the line through α is a half-integral multiple of α.

In view of property 3, the integrality condition is equivalent to stating that β and its reflection σα(β) differ by an integer multiple of α.

The rank of a root system is the dimension of the Euclidean space V in which it resides. Here are examples of rank 2 systems.

In the Plant Kingdom:

In vascular plants, the root is the organ of a plant body that typically lies below the surface of the soil. But, this is not always the case, since a root can also be aerial (that is, growing above the ground) or aerating (that is, growing up above the ground or especially above water). On the other hand, a stem normally occurring below ground is not exceptional either (see rhizome). So, it is better to define root as a part of a plant body that bears no leaves, and therefore also lacks nodes. There are also important internal structural differences between stems and roots. The two major functions of roots are 1.) absorption of water and inorganic nutrients and 2.) anchoring the plant body to the ground. Roots also function in cytokinin synthesis, which supplies some of the shoot’s needs. They often function in storage of food. The roots of most vascular plant species enter into symbiosis with certain fungi to form mycorrhizas, and a large range of other organisms including bacteria also closely associate with roots.

On TV:

Roots is a 1977 American television miniseries based on Alex Haley‘s work Roots: The Saga of an American Family, his critically acclaimed but factually disputed genealogical novel.

Roots was made into a hugely popular television miniseries that aired over eight consecutive nights in January 1977. Many people partially attribute the success of the miniseries to the original score by Quincy Jones. ABC network television executives chose to “dump” the series into a string of airings rather than space out the broadcasts, because they were uncertain how the public would respond to the controversial, racially-charged themes of the show. However, the series garnered enormous ratings and became an overnight sensation. Approximately 130 million Americans tuned in at some time during the eight broadcasts. The concluding episode was rated as the third most watched telecast of all time by the Nielsen corporation.
The cast of the miniseries included LeVar Burton as Kunta Kinte, Leslie Uggams as Kizzy and Ben Vereen as Chicken George. A 14-hour sequel, Roots: The Next Generations, aired in 1979, featuring the leading African-American actors of the day. In 1988, a two-hour made-for-TV movie, Roots: The Gift, aired. Based on characters from the book, it starred LeVar Burton as Kunta Kinte, Avery Brooks as Cletus Moyer and Kate Mulgrew as Hattie, the female leader of a group of slave catchers.

In the Charts:

Roots is the sixth studio album by Brazilian thrash metal band Sepultura, released in 1996 through Roadrunner Records. It was the band’s last album to feature Max Cavalera. The majority of the themes presented on Roots are centered on Brazilian politics and culture.

The inspiration for Sepultura’s new musical directon was two-fold. One was the desire to further experiment with the music of Brazil, especially the percussive type played by Salvador, Bahia samba reggae group Olodum. A slight influence of Northeastern Brazil‘s native music is also present in the guitar riffs, especially baião and capoeira music. Another innovation Roots brought was the inspiration taken from the (then) cutting-edge nu metal sound of the Deftones and KoЯn – especially the latter’s debut, with it’s heavily down-tuned guitars.

Roots was released in February 1996 and received with unprecedented enthusiasm. Even the popular press, that usually doesn’t pay a lot of attention to metal records, halted the presses to appreciate the unusual rhythms mixture of Sepultura. American newspapers like The New Times, the Daily News[disambiguation needed] and the Los Angeles Times reserved some space for the Brazilian band: “The mixture of the dense metal of Sepultura and the Brazilian music has a intoxicating effect”, wrote a Los Angeles Times’ reviewer. The Daily News went even further: “Sepultura reinvented the wheel. By mixing metal with native instruments, the band resuscitates the tired genre, reminding of Led Zeppelin times. But while Zeppelin mixed English metal with African beats, it’s still more moving to hear a band that uses elements of its own country. By extracting the sounds of the past, Sepultura determines the future direction of metal”.

Acknowledgements: This post would not have been possible without the untiring effort of all of those kind folk at Wikipedia who are up at all hours of the day and night writing entries.

I’d also like to thank they keys Ctrl, C and V

The Brazil Nut Effect

Posted in Manchester, maths with tags on May 13, 2008 by Philonous

Ever wondered why on packets of cereal it sometimes says something to the effect of “Please turn packet on its side to redistribute nuts evenly”? I’d always thought that this was because any of the larger particles would have sunk to the bottom. It turned out that this was in fact not the case.
Yesterday was the day of the MRSC08 conference for postgraduate maths students in Manchster. One of the talks explained the so called “Brazil nut effect”, the phenomenon of larger nuts rising to the top of packets of mixed nuts during transport. The following video (taken from here) shows the effect working on a brass cylinder among polystyrene foam beads.

It turns out that although gravity pulls the cylinder down relative to the polystyrene, there are granular flow effects that also push the cylinder up which are far greater than expected. I can’t claim to be an expert, but the talk was incredibly interesting. (See Prof. Nico Gray’s website for research about granular flows in Manchester).

Here’s a slow motion version of the above video to maybe give some intuitio for what’s going on. I have to admit, I still don’t quite understand from where the upward force comes.

Mathematical duality

Posted in maths with tags on May 5, 2008 by Philonous

Let me set the scene. A brooding overcast morning in post-industrial Manchester. A Mancunian café underneath an old railway arch. Two large lattés. On my side of the table a physics book for mathematicians, on Le Fox’s the Economist. I opened my book on at a chapter on the basics of quantum mechanics. It stated:

“A description of physical reality is made in terms of two set of objects: observables and states. A set of obsevables A, B,… will be denoted by A and states w, x, y,… by W. Each state assigns to each observable its probability distribution on the real line. This pairing (mean value) defines a duality between A and W.”

I just love how physicists like to use phrases like physical reality… It reminds me of Arnold’s classic book Mathematical Methods in Classical Mechanics, which I seem to remember states in no uncertain terms

Definition: The universe is a four dimensional affine space.

Which was perhaps inspired by Wittgenstein’s

1 The world is all that is the case
1.1 The world is the totality of facts, not of things. etc.

In any case, I started to wonder about the notion of duality. In mathematics at least, duality seems to be reasonably well defined. The modern idea I suppose started with the idea of a vector space having a dual which might have been somehow inspired by duals of platonic solids through the reflexive property of taking duals (at least for finite dimensional spaces). But then there’s also Poincaré duality for compact oriented manifolds which says that the kth and (n-k)th Betti numbers are equal (for coefficients in Z2, orientability isn’t required).

The picture of foolhardiness, I then tried to give Le Fox some idea of what Poincaré duality might mean through a couple of badly explained examples messily scribbled on a napkin – decomposing a sphere into a 0-cell (point) and a 2-cell (plane) and a torus into a 0-cell, two 1-cells and a 2-cell.

As her eyes slowly glazed over, I tried to think of a more concrete way of thinking of duality. Suppose that I have a bag containing ten balls. I then take four balls out of the bag. If I know I have four balls in my hand then I know that there are six balls in the bag. Knowledge of how many balls are outside the bag is therefore somehow equivalent to knowledge of how many balls are in the bag as long as we know the total number of balls. It seemed to me that these two statements were therefore in some sense dual to each other. In the back of my mind I had Poincaré duality which could be rephrased along the lines of “Knowledge of k-cells is equivalent to knowledge of (n-k)-cells as long as the space is compact and oriented…”. Here it seems that knowing that there are ten balls in total and knowing that the manifold is compact play the same role, setting up the bridge between the two dual statements in each case.

For finite dimensional vector spaces, the notion of the dual space can be formulated in a similar way. Given a set with the structure of a finite dimensional vector space, the space of linear functionals can be formed. This is non-canonically isomorphic to the original space, just by mapping a given basis to its dual basis. On the other hand, given the dual space, I can construct the original space again by applying the dualising process. It’s one of those well known facts that a finite dimensional vector space is canonically isomorphic to its double dual (reflexivity).

On one level, the analogy with the balls fits quite nicely into the first half of this: knowledge of the dual is “the same as” knowledge of the space. On another, it doesn’t really work so well – how do I get a notion of ‘canonical isomorphism’ in this setup?

I still haven’t thought of an example… any ideas?

100 hat solution

Posted in maths, puzzles with tags on April 16, 2008 by Philonous
I suddenly realised that I haven’t yet posted the solution to the hundred hat problem.

In slightly mathematical language, here’s the solution. Replace black by 0 and white by 1. Let the ith prisoner have hat colour c(i). The first person shouts out the value of

The next person does the calculation

and shouts his answer out. In general, since the kth prisoner knows c(2), . . . , c(k-1), (having heard them shouted), c(k+1), . . . , c(100) (being able to see them) and

courtesy of the first prisoner’s shout, he can calculate c(k) = (c(k) mod 2). Of course, here we’ve set the number of prisoners to 100, but this isn’t necessary.

For non-mathematically literate folk:

The idea really isn’t complicated at all – if it seems so, then I’ve explained it badly. First of all, knowing if a hat is white or not is equivalent to knowing its colour since there are only two. The first person to go counts up the number of white hats he can see in front of him. If it’s odd, he shouts ‘white’ and if it’s even, he shouts ‘black’. Prisoner number 2 then counts the number of white hats in front of him. He then figures out if it’s odd or even. We know that

  • (odd number) – (odd number) = even number
  • (odd number) – (even number) = odd number
  • (even number) – (odd number) = odd number
  • (even number) – (even number) = even number

Prisoner 2 can use prisoner 1’s shout as the first number and his own tally as the second. This spits out something on the right hand side. If he gets ‘even’ he shouts black, if he gets ‘odd’ he shouts white. Prisoner 3 has heard prisoner 1 and 2 and also knows whether the number of hats in front is even or odd. He then does a similar calculation and shouts out the right answer…and so on.

Perhaps this isn’t perfectly explained. Never mind.

100 Lamp solution, 100 Hat problem.

Posted in maths, puzzles with tags on April 12, 2008 by Philonous

Yesterday, I posted a little puzzle about a hundred blind lamp lighters. Don’t read on if you’d like to give this a go for yourself.

Here’s the solution: The squared number lamps will be on at the end, i.e. 1,4,9,16,25,36,49…etc.

We need to find the number of lamps which are on at the end. So let’s try and figure out in what circumstances a given lamp will be on. As an example, think about whether the 6th lamp will be on. It’s turned on by the 1st guy, off by the 2nd, on by the 3rd, and then off by the 6th. Notice that we’ve just listed all of the numbers that divide 6, i.e. 1,2,3,6. In general, for a lamp to be off, the button should have been pressed an even number of times; for it to be on, the button should have been pressed an odd number of times. For the kth lamp, the number of times it is pressed is the number of numbers dividing k.

Now let’s look at another example. The 12th lamp will be pressed by guys number 1, 2, 3, 4, 6, 12. We can split these divisors into pairs: (1,12), (2,6), (3,4) where multiplying the numbers together gives 12. Since we can pair these things up, there is an even number of divisors.

Suppose now for example, that we take the number 36. Here we have divisors 1, 2, 3, 4, 6, 9, 12, 18, 36. In this case, we have the pairs (1,36), (2,18), (3,12), (4,9), and 6 is paired with itself. Here we have an odd number of divisors, the reason being that one number (6) is paired with itself. So a general number k has an odd number of divisors if (and only if) one divisor is paired with itself, that is, there is some number which multiplied by itself gives k. In less obtruse words, k is a square number.

And there’s the proof. (Note that actually, the number of lamps doesn’t matter as long as it’s the same as the number of lamp lighters.)

Last night after the Pure Postgraduate seminar, a post-pub discussion brought up the following hundred hat problem:

Here’s another problem, this time with a hundred hats. Suppose that there is a strange kingdom where logical prowess is prized above all other things. 100 prisoners languish in prison all sentenced to death. Since the king is a nice sort of fellow, he decides that he will set the following challenge: The prisoners are all lined up facing along the line so that a given prisoner can see all the people infront of him, but not himself or the people behind him. A hat is put on each prisoner, either white or black. The prisoner at the back (who can see everyone else’s hat but not his own) then shouts out the colour he thinks his hat is. If he gets the answer right, he is allowed to live. If he gets the answer wrong, he dies. The same process is repeated by the prisoner second from the back and so on.

The prisoners are allowed to confer before the whole process and so can decide on a strategy. How many prisoners can defintitely be saved by an appropriate strategy?

(Hint: At least half can be saved. The guy at the back can’t be helped: he may as well shout randomly. He could however, help the guy in front of him. If they decide beforehand that the first guy will shout the colour of the hat in front of him, then at least he can save the second guy who will then know his own hat colour. Continuing the process, we can save for certain at least every other man. In fact, they can do a lot better…)

Too many Lamp Lighters

Posted in maths, puzzles with tags on April 11, 2008 by Philonous

Yesterday, as on every Thursday, Manchester University’s geometers, mathematical physicists and topologists gather for the Geometry Seminar. The talk itself was a really great proof of the Mischenko-Fomenko conjecture about the existence maximal complete commutative subalgebras of finite dimensional Lie algebras. Alexei Bolsinov gave a geometric proof of one of the subcases of the proof using some quite neat parts of the theory.

As per usual, we decamped to the pub afterward to discuss anything and everything. Soon enough, one of my supervisors started posing interesting little mathematical problems. Here is one which “of course, the school children know how to solve” which occupied a some minutes in an otherwise boring sleepless night.
Suppose there are a hundred push-button lamps and a hundred blind lamplighters. All lamps are initially off. The first lamplighter passes along the row and pushes each button in turn thereby turning every lamp on. The second lamplighter passes along the row afterward and pushes every other button regardless of whether the lamp is on or off. (So now all the even numbered lamps are off.) The third lamplighter passes along the row and pushes every third button, the fourth pushes every fourth button and so on. After all of the hundred lamplighters have passed by, which of the lamps remain lit?

For the answer and an explanation, see tomorrow.

Axiomatic Sheaf cohomology/Aural Objets Trouves

Posted in art, maths with tags , on November 12, 2006 by Philonous

… Updates, updates… Well, right now I’ve learnt the axioms of axiomatic sheaf cohomology. Which is nice. The only problem is that everything is rather esoteric at the moment – apparently every fine torsionless resolution of the constant sheaf defines canonically a sheaf cohomology theory. Not only that, but later in the book, it proves that every sheaf cohomology theory is isomorphic for the same choice of the base principal ideal domain K (the sheaves are sheaves of K-modules). This seems nuts right now, but I guess it must be a little less crazy than it sounds. Actually a sheaf cohomology theory seems to be rather a big beastie and so if you pick K as the field of real numbers, the cohomology theory that you get must encapsulate all the classical cohomology theories like de Rham and Cech. Crazy eh?

On a lighter note, I found out about an awesome experimental music project called Milkcrate. The idea of the whole thing is to make, I guess, unconventional music. It’s awesome. The rules are essentially that all of the objects that you use have to be explicitly non-musical (egg cartons and yoghurt pots?) and they have to all fit inside a standard – presumably Australian – milkcrate. Oh, and it all has to be completed in 24 hours. It seems to me that this is bringing the idea of objets trouves to music in a whole new electronic way. Hurrah for the internet!