Archive for the maths Category

Sporadic Simple Games

Posted in maths, Uncategorized on September 16, 2008 by Philonous

Are you au fait with finite simple groups? Well I know I’m certainly not.

For the uninitiated, a group is a mathematical object which can be thought of as the collection of symmetries of some object. As for simple groups, I guess they can sort of be thought of as the basic buidling blocks of groups (whatever that means…)

There’s a classification of finite simple groups which is one of the great results of the 20th/21st century. It shows that there are four categories (not in the mathematical sense) into which the finite simple groups may be divided, the most romantic of these consisting of the 26 sporadic groups. I say romantic, what I really mean is quirkily named – one of the groups contained therin is the Monster group while another is called the Baby Monster… Ah those finite group theorists.

Anyway, as someone who finds finite simple groups particularly scary, I was rather pleased to see a rather interesting page at Scientific American. There you can find a couple of games that give some idea of the structure of the groups M12, M24 and the Conway group without actually knowing any mathematics whatsoever. Click on the link below to investigate. Happy playing!

The finite simple groups at play

Incidentally, if you’re feeling slightly fobbed off by my lack of a definition of a finite simple group, then check out the articles on group, normal subgroup, quotient group and simple group.

Irrational number

Posted in maths on September 1, 2008 by Philonous

So it turns out that it’s unknown whether pi + e is irrational. Crazy eh?

For those not in the mathematical loop, pi is the ratio of a circle’s circumference to its diameter.

e is a little trickier to explain. Perhaps the easiest way of learning about e is to look at the Wikipedia page, which gives some explanation and a great number of equivalent definitions.

A number is said to be rational if it can be written as fraction of two whole numbers. For instance 12 = 12/1, -1.5 = (-3)/2 etc… It turns out that neither pi nor e can’t be written as a fraction, but it’s certainly not obvious (indeed unknown) whether their sum can or can’t.

Proofs on a postcard to…


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?