The second law of thermodynamics and the history of the universe

Oxford Physicist Roger Penrose to Speak at Bro...
Image via Wikipedia

I had to go on quite a long plane journey yesterday and I bought a book to read – Roger Penrose‘s work on cosmology: Cycles of Time: An Extraordinary New View of the Universe

I bought it on spec – it was on the popular science shelves: somewhere I usually avoid at least for the physical sciences, as I know enough about them to make hand waving more annoying than illuminating, but it seemed to have some maths in it so I thought it might be worthwhile.

I have only managed the first 100 pages of it so far, so have not actually reached his new cosmology, but already feel it was worth every penny.

Sometimes you are aware of a concept for many years but never really understand it, until some book smashes down the door for you. “Cycles of Time” is just such a book when it comes to the second law of thermodynamics. At ‘A’ level and as an undergraduate we were just presented with Boltzmann’s constant and told it was about randomness. If anybody talked about configuration space or phase space in any meaningful sense it passed me by.

Penrose gives both a brilliant exposition of what entropy is all about in both intuitive and mathematical form but also squares the circle by saying that, at heart, there is an imprecision in the law. And his explanation of why the universe moves from low entropy to high entropy is also brilliantly simple but also (to me at least) mathematically sound: as the universe started with such a low entropy in the big bang a random walk process would see it move to higher entropy states (volumes of phase space).

There are some frustrating things about the book – but overall it seems great. I am sure I will be writing more about it here, if only to help clarify my own thoughts.

In the meantime I would seriously recommend it to any undergraduate left wondering what on earth entropy really is. In doing so I am also filled with regret at how I wasted so much time as an undergrad: university really is wasted on the young!

(On breakthrough books: A few years ago I had this experience with Diarmaid MacCulluch’s Reformation and protestantism. People may think that the conflict in the North of Ireland is about religion – but in reality neither ‘side’ really knows much about the religious views of ‘themuns’. That book ought to be compulsory reading in all Ireland’s schools – North and South. Though perhaps the Catholic hierarchy would have some issues with that!)

The future of Britain’s coalition government depends on “agile” methods

Iain Duncan Smith, British politician and form...
Image via Wikipedia

I don’t write about politics here, much, but I am not going to shock anyone who knows me if I admit to not being a supporter of the current Conservative – Lib Dem coalition government.

But I will give them credit for some of the things they have said and done on software use and development in government – at least in their willingness to abandon the Labour administration’s over-reliance on proprietary software and vendor lock-in.

What has been less impressive has been their willingness to look for other forms of snake-oil as a replacement. Not so many years ago the Conservatives claimed that they could replace Labour’s plans for a patient record system in the NHS with an off-the-shelf alternative from Google. They don’t talk about that any more, as Google’s offer itself folded.

And there is certainly a whiff of snake oil about the idea that the “Universal Credit” proposed to replace Britain’s current plethora of benefits for workless and low income households – probably the biggest IT project Britain has ever seen – can be delivered without problem simply because government has moved from “waterfall” methods of development to “agile” methods.

(For anyone wondering, a “waterfall” method is essentially linear – one bit of the project is completed and you move on to the next: like water falling down a cliff. “Agile” methods are iterative – you keep going back to the client trying to improve the product. The idea is that agile methods deal better with the great bugbears of software development – changing specifications and user needs.)

I am no great expert on these matters, but I did read a few textbooks about them in the last year, and they were all are pretty clear that “agile” methods are great for small to medium sized projects, but no so great for big, mission-critical projects. And surely the income of the poorest families and the need to have a benefits system that gets them into work is just such a project?

But, of course, all this is bound up in politics. Iain Duncan Smith, the Work and Pensions Secretary has already threatened to walk out of the government unless his project is allowed to proceed. Inside the DWP this seems to have become an Emperor’s-New-Clothes affair, and no one is allowed to say it will not work, at least not on a timetable that says it will happen by 2013.

(As an aside, it was good to see Sir Brian Urquhart quoted in the papers last week – he is the man who, as shown in A Bridge Too Far, warned that Operation Market-Garden was set to fail, but was told his views were not acceptable.)

The UK government has minimal experience of managing an agile software project, but has decided to bet everything on getting it right more or less the first time.

I hope they do – the alternative is too hideous. But the security of some of the least well off ought to come before the political career of Iain Duncan Smith.

Copyright in students’ work

© is the copyright symbol in a copyright notice
Image via Wikipedia

Slashdot has an interesting discussion on copyright in students’ work.

I raised this issue with faculty staff in Birkbeck a year ago and got a very vague answer – which sounded pretty much like “the university has copyright in your work but we won’t enforce it.”

I honesty do not know whether that is true or not and, in fact, the phrase I was required to apply to my MSc project report – “the report my be freely copied and distributed provided the source is explicitly acknowledged” – would represent a breach of the GNU GPL if the university sought to apply it to the Linux patch code which currently sites in my Git repo. (Of course we can argue as to whether or not I have actually distributed anything, and the answer is probably no).

I was left slightly dissatified by the official response to my question, because it suggested the college did not really take the issue seriously, which, given that it is at the heart of some long-running and extremely important debates in software development and distribution seems odd. Though, to be fair, I think the concern was more that someone would try to “go proprietary” on their software rather than allow it to be shared as part of the academic commons.

I would share that concern, of course. But it would not take much effort to allow students to pick from a range of licensing terms that would both protect the college’s desire to share and reuse any software or ideas present in the work while ensuring that other, necessary, licensing constraints are met.

If it wasn’t for those pesky neutrinos

Reactions in the proton-proton chain. The % va...
Image via Wikipedia

Neutrinos have proved to be nothing but trouble for scientists over the years.

They could not detect them from the Sun (where they are produced as a by-product of fusion), then they did or did not have mass. Now, it seems, they travel faster than light and are threatening to overturn the apple cart of relativistic space-time. If my dimly recalled understanding of relativity is correct, this would imply that, from the netrino’s point of view, it travels in the opposite direction to the way we see it moving in our reference frame: plainly, either the experiment is giving the wrong results or our theory of space-time is very seriously flawed.

Of course, what these troubles mean is that neutrinos have been huge allies in our search for a better understanding of physical reality. Though this new finding – which has plainly caused consternation amongst those who have been conducting the experiment – would be truly shocking if confirmed.

More on the Erdős–Straus conjecture

Number of ways to write an even number n as th...
Image via Wikipedia

As a reminder, the Erdős–Straus conjecture is: \forall n \in \mathbb{N}_1:\frac{4}{n+3}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} .

Let {n+3} = k , we can see if k is even, then k=2j and \frac{4}{k} = \frac{4}{2j} = \frac{2}{j} =\frac{1}{2j} + \frac{1}{2j} + \frac{1}{j} so the conjecture holds for all even k . In fact this holds for k=2, j=1 also.

For odds it’s not so simple, though an expansion does exist for the odd numbers of the form k=4j+3


The Erdős–Straus conjecture

An Erdős-Diophantine graph with five points.
Image via Wikipedia

I came across this as a result of links to stories about Hilbert’s tenth problem, and it looks fun, so I thought I’d write a little about it.

The Erdős–Straus conjecture is that for any integer n \geqslant 4 then \frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z} where x, y , and z are positive integers.

This is equivalent to a diophantine equation:

xyn^1 + xzn^1 + yzn^1 - 4xyz = 0

Which is, apparently, trivially solvable for composite (non-prime) numbers. And we can obviously see that if n=pq then if we had an expansion \epsilon for \frac{4}{p} then the expansion for \frac{4}{n} would simply be \frac{\epsilon}{q} – so if we found the conjecture true for a prime then it would be true for all that prime’s multiples. Hence to disprove the conjecture one needs to find a prime for which it is false.

And, indeed, the truth of the conjecture is an open question, though computationally it has been verified as true up to 10^{14} .

I bought a book

“Programming Android: Java Programming for the New Generation of Mobile Devices” to be precise.

Now I just have to decide what I want to do with it! I had thought about a LaTeX based application, as both LaTeX and LyX are thinly supported on Android. But that looks very ambitious at the moment.

We’ll see.

More on Hilbert’s tenth problem

Mathematician Julia Robinson in 1975
Image via Wikipedia

One of the best things about is that it gives me, as a blog author, the opportunity to link in other relevant stories and so it was with my last post on diaphantine sets and the integers.

The top linked item there gives a lot more on David Hilbert‘s “tenth problem” – which was solved (negatively), in its original conception for the integers, in 1970 but which remains open in wider domain of the rational numbers.

One of the items is a short film on Julia Robinson – a truly remarkable woman who contributed so much this field. Despite once being president of the American Mathematical Society and a truly world leading mathematician, Robinson, as a woman, often struggled to find work in the field.

But getting to the top had always been a struggle for her – as The Honors Class: Hilbert’s Problems and Their Solvers relates her (adoptive) mother was keen for her to get a teachers’ diploma and forget maths, she lost a years schooling due to childhood illness and she also, as an adult, suffered from a serious heart complaint and at one time was given a very gloomy medical prognosis as a result of that same illness.

It’s also worth noting that Constance Reid, described in the film as Julia Robertson’s biographer (and indeed a distinguished biographer of mathematicians) was rather more than that: she was also Julia’s sister.

Diophantine sets and the integers

Hilbert's problems
Image via Wikipedia

This is not some great revelation, but it interested me, so might interest some readers (I got it from The Honors Class: Hilbert’s Problems and Their Solvers).

A diophantine equation is one of the form ax^4+bx^3+cx^2+dx+e=0 and a diophantine set is a set of numbers to solve a diophantine equation (Hilbert’s 10th problem was to find an algorithm to solve these in the general case – a task we now know to be impossible).

One diophantine set is the integers – which are a solution to x=a^2+b^2+c^2+d^2





and so on…