16 March 2016

Chartres cathedral (again)

This was my second time in Chartres (I have already taken the same photo three years ago), but now I was more interested in the juxtaposition of the various textures (hence the B&W rendition).

10 March 2016

Social reform as arbitrage

Changes in laws and regulations (including for instance tax rates) are presented by their proponents as improvements to society in general, but it is generally obvious (and accepted) that they favor one group to the expense of another: for instance, changes in labor law can be to the advantage of the employer or the employee.

Sometimes, however, these measures are advertised as benefitting all involved, with no negative effects. Instead of altering the balance between the various players, the new laws are expected to eliminate inefficiencies in the social and economical system, as the practice of arbitrage levels price differences between markets.

This is counterintuitive, and so are the arguments used by the advocates of change, who forecast an outcome exactly opposite to the one expected in normal settings, because the situation is supposedly very unbalanced, even untenable (hence the urgency of the changes.) The problem is that the system to be changed has often existed for decades: the claims that some extreme imbalance has persisted (or has gradually evolved) and that something needs to be done right now (before the next election cycle, preferably) are difficult to take seriously.

A recent example are the proposed ammendments to French labor law: they are supposed to reduce unemployment by relaxing conditions on layoffs and overtime. I guess this could only work if the labor market were completely unbalanced in France.

This is similar to Reagan's proposal of increasing tax revenue by reducing tax rates, which would only have been possible in very specific circumstances (on the decreasing side of the Laffer curve.)


13 February 2016

Water is HO2 (for at least one philosopher)

This evening I've been listening to some podcasts of talks given at the Nietzsche on Mind and Nature conference (Oxford, 2009).

There are some interesting points to be made, so I'll probably write a couple more posts on this, but what I found striking is Günter Abel's affirmation (about 08:55 into his talk; see the video) "[...] saying for example (famous example) 'water is HO2'..." Now, everyone can misspeak, but he did not correct himself and there were no reactions from the audience (then again, this was a philosophy conference held in Oxford, so maybe all attendants were being exceptionnally polite). Is this the general level of scientific education in the population of philosophy professors?!

The topic of Abel's talk is not directly related to science, although he does address the tension between consciousness and neurobiology (and how one should not identify conscious states and physical processes etc.)

11 February 2016

24 January 2016

Sample median for a Lorentz (Cauchy) distribution - part 2

In a previous post I derived the distribution function for the median of a sample of size \(n = 2k +1\) (with \(k \geq 0\) integer) drawn from a Lorentz (or Cauchy) distribution:
\begin{equation}
\label{eq:result}
g(x) = \frac{n!}{(k!)^2 \, \pi ^{n} \, \gamma} \left [ \frac{\pi ^2}{4} - \arctan ^2 \left (\frac{x - x_0}{\gamma} \right )\right ]^k \frac{1}{1+ \left (\frac{x - x_0}{\gamma} \right )^2 }
\end{equation}
I will now consider some of its properties.

23 January 2016

Sample median for a Lorentz (Cauchy) distribution

The Lorentz (or Cauchy) distribution
\begin{equation}
\label{eq:def}
f(x) = \frac{1}{\pi \gamma}\frac{1}{1+ \left (\frac{x - x_0}{\gamma} \right )^2 }
\end{equation}is pathological, in that it has no finite moments apart from the zero-order one (which ensures proper normalization of the density function.) Many results we usually take for granted (e.g. the central limit theorem) do not apply and, when sampled, the sample mean and sample variance are not good predictors for the position parameters \(x_0\) and \(\gamma\). This should not come as a surprise, since the distribution mean and variance do not even exist. How can we then estimate the position parameters?

19 January 2016

Scattering from a bunch of parallel wires

I got interested in this problem by trying to understand a result in [1], and also because it may be useful for some stuff I'm currently working on. Consider a collection of \(N\) very (infinitely) long objects, parallel to the \(z\)-axis and whose centers have positions \(\mathbf{R}_{j}\) in the \((x,y)\) plane. We are interested in the orientationally averaged scattering signal.

19 December 2015

What is an order parameter?

For those of us working with liquid crystals, the answer tends to be fairly automatic: "the average of the second Legendre polynomial over the orientation distribution". Only in a second step do we think to qualify the definition: it concerns the quadrupolar order parameter (call it S) in a three-dimensional system. S = 0 for isotropic orientation, S = 1 when the molecules are perfectly oriented along an axis (the director) and S = –1/2 when they are all perpendicular to the director.

Whether it is the appropriate one depends on the problem at hand: if the particles constituting the system do not have inversion symmetry we should probably use the dipolar order parameter. It is less obvious that, although S describes well the tendency of molecules to align along an axis or perpendicular to it, it is not appropriate for any preferred angle: if all molecules make the "magic angle" θm = arccos(1/√3) ≅ 54.7° with the director, S is again zero so it cannot help distinguish between this situation and a purely isotropic distribution. One should then resort to the octupolar order parameter (or possibly a combination of quadrupolar and octupolar terms, for other angles?)

It may be useful to look at order parameters pragmatically, as Landau did for his theory of phase transitions: they are constructed such as to be zero in one phase and finite in another one. It is up to us to identify the phases and to define the most convenient parameter with regard for the particular system but also for the information we want to extract.

12 December 2015

Note to writers

I do not want to read about your personal history. Unless you have led a very interesting life (literary or otherwise) or you have such an uncommon talent that everything you write is captivating, no matter the topic, do not bother. To wit, if you are not Goethe (or Feynman, or Leigh Fermor) or Proust, your recounted experience had better stand for something larger than itself.