28 August 2014

RIP, Jacques Friedel (1921-2014)

Jacques Friedel passed away yesterday. He was among the pioneers of the study of defects in solids and of the electronic structure of matter, one of the founders of my lab and my scientific great-grandfather.

25 August 2014

The arbitrariness of words

(via phys.org) A statistical analysis of English terms, recently published in Phil. Trans. R. Soc. B (free preprint on the corresponding author's site), finds systematic relations between sound and meaning, refuting a pure arbitrariness of the linguistic sign.

Of course, such relations have been sought for —and found— ever since Plato (last year, I reviewed on this blog Genette's Mimologics, a great exploration of the topic.) The novelty is the quantitative aspect of the analysis: the authors define phonetic and semantic distances between pairs of terms and then measure the correlation of these distances, which is higher than expected by pure chance. Unfortunately, they give no intuitive illustration for the amplitude of the effect, expressed as an \(r\)-factor. So, finally, how systematic is the English language?

Another interesting result is that more systematic words are acquired earlier. The authors speculate that systematicity helps language development in its early stages but might hinder it later, when (the vocabulary being larger) it can lead to confusion.

24 August 2014

Moments of inertia of triangular prisms

Now that we have determined the moments of inertia of regular and truncated equilateral triangles, it is time to calculate them for the corresponding right prisms. These bodies, with mass density \(\rho\), can be seen as stacks of infinitesimally thin triangles of thickness \(\text{d}h\) and surface density \(\text{d} \mu = \rho \text{d}h\) (we preserve the notations from the previous posts and introduce the height of the stack, \(H\). The inertia moments of the prisms are denoted by \(P\), instead of \(I\).)
The centers of mass of these sheets are all situated on \(z\), so the total moment of inertia about this axis is simply the sum of the individual ones. We must simply replace \(\mu\) by \(\rho H\) in (1) and (3):
P_z(L,H) &= \rho H L^4 \frac{\sqrt{3}}{48}\\
P^{\text{tr}}_z(L,a,H) &= \rho H \frac{\sqrt{3}}{48} [L^4 - 3 a^4 - 12 a^2 (L-a)^2]
The derivation is slightly more complicated for axis \(y\), since we need to account for the variable distance between it and the centers of mass of the sheets (using, of course, the overworked parallel axis theorem!) Fortunately, we only need the integral \( \displaystyle \int_{-H/2}^{H/2} \text{d}h \, h^2 = \frac{H^3}{12}\) to get:
P_y(L,H) &= \rho H L^4 \frac{\sqrt{3}}{96} \left [ 1 + 2 \left ( \frac{H}{L} \right )^2 \right ] \\
P^{\text{tr}}_y(L,a,H) &= \rho H \frac{\sqrt{3}}{96} \left [ L^4 - 3 a^4 - 12 a^2 (L-a)^2 + 2 H^2 (L^2 - 3 a^2) \right ]\\
 &= \rho H L^4 \frac{\sqrt{3}}{96} \left [ 1 + 2 \left ( \frac{H}{L} \right )^2 - 12 x^2 \left ( (1-x)^2 + \frac{x^2}{4} + \frac{x^2}{2} \left ( \frac{H}{L} \right )^2 \right ) \right ]
\end{array}\tag{6}\]where \(x = a /L\).

23 August 2014

Moment of inertia of a clipped triangle

After calculating the moment of inertia for an equilateral triangle, let us consider the same shape, but with clipped corners, as in the drawing below:

We will preserve the notations of the previous post, adding the superscript "tr" for the truncated shape: \(I_{z}^{\text{tr}} (L,a)\) is the moment about the \(z\) axis of the equilateral triangle with side \(L\), clipped by \(a\) at each corner (with \(a \leq L/2\)). We will also use the same strategy, writing the moments of the complete shape as a combination of its four fragments:
\[I_{z}(L) = I_{z}^{\text{tr}} (L,a) + 3[I_{z}(a)+m(a)d^2] ,\]
where \(d=(L-a)/\sqrt{3} .\) Using the results obtained for the full triangle immediately yields:
\[ I_{z}^{\text{tr}} (L,a) = \frac{\sqrt{3}}{48} \mu \left [ L^4 - 3a^4 - 12 a^2 (L-a)^2\right ] \tag{3}\]
Similarly, from:
\[I_{y}(L) = I_{y}^{\text{tr}} (L,a)+I_{y}(a) + 2[I_{y}(a)+m(a)(L-a)^2/4]\]
we get:
 \[ I_{y}^{\text{tr}} (L,a) = \frac{\sqrt{3}}{96} \mu \left [ L^4 - 3a^4 - 12 a^2 (L-a)^2\right ] = I_{z}^{\text{tr}} (L,a)/2 \tag{4}\]
The clipped shape preserves the threefold symmetry of the original one, so the same conclusion as to the in-plane isotropy of the inertia tensor holds. Also, \( I_{z} = 2  I_{y}\) in both cases; I'm sure there is some elegant way to explain this, but I can't find it.
A quick check of results (3) and (4) is that \( I_{y,z}^{\text{tr}} (2L,L) = I_{y,z}(L) .\) In this case, one retrieves the situation shown in the illustration to the previous post.

22 August 2014

Career advice

(via Slashdot) Infoworld gives some career advice to young programmers. I'm surprised at how much sense these short observations also make for beginning scientists. OK, maybe point 7 is not that relevant, and point 4 should be rephrased as Do not re-invent the wheel. Otherwise, they are spot-on.

21 August 2014

Mass moment of inertia of an equilateral triangle

As in previous posts, I would like to determine the moments of inertia of a solid body, this time an equilateral triangular prism. I will start in this post by a (very thin) equilateral triangle. The challenge is getting the result in the simplest way, making the most of the symmetry elements and taking advantage of the parallel axis theorem.

Around the \(z\) axis

The \(z\) axis goes through the center of mass of the triangle of interest (gray central area of side \(L\) in the illustration above) and is perpendicular to its plane. We denote the corresponding moment by \(I_z(L)\). The moment of the large triangle, with side \(2L\), is \(I_z(2L)\). We can relate these two parameters in two ways:
  • For a given shape and surface mass density, the moment of inertia scales as the size to the fourth power, on dimensional grounds. Thus, \(I_z(2L) = 16 I_z(L)\).
  • The large triangle can also be described as the rigid assembly of the small central triangle and the three adjacent ones. The parallel axis theorem yields:
\[I_z(2L) = I_z(L) + 3 [I_z(L) + m(L) d^2] \]
where \(m(L)= \mu L^2 \sqrt{3}/4\) is the mass of the small triangle, with \(\mu\) the surface mass density, and \(d=L/\sqrt{3}\) is the distance between the centers of mass of the side triangles and the \(z\) axis.

Combining these two expressions for \(I_z(2L)\) immediately yields:
\[ I_z(L) = \mu L^4 \frac{\sqrt{3}}{48} \tag{1}\]

Around the \(y\) axis

The \(y\) axis is contained in the plane of the triangle and goes through its center of mass and one vertex. Using the same strategy as above, we get:
\[ \left\{
I_y(2L) &= 16 I_y(L)\\
I_y(2L) &= 2 I_y(L) + 2 [I_y(L) + m(L) (L/2)^2]
\end{array} \right. \]
where on the right-hand side of the second equality the first term corresponds to the central and top triangles (both their centers of mass are on axis \(y\)) and the second one to the side triangles, whose centers are shifted by \(L/2\). Finally:
\[ I_y(L) = \mu L^4 \frac{\sqrt{3}}{96} = \frac{I_z(L)}{2} \tag{2}\]

How about the \(x\) axis? To answer this question, we start by noting that there are three equivalent directions within the plane of the triangle: \(y\) and the axes (say, \(y'\) and \(y''\)) going through the other two vertices: \(I_y(L) = I_{y'}(L) = I_{y''}(L)\). This third-order symmetry in a two-dimensional space means that the inertia tensor is in fact isotropic in the plane of the triangle, with the same value \(I_{\bot}(L) = I_{y}(L) = I_{x}(L)\) for any axis in this plane. The inertia tensor is then:
\[\mathrm{I} = \left (
I_{\bot}(L) & 0 & 0 \\
0 & I_{\bot}(L) & 0 \\
0 & 0 & I_z(L)
\right ) \]
This isotropy of a tensorial property for a system that does not in fact have full rotational symmetry is a very useful result (albeit somewhat counterintuitive). To give only one example from a completely different area of physics: a cubic crystal cannot be birefringent!

3 August 2014

Madrid - the Prado Museum

Fra Angelico's Annunciation
Fra Angelico's Annunciation (see in high resolution)
Samuel Menashe's Annunciation is the best comment.

18 July 2014

French research grants - the 2014 campaign

The main research funding organization in France is the ANR: "Agence Nationale de la Recherche" (national research agency). Every year, it finances a variable number of projects, in all scientific fields.

The results for 2014 were announced today, and the presentation text (in French only) is very upbeat: the success rate of 28% is more than 11% higher than that of 2013 ! A historical increase, one might conclude.

Unlike last year, however, the 2014 selection was done in two steps, and the 28% figure only accounts for the second one. The first step had already selected only 33% of the initial submissions, reducing the overall success rate to 9.4%. The only historical event is that this value dipped below 10% for the first time in the ten years since the ANR was created.

I make two predictions:
  • The 9.4% figure will never appear in official documents.
  • The 2015 campaign will consist of three steps, the last of which will select 100% of the projects that made it through the second round. 

    [UPDATE 19/07/2014] : For comparison, I plotted below the yearly success rates and total amounts distributed by the ANR since 2005.

      11 July 2014

      Reproducible experiments

      Yesterday evening, after having spent my day trying (and failing) to reproduce somebody's published research, I stumbled (via Soylent News) upon a psychologist's essay on "the emptiness of failed replications". Jason Mitchell, psychology professor at Harvard, states that failing to replicate somebody else's experiment does not represent a meaningful scientific contribution. Well, thank you, Prof. Mitchell !

      All jokes aside, it took me quite some time to parse the text, and even more time to realize that this difficulty is likely due to the implicit assumptions that I brought from my own field of work (experimental physics), which are quite different from those of the author, an experimental psychologist. Ultimately, I learned more from trying to separate these two viewpoints than from the text itself, which makes a rather simplistic argument.

      The argument

      Mitchell's main point appears to be that one cannot learn from negative arguments, since not finding something cannot prove it doesn't exist. This sounds entirely reasonable, and is certainly true in the case of the "black swan" example the author uses, but is completely wrong in usual scientific experiments: learning that the correlation between two variables is zero (within the uncertainty) is as strong a result as saying that it is significant and positive. Of course, the first outcome is less likely to lead to a high-profile paper.

      The assumptions

      A basic assumption in physical sciences is that of "homogeneity": the outcome of an experiment should not depend on its location, time or the personality of the scientist. Mitchell does not address this point directly, but seems to imply that getting all the details right for precisely replicating an experiment is next to impossible. He then blames this on the replicators' lack of some sort of "core competence". This is a valid point: if Nature is the same everywhere but the experimentalists are sloppy, their results will of course differ. From this I would however draw two uncomfortable conclusions:
      1. This sloppiness may just as well affect the initial experiment as the attempt to reproduce it.
      2. It also undermines an entire field of study if there is no way of distinguishing careful scientists from the careless (or incompetent) ones.
      In "tabletop" physics, replicating an experiment is relatively cheap1. It is also crucial: our research builds on someone else's results, and very often the replication is a necessary step before being able to go further. Chemists sometime spend weeks or months in order to reproduce published protocols. Needless to say, this is not done to prove the original author wrong ! Neither of these points seems to apply in psychology, as presented by Mitchell.

      Finally, I find quite strange Mitchell's attitude that replicating experiments is almost morally wrong: "One senses either a profound naiveté or a chilling mean-spiritedness at work." This goes beyond mere scientific debate and sounds more like responding to a personal offense.

      1. Even in large scale experiments, reproducing the results may be necessary, albeit very expensive. A good example is the search for the Higgs boson, with the two experiments, ATLAS and CMS, working side-by-side but without communicating (see for instance Jon Butterworth's "Smashing Physics".)

      6 July 2014

      ILCC2014: days four and five

      The plenary lectures of these last two days were given by chemists (Carsten Tschierske and Tadashi Kato) and they were both impressive for the results but also for the huge amount of work these clearly required.

      Overall, I think the scientific level was higher than two years ago, maybe because of the smaller number of oral presentations: there were only three parallel sessions (four on Tuesday) compared to five in Mainz. Finally, the next (26th) edition of the ILCC will take place at Kent State!