Power-law distribution of war magnitudes

While reading Steven Pinker's The better angels of our nature I stumbled upon the following argument for the magnitude of war (number of casualties) following a power-law distribution (page 220):

Recall the mathematical law that a variable will fall into a power-law distribution if it is an exponential function of a second variable that is distributed exponentially. My own guess is that the combination of escalation and attrition is the best explanation for the power-law distribution of war magnitudes.

The efficiency of molecular motors

We know since the work of Sadi Carnot that the efficiency \(\eta\) (the fraction of heat converted into work) of a thermal engine cannot exceed a surprisingly simple maximum value, \(\eta_{\text{max}} = 1 - T_c/T_h\), defined in terms of the absolute temperatures of the cold and hot heat sources, \(T_c\) and \(T_h\). This limitation applies to a wide variety of devices, from combustion engines to solar cells: in the latter case, \(T_h \simeq 5800 \, \text{K}\) is that of the Sun and \(T_c\) is the ambient temperature, yielding \(\eta_{\text{max}} = 93\%\) [1].

We also know that living organisms can operate at (or even below) the temperature of their environment. In these conditions, Carnot's formula would yield zero efficiency, and thus no work production. And yet, our muscles can reach an efficiency of above 50% [2], higher than that of our cars! How can we solve this paradox?

The availability of research data

A recently published paper [1] (free preprint here) warns that research data becomes less accessible with time. The authors tried to retrieve email addresses from articles 2 to 22 years old, sent standard messages requiring the data sets and followed up on the responses. In most cases (63%) the addresses were not working or they received no response. The other outcomes were: no information on the status of the data (6%), claim of data loss (7%), refusal to share (4%) and receipt of data (19%).

Heidegger's "black books"

[UPDATE 02/03/2014] (via enowning) A short piece in The Chronicle of Higher Education covering pretty much the same ground as the articles below.
[UPDATE 27/01/2014] An interview with Peter Trawny (editor of the "black books") appeared in Die Zeit (in German). An adapted version in French was published in Le Monde (in French).

[First seen here.] The controversy around Heidegger's political position is rekindled in anticipation of the philosopher's personal notebooks being published next spring. Supposedly, they contain clear antisemitic remarks. The debate has already started in the French press [1,2] and on the radio [3].

The more general question is: to what extent is the quality of the work affected by the morals of the author? There are many possible (and partially overlapping) answers, depending on the precise failing imputed to the author. This is where Heidegger's example is instructive, since his relation to national-socialism and antisemitism is not totally clear (among other things, because his work has not yet been completely published).

Are espressos fast?

Another example of an abstract term with a very concrete Latin origin: express.

The weight of an hourglass

This seems to be a classical problem [1]: what is the weight of an hourglass? Careful consideration shows that the weight is larger while running than at rest!

The Kramers-Kronig relations - part 2

In part 1, we had stopped before going to the frequency domain  because we needed the Fourier transform of the sign function. This is where the technical difficulty appears, because we cannot simply write:

\begin{equation}
\label{eq:sgnTF}
\operatorname{sgn}(\omega) = \int_{-\infty}^{\infty} \text{d} t \exp (-i \omega t) \operatorname{sgn}(t) \tag{5}
\end{equation} as the integral does not converge. One can however define\begin{align}
\label{eq:sgnvp}
&\operatorname{sgn}(\omega) = \lim_{\epsilon \to 0} \int_{-\infty}^{\infty} \text{d} t \exp (-i \omega t - \epsilon |t|) \operatorname{sgn}(t) = \nonumber \\
&- \lim_{\epsilon \to 0} \left [ \frac{1}{i \omega + \epsilon} + \frac{1}{i \omega - \epsilon} \right ]= \lim_{\epsilon \to 0} \frac{2i \omega}{\omega^2 + \epsilon ^2}= \mathcal{P} \left ( \frac{2i}{\omega}\right ) \tag{6}
\end{align}

The Kramers-Kronig relations - part 1

[See part 2 for some technical aspects]
Very nice derivation of the Kramers-Kronig relations (on Wikipedia, of all places), exploiting the relation between the even and odd components of a function \(\chi (t)\) and the real and imaginary parts of its Fourier transform \(\chi (\omega) = \chi ' (\omega) + i \, \chi '' (\omega)\).

One usually invokes the analyticity of \(\chi (\omega)\) in the upper half-plane, which must first be derived from the causality: \(\chi (t) = 0\) for \(t < 0\). Complex integration along a well-chosen contour then yields the Kramers-Kronig relations in their standard form [1]:

\begin{align}
\label{eq:KK}  
\chi (\omega) &= \frac{1}{i \, \pi} \mathcal{P} \int_{-\infty}^{\infty} \text{d} \omega ' \frac{\chi (\omega ')}{\omega ' - \omega} \quad \text{or, for the components:} \nonumber \\
\chi '(\omega) &= \frac{1}{\pi} \mathcal{P} \int_{-\infty}^{\infty} \text{d} \omega ' \frac{\chi ''(\omega ')}{\omega ' - \omega} \\
\chi ''(\omega) &= -\frac{1}{\pi} \mathcal{P} \int_{-\infty}^{\infty} \text{d} \omega ' \frac{\chi '(\omega ')}{\omega ' - \omega} \nonumber
\end{align}
where \(\mathcal{P}\) denotes Cauchy's principal value.

Statistical and subjective evidence

Are courts of law more receptive to subjective evidence (e.g. witness testimony) than to naked statistical evidence? This is the topic of a recent article [1], selected as one of last year's best philosophical papers. This "Blue Bus" problem seems to have a rather long history in the legal and psychological literature [2,3,4] and is loosely based on a real case ([2], n. 37). Wells [3] gives a particularly clear exposition.

The problem statement: A bus causes some harm, and we know for sure that it necessarily belongs either to the Blue Bus Company or to the Red Bus Company. Should the Blue Bus Company be held liable?

Two scenarios are put forward:

1.  A witness testifies that the bus does indeed belong to the Blue Bus Company, but we have good reason to believe the witness is only 80% accurate.
2. The Blue Bus Company accounts for 80% of the traffic in the relevant area.

In both cases, the probability we can assign to the offending bus belonging to the Blue Company is 80%. Nevertheless, the courts are unlikely to accept the second type of evidence.

Les hautes bergères