20 November 2013

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}

17 November 2013

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.

3 November 2013

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