30 December 2012

Death by firearm and its relation to health expenditure

An argument that often appears in the discussion of gun casualties is that adequate care for the mentally ill can reduce crime to a certain extent (see e.g. Joe Nocera's column in the New York Times.) This sounds intuitively reasonable, but the importance of the effect is less clear.

It is also difficult to find a relevant quantitative measure of health care quality, but I assume that health expenditure (per capita) is a reasonable proxy. Below, I plot the number of casualties (same data as in the previous post) as a function of expenditure, as a total amount and as GDP percentage (data from Wikipedia.)
There might be a very slight descending trend as a function of total expenditure, but scarcely any effect of the GDP percentage. The healthcare argument is mainly invoked with respect to the USA, but the graph tells a different story: high expenditure and high casualty rate. It is of course possible that the expenditure on mental healthcare is disproportionately low in the US, but indiscriminately throwing money at the problem will not make it go away.

Another interesting region is the lower left corner of the first graph: it is populated by Eastern European countries and former Soviet republics where the number of casualties and the health budget are both low. Are the mentally ill better taken care of in Azerbaijan than in the US ? I would not bet on it. What these countries have in common, however, is that they have fewer guns and lower inequality (see the previous post).

It is tempting to find a causality relation between two social problems, if only because solving one would automatically get rid of the other, but in the present case I have not seen a solid, data-based  argument.

29 December 2012

27 December 2012

Death by firearm, gun possession and the GINI coefficient

The recent tragic events in Newtown rekindled the debate on gun control and the interest in the correlation between gun ownership and gun-related deaths. The issue is clearly very complicated, and a single variable will not explain much, but it would still be interesting to plot the interdependence of the various parameters. Of course, I claim no causality relation between them.

The most obvious variable couple to plot is the number of gun-related deaths vs. the number of guns per capita (both retrieved from Wikipedia). When I was halfway through the data treatment I noticed that a similar graph was made by Mark Reid. The plot is below, in log-log representation, since both the x and y axes cover more than two decades:

The three-letter country code is retrieved from here. The data seems to follow a linear tendency (dashed line), but even in a log-log plot some deviations are clearly visible, such as the cluster of values at top center (dashed frame).

I decided to consider some additional (economical) parameters, and the results are quite interesting.  Another variable that could be correlated with gun violence is the economical inequality, quantified for instance by the GINI index. Plotting the number of firearm casualties versus this variable (I used the coefficient defined by the World Bank, fourth column in the table) yields the following graph:
The "anomalous" points in the first graph now follow more or less the same tendency as the other countries (the dashed line is a guide for the eyes). The gun ownership is used as color code (see the legend), but values between 2 and 20 are difficult to distinguish in this log scale. A nice feature is that the countries with low gun ownership (in blue) which were at the bottom left in the first graph also follow the tendency in the second one.

Visually, I would say that the GINI coefficient explains the data better than gun possession. What other variables might be correlated with gun-related deaths ? For instance, below is the GDP dependence (data from here):
For the USA, the relation between firearm deaths and gun ownership can be seen, for instance, here. How about the economical indicators ? I retrieved GINI values for the US states in 2010 and the number of firearm murders in 2011. The graph is below, in lin-log representation:

The GINI range (41-50) is much smaller than the global scale (12-65 percentage points) but there is a clear correlation. In contrast, there is almost no dependence on the GDP per capita:
In both graphs I left out DC, which is far to the right of the other points in both inequality (53.2)  and GDP (174500) with a y value of 77.

It would therefore seem that the inequality is strongly correlated to firearm casualties, both for the US states and for world nations.

23 December 2012

All the Pretty Horses

I have just finished reading Cormac McCarthy's All the Pretty Horses. I would have thought that the sequence of stark dialogue and world-encompassing metaphor is a sure recipe for kitsch, but he manages to pull it off every time. And some of the sentences are as polished as one of Menashe's poems, with subtle rhymes, consonances and alliterations. This is how the last paragraph starts:

The desert he rode was red and red the dust he raised, the small dust that powdered the legs of the horse he rode, the horse he led.

5 December 2012

Latour and Woolgar's "Laboratory Life"

With Laboratory Life. The Contruction of Scientific Facts, published in 1979, Latour & Woolgar more or less launched the field of scientific anthropology. I think their constructivist view is a bit extreme, but I did find some very interesting observations, in particular in Chapters 4 and 5.

Here, however, I'm interested in the validity of the anthropological approach (as defined by the authors, pp. 27-30 in the 2nd edition, Princeton University Press, 1986) when applied to highly specialized fields.  In particular, they state explicitly (p. 28):

 We envisaged a research procedure analogous with that of an intrepid explorer of the Ivory Coast, who, having studied the belief system or material production of "savage minds" by living with tribesmen, sharing their hardships and almost becoming one of them, eventually returns with a body of observations which he can present as a preliminary research report.

On the face of it, Latour's claim that he can provide new insight into scientific activity even though (or, rather, precisely because) he is not trained in that specific discipline  is contradictory. The initial "bracketing" he invokes (p. 29) must at some point give way to familiarity, and this is not possible without (at least) the common language of the particular scientific domain.

Does the anthropological approach require a certain "simplicity" of the population under study? The anthropologist must on the one hand be "naive" enough, but on the other hand still be able to immerse himself in the culture in a few years. What happens if the subjects have a kind of specialized knowledge that is essential to their society ?


2 December 2012

Untranslatable concepts

In French, one can say Je suis seul (I am alone) or Je me sens seul (I feel alone), but nothing as baldly distressing as “I am lonely” or “I am lonesome.”
says Henry Cole in the New Yorker. Suggesting other lonely French words such as esseulé or délaissé would probably not change his mind, since they are arguably not an exact translation for lonely. Can one then say that there are words in one language without an identical counterpart in another language? As a second example, several people told me (with patriotic pride) that the Romanian dor cannot be translated in any other language. I have trouble seeing how it is substantially different from longing.

I believe this kind of affirmation (like many declarative sentences) can be read in a strong and a weak sense.
  • Strong interpretation: words in one language can convey meaning that cannot be expressed in another language. I would say that this position is obviously false, at least for living languages with a similar level of complexity.
  • Weak interpretation: words come with their own subtext and connotations, which do not survive translation. This is obviously true, but not particularly interesting. For instance, the gender of inanimate objects varies from language to language; so does the emphasis placed on it. Words are also colored by the literature that uses them (Proust's madeleine comes to mind; I do not believe finger cake has similar baggage).
Is there some intermediate and valid position that is strong enough to be relevant ? Where is Cole's own point of view on this continuum ?

28 November 2012

The Carvallo paradox and femtosecond lasers - part I

Fourier analysis amounts to writing any signal (including those limited in time) as a sum of infinitely extended harmonic functions. One way of framing this apparent contradiction is the Carvallo paradox:

Since a spectrograph only selects one component of the signal to be analyzed (and this component is infinite in time), it should detect that component both before and after receiving the signal.

The standard answer is that spectrographs have a finite resolution: when selecting light with a given wavelength \(\lambda\), the result is in practice a finite interval \( ( \lambda - \Delta \lambda,  \lambda + \Delta \lambda)\), defining the spectral resolution \( R = \lambda / \Delta \lambda \). Let us assume a (respectable) value \( R = 10^5\) in the visible range, at \(\lambda = 500 \, \text{nm}\).

The uncertainty relation connects the resolution to the lenth of the pulse by: \[ \Delta \omega \Delta t \geq \frac{1}{2} \Rightarrow \Delta t = \frac{R}{2 \omega} \geq 8 \; 10^{-11} \, \text{s} \simeq 100 \, \text{ps}\] where in the second step I used \( \left | \Delta \omega / \omega \right | = \left | \Delta \lambda / \lambda \right |\). A pulse with the spectral sharpness \(\Delta \lambda\) must then be at least 100 ps long.

This is a very short interval for classical spectroscopy measurements. However, using modern optical techniques one can create ultrashort pulses, down to \( \Delta t \simeq 10 \, \text{fs}\). When observing such a pulse at a resolution \( R \) we should then see it as spread out to 100 ps. Where is the error?

More about in in the next part, where I'll also try to give a version of the paradox that is not affected by resolution.

27 November 2012

Gaussian integral of an error function

In Surely You're Joking, Mr. Feynman!, Richard Feynman mentions a useful technique he used for evaluating integrals, namely taking the derivative under the integral sign. I will show here how this trick works in calculating the Gaussian integral of an error function. Averages over Gaussian distributions are omnipresent in physics, and the error function is just the primitive of the Gaussian, making the calculations relatively easy (and the result quite elegant.) Nevertheless, Mathematica (version 8) cannot perform this integral, and I could not find it in Gradshteyn & Ryzhik. I needed it to describe the interaction of a phase front with an external field, see the paper here.

Let us define: \[ I(\alpha, \beta, \gamma ) = \int_{-\infty}^{\infty} \text{d}x \exp (-\alpha x^2) \,\text{erf}(\beta x + \gamma) \] with \( \alpha, \beta \, \text{and}\, \gamma\) real and \( \alpha \) positive. For \( \gamma = 0\) the integrand is an odd function, so \( I(\alpha, \beta, 0 ) = 0\). We can also estimate \[ I' (\gamma) = \frac{\partial}{\partial \gamma} I(\alpha, \beta, \gamma) = \frac{2}{\sqrt{\pi}} \int_{-\infty}^{\infty} \text{d}x \exp (-\alpha x^2)\, \exp \left [-(\beta x + \gamma)^2\right ] \] which is a simple Gaussian integral: \( \displaystyle I' (\gamma) = \frac{2}{\sqrt{\alpha + \beta^2}} \exp \left ( - \frac{\alpha \gamma ^2}{\alpha + \beta^2}\right )\)

Finally, \[ I(\alpha, \beta, \gamma ) = \int_{0}^{\gamma} \text{d}u \, I' (u) = \sqrt{\vphantom{\beta}\frac{\pi}{\alpha}} \,\text{erf} \; \left ( \gamma \sqrt{\frac{\vphantom{\beta} \alpha}{\alpha + \beta^2}} \right )\] The reader can check that all derivatives exist and all integrals converge. What happens if we replace the linear term in the error function by a quadratic one?

26 November 2012

An aesthetic argument against solipsism

In his short story The Other, from the volume The Book of Sand, Borges uses an artistic proof to convince his interlocutor (his younger self, actually) that their encounter is real and not a mere dream. He quotes a verse from Hugo (that the young Borges had not yet read), which is so striking that it could not have been dreamed up; the two then communicate across half a century.

This being a Borges story, things are more complicated than the summary above. In particular, the older Borges concludes that, although they had met, the younger one had in fact been dreaming. What I am interested in here is whether the story puts forward a successful argument against solipsism.

Such an argument requires demonstrating the existence of another person, who is:
  • essentially similar, enough to achieve meaningful communication, and
  • sufficiently different, so that it cannot be a figment of the subject's imagination
Does a successful encounter with a work of art fulfill these two conditions?

25 November 2012

Power laws in small-angle scattering - part II

In the first part I showed that the SAXS intensity scattered by a platelet system goes like \( I(q) \sim q^{-2}\), at least in some intermediate (but as yet unspecified) q range. Here I will show that for thin rods this dependence becomes \( q^{-1}\), I will then derive the terminal (Porod) behaviour \( q^{-4}\) and briefly consider the transition between these two regimes.

Rods: α = 1

For a rod, the electron density profile:
\[ \rho (\mathbf{r}) = \delta(x) \delta(y) \mathrm{Cst}(z)\] with the same notations as in part I. Its Fourier transform \(\tilde{\rho} (\mathbf{q}) = \mathrm{Cst}(q_x) \mathrm{Cst}(q_y) \delta(q_z)\) and, as in part I, I will assume the same form for the intensity \( I(\mathbf{q})\). The "dual" object of a rod under Fourier transform is a platelet:
When spreading this intensity over reciprocal space we must keep in mind that the intersection of the plane with a sphere of constant \(q\) is a great circle (shown in red in the figure above). Thus, the total contribution \( 2 \pi q I_0\) increases with \(q\), but it must also be divided by the surface area of the sphere, yielding for the two spheres: \[ I(q_0) = \frac{2 \pi q_0 I_0}{4 \pi q_0^2} = \frac{I_0}{2 q_0} \quad \mathrm{and} \quad I(2 q_0) = \frac{4 \pi q_0 I_0}{4 \pi (2 q_0)^2} = \frac{I_0}{4 q_0} \] so that \( I(2 q_0) = I(q_0) /2\) and \( \alpha = 1\).

Interfaces: α = 4 (Porod)

The density profile of an interface is invariant under x and y translations and is a Heaviside (step) function along z: \[ \rho (\mathbf{r}) = \mathrm{Cst}(x) \mathrm{Cst}(y) H(z)\] The Fourier transform of \( H(z)\) is \( 1/q_z \), as one can see either by direct evaluation or by noting that the derivative of \( H(z)\) is the Dirac delta etc. Finally, \[ \left | \tilde{\rho}(\mathbf{q}) \right |^2 = \delta(q_x) \delta(q_y) {q_z}^{-2}\] As for the platelet in part I, the scattering is confined along a rod perpendicular to the interface, but its intensity, instead of being constant, decreases along its length. Further spreading this signal over the sphere adds an additional \( {q}^{-2}\) factor, for a final \( {q}^{-4}\) dependence.

Crossover

Although both are infinitely extended in the plane, a single interface (Porod) and two interfaces very close together (thin plate) exhibit very different scattering laws. However, at high enough scattering vector, all objects reach the Porod regime. We will discuss this crossover for the case of a plate with finite thickness. The Fourier transform of this object along its normal is easily shown to be a cardinal sinus, so that: \( \left | \tilde{\rho}(q_z) \right |^2 = \left [ \frac{\sin(q_z a)}{q_z a} \right ] ^2\). Close to the origin, ie. for \(q_z a \ll 1\), this function is flat, hence the approximation \(\sim \mathrm{Cst}(q_z)\) we used for the platelet in part I. At high \(q\), on the other hand, it behaves as \(q^{-2}\), yielding the Porod law. To put it differently, when the typical scale \(L = 2\pi/q\) over which we observe the object is much larger than its thickness, we are in the "platelet" regime and do not resolve the two interfaces. When \(L \ll 2a\), on the other hand, we only observe "one interface at a time", justifying the treatment above and thus the \( q^{-4} \) law. This regime is "terminal" insofar there is no smaller typical length scale in the system, until that of the composing atoms or molecules.

24 November 2012

DIC microscopy image


The edge of a spin-coated droplet (reflection DIC image).

23 November 2012

Power laws in small-angle scattering - part I

The small-angle X-ray scattering (SAXS) spectrum of particles with a well-defined shape (such as rods or platelets) is often characterized by a power-law dependence: \( I(q) \sim q^{-\alpha}\), where the exponent \( \alpha \) is directly related to the particle geometry. For "compact" particles, the large-\( q \) intensity scales as \( q^{-4}\) (Porod regime). Below, I'll give the most compact and yet -hopefully- understandable derivation I can think of for these power laws.

To simplify the derivation, we'll consider these objects as infinitely thin and infinitely large, meaning that we'll be looking at them on length scales much larger than their thickness and much smaller than their lateral extension. The approximation is legitimate, since it is in this range of length (or, conversely, scattering vector) that the power-law regimes are encountered.

Platelets: α = 2


For a platelet, this simplification yields for the electron density profile:
\[ \rho (\mathbf{r}) = \mathrm{Cst}(x) \mathrm{Cst}(y) \delta(z)\]
where "Cst" (constant) means that the density does not vary as a function of \(x\) and \(y\). Of course, a constant does not need an argument, but we will specify it in order to keep track of the space dimensions.

We now need the Fourier transform of \(\rho (\mathbf{r})\), \(\tilde{\rho} (\mathbf{q})\). Since the Fourier transform of a constant is a Dirac delta and viceversa, we simply have:
\[ \tilde{\rho} (\mathbf{q}) = \delta(q_x) \delta(q_y) \mathrm{Cst}(q_z)\]
The intensity scattered at a given wave vector \( I(\mathbf{q}) = | \tilde{\rho} (\mathbf{q})|^2\), and we'll admit that the latter function has the same form as \( \tilde{\rho} (\mathbf{q}) \). This is plausible: if a distribution is perfectly localized, we expect its square to share this property. Rigorously speaking, however, the square of a Dirac delta makes no sense (within classical distribution theory) so a proper derivation involves considering a finite size for the object, taking the modulus square of its Fourier transform and only letting the size go to infinity in the last step. Finally, the intensity scattered by a platelet perpendicular to the \(z\) axis is a thin rod parallel to \( q_z\), as shown in the figure below.
Fourier transform of a platelet

In solution, colloidal particles assume all possible orientations, so that this intensity is spread evenly over the sphere of constant \( q\). Consider two such spheres, with radii \( q_0\) and \( 2 q_0\). The rod contributes to each sphere the same amount, namely twice its (constant) intensity \( 2 I_0\), shown as red dots at the poles. This signal must however be divided by the surface area of the sphere, an area that increases as the square of the radius: \[ I(q_0) = \frac{2 I_0}{4 \pi q_0^2} \quad \mathrm{and} \quad I(2 q_0) = \frac{2 I_0}{4 \pi (2 q_0)^2}\] so that \( I(2 q_0) = I(q_0) /4\) and, finally, \( \alpha = 2\).

More power laws coming in part II of this post...

22 November 2012

Physics and Photography

Short review of a short but very well-written introduction to the principles of photography: Science for the Curious Photographer: An Introduction to the Science of Photography by Charles S. Johnson Jr (author's blog is here). This slim (180 p.) volume covers a lot of ground, not only the obvious topics such as image formation or the physiology of vision but also, for instance, the operation of CMOS and CCD detectors and a discussion of polarization. The presentation is fairly technical, so an adequate background in physics is necessary.

The only critique would be that, for an introduction, it is somewhat light on the references (many of which are simply Wikipedia articles).

21 November 2012

Electrical circuits and Euler's polyhedron formula

When solving an electrical circuit consisting of impedances and voltage sources, one needs to apply Kirchhoff's laws:
  • The current law, yielding N-1 equations (with N the number of nodes)
  •  The voltage law, for an additional L equations (where L is the number of elementary loops)
The unknowns are the currents flowing in each branch (B of them). We also assume that the branches do not cross. Note that writing the current law for the N-th node or the voltage law for a composite loop (consisting of several adjacent elementary loops) does not provide any further information, the resulting equations being linear combinations of the previous ones.

For the problem to be well-posed the number of unknowns and equations is equal, which we can write as:
\[ N + L - B = 1\] This relation is easily proven in plane geometry, but here I would like to show its intimate connection with Euler's formula, which states that, for a convex polyhedron, \[ V + F - E = 2\] where V, F and E are the numbers of vertices, faces and edges, respectively.

Let us start by establishing a correspondence between circuits and polyhedra, as shown in the figure below. Place a sphere on top of the (planar) circuit diagram and connect each node to the North pole by a line segment (this is known as a stereographic projection.) We define the vertices as the intersections of these segments with the sphere; the result is a convex polyhedron.



It is easily seen that, with the notations above, we have the straightforward equivalences V = N and E = B. The number of faces, however, F = L + 1, since the "topmost" face corresponds to the open area surrounding the circuit. Substitution in either of the equations above yields the other one.

18 November 2012