# A Candle in the Dark

A look on science, politics, religion and events

## Archive for April 2008

The HRW released this report today: Human Rights Abuses Stemming from Male Guardianship and Sex Segregation in Saudi Arabia

The Saudi government has instituted a system whereby every Saudi woman must have a male guardian, normally a father or husband, who is tasked with making a range of critical decisions on her behalf. This policy, grounded in the most restrictive interpretation of an ambiguous Quranic verse, is the most significant impediment to the realization of women’s rights in the kingdom. The Saudi authorities essentially treat adult women like legal minors who are entitled to little authority over their own lives and well-being.

Every Saudi woman, regardless of her economic or social status, is affected by these guardianship policies and the deprivation of rights that their enforcement entails. Adult women generally must obtain permission from a guardian to work, travel, study, or marry. Saudi women are similarly denied the right to make even the most trivial decisions on behalf of their children.

As the HRW report details, there’s the almost criminally mislabeled institution called the Permanent Council for Scientific Research and Legal Opinions (CRLO), whose legal opinions on interpretation of religious statements are often implemented as law. When asked about women’s employment, they said

God Almighty … commended women to remain in their homes. Their presence in the public is the main contributing factor to the spread of fitna [strife]. Yes, the Shari’ah permits women to leave their home only when necessary, provided that they wear hijab and avoid all suspicious situations. However, the general rule is that they should remain at home.

If you have the time, and are not aware of the extent, I strongly suggest you go through the entire report. It summarizes the absolutely pathetic “rights” of women, and how truly messed up the situation is.

Written by parseval

April 21, 2008 at 8:00 pm

Posted in politics, religion

## What harm is there in letting them have their beliefs?

A powerful message from cectic, the webcomic.

Image from cectic

On a related note, with the creationist propaganda movie Expelled crying about “oppression” in academics, Blake Stacey has a much more informative and saddening post, titled “Creation, Power and Violence”, about the actual discrimination that educators face in the United States for teaching evolution in science classes.

Written by parseval

April 19, 2008 at 11:57 am

## How the leopard may have really got her spots

A mechanism of pattern formation in many animals, (including the zebra, leopard and the giraffe), was first suggested by Alan Turing in 1952.

To understand his mechanism, we’ll first look at a system where two reactive chemical species (morphogens) are present and they do not move about in space(ie, they don’t diffuse). If the concentration of the species are denoted by $u$ and $v$, the rate at which the concentrations change will simply be the rate at which they react. That is,

$\frac{\partial u}{\partial t} = f(u,v)$

$\frac{\partial v}{\partial t} = g(u,v)$

where, $f(u,v)$ and $g(u,v)$ describe the rate law kinetics which the species obey. If we want to find the steady state concentrations of the species, we simply set the partial derivatives with time as 0, and solve for $u$ and $v$.

What happens if $f(u,v)$ and $g(u,v)$ are non-linear equations with multiple solutions? The stability of each solution can be determined by a linear stability analysis, and the final stable solution will depend on the initial condition. Indeed, because of the homogeneity in the initial constraint, that the initial concentrations of the species are uniform everywhere and the they are spatially fixed, the final concentrations at steady state are again uniform in space.

Linearizing the above system of equations (ie, set $u=u_{ss}+\tilde{u}$ and $v=v_{ss}+\tilde{v})$, one obtains

$\left(\begin{array}{c} \frac{d \tilde{u}}{dt} \\ \frac{d \tilde{v}}{dt} \end{array}\right)=\left(\begin{array}{cc} f_u & f_v \\ g_u & g_v \end{array}\right) \left(\begin{array}{c} \tilde{u} \\ \tilde{v}\end{array}\right)$

For the steady states to be stable, the condition is that the real parts of the eigenvalues of the jacobian are negative. For a two component system, this is true when the trace is negative and the determinant positive.

Now, what happens if we remove the spatial constraint and allow the chemical species to diffuse? The equations which describe the concentration of the species will be modified to account for diffusion. We’ll now have

$\frac{\partial u}{\partial t} = D_u \nabla^2 u + f(u,v)$

$\frac{\partial v}{\partial t} = D_v \nabla^2 v + g(u,v)$

where, $D_u$ and $D_v$ are the diffusivity of the chemical species. One of the observations which Turing made, was that under certain conditions, steady states which were initially spatially uniform could now show spatial variation due to diffusion. The natural question would be, under what conditions would we get these spatial variations? To answer that, we need to delve a bit into linear stability analysis.

However, before doing that, I’ll have to define what the boundary conditions and the geometry of the system is for the second set of equations. A very simple system would be a one-dimensional strip of space, where diffusion and reaction occurs. To make this system ‘self-contained’, we set the flux of the chemical species at the boundary to be 0.

Therefore, the system of equations are

$\frac{\partial u}{\partial t} = D_u \frac{\partial^2 u}{\partial x^2} + f(u,v)$

$\frac{\partial v}{\partial t} = D_v \frac{\partial^2 v}{\partial x^2} + g(u,v)$

Since we want to find out when the homogeneous solutions are unstable, we need to look at the linearized system in terms of the deviation variables from the steady state solutions, $\tilde{u} = u - u_{ss}$ and $\tilde{v} = v - v_{ss}$. With these variables, the linearized equations are

$\frac{\partial \tilde{u}}{\partial t} = D_u \frac{\partial^2 \tilde{u}}{\partial x^2} + f_u \tilde{u} + f_v\tilde{v}$

$\frac{\partial \tilde{v}}{\partial t} = D_v \frac{\partial^2 \tilde{v}}{\partial x^2} + g_u \tilde{u} + g_v\tilde{v}$

The solutions to this equation can be shown to be of the form.

$\tilde{u} = u^* e^{\sigma t} \sin{\alpha x}$

$\tilde{v} = v^* e^{\sigma t} \sin{\alpha x}$

(Why? Substitute and check!)

Substituting to the equation, we get

$u^* (\sigma +D_u\alpha^2 - f_u) -f_v v^* = 0$

$v^* (\sigma +D_v\alpha^2 - g_v) -g_u u^* = 0$

Here’s the important argument. For a spatially varying pattern, we require non-zero solutions to $u^*$ and $v^*$. The condition for that is

$\left|\begin{array}{cc} \sigma +D_u\alpha^2 - f_u & -f_v \\ -g_u & \sigma +D_v\alpha^2 - g_v \end{array}\right| = 0$

This can be re-written as

$det( \sigma I - A + \alpha^2 D) = 0$

where,

$A= \left(\begin{array}{cc} f_u & f_v \\ g_u & g_v \end{array}\right)$

$D = \left(\begin{array}{cc} D_u & 0 \\ 0 & D_v \end{array}\right)$

Now, the value of $\sigma$ is the eigenvalue of the matrix $R= (A - \alpha^2 D)$, and for the homogeneous solution to be unstable, one requires that the real part of the eigenvalues of R are positive for some $\alpha^2$

This condition can be simplified to,

$\sigma^2 - Tr(R) \sigma + det(R) =0$

For the eigenvalue to be positive, we require that $det(R)<0$ and $Tr(R)>0$ .

But, since $Tr(R) = Tr(A - \alpha^2 D) = Tr(A) - \alpha^2 Tr(D)$ and $Tr(A)$ is less than 0 because the homogeneous steady state is assumed to be stable, $Tr(R)$ can never be positive. So, the criteria for instability of the homogeneous solution is that $det(R)<0$. One can also relate this to the ratio of the diffusivites of the two species u and v.

This means that under this condition, the homogeneous steady state solution is unstable, and the concentrations of u and v will increase till the non-linear terms we have neglected plays a role and causes spatial patterns.

I will edit and add pretty pictures displaying the simulation of this phenomenon once I figure out how to use the pdetool in MATLAB to solve non-linear parabolic partial differential equations in 2 variables.

References
The Chemical Basis of Morphogenisis, A.M Turing, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, Vol. 237, No. 641. (Aug. 14, 1952),pp 37-72

Two-stage Turing model for generating pigment patterns on the leopard and the jaguar, Phys. Rev. E 74, 011914 (2006)

Mathematical Biology, JD Murray, 3rd edn, Springer (2002). chapter 2.4 &2.5

Written by parseval

April 14, 2008 at 4:28 pm