A Candle in the Dark

From the BBC,

Punjab disability ‘uranium link’

Tests on children with cerebral palsy or mental disabilities in the Indian state of Punjab have revealed high levels of uranium.

A charity based in Faridkot city said chemical analyses of hair specimens collected from 149 children in its care showed “unexpected amounts” of toxins.

The result has baffled the authorities as there are no known sources of uranium in Punjab.

Here’s the Punjab health minister Lakshmi Kanta Chawla on the issue

“This is not a health subject. We don’t know how children are showing such high concentrations of uranium”, Punjab health minister Lakshmi Kanta Chawla said.

“It is for the central government in Delhi to deal with the problem.”

A prime example of why politicians are redundant. It’s because they exist only to deflect any sort of responsibility up the hierarchy, before it gets lost in the singularity at the top.

Adapted from notes I copied during a course I attended

Consider a 1+1 dimensional scalar field where x corresponds to the space dimension, and t corresponds to the time dimension. The Lagrangian and the Hamiltonian energy density for the field are given by,

Here, the Einstein summation convention is used, where repeated indices are summed over. is used to denote the four-vector

The evolution of this scalar field is governed by the Euler-Lagrange equation,

Plugging in Lagrangian for the scalar field I’ve written above, the E-L equation reduces to,

where, is the D’Alembertian. ie,

A specific example, where the solutions of the E-L equation have interesting properties, is given by the potential,

The lowest energy solutions which satisfy the E-L equations are known as the classical vaccum. In this specific example, they are given by

You can see that these solutions correspond to the minimum of the potential. Additionally, the total energy is also minimized by these solutions, because the contribution of the space and time derivatives to the Kinetic Energy density term is zero.

Apart from these vacuum solutions, there exist finite energy solutions of the E-L equations for this potential. From the criteria that the energy must be finite when the Hamiltonian energy density is integrated over x, such a solution must tend to the classical vacuua solutions at the extremities.

Such a solution is known as the kink solition, and is given by,

where,

You can check that this solution satisfies the Euler-Lagrange equation by plugging it in. Note that the solution is time-independent. Furthermore, when , . Therefore, the solution interpolates between the classical vacuua solutions.

To understand why it’s called a kink soliton, let’s plot its functional form, and the corresponding Energy density. For the plots below, I’ve set

Plot of the kink soliton solution

Energy density for the kink-soliton solution

As you can see, the solution tends to the vaccum solution (here, a=1) at either end and the energy density appears as a stationary wave packet which is like a kink in a rope.

Now, there are certain global symmetries that Lagrangian has, which are absent in the kink soliton solution. For instance, if we set , the Lagrangian in Eq. (1), and consequently the corresponding action, is invariant. However, the solution , is a new solution and interpolates the other way (ie, as . This is called the anti-kink solition, and has the same energy as the kink soliton.

There are other symmetries where the Lagrangian is invariant which give rise to new solutions. For example, a spatial translation will give a solution , which shifts the location of the center of the kink. A lorentz transformation will give a new time dependent solution, , where . This corresponds to a kink which starts out at the origin and moves to the right with a constant velocity.

Therefore, the most general form of the kink soliton can be written as,

This corresponds to a soliton which is initially centered at x_0 and moves with a velocity v to the right. The anti-kink soliton corresponding to this is simply

Now, consider a box which has both a kink and an anti-kink solition placed away from each other and with velocities in opposite directions. What happens to them as they approach each other? They can’t pass each other, otherwise they’d violate the boundary condition (remember that the kink and anti-kink solitons tend to different vacuum solutions at either end). So you’d expect them to bounce off each other.

In fact, we can even numerically test this, because we know that they are governed by the Euler-Lagrange equation. If represents both the kink and anti-kink soliton, then initially, one has

then, the E-L equation can be rewritten as two first order differential equations

where, is the momentum density

To solve the system of equations above, I discretized the spatial derivative using the central difference method, and used a fourth-order Runge Kutta solver with corresponding intial and boundary condition discussed above. You can see how the kink and anti-kink soliton bounce off each other in the video below.

Kinky, ain’t it?

Wikipedia has a gallery of images on solitons for the Sine-Gordon model, where a Sine potential is used instead of the quartic potential. There are also a lot of nice websites on the fantastic properties that such non-linear PDE’s exhibit.

See Also
Sine-Gordon Solitons and Soliton Collisions:
Pendulum Model
Soliton-Lab Art Gallery

The Nature blog, the Great Beyond links to this interesting article from the NY Times

India’s use of brain scans in courts dismays critics

Now, well before any consensus on the technology’s readiness, India has become the first country to convict someone of a crime relying on evidence from this controversial machine: a brain scanner that produces images of the human mind in action and is said to reveal signs that a suspect remembers details of the crime in question.
….

The woman, Aditi Sharma, was accused of killing her former fiancé, Udit Bharati. They were living in Pune when Sharma met another man and eloped with him to Delhi. Later Sharma returned to Pune and, according to prosecutors, asked Bharati to meet her at a McDonald’s. She was accused of poisoning him with arsenic-laced food.

Sharma, 24, agreed to take a BEOS test in Mumbai, the capital of Maharashtra. (Suspects may be tested only with their consent, but forensic investigators say many agree because they assume it will spare them an aggressive police interrogation.)

After placing 32 electrodes on Sharma’s head, investigators said, they read aloud their version of events, speaking in the first person (“I bought arsenic;” “I met Udit at McDonald’s”), along with neutral statements like “The sky is blue,” which help the software distinguish memories from normal cognition.

For an hour, Sharma said nothing. But the relevant nooks of her brain where memories are thought to be stored buzzed when the crime was recounted, according to Joseph, the state investigator. The judge endorsed Joseph’s assertion that the scans were proof of “experiential knowledge” of having committed the murder, rather than just having heard about it.

In the only other significant judicial statement on BEOS, a judge in 2006 in Gujarat denied the test the status of “concluded proof” but wrote that it corroborated already solid evidence from other sources.

In writing his opinion on the Pune murder case, Judge S. S. Phansalkar-Joshi included a nine-page defense of BEOS.

Sharma insists that she is innocent.

What I’d like to know is whether this technology has been independently tested? How does someone know whether the answers that the test gives are reliable and can be used in court?

…
Despite the technology’s promise — some believe it could transform investigations as much as DNA evidence has — experts in psychology and neuroscience were almost uniformly troubled that it was used to win a criminal conviction before being validated by any independent study and reported in a respected scientific journal.

Publication of data from testing of the scans would allow other scientists to judge its merits — and the validity of the studies — during peer reviews.

“Technologies which are neither seriously peer-reviewed nor independently replicated are not, in my opinion, credible,” said Rosenfeld, a psychologist and neuroscientist at Northwestern University and one of the early developers of electroencephalogram-based lie detection. “The fact that an advanced and sophisticated democratic society such as India would actually convict persons based on an unproven technology is even more incredible.”

Touché (Except for the last sentence)

A week ago, it was reported by the media that a study conducted by a group of US scientists from the Boston Medical Center and published in the peer-reviewed Journal of the American Medical Association, reported that 20% of ayurvedic and herbal “medicines” sold over the internet contained dangerous amounts of toxic heavy metals ^{1 2}.

Now, this isn’t the first time that such a concern has been raised. In 2004, the same group had published a study which concluded that³

One of 5 Ayurvedic HMPs produced in South Asia and available in Boston South Asian grocery stores contains potentially harmful levels of lead, mercury, and/or arsenic. Users of Ayurvedic medicine may be at risk for heavy metal toxicity, and testing of Ayurvedic HMPs for toxic heavy metals should be mandatory.

The new study further looked at the prevalence of toxic heavy metals in ayurvedic and herbal “medicine” depending on the country of manufacture of the drug. The study found that⁴,

One hundred ninety-three of the 230 requested medicines were received and analyzed. The prevalence of metal-containing products was 20.7% (95% confidence interval [CI], 15.2%-27.1%). The prevalence of metals in US-manufactured products was 21.7% (95% CI, 14.6%-30.4%) compared with 19.5% (95% CI, 11.3%-30.1%) in Indian products (P = .86)

That is, the study found that there’s very little difference between the amount of heavy metals found in herbal “medicine” whether it was manufactured in the US or India.

You might be aware that India has a government sponsored national department of pseudoscience, known as AYUSH. In fact, the home page on their website proudly displays,

a) The Law of Similars – It is also called the Law of Cure. This law demonstrates that the selected remedy is able to produce a range of symptoms in a healthy person similar to that observed in the patient, thus leading to the principle of Similia Similibus Curentur i.e. let likes be treated by likes. To give a simple example the effects of peeling an onion are very similar to the symptoms of acute cold.

…

Psora, Syphilis and Sycosis are the three fundamental causes of all chronic diseases that afflict the human race as discovered by Dr. Hahnemann and called them miasms

Er… what year are we living in again? Is the national department of pseudoscience even vaguely familiar with the concept of viruses, bacteria or indeed, modern biology? The page then goes on to blatantly display rubbish as fact. Vital force? Miasm? It’s appalling how they attempt to display such nonsense with such seriousness.

Anyway, the aforementioned AYUSH has responded to the recent published study on heavy metals in herbal “medicines”. Their entire statement can be found here.

The press release by AYUSH makes several claims to serve as evidence that the study by the Boston Medical group was flawed. Let’s look at some of them which I found interesting.

CLAIM 1:

A project for Physico chemical characterization and toxicity studies of 8 widely used Bhasmas (Rasa Aushadhies) was also sanctioned under the Golden Triangle Project which is being carried out by various laboratories of CSIR i.e. Indian Institute of Toxicological Research (IITR), Lucknow, Indian Institute of Chemical Technology (IICT), Hyderabad. Under this project, one of the reputed manufacturers of Rasa Aushadhies was chosen for manufacturing of the selected Rasa Aushadhies as per the classical texts which were made available to CSIR laboratories for physio chemical characterization and their toxicity studies. On the basis of 28 days toxicity studies, all the 8 Rasa Aushadhies have been found to be non-toxic.

CLAIM 2:

Samples of 600 Indian medicinal plants collected from the wild as well as various medicinal plant gardens in India by the Council for Scientific Research in Ayurveda and Siddha were sent to the Indian Institute of Toxicological Research (CSIR), Lucknow, Sri Ram Institute of Industrial Toxicology, New Delhi and Centre for Research in Indian Medicine, Shastra University, Thanjavur. The test reports received from these three laboratories disclose that Lead, Mercury and Arsenic have not been found in these 600 Indian medicinal plants samples above the permissible limits laid down by WHO which is 10ppm for Lead, 1ppm for Mercury and 3ppm for Arsenic. This study clearly indicates that Indian medicinal plants collected from the wild or cultivated have been found to be free from lead, mercury and arsenic contamination contrary to the claim made by Dr. Robert Saper et al in their second article “Lead, Mercury and Arsenic in US- and Indian- Manufactured Ayurvedic Medicines sold via the Internet” published in JAMA, August 27, 2008.

CLAIM 3:

Further, some of the medicines mentioned in Table 3 of the article, namely Akangvir Ras, Agnitundi Bati, Arogyavardhini Bati are herbo metallic compounds which contain these metals as therapeutic agents after purification process. Dr. Saper has visited India and has been in touch with several Ayurveda experts and is fully aware that these herbo-metallic preparations used in Ayurvedic system of medicine contain heavy metals like lead, mercury and arsenic as therapeutic agents after proper de-toxification process and no significant adverse drug reactions have been reported regarding their use in India.

(emphasis mine)

(i) CLAIM 1: AYUSH do not mention if the results of the toxicity studies of the eight (compare this with the peer-reviewed study) herbal “medicines” have been published in a peer-reviewed scientific journal, but they go on to claim that most of the studies are “in progress” and the results will be obtained soon. If so, how do they claim to know that the results are going to prove that the herbal “medicine” will be free of toxic metals? Statements such as the one below makes one wonder if there’s a bias introduced, because they claim to know the results of the tests before they actually get them!

CSIR would be getting the results of this scientific research published to set at rest doubts regarding the safety of Rasa Aushadhies prepared properly as per classical texts.

Also, it’s important to note that we don’t know the methodology of the study and therefore can’t evaluate how credible this study is as there could be many flaws. For instance, while they claim that the 28 day toxic test showed that the herbal “medicine” was nontoxic, they do not mention the actual amounts detected. Compare this with the peer-reviewed studies.

Additionally, there’s a lot we don’t know about the methodology of the study. For example, did they ask the “reputed manufacturers” of Rasa Aushadhies to specially prepare the samples for a test? Could they have been that stupid? Or, did they randomly take samples from the market, which is more representative of the actual product that the consumer buys? You see, details like this are important before one can judge how credible a claim is. Besides, even if this claim was right, it doesn’t invalidate the study published in JAMA. It merely shows the importance for additional tests!

(ii) CLAIM 2: This is a classic straw-man argument. It has absolutely no relation to what Dr Saper’s group are claiming. Just because you tested 600 wild plants for heavy metal content DOES NOT automatically mean that the final herbal “medicine” and plants sold to the customer are free from the toxic heavy metals. What you have to do, is to test the actual ayurvedic “medicine” which is sold! This is precisely what Dr Saper’s group have done. If you’re claiming that Dr Saper’s methodology was flawed, then you need to point out precisely why. Just claiming that their results were wrong isn’t sufficient.

(iii) CLAIM 3: Perhaps, this is the worst one of the lot. It’s criminally ignorant and malicious. There’s no such thing as “detoxifying” heavy metals to make them safe. The toxicity of lead is because of its chemical action once it binds and interacts with enzymes in our human body, and interferes with essential biological activities. You can’t magically “detoxify” lead to remove it’s elemental properties. No sane person would recommend using elemental lead, mercury and arsenic as therapeutic agents. I don’t think there’s a single shred of scientific evidence which backs that claim.

When a governmental organization blindly ignores experimental evidence in a peer-reviewed journal about the dangers of a product using flimsy and non-existent reasoning, and instead states that the authors of the study “are consciously trying to malign the reputation of Ayurveda”, they are displaying the extent of irresponsibility and the unscientific and unprofessional attitude which maligns the organization.

Some Ayurvedic “medicine” may indeed work. But, what is required is stringent scientific tests and quality control to ensure the products are safe to use, and to try to understand the chemical and biological reason for their efficacy. The policy of ignoring information about potentially dangerous products, and attempting to rubbish evidence which does not agree with their viewpoint, may have disastrous consequences for the populace which buys such traditional “medicine”

External Links

[1] – Toxic metals in Ayurvedic remedies
[2] – Indian herbal medicines queried
[3] -R. B. Saper et al., Heavy Metal Content of Ayurvedic Herbal Medicine Products, vol. 292 (Am Med Assoc, 2004).
[4] – Robert B. Saper et al., “Lead, Mercury, and Arsenic in US- and Indian-Manufactured Ayurvedic Medicines Sold via the Internet,” JAMA 300, no. 8 (August 27, 2008): 915-923, doi:10.1001/jama.300.8.915.

For more, check out this nice introductory post by Prof. Arunn Narasimhan on optimization and genetic algorithms.

This is an image the Mars Reconnaissance Orbiter took of the the Phoenix spacecraft landing on Mars. WOW!!!

Mindblowing.

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.

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 and , the rate at which the concentrations change will simply be the rate at which they react. That is,

where, and 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 and .

What happens if and 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 and , one obtains

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

where, and 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

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, and . With these variables, the linearized equations are

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

(Why? Substitute and check!)

Substituting to the equation, we get

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

This can be re-written as

where,

Now, the value of is the eigenvalue of the matrix , and for the homogeneous solution to be unstable, one requires that the real part of the eigenvalues of R are positive for some

This condition can be simplified to,

For the eigenvalue to be positive, we require that and .

But, since and is less than 0 because the homogeneous steady state is assumed to be stable, can never be positive. So, the criteria for instability of the homogeneous solution is that . 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

An excerpt from the message that the pope released and is going to convey during New Year’s Day,

The family, the human community and the environment

…
Humanity today is rightly concerned about the ecological balance of tomorrow. It is important for assessments in this regard to be carried out prudently, in dialogue with experts and people of wisdom, uninhibited by ideological pressure to draw hasty conclusions

Delicious irony.

On 24 November 1859, for the first time ever in the history of the human race, the common person finally had an opportunity to find out the answer to one of the most important questions of all (How did we get here?) for 15 shillings ¹.

It is interesting to contemplate an entangled bank, clothed with many plants of many kinds, with birds singing on the bushes, with various insects flitting about, and with worms crawling through the damp earth, and to reflect that these elaborately constructed forms, so different from each other, and dependent on each other in so complex a manner, have all been produced by laws acting around us. These laws, taken in the largest sense, being Growth with Reproduction; Inheritance which is almost implied by reproduction; Variability from the indirect and direct action of the external conditions of life, and from use and disuse; a Ratio of Increase so high as to lead to a Struggle for Life, and as a consequence to Natural Selection, entailing Divergence of Character and the Extinction of less-improved forms.

Thus, from the war of nature, from famine and death, the most exalted object which we are capable of conceiving, namely, the production of the higher animals, directly follows. There is grandeur in this view of life, with its several powers, having been originally breathed into a few forms or into one; and that, whilst this planet has gone cycling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved.

Notes
[1] – Charles Darwin, Chapter XIV. Recapitulation and Conclusion, On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life