## A sequence related to prime counting

The current note arose as an exploration branching off from the matter discussed in these earlier notes: this one and this one. As we saw before, Carl Gauss, while still in his teens, produced his first estimate of the prime number distribution in the form of the function:

$\pi(n) \sim \dfrac{n}{\log(n)}$

Here is $\pi(n)$ is the prime counting function, which counts the number of prime numbers up to a given number $n$, and $\log(n)$ is the natural logarithm of $n$. The $\sim$ notation indicates that the prime counting function is asymptotic with $\tfrac{n}{\log(n)}$, i.e. as $n\to \infty$ the ratio $\pi(n)\big / \tfrac{n}{\log(n)} \to 1$.

Subsequently, Gauss refined his fit for the prime counting function by using the famed logarithmic integral $\textrm{Li}(x)$. We were curious if there was some arithmetic function, which was actually fitted by $\tfrac{n}{\log(n)}$ rather it being merely a single term approximation of the $\pi(n)$. In course of some arithmetic experiments, we stumbled upon a sequence, which we believe, without formal proof, is fitted by $\tfrac{n}{\log(n)}$ in terms of average behavior.

This sequence $f$ is defined thus: $f[1]=1$. Thereafter, add $n-1$ to all terms $f[1:(n-1)]$. Count how many of $f[1:(n-1)]+(n-1)$ are primes. This count is $f[n]$. For example when $n=2$ we add 2-1=1 to 1 we get 2. Which is a single prime; hence, $f[2]=1$. Now for $n=3$ we add 3-1=2 to the first two terms and we get 3, 3. Thus, we have 2 primes; hence $f[3]=2$. For $n=4$, we add 4-1=3 to the prior terms and get 4, 4, 5, which yields a single prime, 5; hence, $f[4]=1$. Thus, the first few terms of the sequence goes: 1, 1, 2, 1, 3, 1, 4, 1, 1, 2, 7, 2, 7, 1, 1, 4, 11, 3, 9, 2, 4, 4, 11, 0, 2, 4, 4, 11, 11, 6. Figure 1 shows a plot of the first 20000 terms of the sequence.

Figure 1

The blue line is the plot of this sequence and we notice right away that despite the fluctuations the average tendency is to grow with $n$. Via numerical experiments we were able to establish that this average growth is fitted best by the function $\tfrac{n}{\log(n)}$ (red line in Figure 1). The green line in Figure 1 is the count of primes $\pi(n)$. We observe that though some extreme values of $f$ exceed $\pi(n)$, the average behavior of $f[n]$, i.e. $\tfrac{n}{\log(n)} < \pi(n)$. This relates to a central development in the number theory: when Gauss conjectured the asymptotic relationship between $\tfrac{n}{\log(n)}$ and $\pi(n)$ the mathematical apparatus was not yet in place to prove it. This was finally developed by his last student Bernhard Riemann. Using those ideas, nearly century after Gauss’ conjecture, Hadamard and de la Vallée-Poussin proved it and it became known as the Prime Number Theorem. Further, de la Vallée-Poussin showed that $\pi(n)$ was related to $\tfrac{n}{\log(n)}$ thus:

$\pi(n)=\dfrac{n}{\log(n)}+O\left(\dfrac{n}{\log^2(n)}\right)$

Here, the second term is gives the error and is denoted using the big-O notation which was explained in an earlier note. This indicates that indeed $\tfrac{n}{\log(n)}$ would be less than $\pi(n)$. Thus, as can be seen in Figure 1 the average growth of $f[n]<\pi(n)$.

We then used $\tfrac{n}{\log(n)}$ to ‘rectify’ $f[n]$ i.e. obtain:

$f[n]-\dfrac{n}{\log(n)}$

Figure 2

This rectified $f[n]$ is plotted in Figure 2 and provides a clear picture of fluctuations in $f[n]$ once we have removed the average growth trend. We observe right away that the amplitude of the fluctuations grows with $n$. To determine this growth trend of the rectified $f[n]$, we first noticed from Figure 1 that $\pi(n)$ tends to run close to the maxima of $f[n]$. Hence, we utilized the asymptotic expansion of $\textrm{Li}(n)$, which is a better approximation of $\pi(n)$ and captures the behavior beyond the basic $\tfrac{n}{\log(n)}$ term:

$\textrm{Li}(n) \sim \dfrac{n}{\log(n)} \displaystyle \sum_{k=0}^\infty \dfrac{k!}{(\log(n))^k}$

$\textrm{Li}(n) \sim \dfrac{n}{\log(n)}+\dfrac{n}{\log^2(n)}+\dfrac{2n}{\log^3(n)}+\dfrac{6n}{\log^4(n)}...$

Using the first 4 terms to approximate the growth of the amplitude of rectified $f[n]$ we get the red bounding curves shown in Figure 2. Thus, we conjecture that while $f[n]$ grows on an average as $\tfrac{n}{\log(n)}$, the amplitude of its fluctuations is roughly approximated by $\textrm{Li}(n)-\tfrac{n}{\log(n)}$ (Green bounding curves in Figure 2).

Posted in Scientific ramblings |

## Convergence to a palindrome

This is a brief account of a sequence we constructed inspired by Dattatreya Ramachandra Kaprekar. It is not known to us if he had discovered it in one of his obscure publications from a small town in the Maharatta country. In any case we explored this sequence independently upon hearing of some procedures he used in his work. Consider a number like $n=100$. Its reverse is $r(n)=1$. Then, $n+r(n)=101$. We find that 101 is a palindrome. Consider another case $n=155$, then $n+r(n)=155+551=706$. This is not a palindrome so we continue the same process $n+r(n)=706+607=1313 \rightarrow n+r(n)=1313+3131=4444$. Thus, after 3 iterations of the process we have a palindrome. Thus, if we take any number and perform this operation of adding it to its digital reverse iteratively till we get a palindrome then our sequence $f$ is defined as the palindrome to which each number $n$ converges. Thus, $f[100]=101; f[155]=4444$. One question which arose was whether there are $n$ that never converge to a palindrome. Between 1:99 all numbers converge to a palindrome even if a large one. Hence, we explored 100:999 in greater detail. Figure 1 shows a plot of $f[100:999]$

Figure 1. y-axis: $\log_{10}(f[n])$

Of these our experiments suggested that the following 13 numbers in the range 100:999 never converge to a palindrome (marked by red dots in Figure 1): 196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986. One can see that barring 790 all of them come in pairs. The maximum value attained by $f[n]$ in this range is the 13 digit number: 8813200023188. This value is attained when $n$ is 187, 286, 385, 484, 583, 682, 781, 869, 880, 968. Comparable to the non-converging cases, here all ten cases come as pairs. There is also an interesting pattern for both the maxima and the non-converging cases: we observe that in each century it comes one number earlier but at some point a “new line of descent” emerges which then perpetuates the same pattern along with the older one. The same maximum value is attained for $n$ in the range 1:99, with $f[89], f[98]=16668488486661$.

Omitting the single digit numbers, 11 is the primordial palindromic number and its multiples often tend to have a palindromic structure. Its multiples for $k=1:9$ are the only double digit palindromes. Multiples of 11 are also found among the triple digit palindromes: 121, 242, 363, 484, 616, 737, 858, 979. For $n=1:99$ all $n>4$ converge to a palindrome which is a multiple of 11. For $n=100:999$, $f[n]$ have 64 unique prime factors ranging from 2 to 18209090957. Notably, the largest number of $f[n]$, 641, are divisible by 11. Thus, the most frequent convergence even in this range is to a multiple of 11 (Figure 2).

Figure 2. The number of $f[100:999]$, which are divisible by a given prime divisor from the set of all unique prime factors of $f[100:999]$.

Figure 2 shows that other than than the first few primes (2, 3, 5, 7 etc) there are some anomalous standout values. Notable among these are 37 which with 3 reaches a palindrome $3 \times 37= 111$ and other palindromic primes, most notably 101 and 131. The convergent might itself be a palindromic prime and for $f[100:999]$ we have 101, 727, 929, 181, 383, 787. As can be seen in Figure 1 (Violet points) the $n$ for which these palindromic prime convergents are reached have a distinctive pattern of distribution reminiscent of the maxima and non-converging values.

Finally, this sequence is notable for the very large values that are attained amidst otherwise pedestrian values (Figure 3).

Figure 3. x-axis in $\textrm{arcsinh}(f[n])$ scale. Mean of $f[n]$ is the red line while the median is the blue line

This makes for an interesting distribution where extreme events on the right end are rare but enormous in magnitude. This is reflected in the difference of several orders of magnitude between the mean and the median of $f[n]$. However, at least the actual occurrence of these extreme values is quite regular (Figure 1).

Posted in Scientific ramblings |

## A problem from 600 CE and some curiosities of Āryabhaṭa’s kuṭṭaka algorithm

Around 600 CE in the examinations of one of the Hindu schools of mathematics and astronomy one might have encountered a problem such as below (given by Bhāskara-I in his commentary on Āryabhaṭa’s Āryabhaṭīya):

dvayādyaiḥ ṣaṭ-paryantair ekāgraḥ yo ‘vaśiṣyate rāśiḥ |
saptabhir eva sa śuddho vada śīghraṃ ko bhaved gaṇaka ||

Quickly say, O mathematician, which number when divided by the numbers starting with 2 and ending in 6 (i.e 2:6) leaves 1 as the remainder, and is exactly divisible by 7?

This problem was given to illustrate the use of the kuṭṭaka algorithm first provided by Āryabhaṭa. Before we actually solve the above problem we will briefly examine the kuṭṭaka. The kuṭṭaka is a general algorithm deployed to obtain integer solutions for the indeterminate linear equations of the form $ax-by=c$, where $a, b, c$ are positive integers. Thus, it is essentially the problem of finding the coordinates of the the integer lattice point through which the line $ax-by=c$ passes. Of the three constants in the equation, $a$ and $b$ are given; $c$ is not given but we have to find the smallest $c$ for which the equation can be solved in integers. From that we can build other valid $c$ For computational simplicity (with no loss of generality) we take $a$ to be the bigger number and $b$ to be the smaller number. The below presentation of it follows the matrix representation of Āryabhaṭa’s operation given by the mathematician Avinash Sathye:

Let us consider as an example the following equation $95x-25y=c$. From $a=95$ and $b=25$ we can generate the below matrix which is the result of the kuṭṭaka procedure. We have written a function ‘kuṭṭaka’ in the R language that computes this matrix and few other details given $a,b$.

$K=\begin{bmatrix} 19 & 4 & 95 & NA \\ 5 & 1 & 25 & 3 \\ 4 & 1 & 20 & 1 \\ 1 & 0 & 5 & 4 \\ 0 & 1 & 0 & NA \\ \end{bmatrix}$

The process is initiated with column $K[,3]$. Write $K[1,3]=a$ and $K[2,3]=b$. Kuṭṭaka in Sanskrit means ‘to powder’, common translated as ‘pulverizer’. We start ‘pulverizing’ $a=95$ with $b=25$, which means finding the $K[3,3]=K[1,3] \mod K[2,3]$. Then $K[4,3]=K[2,3] \mod K[3,3]$. We iterate this procedure until we get $K[n,3]=0$; that completes the column $K[,3]$. We call $n$ as the number of iterations for convergence. Thus, the matrix will have $n$ rows and 4 columns. The quotient of the division of $K[1,3] \div K[2,3]=3$ is written as $K[2,4]$, that of $K[2,3] \div K[3,3]=1$ as $K[3,4]$, so on till convergence. Thus, the cells $K[4,1]$ and $K[4,n]$ will always be empty (NA in the R language).

Then we fill in $K[n,2]=1$ and $K[n-1,2]=0$. There after we compute the remaining elements of this column working upwards from $K[n-2,2]$ with the formula:

$K[j,2]=K[j+1,2]\cdot K[j+1,4]+K[j+2,2]$

We then fill in $K[n,1]=0$ and $K[n-1,1]=1$ and complete the column starting $K[n-2,1]$ upwards with the formula:

$K[j,1]=K[j+1,1]\cdot K[j+1,4]+K[j+2,1]$

With that we have our kuṭṭaka matrix $K$ and all the needfull stuff to solve the said indeterminate equation:
1) The greatest common divisor $\textrm{GCD}(a,b)=K[n-1,3]$. This is also the smallest positive $c$ for which our indeterminate equation has integer solutions.
2) The least common multiple $\textrm{LCM}(a,b)=K[1,1]\cdot K[2,1]\cdot K[n-1,3]$
3) The integer lattice points through which the line passes are obtained from $(K[2,2],K[1,2])$ by assigning the appropriate signs. Thus, for the above equation we have the solution $95\times(-1)-25\times (-4) = 5$. Thus, $95x-25y=5$ will pass through the integer lattice at the point $(-1,-4)$
4) If we enforce the need for positive solutions then we can use $(K[2,1]-K[2,2], K[1,1]-K[1,2])$ to obtain the minimal integer solution: $95 \times 4 -25 \times 15 =5$. Thus, $95x-25y=5$ will pass through the integer lattice at the point $(4,15)$ in the first quadrant.
5) We can write the following relationship, which helps us to more generally get the lattice points through which the line $ax-by=c$ passes even if the values of $a$ and $b$ are interchanged or for $c$ other than the minimal $c$:

$b(K[1,1]\cdot p+K[1,2] \cdot q)-a(K[2,1]\cdot p +K[2,2]\cdot q)=K[n-1,3]\cdot q$

Thus, if we set $p=-1, q=1$, we get $(-4,-15)$ as further lattice points through which the line $95x-25y=5$ passes.

Now we can tackle the original problem: Since it says that 7 divides the number $r$ perfectly it can be written as $r=7y$ where $y$ will the y-coordinate of the lattice point. The numbers 2, 3, 4, 5, 6 leave a remainder of 1. Of them 4 is divisible by 2, and 6 by both 2 and 3. So all we need to consider are the numbers 4, 5, 6. Using the above kuṭṭaka or any other means we can show that $\textrm{LCM}(4,5)=20$ and $\textrm{LCM}(20,6)=60$. Thus, we can write $7y=60x+1$, where $x$ will the $x$ coordinate of the integer lattice. Using kuṭṭaka on the equation $7y-60x=1$ we get the matrix:

$K=\begin{bmatrix} 60 & 17 & 60 & NA\\ 7 & 2 & 7 & 8\\ 4 & 1 & 4 & 1\\ 3 & 1 & 3 & 1\\ 1 & 0 & 1 & 3\\ 0 & 1 & 0 & NA\\ \end{bmatrix}$

From this we can compose $(60-17) \times 7 - (7-2) \times 60 = 1$. Thus, our number is $r= 43 \times 7 = 301$, which is divisible by 7 but leaves a remainder of 1 for all integers from 2:6. More generally, if we say that $r \mod 2:6 \equiv 1$ then we can use $K$ to compose the negative solutions $r=-17 \times 7=-119$ or $r=-77 \times 7 =-539$. Such triplets of solutions correspond to symmetric lattice points along the line.

In the final part of this note we shall consider the following operation: Take a number $n$ and perform the kuṭṭaka operation with it (i.e. $a=n$) and all integers lesser than or equal to it (i.e. $b=1:n$). Then we count the number of iterations it takes with each of these integers to reach convergence. From above it is clear that the minimum number of iterations for convergence will always be 3. We term the result the kuṭṭaka spectrum of a number and plot this spectrum for the numbers 120, 123, 127 and 128.

Figure 1

The kuṭṭaka spectrum displays several notable features:
1) It is pseudo-symmetric about the mid-point, i.e. either side of $a/2$ is an approximate mirror image of the other side but they differ in “height” by one iteration.

2) The number of times the kuṭṭaka spectrum hits a minimum (i.e. converges in 3 iterations) is equal to the number of divisors of $a$, $D(n)$. Thus, for a highly composite number, as defined by Ramanujan, we get the record number of minima in the kuṭṭaka spectrum for any number less than it. Thus, in our example the highly composite number $a=120$ has 16 minima with the first six integers 1:6 giving a run of 6 successive minima. A minimally composite number like $a=123=3 \times 41$ in our figure we get 4 minima, namely 1 and the number itself and its two prime factors. The prime number in our figure, $a=127$, as expected has only 2 minima.

3) The more composite a number the lower its mean value of iterations (red line in Figure 1) than other integers in its immediate neighborhood. Thus, the highly composite number 120 has the lowest mean value in our set. In contrast, the primes have higher mean values than the integers in their immediate neighborhood. This is can be seen with $a=127$ in Figure 1.

4) A curious feature of the spectrum are the maxima, i.e. the value of $b$ for which the maximum number of iterations are required for pulverizing $a$ to convergence. For example the spectrum of $a=128$ shows 2 maxima: $b=79$ pulverizes it via the pathway: 128, 79, 49, 30, 19, 11, 8, 3, 2, 1, 0. The other one $b=81$ via 128, 81, 47, 34, 13, 8, 5, 3, 2, 1, 0. One immediately notices that the convergence in each of these two cases enters the Golden ratio convergent sequence. This feature can be investigated further by examining the distribution of the values of $a/b$ for those $b$ which result in maxima in the kuṭṭaka spectrum for a given $a$. In order to have have clear discrimination of these fractional values of $a/b$ corresponding to the spectral maxima we chose a set of relatively large $a$, namely all integers from 500 to 1000. We then determined the kuṭṭaka spectrum for each of those numbers and extracted the maxima for each $a$ and plotted a distribution of the $a/b$ values (Figure 2).

Figure 2.

First, the maxima always occur in second half of the spectrum (Figure 2), i.e. $b>\tfrac{a}{2}$. This makes sense because smaller $b$ would reach close to $a$ in the first division itself and could pulverize it to a relative small number. However, $b>\tfrac{a}{2}$ would fit only once in $a$ and could leave a relatively large remainder that could need more steps for pulverizing. Second, strikingly, the dominant peak in this distribution is the Golden ratio $\phi$ (Figure 2), suggesting the maxima tend to occur where $\tfrac{a}{b} \approx \phi$. Indeed in our above example $\tfrac{128}{79}=1.620253$. This can be intuitively understood as the $b$ which generates a maximum may be seen as a Golden cut of $a$: if $b>\tfrac{a}{2}$ is too big then the remainder generated will be small and might be pulverized quickly. If $b>\tfrac{a}{2}$ is too small then it will leave a big remainder relative to $b$ which might be quickly pulverized in the next step. Thus, the $\phi$ could give you the cut that is just right. The next dominant peak is at $3-\phi \approx 1.381966$. This is similar to $\phi$ in its operation. These two are marked by a red dashed line in Figure 2.

There are further peaks in the distribution corresponding to other fractions on either side of $\phi$ following a certain pattern of declining heights. Further they show the same symmetry as $\phi$ and $3-\phi$, with each peak $m$ having a counterpart $3-m$. We have thus far not been able to determine a more
general expression describing all these peaks or prove why they should be peaks but we were able to account for a subset of them as corresponding to other quadratic irrational numbers (marked by grey dashed lines in Figure 2). These include:

$\sqrt{35/11} \approx 1.783765, 3-\sqrt{35/11} \approx 1.216235$
$\sqrt{3} \approx 1.732051, 3-\sqrt{3} \approx 1.267949$
$\sqrt{24/9} \approx 1.632993, 3-\sqrt{24/9} \approx 1.367007$
$\sqrt{2} \approx 1.414214, 3-\sqrt{2} \approx 1.585786$

Of these the pair $\sqrt{2}, 3-\sqrt{2}$, especially the former is not the best fit to the peak but given the breadth of that peak it is possible that more than one attractor fraction is merged in that peak. It would be a good mathematical quest to discover the general expression for the peaks, their dominance and prove why they tend to be peaks. There might be a subtle fractal structure to them that might become apparent at large values of $a$.

Posted in Heathen thought, Scientific ramblings |

## A brief note on some new developments regarding the genomics of Indians

When we wrote a previous article on this matter we had stated that new data will alter the details of our understanding of picture discussed therein. Indeed, two new manuscripts which were deposited in the past month by McColl et al and Narasimhan et al have done so. These are still deposited manuscripts and have not been formally published. Further other data might also come in the near future. Hence, we are not launching into any detailed presentation of the revised scenarios in this note. What we intend here is to simply provide a few illustrations of the authors’ results without much critical investigation.

A screen shot of McColl et al Figure 4

First, the study of McColl et al focuses on the far east bringing in new ancient DNA data. The main point of interest to the Indian scenario is that the Andaman Onge are part of a major push of hunter-gatherers into the far east and Pacific, which spawned several branches that in turn mixed among themselves in various combinations giving rise among others to the Austronesian groups and East Asians. Further, in deep Pacific there were admixtures of the basal branches of this radiation with the Denisovans, the signal of which is very clearly seen in Papuans and Australian aborigines. The basal-most branch of this group analyzed by McColl et al is the 40000 YBP Tianyuan man, suggesting that these populations were in the east by then. A basal branch of this radiation group also seems to have contributed to the ancestry of only a subset of native Americans (independently of the East Asian branch that also originated from this group). This suggests that they might have reached the New World independently in an earlier wave or mixed with one strand of the main East Asian line of Native American ancestry as they entered the New World. A deep sister group of the Onge and probably a basal member of this Eastern radiation was an ancient hunter-gatherer group that settled India, where they might have undergone admixtures with one or more preexisting non-sapiens species of Homo. This population is now defined as Ancient Ancestral South Indian (AASI) by Narasimhan et al, refining the earlier definition of “Ancestral South Indian” by Reich et al. We may term them the Indian hunter-gatherers.

The key point which Narasimhan et al make is that Neolithicization of the North-Western Indian Subcontinent proceeded via the entry of Iranian farmers from the west. Thus, this clarifies a previously uncertain situation based on archaeology alone. The entry of these Iranian farmers could have happened as early as the Mehrgarh Neolithic or in more than one wave of closely related western populations. In any case the authors posit that it had happened by 6700-5000 YBP. This Iranian farmer group mixed with the AASI in the NW of the Indian subcontinent and this admixture was likely the form of the population of the Harappan civilization that arose in this region. They term this population Indus periphery. Narasimhan et al also show that the Bactria-Margiana complex (BMAC) received some admixture from this population, likely of Harappan provenance, but did not contribute notably to the ancestry of the Indian subcontinent. Starting around 4100 YPB they start seeing Sintashta Steppe contributions appear for the first time in BMAC. This ancestry appears to have filtered south and reached the core Indian subcontinent thereafter. By 3700-3500 YBP they start seeing East Asian admixture on the Central Asian steppes, which continues down to the Scythian Iron Age. However, this East Asian ancestry is not visible in Indian populations. Hence, it appears that we are left with a window of 4100-3500 YBP when the Aryan invasion of the subcontinent happened. This is at the upper end of the mainstream invasion scenarios. Further, it is not inconsistent with the possibility that the invasion triggered the collapse of the Harappan urbanization around 3900 YBP. But it is also possible that the Aryans entered and occupied a landscape where the Harappan urban civilization had already collapsed or was in its last throes. It also provides support for the young age of the Veda, especially if one chose to place the Ṛgveda in the Panjab. Further, it lends some support to the scenario that the Soma cult was acquired by the Indo-Iranians and integrated with the older fire-cult as they reached the BMAC sites. It is notable in this context that one of the main proponents of the Soma cult in the Vaidika system, the Kaśyapa clan, was the default gotra for a brāhmaṇa who did not know his. There are issues with each of these points and interesting complications but we desist from discussing any of these now.

Interestingly, there was another recent publication by Vishnupriya et al applying the Bayesian phylogenetic methods to Dravidian languages. The results suggested a possible expansion of Dravidian happening around 4500 YBP. Narasimhan et al seem to mildly favor a Harappan origin of Dravidian. However, both the linguistic date estimate and several other linguistic arguments are against the Harappan civilization being that of Dravidian speakers. Rather, we suspect the Dravidians arose in the South as part of the Southern Neolithicization – this might have had genetic and memetic contributions from the Indus periphery but the Dravidian languages themselves were likely of Southern provenance, probably in the upper Godavari valley. In the aftermath of the Indo-Aryan reconfiguration of the north, it is likely that the Dravidians had their own expansions both South and North adopting various Indo-Aryan technologies and ideologies. This led to the Dravidianization of many AASI hunter-gathers, who might have earlier spoken other languages.

Narasimhan et al model extant Indian populations as a three-way mixture of the Indus-periphery, the Indian Hunter-gather (AASI) and the Steppe population related to the Sintashta complex. Below are some figures based on their model to illustrate the situation.

Figure 1. A box plot showing the three modeled components of Indian Ancestry for the 140 populations studied by the authors. The gray line indicates the position of the genuine brāhmaṇa population with the lowest steppe ancestry (i.e. leaving out some groups which are not conventional brāhmaṇa, e.g. viśvakarman). It is clear that the brāhmaṇa-s show above average steppe ancestry and below average Indian hunter-gatherer ancestry.

Figure 2. The same data is represented as a histogram. It is clear that whereas the Indus-periphery and Indian Hunter-gatherer ancestry is unimodal, the steppe ancestry is not with groups showing low and high steppe ancestry. This explains the authors’ earlier model of ASI and ANI.

We then sorted the populations into five categories: 1) braḥmaṇa-s (here we retained the viśvakarman); 2) Warrior caste (traditional kṣatriya-s) and their equivalents; 3) Middle castes: vaiśya-s, cattle-breeders and agriculturalists; 4) service castes: traditional service jati-s often included as other backward, backward and scheduled castes; 5) tribes. For this we had drop generic groups like Gujarati, Punjabi, Muslim and the like. This left us with 124 populations. These are plotted as a ternary diagram.

Figure 3. Ternary diagram of the 3 strands of Indian ancestry. The 5 caste-tribal groups defined above are colored: 1-red, 2-orange, 3-aquamarine; 4-blue; 5-violet. One can see the effect of the two admixtures with the steppe ancestry’s effect being predominant in the varṇa populations.

A closer examination of this is seen the next three figures:

Figure 4. A box plot of the inferred steppe ancestry in the above-defined five groups. The steppe ancestry is arrayed in accordance with the caste ladder and tribals have the least of it on an average.

Figure 5. A box plot of the inferred Indus-periphery ancestry in the above-defined five groups. It is interesting to note that unlike the steppe ancestry’s the Indus-periphery ancestry is greater in the warrior and middle caste groups than in braḥmaṇa-s, who have a lower median value of this component. However, this difference is only mildly significant in the current data (p=.033) and sampling bias cannot be ruled out.

Figure 6. A box plot of the inferred Indian hunter-gatherer ancestry in the above-defined five groups. Here for the four groups from the warrior castes to the tribes we see a reverse of the scenario seen for the steppe ancestry. However, the braḥmaṇa-s show a slightly higher median value of this component. While again we should be clear that this could be due to sampling bias, taken together with the above plot, it might reflect some sociological reality. The braḥmaṇa-s probably to start with did not mix much with the preexisting populations of the subcontinent but as they expanded, especially while moving south, they mixed with directly with populations with lower Indus-periphery and higher hunter-gatherer components.

Together, these plots suggest a picture, which was long suspected from the physical appearance of Indians. The Indo-Aryans established themselves in the subcontinent entering via the NW, where they mixed with the older Indus-periphery populations that were likely part of or survivors of the old Harappan civilization. The groups with a wide-range of older Indian hunter-gatherer-Iranian farmer mixture were incorporated across the upper caste ladder but especially in warrior and middle castes where we see considerable dispersion (e.g. southern agnikula-kṣatriya with low steppe ancestry). The movement of braḥmaṇa-s into the south possibly also involved admixture with these groups.

Finally, a brief political note. The pro-Hindu pakṣa had acquired an aberration mainly in the past 3 decades known as OIT or the out of India theory for the origin of the Indo-Europeans. This never had a leg to stand on but is now dead and cremated. Unfortunately, the pro-Hindu side and mainstream H nationalism has invested so much in making Indo-Aryan autochthonism a centerpiece of their thought that it mostly ceded the writing of data-based Hindu prehistory to parties who are never going to be favorable to them. Even more tragically they do not even seem to recognize how wrong they were – there is a finite probability that most of the OIT proponents are going to continue that way. Further, there is an unsubstantiated rumor making rounds that the Indian side might have prevented the use of Indian aDNA in the current analysis fearing the inevitable end to OIT. If this were true then it would add to the scandal and only provide more fuel for the usual enemies of the Hindus. This intellectual failure of mainstream Hindu nationalism in framing its foundations is quite worrisome as it might reflect a deeper systematic failure in thought.

References:
The Genomic Formation of South and Central Asia, Narasimhan et al. https://www.biorxiv.org/content/early/2018/03/31/292581

Ancient Genomics Reveals Four Prehistoric Migration Waves into Southeast Asia, McColl et al. https://www.biorxiv.org/content/early/2018/03/08/278374

A Bayesian phylogenetic study of the Dravidian language family http://rsos.royalsocietypublishing.org/content/5/3/171504

Posted in History, Politics, Scientific ramblings |

## The remarkable behavior of a map displaying derived from a simple model for a biological conflict

One of the simplest yet profound mathematical models for biological growth emerged sometime in the middle of the 1800s due to the work of Verhulst. It describes population growth thus: let $x$ be the population of the organism at time $t$. $r$ is the Malthusian parameter or the rate of maximum population growth. $K$ is the carrying capacity or the maximum sustainable population in a given environment. Then we have the following ordinary differential equation (ODE) as a descriptor of population growth:

$\dfrac{dx}{dt}=\dfrac{rx(K-x)}{K}$

This is the logistic model that profoundly informs us about various aspects of biology. By define a new $x \equiv \tfrac{x}{K}$ we get:

$\dfrac{dx}{dt}=rx(1-x)$

This simplified form can be next converted into a discrete equation thus:

$x_{n+1}=rx_n(1-x_n)$

This is the famous logistic map. Simple as it looks, starting with von Neumann’s work and then that of Ulam and Feigenbaum thereafter, it became clear that it exhibits notoriously complex behavior that have considerable implications for dynamics of systems. We had earlier described an exploration of some of this via the vehicle of Kaneko’s coupled-map lattices. Since our teenage years we have been fascinated by the complexity in simplicity of this map and wondered if we might uncover other such cases in simple models of biological systems. Here we describe an example of such, which we discovered.

Consider a two species-system where one species $x$ is a parasite or predator, whose growth rate depends on the number of the host/prey species that it utilizes. The host/prey species $y$ is communal in that it produces certain public goods, which all members of its community share. It incurs a certain cost for producing these. More its population less is this cost because it gets spreadout over more individuals. It grows like any other free-living organism otherwise with growth rate proportional to the number of individuals at a given instant. Its growth is further negatively affected by the predator/parasite which kills all infected individuals. In principle, this system might be described by the below ODEs:

$\dfrac{dx}{dt}=k_1 y$

$\dfrac{dy}{dt}=\dfrac{k_a}{y}+k_b y-k_2 x$

By simplifying this by taking $k_1=k_2=1$ and expressing it as a discrete equation we get:

$x_{n+1}=y_n$

$y_{n+1}=\dfrac{k_a}{y_n}+k_b y_n-x_n$

Since $k_a$ is a cost of public goods production by the communal organism it will always be negative, while $k_b$ being a regular growth constant will be positive. This is the form of the map, which we shall explore further for its dynamics. A few things are readily apparent if $y_n=0$ then the first term of second equation of the map goes to $-\infty$ resulting in a singularity. Second both $|y_n|$ and $|x_n|$ can wander away towards $\infty$; hence in out experiments we terminate our runs if certain arbitrarily defined limit is reached.

Figure 1

Figure 1 shows a run of this system for 100 generations with $k_a=-0.288, k_b=0.51$ and initiated with $x_0=-0.0001, y_0=0.04997$. The expected pattern of $x_n$ tracking $y_n$ is seen in the oscillatory pattern typical of predator/parasite-prey/host system. We can better understand this by plotting a phase diagram of $x_n,y_n$ as can be seen in Figure 2.

Figure 2

In Figure 2 we have 9 different runs of the map with different starting $x_0, y_0$ and $ka=-0.188, kb=0.45$ evolving up to 10000 iterations or terminated if it hits a singularity (in practice we set some absolute value cut off of say 100 above which we stop further iterations). The maps are symmetric about the $x=y$ line, which arises from the form of the first equation in the map. The map is always bounded by an ellipse of the form $x^2+y^2-kxy=1$, where $0 \le |k| < 2$. Hence, we call it the 2-parameter elliptical map. Figure 2 reveals that the map is extremely sensitive to changes in the initial conditions: changes in the 3rd to 5th place after the decimal point can result in dramatic differences in the evolution of the $x_0,y_0$.

Figure 3

Figure 3 shows 9 runs of the evolution of $x_0=-0.0001, y_0=0.04997$ for different $k_a, k_b$. We gain see that the map is very sensitive to changes in both the parameters and the evolution of the initial point dramatically varies with these changes.

Figure 4

In Figure 4 we run the following experiment: We chose a square region defined by the diagonal (0.2, 0.2):(-0.2, -0.2). Within this square we chose 3 different sets of 100 random points and allow each set of these points to evolve for a given $k_a, k_b$. Thus, each row in Figure 4 is the evolution of the three different sets of random points for a given $k_a, k_b$. The evolution of each of the 100 random points for a given run is plotted in a different color. We observe that each run shows a different color pattern because the points are random and evolve very differently, as we know from the single point runs shown in Figures 2 and 3. However, remarkably, the overall structure of the map for a given $k_a, k_b$ converges to a constant form. This suggests that even though the map is very sensitive to the initial conditions, the average evolution of a given region, i.e. a random set of points drawn from the region, is conservative for a given parameter set.

Figure 5

In the next experiment (Figure 5) we try to understand the effect of the $k_a, k_b$ on the average evolution of a given “region”. However, we use a slightly variant definition of the region in this case: we chose 64 points on a circle of radius $r=0.25$ each separated from the next by an angle of $\pi\big/32$ radians from 0 to $2\pi$. We then let these points evolve for 5000 iterations or until they explode to a singularity. The evolution of each point is plotted in a different color as above. For each row $k_a$ is constant while $k_b$ changes. For each column $k_b$ is constant while $k_a$ changes. The effect of increasing $k_a$ is like that of a magnifier. Thus, the increasing $k_a$ acts like zooming in on the structure of the interior of the map. The obvious effect of changing $k_b$ is immediately apparent: with increasing $k_b$ the eccentricity of the bounding ellipse of the map increases. As the bounding ellipse is of the form $x^2+y^2-kxy=1$, essentially $k=k_b$. $k_b$ also defines the internal structure of the map, such as its basic geometry as well as the zones of complete occupancy and the zones of exclusion. One notices these changing with changes in $k_b$.

Figure 6A. $k_b=$ 0.61, $\tfrac{1}{\phi}$, .62, .99, 1, 1.01, 1.24, $2\cos\left(\tfrac{2\pi}{7}\right)$, 1.2475

This effect of the role of $k_b$ on the internal structure is explored at greater depth in the several parts of Figure 6, which were essentially rendered by running the map as in Figure 5 with a constant $k_a=0.618$ and various $k_b$. Remarkably, at $k_b=\tfrac{1}{\phi}$, where $\phi$ is the Golden ratio, the map assumes a 10-rayed form with a tendency to diverge rapidly only along those rays (panel 2 of Figure 6A). This value is the ratio of the side of a regular pentagon to its diagonal. This $k_b$ is the central value of the “pentad” state, i.e. $k_b$ close to it on either side will show the characteristic pentad geometry as can be seen in panels 1 and 3 of Figure 6A. Similarly, we observe that $k_b=1$ is the central value of the hexad state (panel 5 of Figure 6A). At values close to it on either side the map assumes a hexad form as can be seen in panels 4 and 5 of Figure 6A. Another central value is $k_b=2\cos\left(\tfrac{2\pi}{7}\right) \approx 1.2469...$. This number is derived from heptagonal equivalents of the Golden ratio: it is the ratio of the length of the larger diagonal to the smaller diagonal of a regular heptagon. Here again we see a radiating structure with 14 rays (panel 8 of Figure 6A). As $k_b$ approaches this value we see the interior of the map assuming a heptad structure (panel 7 and 9, Figure 6A). These cases illustrate that for values close to the central value but less than it the map displays a $n$-ad polygonal interior structure. For the values close to but greater than the central value the $n$-ad interior structure assumes a rosette form with zones of exclusion defined by the rosette.

Figure 6B. where $k_b=$ 1.4, $\sqrt{2}$, 1.53, $2\cos\left(\tfrac{2\pi}{9}\right)$, 1.61, $\phi$, $2\cos\left(\tfrac{2\pi}{11}\right)$, 1.73, $\sqrt{3}$

Continuing this way, we can find further central values associated with higher $n$-ads. At $k_b=\sqrt{2}$, the map displays 8 rays (panel 2, Figure 6B). This corresponds to the ratio of the largest to the shortest diagonal of a regular octagon. As we approach this value we get a map with an octad structure (panel 1, Figure 6B). This is followed by the central point defined by $k_b=2\cos\left(\tfrac{2\pi}{9}\right)$, which displays a map with 18 rays (panel 4, Figure 6B). This $k_b$ corresponds to the ratio of the largest to the shortest diagonal of a regular nonagon. Thus, as we approach this value the map takes on a nonad structure (panel 3, Figure 6B). The next central value is $k_b=\phi$ (panel 6, Figure 6B), which again results in a 10-rayed map and values approaching it have a decad structure (panel 5, Figure 6B). The hendecad (11-fold) central value is $k_b=2\cos\left(\tfrac{2\pi}{11}\right)$, which results in a 22-rayed map (panel 7, Figure 6B). Similarly, $k_b=\sqrt{3}$ is the central point for the dodecad structure (panels 8 and 9, Figure 6B). Studying these central values of $k_b$, which mark the $n$-ad geometry of the map, we can derive a general relationship between these $k_b$ to the ratio of the diagonals of the corresponding regular $n$-sided polygon thus: Given a vertex of the regular polygon with $n$ sides, let $d_1$ be the
first side of the polygon emanating from that vertex. Then $d_2, d_3, d_4,...d_{n-2}$ will be the successive diagonals and $d_{n-1}$ will the second side emanating from that vertex. Then,

$k_b=\dfrac{d_4}{d_2}$

Using some trigonometry, we can show that for a $n$-sided polygon with unit sides the length of the $j$th diagonal (including sides) connected to a given vertex is:

$d_j=\dfrac{\sin\left(\dfrac{j\pi}{n}\right)}{\sin\left(\dfrac{\pi}{n}\right)}$

$\therefore k_b= \dfrac{\sin\left(\dfrac{4\pi}{n}\right)}{\sin\left(\dfrac{\pi}{n}\right)} \Bigg / \dfrac{\sin\left(\dfrac{2\pi}{n}\right)}{\sin\left(\dfrac{\pi}{n}\right)}=\dfrac{\sin\left(\dfrac{4\pi}{n}\right)}{\sin\left(\dfrac{2\pi}{n}\right)}=\dfrac{2\sin\left(\dfrac{2\pi}{n}\right)\cos\left(\dfrac{2\pi}{n}\right)}{\sin\left(\dfrac{2\pi}{n}\right)} = 2\cos\left(\dfrac{2\pi}{n}\right)$

This defines the $k_b$ values at which the 2-parameter elliptical map will take a divergent form with $n$-rays (if $n$ is even) or $2n$-rays (if $n$ is odd). At values approaching these $k_b$ it will show a pronounced $n$-ad internal structure. This explains why the pentad is centered at $\tfrac{d_4}{d_2}=\tfrac{1}{\phi}$ where $d_2=\phi$ and $d_4=1$ for a unit pentagon. The tetrad has no $d_4$; hence its central value would correspond to $k_b=0$: that’s why we see the tetrad structure at $k_b$ closer to zero (Figure 5). For $n=12$ we already reach $k_b=\sqrt{3} \approx 1.73205$. As $n$ increases we have:

$\displaystyle \lim_{n\to\infty} k_b= 2\cos\left(\dfrac{2\pi}{n}\right)=2$

Thus, the higher $n$-ad structures are closely clustered near 2. The derivation of the above formula for $k_b$ enabled us to investigate other central points, where instead of integer $n$ we have a rational number $\tfrac{p}{q}$; $p,q$ being mutually prime. Thus we consider $k_b$ of the form:

$k_b=2\cos\left(\dfrac{2\pi}{p/q}\right)$

Figure 7A. $k_b=$ 0.5165, $2\cos\left(\tfrac{2\pi}{24/5}\right)$, 0.5195, 0.8672, $2\cos\left(\tfrac{2\pi}{28/5}\right)$, 0.8683, 0.39, $2\cos\left(\tfrac{2\pi}{32/7}\right)$, 0.3911

The transition points associated with such rational values are explored in the various parts of Figure 7. In Figure 7A, we show the maps associated with $\tfrac{p}{q}= \tfrac{24}{5}, \tfrac{28}{5}, \tfrac{32}{7}$ (panels 2, 5, 8). Each of these shows the $n$-rayed structure typical of the the central points. Further, the number of rays corresponds to the value of $p$. Again as in the case of the $k_b$ corresponding to integer $n$ as we approach the $k_b$ corresponding to $\tfrac{p}{q}$ we get $p$-ad structures with a similar pattern as with the former. Here we have chosen 24, 28 and 32 (the syllable counts of the gāyatrī, uṣnik and anuṣṭubh meters).

Figure 7B. $k_b=$ $2\cos\left(\tfrac{2\pi}{7/2}\right)$, -0.44, $2\cos\left(\tfrac{2\pi}{9/2}\right)$, 0.342, $2\cos\left(\tfrac{2\pi}{13/3}\right)$, 0.24, -0.28, $2\cos\left(\tfrac{2\pi}{11/3}\right)$, -0.2855

In figure 7B we explore further cases of $k_b$ corresponding to different $\tfrac{p}{q}$. Notably, using these $\tfrac{p}{q}$ we get lower values of $k_b$ at which we can get central points of hept-, hendec- and 13- ad structures. Interestingly, at some of these values $k_b$ is negative (Figure 7B, panels 1,2 and 7,8,9). When $k_b$ is negative, we see that the map becomes a mirror image of the positive $k_b$, now being symmetric along the $y=-x$ line and maintaining an elliptical form between $(-2,0)$. A negative $k_b$ is not quite meaningful in the original biological model in which the map was defined because it is a growth constant and positive. However, we could still contrive a definition wherein the negative $k_b$ specifies a decaying population, which has a positive effect by reducing the parasite load, which is directly dependent on it, and thereby allowing relative growth.

Figure 8. $k_b=$ $2\cos\left(\tfrac{2\pi}{64/15}\right)$, $2\cos\left(\tfrac{2\pi}{128/31}\right)$, $2\cos\left(\tfrac{2\pi}{96/23}\right)$, $2\cos\left(\tfrac{2\pi}{100/23}\right)$

In Figure 8 we explore the more complex effects of the fraction $\tfrac{p}{q}$ on the structure of the map. The outer structure of each of these maps respectively shows 64, 128, 96 and 100 rays as would be expected from value of $p$. However, in the case of $\tfrac{64}{15}$, a closer look at the interior of the map reveals first a 17-ad and then closest to the center a tetrad structure (Figure 8, panel 1). Similarly, with $\tfrac{128}{31}$ we see an inner 33-ad and innermost tetrad structure (Figure 8, panel 2). For $\tfrac{96}{23}$ we likewise see a inward progression first to a 25-ad and then a tetrad structure (Figure 8, panel 3). For $\tfrac{100}{23}$ we see an inward progression to a 13-ad then tetrad structure. In each of these cases the innermost state is a tetrad. The corresponding value of $k_b$ is: 0.1960, 0.0981 0.1308, 0.2506. In each case is much smaller than $\tfrac{1}{\phi}$ and closer to 0, which is the center of the tetrad state. Hence, it in each of these cases at the innermost level the effect of tetrad central value i.e. 0 dominates. Beyond that we see that $k_b=0.1960$ is close to $2\cos\left(\tfrac{2\pi}{17/13}\right)$, thus resulting in the emergence of 17-ad structure in next level. Similarly for $k_b=0.0981$, we see that it is close to $2\cos\left(\tfrac{2\pi}{33/25}\right)$ which results in the 33-ad structure at the second level. It is notable that both $\tfrac{p}{q}=\tfrac{64}{15}, \tfrac{128}{31}$ have a comparable second layer $n$-ad of the form $\tfrac{p}{4}+1$. Now in the case of $k_b=0.1308$ we have $2\cos\left(\tfrac{2\pi}{25/19}\right)$ close to it resulting a 25-ad second layer. Finally, with $k_b=0.2506$ we have $2\cos\left(\tfrac{2\pi}{13/3}\right)$ near it resulting in a 13-ad second layer. Thus, the value of $p$ in the fraction $\tfrac{p}{q}$ in the formula $k_b=2\cos\left(\tfrac{2\pi}{p/q}\right)$ determines the general $n$-ad structure of the map and actual value of $k_b$ determined by the fraction $\tfrac{p}{q}$ determines its fine structure. In general terms, the resultant map at a certain $\tfrac{p}{q}$ may be viewed as a superposition of the corresponding $p$-ad structure in the outer layers with all the $p$-s of the nearby fractions competing to contribute to the inner layers. Thus, away from the strong central values, such as those from integer $n$ we get more complex maps.

The lower $k_b$ values at which we get higher $n$-ad structures result in lower eccentricities of the bounding ellipse of the map. This allows us to visualize them better than at $k_b$ values derived from integer $n$ because the latter result in bounding ellipses with high eccentricity. The absolute value of $k_b$ determines the eccentricity $e_e$ of the bounding ellipse of the map: As $|k_b| \to 2, \; e_e \to 1$ (the eccentricity of an ellipse $0; at eccentricity 0 we get a circle and at eccentricity of 1 a parabola). We can derive the below formula giving the relationship between the the parameter $|k_b|$ and $e_e$. For the bounding ellipse with $k_b$ corresponding to the $\tfrac{p}{q}$ discussed above we can write the eccentricity thus:

$e_e=\sqrt{1-\tan^2\left(\dfrac{\pi}{p/q}\right)}$

By expressing $\tan^2\left(\tfrac{\pi}{p/q}\right)$ in terms of $k_b$ using the above formula we derived for it and doing some trigonometric manipulations we get:

$e_e=\sqrt{\dfrac{2|k_b|}{|k_b|+2}}$

Thus, one can see how only for $|k_b|<2$, $e_e<1$ and the map is elliptical.

In a final experiment we performed a Julia-like operation: $k_a$ was kept constant and $k_b$ was changed. The evolution was studied for all points separated from each other by 0.001 in the square defined by the diagonal (-.2,-.2):(.2,.2). The points were allowed to evolve to a maximum of 5000 iterations if they did not crash to a singularity or reach an absolute value greater than (100,100). Each point in this square is colored by the number of iterations it survived on a color scale from black to gray via the colors dark blue, dark red, burlywood, coral, chartreuse, and cadet blue. Thus, the points that crash out after the first iteration are black and those that last all 5000 iterations are gray. Figure 9 shows the results of this run for selected $k_b$

Figure 9. $k_a=-0.24$, $k_b=$ .25, $2\cos\left(\dfrac{2\pi}{100/23}\right)$, .26, .6, $2\cos\left(\dfrac{2\pi}{5}\right)$, 0.62, 0.99, $2\cos\left(\dfrac{2\pi}{5}\right)$, .62, .99, $2\cos\left(\dfrac{2\pi}{6}\right)$, 1.05.

From this figure it becomes clear that beyond the basic symmetry axis along the $y=x$ line the plots are rather “entropic”, i.e. there is not much sign of order in arrangement of the points in terms of the number of iterations they survive. However, one tendency becomes clear, consistent with the actual structure of the maps plotted above: If $k_b$ takes the form $2\cos\left(\tfrac{2\pi}{n}\right)$ with integer $n$, like $n=5,6$ (Figure 9, panel 5, 8) then the majority of points do not survive the entire 5000 iterations — we see little gray. Thus, when $k_b$ corresponds to $n$ or fractions of the form $\tfrac{p}{q}$ with relatively small $p$ then most points are not stable in their evolution and crash to a singularity or diverge rapidly. On the other hand, if $k_b$ is defined by a $\tfrac{p}{q}$ with relatively large $p$ (Figure 9, panel 2) or is not defined by a rational number in the formula, then we get many more points evolving stably over a large number of iterations (the remaining panels of Figure 9) — here we see much more gray. The fraction of points surviving at least 5000 iterations is approximately half the total number of points (fraction ranges from 0.509 for Figure 8, panel 7 to 0.599 for panel 2). The remaining points bail out at fewer iterations in decaying fractions when grouped by bins of 100 from 100 to 4900 iterations (Figure 10). However,
the mostly disordered distribution of these points, relates to the mostly convergent structure of the map for a given region that was illustrated above.

Figure 10. Histogram of bailout or survival till 5000 iterations with same parameters as in Figure 9.

In conclusion, while this simple map shows astounding visual beauty, interesting mathematical structure, and deep links to polygons and coprimality of numbers, one may ask if it provides any biological insights at all. At first sight it does seem like a rather unrealistic model for a biological system given the negative values of population that it generates. But perhaps even this can be remedied by the suggestion that these negative values are not absolute population sizes but only represent the part of a larger population which actually participates in the dynamics. But the key lessons the map provides for biological systems are of a more qualitative type: 1) On one hand it shows how systems even with just two species locked in biological conflict can show chaotic dynamics, where the long term out come is highly dependent on the initial conditions. Even small changes can push the system, for the same parameter set, from long-term stability to instability (defined as singularity or blowout divergence). 2) On the other hand, despite the above, it shows that, as an aggregate, a set of different initial conditions strongly tend to converge to dynamics of a constant form for a set of parameters. Thus, even if individual starting populations have very different fates, a population of populations tends to have predictable dynamics when taken as an aggregate. Such analogies might help understand the stable conflict despite ongoing arms races that have lasted for few billion years (e.g. between bacteria and their phages). 3) The high instability of certain points such the $k_b$ corresponding to integer $n$ and certain rational fractions suggests that the dynamics of the system might have some deep numerical constraints and natural selection might essentially have to play the role of selecting between parameters that conform to underlying numerical constraints that favor long terms stability.

Posted in Scientific ramblings |

## A day at school

It was the English class in school. The teacher, a swarthy man with somewhat liberal political leanings, strode into the class: “Today we shall be studying a poem by the white American poet Robert Frost, Fire and Ice. Dandadipa, stand up and read the poem aloud.”
Dandadipa did as directed by the teacher:
Some say the world will end in fire,
Some say in ice.
From what I’ve tasted of desire
I hold with those who favor fire.
But if it had to perish twice,
I think I know enough of hate
To say that for destruction ice
Is also great
And would suffice.

The teacher then turned to Lootika and said: “Lootika, stand up and tell the class what you think about this poem you just heard.”
Lootika: “I think, if by world we mean the earth, undoubtedly, the end will come by fire. By fire I do not mean it in the real sense of a conflagration but more in the sense of intense heat.”
The teacher: “Why so young lady?”
Lootika: “The sun is an average yellow star and we know enough of astronomy today to accurately chart its course in time. It will evolve into an enormous distended star known as a red giant, about 100 or more times its current diameter. As this is happening all water on earth will evaporate and there will be lakes of molten aluminum and copper. Then heat of the sun will eventually bring the earth to an end by either vaporizing it completely or leaving behind just a little ball of iron.”
The teacher: “Lootika, that is very apocalyptic. I now see why the some of your science teachers say you are a special girl, while the rest declare you as a showoff.”
Lootika: “I am grateful to those who hold the former opinion but all this is common knowledge, sir.”

Tumul: “Sir, I think Lootika is simply showing off by talking nonsense. This poem is not about science as she tried to make it appear. Robert Frost is presenting two metaphors. In the first one he says that desires of people would lead to the end of humankind. That is what he means by the world ending in fire, where fire is the metaphor for all the conflicts arising from desires. When he says ice, he means the all the hatred that exists in the world. As we can see from events around us that could also end humankind.”

Having delivered his explanation, Tumul proudly surveyed his class, taking a glance at Lootika with the corner of his eye. The teacher almost sensing that Lootika might strike back raised his voice and said:
“Alright girls and boys. Keep quite and listen carefully. Tumul’s explanation about the metaphors is what the poet seeks to convey. However, let me tell you some history about this poem. The poet had met the astronomer Harlow Shapley. He told the Frost about the possible end of the earth stating that either the sun would explode and burn the earth or the earth would freeze in deep space. I believe what Lootika told us is something along the lines of the astronomer’s first explanation. Hence, while you are right Tumul, I don’t think you should be casting aspersion at Lootika’s literal explanation for something astronomical was the apparently inspiration for the poet’s opening lines about the end of the world in fire or ice. In this regard I would advise the class to watch the great serial presenting cosmology for the layman on Sundays by the black American scientist Floyd Mayweather.”

Somakhya was folding an origami model for their classmate Bobon of dark grayish yellow complexion from an erstwhile tribal category, who was a connoisseur of his artwork. The teacher turned to him and said: “Somakhya! This is not the craft class: pay attention!”
Somakhya stood up and said: “Sir it is just my hands; my mind and ears are all into your lesson on Fire and Ice.”
The teacher: “Then tell me what you think of this poem?”
Somakhya: “Lootika is correct insofar as the ultimate fiery end of the earth is concerned. There is also the snowball earth hypothesis, for which support is limited, that the earth froze over more than once. However, there are more distant icy scenarios, which we understand better since the days of Harlow Shapley. First, the sun after distending in to red giant, just as Lootika said, would eventually blow off its outer layers as a planetary nebula and be reduced to a hot small star called a white dwarf. Since the Sun is not a binary star, that white dwarf might in principle cool endlessly to reach a minimally thermal state called a black dwarf, which is a few degrees above absolute zero at some very, very distant time into the future. In one sense we could call this fate an icy end, like the second end the poet talks about.”

The teacher: “Left to yourself, you and Lootika would turn this into a science class. That said, as a rational person, I am happy that today we can derive our metaphors from science rather than religion. That is what people did in the past and Robert Frost might have also been influenced by the superstitious imagery of hell, which was described in western religious literature. Now, Jukuta, stand up and elaborate on the metaphorical aspects of the poem.”

Jukuta: “I think when he talks about desire he is referring to all the passions, like greed, gluttony and lust, which fuel anger and lead to a never-ceasing cycle of violence. The desire for territory, illicit drugs, taking other peoples’ wealth and the patriarchal oppression women leads to more warfare and oppression, which can only be compared to a conflagration. The lack of inter-personal amity and empathy is what he is paraphrasing as ice. We use the metaphor ‘cold’ to describe people with such qualities and it is root of hatred in this world. We see much of that too in this world and I am afraid it will be the cause for our end.”

The teacher: “That is excellent Jukuta. Do you all get the gist of this poem?”
Most of the class answered back in a chorus: “Yes”.
The teacher: “Somakhya I catch you roll your eyes as though all this is a joke. Stand up and explain clearly if you get the metaphorical point or not.”
Somakhya: “I hear what has been said but I think the poet lived in a different space and time. Methinks his diagnosis of the fire and ice were hence superficial.”
The teacher: “That is very arrogant on your part. I would rather see you back those flippant comments with more of an explanation.”
Somakhya: “What were listed by the other student as desires, namely lust, gluttony and the like, stem from basic biological imperatives. Long before Frost the buddha had also diagnosed these as ills, whose removal will bring us closer to a state of upliftment. I would say that for a living organism the elimination of biological imperatives cannot lead to a better state nor will they cause the end of the organism. We will rather see natural processes cause a convergence to a stable state. As for hatred, again, I do not see an end to biologically driven human conflict. If there were no other complicating factors, we will continue to see an arms race or some state of stabilization but not complete destruction. Rather, I would say that a pathological form of conflict engendered by the Abrahamistic religions is what might bring us to the brink of destruction.”

The teacher: “Somakhya I appreciate your articulation but I think this precisely the reason we need to study poetry. Your worldview seems quite set even at this young age and is inordinately naturalistic. We cannot be human with such a worldview. It is things like viewing matters in a more empathetic manner that will help you find your humanity. I should also admonish you for bringing in specific religions. All religions cause conflict. You are only showing prejudice by singling those which are not yours. Ours is a secular country and you should show the same respect you have for your religion to the religions of others.”

Lootika: “But sir history illustrates that all religions are not the same and some lie at the root of truly genocidal conflict.”
The teacher: “Alright, Lootika, don’t answer out of turn and keep that discussion for your history class. Students, we have had a good discussion. Now read the questions at the end of the poem and raise your hands if you do not understand them or have and doubts regarding the poem.”

Hemaling raised his hand: “Sir, I do not understand what metaphor means.”
Gomay raised his hand: “Sir, the first question: ‘what is the figure of speech in the opening lines of the poem?’ How to answer that one?”

The teacher busied himself answering these and other questions till the bell rang and the class came to a close for the recess. As the students spilled out of the room to head for lunch and take a break from the monotony of the classes, Lootika caught Somakhya’s eye and they exchanged a hand gesture. She shouted out to him: “We didn’t even get started with the aliens.” Somakhya: “May be that would have been less traumatic than my mention of ekarākṣasavāda!”

Then they moved on with their respective groups. Somakhya headed to a secluded spot for lunch along with Vidrum and Sharvamanyu. Sharvamanyu noted that Lootika and her gang similarly ensconced themselves in another secluded spot away from all other schoolmates: “That new girl seems like our Somakhya in more than one way. She seems to seclude herself while at lunch with her little gang which seems to be gathered from across different grades.”
Vidrum: “Those are her sisters and she is brāhmaṇa just like our guy”.
Sharvamanyu: “But then we are not brāhmaṇa-s. I believe Somakhya’s reasons are little more complex than that. He was quite suspicious I recall years ago before letting us dine with him.”
Somakhya: “I have told you’ll that before. It is more in the spirit of being a brāhmaṇa than the word of bhojyaprakaraṇa. The alien smells of meat, eggs and other items we perceive as being unclean, abhojya, are something that do not go well with the act of eating, which is one of sacrificing to the god Vaiśvānara within you. Also, the majority of our schoolmates do not understand the concept that we do not stand the act of eating from each other’s plates, given that they so indiscriminately share food, or drinking from others water bottles.”

Vidrum: “Yes. Yes. But I heard that you and Lootika will apparently be given a prize for your submission to the science essay contest on the evolution of human olfactory interest in Maillard reaction products. Does this have something to do with your brahminical sensibilities?”
Sharvamanyu: “First, what is this Maillard’s reaction?”
Somakhya: “It is a reaction in which the carbonyl groups, mostly from sugars, react with the amino groups coming from amino acids among other compounds and the oxygen and hydrogens go away as water leaving a compound with a C-N bond. It happens when food is heated a specific temperatures and the Maillard products are odoriferous imparting many of the favored flavors that are sought after in food. Our basic thesis was that human olfactory interest in Maillard products primarily arose after the use of fire became prevalent. The use of fire to prepare food we posit offered an advantage in making humans more likely to escape food-borne infection. Hence, we posited that those who evolved a liking for Maillard products were likely have a survival advantage due to that conferred by fire-cooked food.”

Sharvamanyu: “That is interesting. Somakhya, I have never seen you too interested in girls before. But I sense you have something for that new pretty girl?” Somakhya ignored his companion’s question and kept focusing on finishing his cold meal but Vidrum waded in: “Well, you may not know it but that is true. If she were to come here I am sure he would drop us and the lunch and they would lost in conversation on all manner of arcana.”
Sharvamanyu: “So may be that is why he said desire is not bad after all?”
Somakhya: “Desires may come and go but there are things deeper than that.”

Vidrum: “OK, OK. You know we do not read all this sciency stuff like you guys do. Nor do we watch Floyd Mayweather as the teacher asked us to do. He is so boring with his cliches of everything wanting to kill you.”
Sharvamanyu: “Still worse is his characterization of the reproductive system as an entertainment center in a latrine. We instead learn most of our stuff from movies. You said world might freeze into a snowball – saw movie like that. Could it really happen?”
Somakhya: “ That is entirely unclear. There are some lines of evidence that ice used to form in the tropics or even close to the equator at two points in the past history of the earth. But the evidence that it resulted in extinction of life is negative.”
Vidrum: “Just as we were coming out you guys seemed to suggest that aliens might bring our end? We saw a movie last weekend which showed something like that.”
Sharvamanyu: “Do you seriously think that would happen?”
Somakhya: “A very low probability event. In the 3 or more billion years of life on earth we don’t see evidence for single alien visitation beyond to two initial visitations that seeded the world with bacteria and archaea. But it if were to happen it could be catastrophic like for the animals of continents which had not encountered humans before. Or for that matter like the fate of the humans like the first Australians and Americans.”

The bell just rang then and they had to rush to back to class. That evening as Vidrum was nearing his home on the way back from school he saw Lootika and her sister Vrishchika hanging out at the corner of the street leaning on their bikes. Vidrum approached them : “So it seems you and Somakhya think that the end of the world could come from aliens? I just saw a movie last weekend which had such a plot.”
Lootika: “Possible but not very probable. Unlike what our teacher would want us to think, the end of our civilization could come sooner due to the unmatta-s. So I don’t think we need to keep an eye for those spaceships.”
Vrishchika: “And if it does happen I would place my bet not on a little green man or a furry abomination of an alien but a contagion like a virus or perhaps less likely some single-celled prokaryotic form.”
Vidrum: “So we should be guarding against those?”
Lootika: “Somakhya and I have been pondering about that of late. Especially on lines of why bacteria cause so many diseases but archaea hardly any. So at this stage we still seem to think this end, while possible, is certainly less likely than the fiery crash of an asteroid or a comet from space.”

Posted in Life |

## Mongolica: Qubilai Khan’s campaign to destroy the Southern Song

The final act in Mongol conquest of China shows the military capability of Qubilai and why his grandfather, the great Khan had singled him out as the one who someday would adorn his throne. We shall place here a very brief account of this war. In late summer of 1259 CE the Mongols faced a major crisis when the supreme leader Möngke Khan died of dysentery while conducting the siege of the fort of Diaoyu of the southern Song. The next in line for the throne of Chingiz was his brother Qubilai who was supported by his younger brother the Il-Khan Hülegü. Hülegü decided to send about half his men from Iran to aid Qubilai continue the campaign against the chīna-s. However, he was thwarted in this plan due to a squabble with his cousins of the Jochid line, who were in state of ferment after the death of their great leader Batu, the grandson of Chingiz via Jochi. Nevertheless, Hülegü manage to send the young Bayan of the Baarin tribe (one of the early tribes that elected Temüjin as Khan), a rising general in his army, to help his brother Qubilai. He was given a command at a young age after his father fell in battle while taking the Hashishin forts under Hülegü. Qubilai pressed on and established a bridgehead south of the Yangtze river near Ezhou. But soon thereafter he had to move back north due to his brother Ariq Böke claiming to be the great Khan. In the mean time the Song under their leader Jia Sidao regained the ground lost to the Mongols south of the Yangtze.

In 1260 CE after Qubilai Khan had settled the rebellion of his brother Ariq Böke, set his eyes on destroying the last Chinese kingdom, the Southern Song, once and for all. The Song were short on horses and thus lacked a swift-moving cavalry. However, Qubilai from his prior experience knew that the terrain meant that Song could still nullify the traditional cavalry tactics of the Mongols. Hence, he decided not to hurry and made elaborate preparations. The Mongols amassed extensive pyrotechnics as well also a powerful array of rockets and explosive bombs for sieges. But then Song had the largest army in the world of over a one million men and also had an impressive array of torsional artillery to deliver fireworks against a besieging force: large trebuchets with good accuracy and less accurate mangonels and also giant crossbows. Moreover, they were the first Chinese kingdom to have a permanent and strong navy fully capable of diverse maneuvers on rivers and the seas. It had an array of some of best ships of the time equipped with naval pyrotechnics for attacking coasts and other ships. They also had strategic depth in the form of the alliance with the kingdom of Campāvati to the south.

In 1265 CE Qubilai tested the Song land forces at Sichuan and comprehensively defeated them. This gave impetus to the Mongol morale to enter into a decisive war with the cīna-s. However, Qubilai knew fully well that this was not enough as Song from then on were going play a defensive strategy banking on their vast army. They were going block the key invasion routes that the Mongols would be able to take with a string of well-stocked forts. Thus, the Mongols would be forced to take routes through difficult terrain in central and south China or risk a completely new form of warfare, naval, against a force of over 100,000 Song marines who were stationed along the coast and instructed to preemptively to thwart any Mongol attempt at building a navy. Qubilai decided that he would first focus on the land campaign and then build up his navy over a period so that they could match up with the Song. He also raised a massive stock of pyrotechnics and bombs specifically designed to attack ships with the Song navy in eye.

Thus, in 1267 CE Qubilai directed his army to systematically proceed along the Han river, the left tributary of the Yangtze in central China with the Song capital of Hangzhou as the focus , even if the progress was slow. Putting this plan into action, the 30 year old general Bayan led the Mongols to a series of victories using heavy bombardment with the pyrotechnics and cutting off supplies with the mobile cavalry units, thus taking all the smaller forts which could be taken with just the land army. Then they besieged the second strongest Song fort of Fancheng and after a prolonged siege captured it. Provisioning the army with this captured fort, they advanced against the strongest and best defended Song fort Xiangyang in 1273. In the meantime Qubilai had readied the phase-I of the Mongol navy, put it on the river and carried out a continuous naval blockade of Xiangyang. Finally, with the land army converging, this mighty fort was captured by 1274. With that the Mongols broke into the core Yangtze region and closed in on Hangzhou. Qubilai promoted general Bayan for his successes as the commander-in-chief of the Mongol army. The Song lord Jia Sidao charged at the Mongols stationed at the Yangtze with a force of 130,000 but Bayan smashed his advance and forced him to retreat. He tried to negotiate a peace treaty but Bayan rebuffed it and continued the attack forcing Jia to flee. Then the Mongol generals Bayan and Aju punched their way forward aiming at Nanking, Changzhou and Wuxi. In the first city the Song army fled at the approach of the Mongols without offering much of a fight. In the subsequent encounters the Song lost heavily against the quick moving Mongol forces and had to surrender the cities. Several top Song generals were targeted and killed by Bayan and Aju in these battles thus denuding their command capacity.

Qubilai then sent three Chinese emissaries to discuss surrender terms with the Song but the Song killed those emissaries. Qubilai immediately ordered strikes on the city of Yangzhou on the Yangtze and Jiading and destroyed the Song units which were positioned there. He then used these as a base to launch a surprise attack with fast-moving boats equipped to hurl bombs and pyrotechnics on the first Song river fleet. The Song admiral taken by surprise was killed and his fleet was rapidly brunt and sunk. The veteran Song lord Jia Sidao was killed shortly thereafter by a fellow Chinese perhaps employed by the Mongols. Then the Mongols swept up the Jiangsu province where the Song population resisted strongly but was massacred upon being defeated. Hunan and Jingxi were taken next by Bayan. Then Qubilai launched a three pronged amphibious assault on the Song. A western wing under general Ajirghan marched to take the fort of Jiankang and secure the key Dusong pass. In the east Qubilai unfolded the second phase of his navy under admirals Dong and Zhang Hongfan to sail along the Yangtze to reach the sea and then secure the coast for a naval showdown with the main Song sea fleet. Bayan led the central wing to march straight to the capital.

Seeing the rapidly unfolding of the Mongol plans, and being reduced to a patch around Hangzhou, the Song sent a emissary stating that they were willing to be a protectorate under the Mongols. Bayan sent him back saying that the Mongols were now aiming for complete conquest of the Song. Finally, in March 1275 the Mongols closed in on the Song capital and launched a simultaneously attack with two land armies and one naval force. The Song thought they would stave off the Mongols with their fire-arrow giant crossbows as they had done to the elephant corps of the Han several centuries before. However, they came up against an overwhelming bombardment by the Mongols with thousands upon thousands of iron-cased and earthen bombs hurled from trebuchets. The Song crossbows and seige engines caught fire and parts of the capital province were were hit from the bay by the naval attack of Mongols. The superior pyrotechnics of the Mongol destroyed the Song naval defenses and allowed their ships to close in on the capital. Then the Mongols launched their main land assault with the cavalry division under Bayan. The Song while having a larger force numerically could not match up to the tulughama-like sorties of the Mongols and eventually folded up. With that the main Song land resistance was over and the Song queen surrendered the capital to the Mongols without any resistance in February of 1276 CE.

There was still a mighty Song fleet and the loyalists taking the two young surviving princes with them sailed down the sea and used Macau and nearby islands as a base. When Qubilai had to move north to face Qaidu who was challenging him as rival Khan the Song tried to reestablish themselves by fomenting rebellion in Fujian, Guangdong and Guangxi in 1277 and 1278 CE. Qubilai having settled the issue of Qaidu for the time being returned to finish the Song rebellion. The Mongols first flushed them out of Fujian and Guangxi by repeated land attacks and then corralled them in Guangdong where one of the Song princes died leaving the last one as the emperor. Seeing the Mongols close in, he and his supporters realized that there was no hope of fighting a land battle. But their navy of nearly 1000 excellent ships and several hundred supporting boats was intact and they decided to retreat to the island of Yaishan off Macau. It was here that the final battle was fought in 1279. The Song arrayed their ships in a rectangular formation and placed several palisades tied to boats to form a perimeter. Thus, the whole array was like a floating fortress from behind which the Song troops could fight. They also kept close to land so that they could supply their men with weapons, food and material for repair. This effectively cost their mobility against a Mongol fleet with smaller but much faster vessels. At first the Mongols took some high positions facing the Yaishan coast and launched a bombardment in February of 1279 with incendiary shells and stones. This damaged several ships of the Song and demoralized them to a degree.

However, when the Mongol fleet finally assembled for battle in March 1279, it was still only half the size of the Song fleet. On March 19th the Mongols calculated that the tide would provide two opportunities that day. The tide receded towards the south early in the morning creating a rush between the two islands where the Song fleet was stationed. Since they had built highly maneuverable ships, the Mongols used the momentum of the tides to launch rapid attack on the Song from the north. For this assault the Mongols chose the admiral Li Heng, said to be a descendant of the old emperor Tai-zong of the Tang. This attack brought the Mongols close to the Song ships and they began hand to hand fighting at which they excelled with the corps on the support boats. Taking advantage of this encounter the Mongols moved a second formation of about 100 ships to the south and launched an attack by mid-morning from the south. The Song turned the full force of their naval artillery against this formation. But by noon the Mongols made an amphibious landing on the island of Yaishan and made a pyrotechnic attack against the Song ships closest to land. By then the fourth Mongol fleet of around 100 completed the encirclement of the large but immobile Song fleet. Shortly after noon the Mongols breached the Song palisade and were now able to attack ships inside it. They then used the momentum of the afternoon tides to launch another rapid attack to close in on the Song before they could deploy their next round of naval artillery. In this attack they broke up the defensive rectangle of the Song completely and got to the main ships. Having boarded them they engaged in fierce closen combat. By evening the Mongols had killed 100,000 Song marines and the Song prince’s corpse was seen floating in the sea. By then Mongols had captured 800 Song ships. Seeing this, the Song navy lost heart and surrendered. With that the Song empire of China came to a close and the survivors fled to Campāvati. This closing battle also showed how far the Mongols had come from a horse-borne steppe power to one which could defeat one of the best navies of the time at sea.

In my childhood, one nice afternoon, I leaned against the cot in my room and lapsed into a reverie. This war flashed in before my eyes in great detail like a movie. Inspired by that we staged an enactment of the same which gave us much pleasure.

Posted in History |