Zeros of the Zeta function and such metaphysical wanderings

•June 15, 2007 • Leave a Comment

The spectrum formed by the first 200 zeros on the critical line of the Riemann zeta function: The Re(s)=.5 while the lines of the spectrum above are the values of Im(s) producing the zeros. These values were obtained from Odlyzko’s table of zeroes.

Many years ago when we were still not seized by the grAhI, we sat beside a pond in which the then arrogant sachiva of the third hero was raging like an untamed bull. Since he was a mathematician had held placed some books on the shore, which he lent to me to glance at. I was far from being a mathematician but it introduced me to few 1-dimensional distributions or spectra that greatly fascinated me:
1) the prime distribution; 2) the distributions of zeros on the critical line of the Riemann Zeta function; 3) the distribution of eigenvalues of a random Hermitian matrix.

There was something primal about these spectra–some thing hidden that fascinates a layman and mathematician alike. All this suddenly struck a new chord again, when ekanetra told me of Wigner’s conjecture that the distribution of real spectral lines in the nuclear orbitals could be explained as the eigenvalue of a Hermitian matrix.

It brought up the fundamental question: Is the hidden Hermitian matrix or operator behind all these something profound, or semi-trivial like the normal distribution or the power-law in the nature of things.

Endless diversity of cyclical waves

•June 14, 2007 • Leave a Comment

mArtANDa bhairava: mallAri

•June 14, 2007 • Leave a Comment

After visiting the shrine of the fiery ulka, accompanied by his two shaktI-s, we were filled with the urge to pay a visit to the shrine atop the massif of Jejuri. It houses khaNDobA, a pastoral deity, an evolute of the ancient revanta, in Andhra, Karnataka and Maharashtra. A Sanskrit tale is narrated in association with this shrine:
-The brAhmaNa-s inform rudra that the world was being tormented by the demons maNi and malla.
-They go to rudra to ask him to save them from the demons. From his locks he produces a fierce shakti named ghR^itamArI
-He himself manifests as mArtANDa bhairava. He is described as having golden flaming robes, with teeth like flashing rubies and he was surrounded by a troop of fierce dogs.
-mArtANDa bhairava appointed kumAra as the commander of his forces with a horde of skandagraha-s mounted on peacocks.
-kumAra further partitioned his division under indra, agni, yama, nirR^iti, varuNa, kubera, brahmA and viShNu. There was also a horde of vinAyakas mounted on rodent-faced horses led by the fiery ulka.
-The forces of mArtANDa bhairava and those of maNi and malla fought a fierce battle.
-In the final fierce battle the horse-faced maNi fought mArtANDa bhairava riding a white horse. Finally, he was struck by an arrow of the bhairava and thrown into a lake termed maNichal along with his horse. mArtANDa bhairava then stamps him underfoot, when he pleased the god with a hymn, and asks for a boon to be stationed at the door-step of his temple. He also gives his white horse as a vAhana to the bhairava.
-Then malla attacked the deva army with his own personal force by uttering so loud a roar that it caused pArvatI-s crown to fall in kailAsha. mArtANDa bhairava dispatched ghR^itamArI to attack his hordes.
-ghR^itamArI swooping down on them like a kR^ityA swallowed all of them and ground them the dust.
-Then ghR^itamArI and malla fought a terrible battle. He pierced her head with 100 arrows, her breasts with 27 arrows each and another 100 to immobilize her shoulders and neck.
-Then mArtANDa bhairava entered the field riding his white horse. With a cresent-headed arrow he cut off the demon’s bow, but the demon retaliated by hurling his axe to cut the bow of the bhairava. The then two joined a hand-to-hand fight with swords.
-Finally the bhairava struck him down and placed his foot on him. Before he was beheaded, he asked the flaming bhairava that his head should be placed outside the temple of the god and that he should be given sacrifices of goats or sheep. He also asked that his name be mentioned before that of his conqueror. mArtANDa bhairava conferred these boons and dispatched him — he took the name mallAri to satisfy the point of the the asura’s name coming first.

…..

This is the canonized Sanskrit work. However, the shUdra-s of maharAShTra and Andhra also narrate a colloquial version. Several shUdra-s I have spoken to were well aware of the connection between mArtANDa bhairava and khaNDobA, and also generally know the Sanskrit version of the story. The colloquial version is usually narrated by a pious shUdra bard called a vAghyA or in the midst of the shepherds by a sooth-sayer termed the “devarShi”. The Andhra and marAThA version differ in minor points, mainly place names. Some Andhras seem to have located the tale to Mallikarjuna or shrIshailam; in the kaNNaDa country I have noticed some say that it was malleshvaraM, now within Bangalore city. Clearly these places were there in the marAThA mind too, because shivAjI bhosle and his clan, great worshipers of khaNDobA, associated him with shrIshailaM and malleshvaraM. shivAjI even wanted to renounce the world and become a mendicant at shrIshailaM, but his men reminded him of his great role to play for svarAjya. This grandson emperor shAhu himself, like a vAghyA came to his court along with his dog on which he placed the royal crown thinking of khaNDoba. But I supply below the story as narrated in the frame of the marAThA country, for that is where I have explored the many shrines of the deity:

The brAhmaNa-s asked shiva to help the people from the depredations of the two demons on earth. shiva told umA that he was leaving kailasa to deal with two asuras.
umA:” You are always taking avatara-s leaving me alone here. I want to come along this time.”
shiva: “Alright, after I kill the asuras you can come along. We will settle there and live for a while. You incarnate as mhALsA the daughter of a tima-sheTTi, a li~NgAyata shopkeeper. After killing the asura-s I will come marry you and take you to my palace atop the hill of jejuri.”
ga~NgA who saw all this, held in the coils of rudra’s hair said: “This time I want to come too”.
shiva: “OK- you take the incarnation of a shepherd girl bAnU among the shUdra-s known as the dhangar (cattle herders). I will come to your hamlet and bring you over to my palace.”
umA: “Alright, but why can’t we come now? You may need us for battle.”
ga~NgA: “You will need us! see when the time comes, you will need us.”
shiva: “OK ladies go and take your incarnations, I really need to be doing this man’s work of fighting asura-s.”
rudra took some haridra and sprinkled it on his snake — it became a metal chain that he wore around his neck. In due course rudra incarnated as mallaNNa or khaNDobA starts a war on maNi and malla. viShNu also incarnates as IThobA or hegaDi pradhAn as bAnU’s brother.

In the fight maNi on being wounded starts spawning new maNi-s from his blood- like raktabIja or andhaka. khaNDobA seeing the trouble sends for the ladies mhALsA and bAnU to come an join him in the battle. mhAlsa taking up a sword and spear rushed to the field. bAnU picked up a magic bow and arrows and ran to the field. Behind bAnU came their dog. First the two ladies try to drink up the blood. But seeing it to be too much they used skulls of beheaded asuras to catch the blood. When even that did not suffice they ran around shouting in consternation and were hit by the arrows of the asuras. Then their dog seeing the predicament came and lapped up all the blood. With his blood drying up maNi was losing strength. Then IThobA also came to the field and gave khaNDobA a chakra, with which he killed maNi.

bAnU and IThobhA killed many demons in malla’s army. Then khaNDobA attacked malla, with mhAlsA mounting his horse and wielding a spear. After a prolonged fight, malla finally rushed on khaNDobA and seized his horse’s reins. But the horse reared and allowed mhALsA to spike him on his flank, even as khaNDobA struck off his head with his sword. The day of victory was a bright ShaShThI day in mArgashIrSha. Then khaNDobA married mhALsA and also brought bAnU along, and settled down with the two ladies in his hill-top palace at jejuri [kaNNaDa-s explicitly call bAnU a concubine].
….
In jejurI, the shrine of viShNu as IThobA or hegaDI pradhan is half-way up the hill. In front of his shrine goat sacrifices are made- he is suppose to convey the goat sacrifices to the recipients. khaNDoba himself only takes vegetarian offerings in the main shrine on the top of the hill. There, on one side in the courtyard, there is a large image of maNi and one of the steps has the image of the head of malla. mhALsA has her own shrine, while bAnU has a little platform on which her image is placed. In the smaller, but venerated, dhamankeL shrine in Junnar near Pune, khaNDobA is seated on a muNDa mancha with the two demons’ heads as the foot-rests. mhALsA is seated next to him, while bAnU is peeping from behind his shoulder. In his hands he has a sword, trident and shield. Besides the 3-peaked tryaMbaka li~Nga, IThobA or viShNu stands with a sword and shield.

Problems in avian evolution

•June 10, 2007 • Leave a Comment

Since the Mesozoic, dinosaurs have become one of the most visible components of the terrestrial ecosystems. Yet, the dinosaurs alive today only represent a small sliver of their diversity during their heydays in the Mesozoic. Nevertheless, these modern dinosaurs, the birds, representing just one lineage of theropods now include about 8000-10000 species, which is more than the number of mammal species, which is around 4500. It is the evolutionary history of these extant dinosaurs, the Neornithes clade of birds, that presents interesting problems: 1) When and where did the neornithes come into being and diversify to spawn the extant lineages? 2) What are the higher-order relationships between the extant lineages.

Since the discovery of Archaeopteryx fossil evidence has revealed a rich history of birds in the Mesozoic. The current analysis of this evidence points to some very basal lineages of birds that are morphologically close to their deinonychosaur sister group and might even lie within it. These include the forms like Rahonavis, Archaeopteryx, Shenzhouraptor(=Jeholornis), and Sapeornis. The pygostylian clade of birds includes those forms which are distinctly closer to the neornithes than to the basal bird lineages. Within the pygostylian clade a basal lineage is seen in the form of Confuciusornis, Changchengornis and relatives. Then the pygostylians show two great lineages that include most of the diversity of birds: the enantiornithines and euornithines. Of these, the enantiornithines are the prominent birds in the Cretaceous fossil record and span a notable diversity adapted to different ecological niches.

The euornithines, the clade which includes all Mesozoic pygostylians closer to crown group neornithes than the enantiornithines, also underwent a remarkable radiation in the late Mesozoic giving rise to several ecologically diverse lineages (See early article for details). These included independent lines of shore birds- the Chaoyangids: Chaoyangia and Hongshangornis and the Songlingornithids like Yanornis and Yixianornis; desert birds like Apsaravis; flightless runners like Patagopteryx; marine lineages – Gansus, the hesperornithids and ichthyornithids among others. Both the enantiornithines and the above basal lineages of euornithines went extinct at the great K/T event which marked the end of the Mesozoic. It is only in the Tertiary that we start seeing numerous fossils representing members of the extant lineages of birds. Strikingly, representatives of most of these extant lineages of birds, the neornithines show hardly any convincing presence in the Cretaceous. This lead to the proposal that the neornithines represent a Tertiary radiation from a single or limited number of survivor[s] of the K/T event, with most major modern lineages emerging after the extinction. In contrast, Cracraft’s biogeographic analysis of neornithines and Sudhir Kumar’s molecular phylogenies first suggest that many of the major higher order splits in neornithines had emerged in the Cretaceous itself in the fragmenting Gondwanaland and and passed through the K/T event as survivors. Their core biogeography was also interpreted as product of vicariance of the Gondwanan fragments. The recently re-described Vegavis from Antarctica was shown to be a definitive member of the galloanseran clade, and possibly even a duck supporting cryptic presence of neornithes in the Cretaceous and their extensive diversification prior to the K/T event.

Though the higher-order phylogeny of the neornithines had earlier confounded both molecular and morphological studies, now several branch points in the tree are becoming much clearer with larger datasets. The basal split in the neornithines separates the paleognaths from the neognaths. The primitive paleognaths in turn are split up into the volant tinamous one hand and the flightless ratites on the other. It is in the evolution of the ratite branch that ancient DNA techniques have brought new data. Several species of tinamous are found in South America. Of the ratites the rhea is from south America, the ostrich in Africa and greater India (till recently), the elephant birds in Madagascar (extinct recently), the emu and cassowaries in Australia and the moa and kiwi in New Zealand. This suggests that the common ancestor of all ratites was flightless and that vicariance during Gondwanan breakup might have had a major role in their distribution. The molecular phylogenies suggest an interesting tree (see below): the rhea is the most basal branch, the moa is the next branch, followed by the ostrich and a crown of the cassowary+emu with the kiwi as a their sister group. Thus, the two New Zealand ratites are not monophyletic suggesting independent invasions of New Zealand. Based on limited sequence data the elephant birds seem to belong to the clade: (ostrich (kiwi(cassowary,emu))).

Assuming the Gondwanan vicariance theory we come with the following scenario: The ancestral ratite split from the tinamous somewhere in South America and spread over the rest of the southern landmass (comprising of Antarctica+Australia+New Zealand and Indo-Madagascar linked by the Kerguelen Plateau), probably just prior to 80 Myrs. As New Zealand broke away from this mass the moa lineage separated and as Indo-Madagascar broke away the the ostrich and the elephand birds diverged respectively on greater India and Madagascar. In the Australian mass were the common ancestor of the kiwi, the emu and the cassowaries. The kiwi somehow seems to have reached New Zealand from here. Given that the inference strongly supports a flightless ratite ancestor this second invasion of New Zealand by ratites should have occurred by land connection between New Zealand and Australia, or at least when the connection to Australia was close enough to allow a float. This points to the late Cretaceous when the Howe Rise and the Norfolk Ridge could have formed a connection to New Zealand including New Caledonia. This would also imply that the ostrich reached Africa from greater India when it joined the Eurasian Landmass, consistent with proposed ostrich fossils from central Asia.

However, there are some earlier fossil paleognaths that are somewhat enigmatic. These are the volant lithornithids found in early Tertiary deposits from Europe and North America. Analysis of the anatomy of the best preserved specimens of the lithornithids suggest a relationship with the tinamous (In the very least like the tinamous it lies outside the ratite clade amongst the paleognaths). They appear to have flown from the Gondwanan continents to colonize the northern landmasses at some point prior to 55 Mya and eventually became extinct. The similarities between the tinamous and the lithornithids suggests the ancestor of the paleognaths was in general morphology a flighted bird similar to these. They seem to have eventually led to the flightless ratites, perhaps via a tinamous like intermediate that spent a lot of time on the ground.

Within the neognaths the fundamental split is between the galloanserans (fowl and water fowl) on one hand, and the neoaves on the other including all other extant birds. Within the neoaves an early distinct branch separate from the rest appears to be the passeriformes. At least two independent molecular analysis suggest that within neoaves the owls and parrots might form a clade, perhaps including the hornbills. This clade might either be an early branch of the remaining neoaves clade after the passeriformes split off, or a sister group of all of other neoaves.

The kea – the foremost of the dinosaurs

•June 5, 2007 • Leave a Comment

Once I asked the yakShiNI bhujaMga-mekhalA which was the greatest of the dinosaurs. The wonderful yakShiNI answered that she felt that it was parrot known as the kea (Nestor notabilis). The kea is one of the most extraordinary birds. Its intelligence is legendary (see the movie below). It is also one of the only known predatory parrot that can take on mammalian prey. It is known to kill and feed on sheep, and for this reason was nearly hunted to extinction by the European invaders of New Zealand. This is pretty amazing for a crow-sized bird, but is not surprising when one factors in its intelligence. It represents one of the members of an distinctive lineage of parrots comprised of 3 New Zealand species- the Kaka, Kea and slightly more distant Kakapo. The remaining few parrots found in New Zealand are all parakeets, namely the Kakarikis and the deep south Antipodean parakeet.

The evolution of parrots is a topic of considerable importance to the more general issue of extant neornithine evolution. Prof. Feduccia, the perpetrator of many falsehoods, had proposed that the major radiation of the neornithes occurred after the great extinction at the K/T boundary. But Cracraft’s analysis suggested that several the major lineages of neornithes including both the paleognaths and neognaths had already radiated by the late Mesozoic and made it past the K/T event. Their basic dispersion was attributed to vicariance associated with the break up and the movement of the fragments of the erstwhile landmass of Gondwanaland. Following this initial dispersion they appear to have radiated extensively during the Tertiary to spawn the modern avifaunas.

To support this theory, Cracraft used a systematic analysis of neornithine radiations to show that the Galloanserae (fowl and waterfowl), Gruiformes (the cranes and allies), Caprimulgiformes+Apodiformes, Passeriformes amongst the neognaths and the paleognaths show a predominantly “Trans-antarctic” distribution – that is in the Cretaceous Gondwanan fragments (assuming that in the Cretaceous there was a land connection between greater India and Madagascar and that greater India was connected via the Kerguelan Plateau to East Antarctica, which was in turn connected to South America. This connection is supported by the biogeography of Abelisaur dinosaurs). Again, in his study Cracraft proposed that the parrots might have had a Gondwanan origin and that their extant distribution might have been influenced by the break up of the Gondwanan landmass. He cited the major radiation of the parrots in South America and Australia in support of this idea. Subsequent molecular analysis of parrot evolution by Miyaki et al and de Kloet et al suggested that indeed that parrots seem to have radiated separately in the main major Gondwanan fragments- Australia and South America, sort of following the same model as the marsupials. These studies also suggested that one of the African lineages of parrots are closer to the S. American forms suggesting that they emerged from an early south American lineage and flew across the much narrow paleo-Atlantic ocean to colonize Africa.

Interestingly, in both Miyaki et al and de Kloet et al studies the Kea-Kaka-Kakapo lineage emerged as the basal-most lineage of all parrots. If this phylogenetic position can be confirmed by further studies, then these parrots of New Zealand possibly represent a remnant of the old lineage of parrots that would have existed on Antarctica to which New Zealand was attached till between 82-90 Myrs ago. One implication of these proposals is that possibly fossil parrots will be found in Antarctica, just as the Cretaceous fossil Galloanseran, Vegavis, was recently described. Molecular studies on avian evolution also strongly suggest that within the neoaves clade the parrots are certainly not particularly closely related to passeriformes, but might instead by closer to owls. So the evolution of high intelligence, over and above the general increase seen in the Neoaves in general, was an independent feature in parrots and passeriformes like the Corvids.

This might also mean that several lineages of intelligent neornithine dinosaurs had probably begun emerging in the Cretaceous itself. This would mean that time alone is not a major factor for high intelligence and practical problem-solving ability to transform to “high-level technological” intelligence as seen in humans — the dinosaurs seem to have possibly arrived close to ape-like intelligence over 65 million years ago but did make the transition the humans made. There are other contingency factors. [As an aside- this is something Carl Sagan completely misses in his book titled Demon-haunted world or something like that]”

skanda prINanaM of the sAmavedic tradition

•June 3, 2007 • Leave a Comment

We had earlier see how the R^igvedic hymns have been adapted for the worship of kumAra. The devI-bhAgavataM states that the worship of ShaShThi is connected with the kauthuma school of the sAmaveda. When one digs into the roots of this tradition one arrives at the sAmavidhAna brAhmaNa, peculiar vidhAna work of the sAmavedins which is termed a brAhmaNa. It provides a series of gAna-s termed the skandasya saMhitA, which are to be used in the pacification or worship of kumAra. The association of skanda with the late sAmavedic tradition is not unprecedented given that he is mentioned in the last section of the chAndogya upaniShat as the teacher of nArada. nArada is also a primary teacher of the sAmavidhAna brAhmaNa lineage.
The skandasya saMhitA consists of 11 gAna-s all drawn from the grAmageya gAna-s:
1-3 (GG246) – the agnervaMrAni
These 3 songs are composed on the following R^ik of vishvAmitra:
A mandrair indra haribhir yAhi mayUra-romabhiH |
mA tvA kechin ni yaman viM na pAshino.ati dhanveva tAnihi || (RV 3.45.1)
come here indra, with your golden horses, exhilarated, with tails like a peacock’s feathers. Let one stop you course like noose-wielders trap the bird, pass over them as though going the coast.
This mantra appears to have been chosen because of the term “mayUra-romabhiH” which reminds one of the peacock vAhana of skanda.

4-6 (GG269) -the shAkrANi
These 3 songs are composed on the R^ik of the kANva-s:
A no vishvAsu havya indraH samatsu bhUShatu |
upa brahmANi savanAni vR^itrahA paramajyA R^ichIShamaH || (RV8.90.1)
May indra, who in battles must be invoked, be on our side;
may the killer of vR^itra, the wielder of greatest power, praised by R^iks, come to the soma pressing and chanting of mantras.
This mantra appears to have been chosen because of reference to battle- kumAra being the war-god.

7-9 (GG533) – the kutsasya-adhIrathIyA
These songs are composed on the R^ik:
pra senAnIH shUro agre rathAnAM gavyanneti harShate asya senA |
bhadrAn kR^iNvan-nindrahavAn sakhibhya A somo vastrA rabhasAni datte || (RV9.96.1)
The hero proceeds in forefront of the chariots, the commander of the army, he captures wealth as his army rejoices. soma dyes his robes with good colors, and makes his friends’ invocation of indra successful.
This mantra appears to have been chosen because of the term senAnI. kumAra is called senAnI and is the leader of the army of the gods. This is the most appropriate of the hymns for this particular context.

10,11 (GG565) – the arka puShpa
These two, also deployed in the pravargya rite are composed on:
pavitraM te vitataM brahmaNaspate prabhur gAtrANi paryeShi vishvataH |
atapta tanUr na tad Amo ashnute shR^itAsa id vahantastat samAshata || (RV9.83.1)
Your soma-filter is spread out, O brahmaNaspati, as the lord you enter its limbs from all over.
The raw, unheated mass does not attain this [filter]; only the well-cooked mass, which bears [the active soma], attains it.

Originally this mantra refers to the process of the boiled soma being filtered for the juice. However, it might have been adapted for kaumara worship because of the word “brahmaNaspate”. After all kumAra is brahmaNya deva.

A neo-vedic composition?

•June 2, 2007 • Leave a Comment

R was giving me ride when we observed some sight that greatly reminded us of a shared juvenile experience, that filled us with much fuzzy nostalgia. At that moment R asked me if anybody could compose vedic mantra-s again or sUkta-s. I asked what the criteria where and we seemed to agree- they should resemble the original sUkta-s or yajushes and should follow vedic grammar and should have correct vedic svaras. They should be in genuine Chandas, adhere to the vedic lexical statistics and deities. Evidently, even though later upaniShad-s have been couched in vedic language over the ages they clear differ from the original. Nor do the purANa-s even superficially attempt revival of the vedic form of expression. Even though the vedic composers themselves saw their works as kAvya, the later day classical kAvya is clearly distinguished from its vedic precursors. So much so that the later day commentators and stylists explicitly exclude the vedic kAvya from the classical kAvya. Yet at some point vedic style composition was prevalent and at least had one representative outside the Indian zone in the form of the avesta of the Iranians. There is pANini, whose grammar describes the vedic language, and by including the unpublished sUtra-s of the ArSha pATha one could theoretically arrive at vedic composition. Yet we see hard any such attempts to imitate to shruti.

But the muni informed me that there was one possible example. These are the details I could gather:
-In 1917 a brAhmaNa named gajAnana sharma aka daivarAta from gokarNa, a student of the great saMskR^ita savant kAvyakaNTha gaNapati shAstrI, composed a set of 50 sUkta-s encompassing 448 R^ik-s.
-He said to have spontaneously composed this while performing tapasya.
-The majority are in the jagati Chandas, followed by triShtubh, then anuShTubh and finally a small set of gAyatrI-s.
-The primary devatA-s of his composition are agni, indra, sarasvatI, savitA, bR^ihaspati, ApaH, vAyu, rudra, aditi and soma. He also has some compositions on puruSha and other resembling the philosophical works like the nAsadIya sUktaM.
-The R^iks are marked with svara-s and the composer also created a pada pATha as seen in the traditional saMhitA-s

Unfortunately, copy of this was in the Telugu script and the gentleman with a copy of this was my only source of obtaining the information. I could not analyze it myself to determine if it met all other lexical and grammatical criteria of a vedic composition. Nevertheless, from what I could see it did seem a work worth considering as a candidate for the neo-vedic genre. If only somebody printed a copy of it in the nAgarI script…

gaNapati shAstrI was a great paNDita himself with a profound understanding of the veda and he composed more modern veda inspired compositions in saMskR^ita like the indra sahasranAmam to indra, indrANI saptashati, gItAmAlA to various vedic deities, and indra gItamu in telugu imitating a vedic praise of indra. It is clear that he was one of those rare individuals who had a genuine grasp of the vedic thought and action. Some people have the action to perform the shrauta ritual and others the inspiration of the vedic mind. A true knower of the veda understands both.

 
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