Alizadeh's Semidefinite Programming page

The link The Semidefinite Programming page is maintained by Farid Alizadeh. Also, the link: Papers on semidefinite Programming has a collection of (quite a few of) the classic papers on SDP and Ellipsoid method. Read the rest of this entry >>

Terror attack in IISc

News link from a dear friend of mine: Terror attack at IISc, Bangalore; 1 killed, 4 injured. My condolences to the family of Prof. Puri, who was killed in the attack. I sincerely hope that all the injured academecians recover and the perpetrators are caught and put to justice.

This blogger knows one of the victims, Prof. Vijay Chandru from his days at IISc, as his first semester faculty advisor, with a warm heart and resourcefulness. Prof. Chandru went on to set up the Simputer.org with others in the CS-dept. Of course, not joining either Simputer (or better, strand genomics) of BLR --and sticking to the organization he was working in-- was one of the most toughest and painful decisions that this blogger had to make.

According to the following news item: Outstanding Social Scientists/Scientists:Science and Society" award to Prof. Vijay Chandru,

"Outstanding Social Scientists/Scientists:Science and Society" award to Prof. Vijay Chandru

The Hari om Ashram Trust Award entitled "Outstanding Social Scientists/Scientists:Science and Society" award for the year 2003 in the recognition of the outstanding work done has been confereed upon Prof. Vijay Chandru. The award carries a sum of Rs.50,000/- and the citation.

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More links:
Abinandanan, a Professor from IISc has the following to say: Terrorist attack in IISc (link via DDD).

IndiaUncut has the following to say: From HITTs to LISTs? and has the following theory, which classifies possible targets of terror attacks: Soft targets for hard terrorists and also the following: Terror, and the game we love.
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PS: A very strongly worded editorial from Indian Express: Congress’s software (link via secular-right).
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Upadate(01/02/06): Subbu says

Prof Chandru is recovering fine. His operation on the shoulder was successful. Another operation would be required in a couple of weeks. Two bullets hit him.

Read the rest of this entry >>

One year from 26th Dec 2004

I thought nukes were a problem of the past,
and that terrorism is a fear that will not last,
nature told me that it never was too kind,
to the masses of mankind,
fury was unleashed across the coast.
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Remembrance Week - 26th December, 2005 - 1st January, 2006 (link via IndiaUncut). Read the rest of this entry >>

Smile for success, tremble at power of information

Smile for Success (link via India Uncut).
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The power of Google Earth that can make govts. shake, vs. the feelings of someone who could reconnect to their place using the same power. Read the rest of this entry >>

Arvind Sharma's book on Sleep as a State of Consciousness

Thanks to a request of a dear friend of mine, I could read parts of this book again and write this review.
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This post, and the ones following it, are on the book Sleep As a State of Consciousness in Advaita Vedanta by Arvind Sharma. As the title explains, Prof. Sharma delves deeply into the avasthatraya aspect of Advaita. Prof. Sharma says in the introduction that "understanding this aspect of Advaita is the first, and possibly most important aspect of Advaita".

The book is organized as to how various philosophers of ancient and moden India --who have cast their long shadows on Advaita-- have thought of, and reasoned about the aspect of avasthatraya(the trichotomy of states), in Advaita.

In concrete, the contents of the book are as follows:

sleep in Prasthanatraya

sleep in Mandukyacharica (or Mandukya Upanishad)

sleep in Sankara Advaita

sleep in Later Advaita

sleep in Moden Advaita

Prasthanatraya is the three texts: Upanishads, Brahmasutras and BhagavadGita. The Mandukya Upanishad is one of the Upanishads, on which Gaudapada who preceeded Sankara (to whom Sankara originally wanted to be a disciple of?), is said to have a written a commentary, which in turn had a special influence of Advaita. Sankara of course, is the exegete from 8th century who cast, possibly the longest shadow on Advaita, Vedanta, Sanatana Dharma, and other Indic religions.

The later Advaita is about the points of view of Sureshwara, Padmapada --who were famous disciples of Sankara and set up the Vivarna school-- and Vacaspati Misra of the 9th and 10th century --who set up the rival Bhamati school-- and others. The two schools differ on various aspects: (1) "where does ignorance originate from": According to Vivarna, Brahman-Atman is the locus of avidya. Bhamati, on the other hand holds that jiva is the locus of avidya. (2)The other difference is the question of "the preexistence of Brahman vis-a-vis the creation" (did the creation of Brahman preceed the creation? If so how?). Another major later-Advaitin is Vidyaranya of the 14th century, who arguing about the characterizations of bliss experienced by human beings, explained on the blissful nature of dreamless-sleep. The chapter has points of views of other later-Advaitins, namely: Sadananda and Dharmaraja.

In the modern Advaita, the discussion is mainly about Swami Krishnananda (of the Divine Life Society), Ramana Maharishi.

In the introduction itself, Prof. Sharma asks,

This monograph deals with the question of sleep in Advaita Vedanta. But the theme presupposes that the phenomen of sleep is an issue of some kind for Advaita Vedanta in particular, or Indian philosophy in general. For the reader who does not share this presupposition, such questions as the following will naturally arise: 'why should philosophers be concerned with sleep as an epistemological or religious problem? Why are Indian philosophers concerned with it? Why do Advaita philosophers view sleep as an important philosophical dilemma, and why are they losing sleep over it?'

Giving a reply to the above question, Prof. Sharma smartly makes an analogy of Advaita with Physics and Chemistry. In Physics, the material world is reduced to either matter or energy. Chemistry on the other hand, reduces substances to the periodic elements. In a similar way, Prof. Sharma says, Advaita reduces the multiplicity of human experience into the avasthatraya, so that some "reasonable analysis" could be done. Prof. Sharma points out that this classification is extremely rational and experiential and not necessarily revelational or scriptural. (My posts on Prof. Sharma's other books: on his book on Advaita Vedanta and on his book on the experiential approach to Advaita.)

from the Sankhya yoga of Gita:

II.69: What is night to all beings, therein the self controlled one is awake. When all beings are awake, that is the night of the sage who sees.

To be continued... Read the rest of this entry >>

An Advaitic interpretation of the Lord's prayer

Swami Prabhavananda gives a wonderful Vedantic interpretation of the Sermon on the Mount. I find it a little too theistic. The following is an attempt to interpret the Lord's Prayer in an Advaitic way:

9 After this manner therefore pray ye:

O Jiva, pray in the following way:

Our Father which art in heaven,
Hallowed be thy name.
Thy kingdom come.

O ParaBrahma which is the highest abode of the Atman,
Whatever names (and forms) are you worshipped, let them be sacred,
and You are everywhere, in the world, in the Maya and in the Atma.


Thy will be done in earth, as it is in heaven.

The realization that the Atman is not different from the Brahman will dawn on all the individual jivas.

Give us this day our daily bread.
And forgive us our debts,
as we forgive our debtors.

Let the grace of seeing through Maya be in every jiva. Let there be cessation of all bonds with Maya.

And lead us not into temptation, but deliver us from evil:
However powerful the avidya maya be -- or the one that has its beginnings in Vidya -- help this Atman stay away from its powerful clutches.


For thine is the kingdom, and the power, and the glory, for ever. Amen.

Let this happen in all the Jivas in the world and Let every Atman realize this truth, consciousness and Bliss forever. Let it be so.

Om. Peace, Peace, Peace
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May the Christ within us grow. May we conquer the evil of avidya-maya, grow beyond maya and obtain God-realization! Read the rest of this entry >>

Monotheistic religions

This is from the foreword of The Way to God by Arun Gandhi:

When Grandfather confessed to his Christian friends how much he was impressed by the Sermon on the Mount he was asked, "Why don't you become a Christian?" "When you convince me that all Christians live according to the Sermon on the Mount, I will be the first to change my religion," he responded.

From an idea I got from reading Atanu's blog, the following is a possible debate about monotheistic religions vs. multi-theistic religions:

She: Isn't monotheistic religion much better than multi-theistic religion?

I: I would assume you like democracy better than communism. Yes?

She: Yes.

I: I would also assume that you like a free market rather than a monopoly of the market. Yes?

She: Yes, but what have any of these got to do with religions?

I: In the religious sense, multi-theism is equivalent to democracy in politics and free-market in economics. The moment I get ready to pray to the idols of two gods, which are beside each other, I understand that there are multiple ways to the same truth. It also makes me (and some other people) question: what is the power behind the gods. What are the qualities that can make me a replica of God. That is one advantage of multi-theism. Mono-theism OTOH, has the disadvantage that you have to accept that God is God, because He is God. You accept that apriori. It (monotheism) is good in the sense that, if you understand the core of what He said, to His chosen people, you are at peace. However, if there is some aberration in the communication, either due to the words of the chosen people, or due to your (mis)interpretation, or due to the fact that words donot have the power to communicate some ideas, then you are a dangerous entity. This is because, you believe in something blindly, and believe that it is the only truth.

She: Yes, that's a good analogy. But, idol worship could also go haywire. It could leave confused masses. It could also leave masses who accept anything less than Truth.

I: Yes. good observation. Idol worship could result in masses being ignorant of the truth. However, it is my opinion that it is a less dangerous way.

She: That's right, but how is it less dangerous?

I: Do you agree that middle way of Buddha is a beautiful concept?

She: Yes

I: Do you also agree that it is extremely difficult to practise it?

She: Yes.

I: Do you agree that multi-theism is more middle-wayed than mono-theism?

She: Yes. So, you are saying that as moderation in everything is what Buddha preached, mono-theism is a non-middle-wayed-way. The opposite of which is multi-theism.

I: Yes multi-theism is inherently more moderate and more tolerant that mono-theism and should be practised when there is a choice between the two.

From Atanu's blog, I got the link to a blog which opposes Monotheistic religions tooth and nail: It is Monotheism is evil. Though some ideas on that blog are extreme, have a set of valid points.

The following is a quote from the movie Gandhi:

I am a Muslim and a Hindu and a Christian and a Jew and so are all of you.

Read the rest of this entry >>

Serpent and the Rope watch: Tristan and Isolde

The references to Tristan and Isolde, one of the classic love stories of the west, in The Serpent and the Rope by Raja Rao are many. This link has the abridged story (one more abridged one here).

The story is being made into a movie. Here is a trailer.

Wagner has an opera on the same. According to this link (again)

Tristan and Isolde is considered one of the most emotionally gripping operas ever written, and is, perhaps, Richard Wagner's greatest work.

To listen to it.
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update(12/21): Shyamala Narayan's Raja Rao: Man ans his works has the story of Tristan and Iseult. Also, it has a comparision of The Serpent and the Rope with The Waste Land by T.S.Eliot.

A much later update(01/14): According to the review in NYTimes and also other places, it seems the movie was not well made. Read the rest of this entry >>

Emerson on Indian texts

Emerson, has the following to say about the texts from India:

"In all nations there are minds which incline to dwell in the conception of the fundamental Unity. The raptures of prayer and ecstasy of devotion lose all being in one Being. This tendency finds its highest expression in the religious writings of the East, and chiefly in the Indian Scriptures, in the Vedas, the Bhagavat Geeta and the Vishnu Purana. These writings contain little else than this idea, and they rise to pure and sublime strains in celebrating it."(Essays, X, p. 120)

More impressions here. Reading the book: Vedic Religion and Philosophy by Swami Prabhavananda. The above quote was written on the book, possibly by the person who donated it to the library of Colorado College.

PS: As it is that time of the year, I have again ordered the book The Sermon on the Mount according to Vedanta by Swami Prabhavananda from library. Will post on the same. Read the rest of this entry >>

Criteria of Knowledge

In the summary of the book The Essential Vedanta : A New Source Book of Advaita Vedanta by Eliot Deutsch and Rohit Dalvi, there is a section in which the authors define the criteria of knowledge:

All knowledge is intrinsically valid. One can falsify a judgement by experience which is contradictory to it, but one can never completely verify a judgement by external means.

All knowledge acquired through the various pramanas [means of valid knowledge] is valid in its own proper sphere, but insofar as it is subject to contradiction by another qualitatively different kind of experience it is necessarily "relative" knowledge. Brahman-knowledge is alone incapable of contradiction.

There definitely are trends of similarity between these two statements and the statements of Godel's incompleteness theorem from logic.

I am not sure if these are general inferences that the authors make, or have translated them from definite sources in Vedanta. Yet to verify. Read the rest of this entry >>

Happy Gita Jayanthi!

From the preface of The Bhagavadgita by S.Radhakrishnan.

aum parthaya pratibodhitam bhagavata narayanena swayam
vyasena gradhitam purana muninam madhye mahabharatam
advaitaamrithavarshinim bhagavatim astadasadhyayinim
amba twam anusandhaninim bhagavadgite bhavadveshinim

Taught by the blessed Narayana Himself to Arjuna, compiled by Vyasa, the ancient seer, in the middle of the Mahabharatha, I meditate on thee, O Mother, O Bhagavadgita, the blessed, of eighteen chapters, the bestower of nectar of non-dualistic wisdom, the destroyer of rebirth.


samasta vedartha sarasangraha bhutam samasta purushartha siddim

This famous Gitasastra is an epitome of the essential of the whole Vedic teaching. A knowledge of its teaching leads to the realization of all human aspirations.

I find solace in the Bhagavadgita that I miss even in the Sermon on the Mount. When disappointment stares me in the face and all alone I see not one ray of light, I go back to Bhagavadgita. I find even a verse here and a verse there and I immediately begin to smile in the midst of overwhelming tragedies -- and my life has been full of external tragedies -- and if they have left no visible, no indelible scar on me, I owe it all to the teachings of the Bhagavadgita: M.K.Gandhi, Young India (1925)

My personal copies of Bhagavadgita (in no particular order):

• The one by Eknath Easwaran: simple just like the other writings of Eknath Easwaran.
  
  
• The extended three volume edition of the above by Eknath Easwaran: Vol. 1: The End of Sorrow, Vol.2: Like a Thousand Suns and Vol.3: To Love Is to Know Me .
  
  
• Bhagavad-Gita as it is by Bhakti Vedanta Swami Prabhupada, the originator of ISKCON: More emphasis on the Love of Krishna as a way to Peace. The book has very nice pictures.
  
  
• God Talks with Arjuna: The Bhagavad Gita by Paramahansa Yogananda : a very scholarly translation, I think the translation is in the Yogic tradition prescribed by the school of Shri. Yogananda. The book has very nice pictures. The introduction to this book has a Yogic interpretation of the story and characters from MahaBharatha war.
  
  
• Bhagavad-Gita: The Song of God by Swami Prabhavananda of the Ramakrishna order and founder of Vedanta society of Southern California. Very simple, very beautiful. A free translation. Has the NeoVedantic-Advaitic theory (this is a term Prof. Grimes and a couple of Philosphers use for the the school of Shri Ramakrishna Math) propounded by the Ramakrishna order.
  
  
• Bhagavad-gita by Sarvepalli Radhakrishnan. Very scholarly. One of my own first copies.
  
  
• Gita Makarandam by Shri Vidya Prakasananda Swami. I own a copy in Telugu. Very refreshing with both Advaitic and Bhakti overtones and is both quite scholarly and simplistic at the same time.
  
  
• The Gospel of Selfless Action or the Gita According to Gandhi by Mahadev Desai. Mahadev Desai, incidentally was described by Gandhi as his Lakshman. I read parts of it long time back.
  
  
• The Song Celestial by Edwin Arnold. A free poetic translation into English. Also said to be one of the first translations of Gita into English. Very simple ala the one by Swami Prabhavananda.
  
  
• The Song Celestial by Ramana Maharishi. Contains a subset of the verses, selected by Shri. Ramana Maharishi himself.
  
  
• The Bhagavad Gita, pocket edition by Swami Nikhilananda.
  
  

Read the rest of this entry >>

SaChin scores record century

Sachin (root: SaChin or Sat-Chin?) scored the 35th century the greatest by any cricketer in the final form of test cricket. More pointers at prem's blog. Read the rest of this entry >>

Shloka on Sankaracharya Bhagavadpada

sruti smriti purananam alayam karunalayam
namami bhagavatpadam sankaram loka sankaram

I salute the divine feet of the great Sankara,
the repository of sacred scriptures,
an abode of immense compassion,
who ever accomplishes the good of the world

Taken from the preface of The Vivekachudamani of Sankaracharya Bhagavatpada: An Introduction and Translation by John A. Grimes. John Grimes, begins the Vivekachudamani with the following Shloka:

Sarva-vedanta-siddanta-gocharam tam agocaram
govindam paramanandam sad-gurum pranto'smy aham

This is what Grimes says about the invocation Shloka:

Traditionally, Indian philosophical treatises begin with an invocation to God and/or one's Guru. Sankara, in this invocation, igeniously, insighfully and subtly reveals the non-duality of Advaita even as he offers his obeisance simultaneously to both God and Guru. He was able to do this because, one of the names for God is Govinda and the name of Sankara's Guru was algo Govinda. Intriguingly, this stanza simultaneously admits both interpretations. Underlying this play of language is the insight that God, Guru and the goal of life for the individual are not different; they are the same.

Read the rest of this entry >>

Surprising results in theory

In an excellent post by Bill Garasch and others on Lance's weblog, the surprising results of CS-theory are listed. The comments section has the opinions of some people on whether a result can be considered BIG or not. Read the rest of this entry >>

Words of great people

Why does the religion whose originator preached "love everyone" practise hate? Why are the followers of a religion which preaches "peace and universal brotherhood", is the cause for the greatest disturbance in today's world? Why is the religion whose originator promised "freedom of the mind", preach the greatest control of mind? Finally, why are people of a religion whose basic tenet is "there are many paths to God - find your own and, be at peace" being quite intolerant?

Without practising a particular religion, how do we go beyond religion? What leads us beyond dogma? How much do we have to progress in religion to be infinitely tolerant?

How and when do we understand the exact meaning of the great words, of the great people, who were originators of the great religions? How do we go beyond the duality that surrounds us, and is within most of our actions?

I donot have answers.

The Mahavakya from Aitareya Upanishad (from Rig Veda) "Pragyanam Brahman" is apt. If the great seers have experienced Brahman knowingly, and want to describe that Brahman to the common people, that Brahman which has been said to be -- among its various attributes -- indescribable, how do the common people, who have not experienced Brahman knowingly, know that the purport of the words of the seers in the right context? How do the common people, the multitudes, know that there is imperfection in description of Brahman by those seers themselves, those who were that wise to know that what they have experienced is Brahman? After all, Isn't Brahman what we experience everyday, whenever we are experiencing Truth, Consciousness or Bliss? This, however does not make us experience it knowingly as, even animals experience it unknowingly. Do the animals know what they are experiencing? Do the multitudes know what makes them happy? Experiencing Brahman knowingly is what differentiates the seers from the rest. How many of the wise seers, who were kind enough to come back and explain to the multitudes, know that there is bound to be imperfectness, some infidelity in their communication? If they did know, how many later tried to contradict their previous words, to explain their experience exactly? If so, how many of those seers were thought by the multitudes to lose their power of communication, lose their special touch, lose their magical words, lose their simplicity and thereby left to be old seers who have no more understanding of their own experience, than any other man? Later, how of the multitudes settled among themselves that, the words of the seers the first time they uttered were sacrosant, more sacrosant than any other human being, including the seers who uttered the words themselves? If so, why did it happen after some more time, that some of the multitudes turned skeptical of the words, the words which they themselves approved of before? Why did the skeptics began to feel that the words had lost their magic touch, lost the power which they had earlier? The seers themselves, either turned into skeptics themselves and thought to be madmen, or turned into sophists and fade into glory, or worst of all, saw the misunderstanding that happened and did not comment so that they can be adulated by the masses, knowing within their heart of hearts, that no one understood the words? What happened to the masses themselves, ones who remained faithful to the words and repeated them, but did not understand the purport? Forget the purport, the masses who did not even understand the meaning of the words as uttered by the seers, the first time they were uttered? The masses who did not even know that they did not know? What happened to all these types of people? How did history judge them? How did they judge themselves, just before their final breath? The seers, of course, being wise, unioned with the Brahman. What happened to the skeptics? What happened to the faithful?

I do not know.

I, however, remember the following passage from Siddhartha by Hesse:

...
Quoth Siddhartha: “I’ve had thoughts, yes, and insight, again and again. Sometimes, for an hour or for an entire day, I have felt knowledge in me, as one would feel life in one’s heart. There have been many thoughts, but it would be hard for me to convey them to you. Look, my dear Govinda, this is one of my thoughts, which I have found: wisdom cannot be passed on. Wisdom which a wise man tries to pass on to someone always sounds like foolishness.”

“Are you kidding?” asked Govinda.

“I’m not kidding. I’m telling you what I’ve found. Knowledge can be conveyed, but not wisdom. It can be found, it can be lived, it is possible to be carried by it, miracles can be performed with it, but it cannot be expressed in words and taught. This was what I, even as a young man, sometimes suspected, what has driven me away from the teachers. I have found a thought, Govinda, which you’ll again regard as a joke or foolishness, but which is my best thought. It says: The opposite of every truth is just as true! That’s like this: any truth can only be expressed and put into words when it is one-sided. Everything is one-sided which can be thought with thoughts and said with words, it’s all one-sided, all just one half, all lacks completeness, roundness, oneness.
...

What is the proof that me, who wrote these words, me who felt all these in a spirit of inspiration, I will understand the message in the above words after some time, whatever it is? I would, I truly would, if the words were written with true Brahman experience. Why do great seers from India pray for smriti? Why do seers pray for everlasting experiences? Read the rest of this entry >>

Geometric puzzle: division of k-d space using (k-1) dim objects

Given 2 points, how can you divide a a line so that you maximize the number of regions?

Given 3 lines, how can you divide the 2 dimensional space (a plane) so that you maximize the number of regions?

Given 4 planes, how can you divide the 3 dimensional space so that you maximize the number of regions?

Can you think of a way to divide a k-dimensional space by (k-1) dimensional objects, so you maximize the number of regions?



Solution: build a k-dimensional solution from a (k-1)-dimension solution. Read the rest of this entry >>

Introduction to Ashtavakra Gita

Once upon a time there was a student of the scriptures. He would work hard all day every day and then read aloud the holy language of sacred verses late into the night. His wife, round of belly with their coming child, would sit beside him in the dim room, listening as her weary beloved chanted the ancient words.

One late night in her eighth month a voice from inside her belly said to the father: "Sir, please be attentive – you are mispronouncing that verse." Tired and short-tempered, without thinking why he would feel so enraged at being corrected by an unborn child, the father cursed the voice- and because the father had built up merit, his curse took hold: the child was born deformed, with eight crooks in his body. That child was called Ashtavakra, a name which means `eight bends'. Everyone who saw him laughed in derision.

That crippled child was an enlightened master who took birth in this family to reveal in simple words the essence of mystical experience. Janaka, king of the known world, father of the bride of God, Sita, daughter of the earth, that very King Janaka became this crippled boy's disciple. The book based on that event is called The Song of the Eightfold Cripple, or Ashtavakra Gita.

Asthavakra was not keen on accepting students, and so had few. When King Janaka came to hear of the wisdom of the crippled child he approached the boy as a humble student, not a commanding king. The boy accepted the king instantly as his disciple. This caused some talk in the sangham. ~Ah, Ashtavakra does have favorites after all, he accepted the king without any of the trials he had all of us face!~ This grumbling became a quiet force, and Ashtavakra knew of it.

One day the King was late and so the boy delayed his discourse. The moment the king arrived, Ashtavakra spoke: `This day I have had a vision, the capitol city will erupt in terrible fires and earthquakes- all there will die. Those who have loved ones or valuables there must hurry now if they wish to save anything!'

All the monks left. As the dust settled, only the boy and the king were sitting. The boy said softly, `Great king, is there nothing you would save?` Janaka replied, ~My lord and my friend, you are my only treasure.~ The cripple nodded and softly said, ~Well then if I am indeed your treasure, mount your horse now and go and gather my students back to me, tell them I have been mistaken, the capitol city is in no danger. Take your horse, and go.~

Rising to do as bidden, the King put his foot into the stirrup, and as he swung up over the saddle, realization dawned in his mind. He swallowed, looked about him at this new earth, heard new birds singing for the first time, and then looked at the cripple at his feet. The two looked at one another, and then the king left to find the other students.

Once back, the other students grumbled at being sent about here and there on foolish errands. One or two howeverd did soon understand why the master had chosen the king as a student in his own way.

This is what was said that day, as all sat about and heard these words of nectarine wisdom. Read the rest of this entry >>

Aryan Invasion Theory, Amitav Ghosh

Now, even the BBC link seems to say so. As if someone believed in it! (link via santosh on the advaitin mailing list)

Amitav Ghosh write a very powerful and moving essay about the 1984 riots (link via Amit Verma).

Incidentally, Amitav Ghosh seems to be quite an interesting author. He is a recipient of the annual award by the Gyanpeeth (Is it the same as The Gyanpeeth award?).
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updated (11/22)
Some very good suggestions on teaching about Hinduism by Rosser. Link by Shadow Warrior. Read the rest of this entry >>

Translations by Swami Krishnanada

The Divine Life Society has lots of ebooks of Vedantic texts. The translations are done by Swami Krishnananda. The translation of BrahmaSutras (PDF link) has a very informative introduction of the schools of Indian philosophy (darsana) and also of the view points of the Acharyas who wrote commentaries of the BrahmaSutras.

The BrahmaSutras are part of the prasthatraya, the three sacred texts. It is said that a person wishing to start a new school of thought must write commentaries on the prasthanatraya as every Vedantic school builds from these. The prasthanatraya consist of the Upanishads (srutis, or revelation), the Bhagavad Gita (being the most important text of the smriti, recollection) and the BrahmaSutras (being of the category sutra, system).

The word sutra itself is etymologically related to sew and thread and hence means some kind of memory aid between different texts or thoughts. The sutra from Vedantic schools does not mean sutra the way Buddhist schools mean it. The BrahmaSutras are said to be terse and initially seem to contain contradictory thoughts. Swami Krishnananda says the following about them:

Sutras are concise aphorisms. They give the essence of the arguments on a topic. Maximum of thought is compressed or condensed into these Sutras in as few words as possible. It is easy to remember them. Great intellectual people only, with realisation, can compose Sutras. They are clues or aids to memory. They cannot be understood without a lucid commentary (Bhashya). The commentary also is in need of further elaborate explanation. Thus the interpretations of the Sutras gave rise to various kinds of literary writings such as Vrittis (gloss) and Karikas. The different Acharyas (founders of different schools of thought) have given their own interpretations of the Sutras to establish their own doctrines. The Bhashya of Sri Sankara on Brahma Sutras is known as Sariraka Bhashya. His school of thought is Kevala Advaita. The Bhashya of Sri Ramanuja who founded the Visishtadvaita School is called Sri Bhashya. The commentary of Sri Nimbarkacharya is known as Vedanta- parijata-saurabha. Sri Vallabhacharya expounded his system of philosophy of Suddhadvaita (pure monism) and his commentary on the Brahma Sutras is known as Anu Bhashya.

Also the following about the need to write commentaries on the classical texts:

Those who wish to study the philosophy of Vedanta should study the Ten Classical Upanishads and the Brahma Sutras. All Acharyas have commented on Brahma Sutras. This is a great authority for every philosophical school in India. If any Acharya wishes to establish his own cult or sect or school of thought he will have to write a commentary of his own on Brahma Sutras. Then only it will be recognised.

Writing the commentaries on the PrasthanaTraya started from Sri Adi Sankara, who started the Kevala Advaitic school. Sri Swami Krishnananda uses the term uncompromising monism, though I think Advaita or non-dualism is possibly a better word. The following is also from the introduction.

According to Sri Sankara, there is one Absolute Brahman who is Sat-chit-ananda, who is of an absolutely homogeneous nature. The appearance of this world is due to Maya - the illusory power of Brahman which is neither Sat nor Asat. This world is unreal. This world is a Vivarta or apparent modification through Maya. Brahman appears as this universe through Maya. Brahman is the only reality. The individual soul has limited himself through Avidya and identification with the body and other vehicles. Through his selfish actions he enjoys the fruits of his actions. He becomes the actor and enjoyer. He regards himself as atomic and as an agent on account of Avidya or the limiting Antahkarana. The individual soul becomes identical with Brahman when his Avidya is destroyed. In reality Jiva is all-pervading and identical with Brahman. Isvara or Saguna Brahman is a product of Maya. Worship of Isvara leads to Krama Mukti. The pious devotees (the knowers of Saguna Brahman) go to Brahmaloka and attain final release through highest knowledge. They do not return to this world. They attain the Nirguna Brahman at the end of the cycle. Knowledge of Nirguna Brahman is the only means of liberation. The knowers of Nirguna Brahman attain immediate final release or Sadyomukti. They need not go by the path of gods or the path of Devayana. They merge themselves in Para Brahman. They do not go to any Loka or world. Sri Sankara’s Brahman is Nirvisesha Brahman (Impersonal Absolute) without attributes.

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Who can show the right way?

Al (of advai.blogspot.com) was intent on proving to a certain community that their thinking of "their way to God is the only way" may not be right. I asked him to stop doing so, as no ordinary person can tell another what the right way is. Also, there is a futility in trying to convince a person who is unwilling to listen.

Initially it seems that, no ordinary man can show us the right way. We feel that we are atleast better that the multitudes who did not even begin their search for realization. So we feel that, only a realized soul, one who is somehow special, can lead us. So most of the quests for realization begin with a search for the Guru. The goal is clear: the Guru, or the Prophet, can lead us by some kind of magic. (This thought is also echoed in the last pages of the The Serpent and the Rope by Raja Rao.)

After some time, it also becomes apparent that, a realized soul cannot show a man the right way, if the man is unwilling to listen, or see things with an open mind. If one realized soul cannot show a man the right way, another realized soul cannot. This is because, no realized soul is greater than another.

It also become apparent that, when a man is willing to see the point and is doing effort to search for the truth, he does not need a realized soul to show him the right way. The right way manifests itself, as it was before him all the time. The moment a man is willing to search for the truth, there comes a time when his veil of ignorance automatically falls off. A realization dawns that a Guru is not doing some magic, but helping to remove the veil.

The where the purpose of a Guru is to just quicken the process. A Guru can reduce the amount of time a yearning soul has to spend in the searching. The purpose of Ishwara is also similar. As Ishwara has control on Maya, He too quicken the process. When I say an Ishwara, I mean the different manifestations of higher powers. They could be from the six Indian theistic schools. Or, they could be the Judeo-Christian-Islamic God.

Hence the sayings of Vivekananda: God is very merciful to those whom He sees struggling heart and soul for spiritual realization. But remain idle, without any struggle, and you will see that His grace will never come.

So, once we understand all this, we have a couple of questions:

What is the right thing for an ordinary man to do with respect to his realization: Should he search for a Guru or not? Should he pray to Ishwara or not?

What is the right thing for an ordinary man to do for the realization of his peers? Should he give them the insights he got? If so, how is he to know whether they are right or wrong? Should he correct them when he feels they are wrong? How does he know for sure whether he is right or wrong? from Plato: What is Good and What is bad, do we need someone to tell us Phaedrus?

What is the right thing for a realized man to do? Is this a trick question? A realized man is beyond right and wrong. What can a realized man do for others' realization? Just guide them, if he wants to.

The Maya of a person can be removed only by Ishwara. Can the Avidya of a person can be removed by a Guru? Possibly. The distinction between Maya and Avidya is possibly quite subtle, as it has caused considerable discussion among the disciples of Sri Adi Sankara. So I will not confuse the picture by posting on them now. Read the rest of this entry >>

Dalai Lama on Science vs. Religion

As a Op-Ed Contributor on the NYTimes, in an article titled Our Faith in Science, His Holiness Dalai Lama (who signs the article as Tenzin Gyatso) has the following to say about religion and science:

If science proves some belief of Buddhism wrong, then Buddhism will have to change. In my view, science and Buddhism share a search for the truth and for understanding reality. By learning from science about aspects of reality where its understanding may be more advanced, I believe that Buddhism enriches its own worldview.

==
Later update (11/14): Also see the letters to the editor in which a writer points out the following:

Instead of focusing on whom to hate and what to fear, these leaders would do well to turn their attention to those "principles we share as human beings" about which the Dalai Lama writes so eloquently.

I am not a Buddhist, but I cannot help noticing how often it is the Buddhists who remind us that we do not have to demonize others in our own search for what is good.

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Thoughts about the AUEE deparment

In an excellent post, SJ recounts some of the experiences he had in the EE department of AU. That is, when the department was in its heydays. I cannot agree with the post more. Read the rest of this entry >>

On Having No Head: Douglas Harding

I have read the book On Having No Head: Zen and the Rediscovery of the Obvious by Douglas Harding, sometime back. Like many Zen concepts, the title is both mysterious and informative at the same time.

This very short book (120 pages: with 12pt font and 1.5 spacing) talks about the experience of realization by Douglas Harding. It (the concept of realization) is explained to a modern day human being with western background using the simplest of Buddhist -- mainly from Zen -- concepts. As the title implies, the essence is in "experiencing" headlessness, with the emphasis on experience. Just like many Zen koans, the book carries the essence in the first few pages. All these reasons make the descriptions in the book simple and striking.

Though I am yet to read the book completely, I felt I need not push myself to do so. I found that couple of first chapters contain the crux of the experience of the author. (This is probably why I could not push myself to complete the book.) I also feel that the two ways of experiencing realization are when you experience zero (shunyata) or you experience infinity (purnata).

The following is the first chapter. I found that there is no point highlighting some part of the text. The text is very short, cohesive and all the text is equally important.

The best day of my life -- my rebirthday, so to speak -- was when I found I had no head. This is not a literary gambit, a witticism designed to arouse interest at any cost. I mean in in all seriousness: I have no head.

It was when I was was thirty-three that I made the discovery. Though it certainly came out of the blue, it did so in response to an urgent inquiry; I had for several months had been absorbed in the question: what am I? The fact that I happened to be walking in the Himalayas at the time probably had little to do with it; though in that country unusual states of mind are said to come more easily. However that may be, a very still, clear day, and a view from the ridge where I stood over misty blue valleys to the highest mountain range in the world, made a setting worthy of the grandest vision.

What actually happened was something absurdly simple and unspectacular: just for the moment I stopped thinking. Reason and imagination and all mental chatter died down. For once, words really failed me. I forgot my name, my humaneness, my thingness, all that could be called me or mine. Past and future dropped away. It was as if I had been born at that instant, brand new, mindless, innocent of all memories. There existed only the Now, that present moment and what was clearly given in it. To look was enough. And what I found was khakhi sleeves, terminating sideways in a pair of pink hands, and a khaki shirtfront terminating upwards in -- absolutely nothing whatever! Certainly not a head.

It took me no time at all to notice that this nothing, this hole, where a head should have been, was no ordinary vacancy, no mere nothing. On the contrary, it was very much occupied. It was a vast emptiness vastly filled, a nothing that found room for everthing -- room for grass, trees, shadowy distant hills, and far above them snow-peaks like a row of angular clouds riding the blue sky. I had lost a head and gained a world.

It was all, quite literally, breathtaking. I seemed to stop breathing altogether, absorbed in the Given. Here it was; this superb scene, brightly shining in the clear air, alone and unsupported, mysteriously suspended in the void, and (and this was the real miracle, the wonder and delight) utterly free of "me", unstained by any observer. Its total presence was my total absence, body and soul. Lighter that air, clearer that glass, altogether released from myself, I was nowhere around.

Yet in spite of the magical and uncanny quality of this vision, it was no dream, no esoteric revelation. Quite the reverse: it felt like a sudden waking from the sleep of ordinary life, an end to dreaming. It was self-luminous reality for once swept clean of all obscuring mind. It was the revelation, at long last, of the perfectly obvious. It was a lucid moment in a confused life-history. It was a ceasing to ignore something which (since early childhood at any rate) I had always been busy or clever or too sacred to see. It was naked, uncritical attention to what had all along been staring at my face -- my utter facelessness. In short, it was all perfectly simple and plain and straightforward, beyond argument, thought, and words. There arose no questions, no reference beyond the experience itself, but only peace and quiet joy, and the sensation of having dropped an intolerable burden.

In the next chapters, Harding takes the concept further and gives vivid details of what happened to him after experiencing headlessness. Also he explains various levels of headless ways. The website The Headless way also has nice information about the experience. The videos are also good.

The following are a set of good "koanistic" quotations from the book:

• Suppose a man were all of a sudden to make his appearance here and cut your head off with a sword! -- HUI-CHUNG
  
• Behead yourself! ... Dissolve your whole body into Vision: become seeing, seeing, seeing! -- RUMI
  
• My Soul has been carried away, and usually my head as well, without my being able to prevent it. --ST.TERESA
  
• Cover your breast with nothingness, and draw over your head the robe of non-existence. --ATTAR
  
• Give yourself utterly ... Even though the head itself must be given, why should you weep over it? --KABIR
  
• Seeing into Nothingness - this is the true seeing, the eternal seeing. -- SHEN-HUI
  
• "I think I'll go and meet her," said Alice ... "You can't possibly do that," said the Rose. "I should advise you to walk the other way." This sounded nonsense to Alice, so she said nothing but set off at once towards the Red Queen. To her surprise, she lost sight of her in a moment. -- Through the Looking Glass.
  
• The soul has now no awareness of the body and will give herself no foreign name, not man, not living being, nor anything at all. -- Plotinus
  
• After the body has been cast off to a distance like a corpse, the Sage never attaches himself to it. -- Sankara
  
• If one opens one's eyes and seeks the body, it is not to be found anymore. This is called: In the empry chamber it grows light. Inside and outside, everything is equally light. That is a very favorable sign. -- The secret of the Golden flower.
  
• Vow to acheive the perfect understanding that the illusory body is like dew and lightning. Zen Master Hsu Yun (on his death-bed, in 1959)
  

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Happy Birthday, Raja Rao!

Raja Rao (08-November-1908 to ?) would be 98 years today. Arguably, he is one of the greatest Indian English authors. He ranks along with Mulk Raj Anand and R.K.Narayan and the three are aptly called the original trinity. The next generation of Indian English writing had to wait for around 20-30 years till Salman Rushdie appeared. The comparision between Raja Rao with Salman Rushdie is reasonable. A better comparision of Raja Rao could be with James Joyce, as both of them created a new way of writing and then expressed themselves excellently in the new way.



His way of writing is unique. The punctuation itself is so close to the the way Indians talk and think. The following, from the preface of his first novel Kanthapura, explains about his way of writing.

...
The telling has not been easy. One has to convey in a language that is not one's own the spirit that is one's own.
...
After the language the next problem is that of style. The tempo of Indian life must be infused into our English expression, even as the tempo of American or Irish life has gone into the making of theirs. We, in India, think quickly, we talk quickly, and when we move, we move quickly. There must be something in the sun of India that makes us rush and tumble and run on. And our paths are interminable. The Mahabharatha has 214,778 verses and the Ramayana 48,000. The Puranas are endless and innumerable. We have neither punctutation nor the treacherous "ats" and "ons" to bother us -- we tell one interminable tale. Episode follows episode, and when our thoughts stop our breath stops, and we move on to another thought. This was and still is the ordinary style of our storytelling....
...

Beyond the use of language in style and punctuation, his scholarly way of writing can put people in his awe. I would like to think that if Sri. Sarvepalli Radhakrishnan would be a novelist, he would write the way Raja Rao. Raja Rao's careful use of Advaitic concepts in The Serpent and the Rope makes it, what Arvind Sharma calls, India's most famous Advaitic novel.

Some biography snippets from the preface of Makarand Paranjape's book.

Raja Rao was born in an ancient and respected Brahmin family in Hassan, Karnataka, on 8th November, 1908. The eldest son in a family of two brothers and seven sisters, he was the centre of the family.
...

The Introduction chapter in the book by Shyamala Narayan gives another date of birth (the text also implies that there has been a confusion regarding the year of birth too). The following is the relevant excerpt from the book.

Raja Rao was born in Hassan, a small town in the state of Mysore (now called Karnataka). There is some confusion about Raja Rao's date of birth: his books mention it as 1909, but he was actually born on November 5, 1908, as M.K.Naik clarifies. There is an interesting story about his birth. He was born at the precise moment when his father was receiving the Maharaja, Krishna Raja Wodeyar of Mysore, at their house, So the child was called Raja and not Ramakrishna, after his grandfather. Raja Rao comes from a family of Brahmins who had been Vedantins and advisors to kings for generations. He was greatly influenced by his grandfather at Hassan, who taught him to love Sanskrit and kindled his interest in Indian Philosophy.
...

The date of birth given in the following link link also agrees with the one given by Makarand Paranjape.

Raja Rao was born on November 8, 1908 in Hassan, in the state of Mysore in south India, into a well-known Brahman family. ...
...

Autobiographical rant:

He has been of huge influence on me. My first memories of any of his works was looking at his classic book The Serpent and the Rope in my home. The cover had a picture of a beautiful European lady. Behind her was a dark man from India with a cigaratte in his hand. The background to this picture was the waters of Benares. It was one of the books that my mother used in her M.A (literature) and had since become part of the family library.

In one of the first classes of first year of engineering, one of our professors, Prof. P.V.Ratnam, who I admire a lot, had suggested us to read this book. I came home and started it right away "I was born a Brahmin. Brahmin is one who knows Brahman and all that..." The magic in the words, however, could not force me to read beyond a couple of pages and so I kept it aside. However, the book was never far away from me or my personal cupboard. I did not keep the the book away probably because I thought this was just a book like M.G.Say or Clayton-Hancock or Van Valkenburg (the classic books of EE) which needed a couple of careful readings to understand it. Alas, it did not occur to me that to understand the book, all that was needed was a mature reader. Also, that the book was recommended by Ratnam-garu made it a top of the list book.

I intuitively understood that, being unable to understand beyond a couple of pages of a novel which presupposes a good understanding of my beliefs is a serious deficiency on my part.

In the summer vacation between first and second years, I had read a couple of novels mostly R.K.Narayan (quite a few of them) and Serpent and the Rope.

When I went for higher studies to Bangalore, this was one of the books that I took with me -- inspite of the fact that I knew I may not have much time for novels -- mainly because I felt that this book had some magic in it. I was enchanted by the first lines.

Later in Bangalore, when I was in a book reading spree, I began Raja Rao with earnst. This time, I did some research and began with Kanthapura. Next I read The Serpent and the Rope. This time, the words in the book started to make sense. I continued and completed the novel. I also read another novel, The Cat and the Shakespeare. When I left to US, I still had my mother's copy with me. I read the book in the summer of 2003. I donot think I read it again in the next two years.

I began with a earnst study of Advaita as such and had read Deutsch, Arvind Sharma and some books on Ramana Maharishi.

In the Sep, OCT-2005, when I had a fracture of the hand and my movement was restricted, I read the book again, and the book made lot of sense and I could understand many of the quotations and monologues.
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How not to still the mind

Once King Janaka was sitting on the bank of a river, repeating 'SOHAM' mantra at the top of his voice.

Sage Astavakra was passing by. The sage was a knower of truth and an enlightened being. He was surprised why king Janaka was chanting the mantra in this loud fashion.

Astavakra wanted to instruct King Janaka the proper way of chanting a mantra. So, he sat down near King Janaka and chanting loudly, :' "This is my Alms Bowl and this is my Yoga Stick " so much so that Astavakra's chanting drowned the chanting of Janaka's chanting!

Now, King Janaka , not to be left behind starting chanting his 'SOHAM' mantra even louder --- this went on for some time -both vyeing with each other in this 'mantra recitation'. ...

Now, King Janaka got really annoyed and asked Asthavakra what the sage was doing. The sage replied "I am repeating - this is my Alms bowl, this is my yoga stick".

King Janaka replied, " Have you lost your mind? Who told you that the Alms bowl and the stick do not belong to you ? Why do you have to keep shouting about it.?"

Sage Astavakra replied , " O Mighty King ! It seems to me you are the one who lacks understanding. who told you that you are not 'THAT '? Why do you to go on shouting that 'I AM THAT' ? 'SOHAM' ?

When king Janaka heard this, he suddenly realized the trut. He understood that he need not go on repeating 'SoHam' mantra , he only needed to understand it and practice 'living' it!

'You are really unbound and actionless, self-illuminating and spotless already. The cause of your bondage is that you are still resorting to stilling the mind.' -- Ashtavakra Gita 1.15 Read the rest of this entry >>

Story of Sudhama as narrated by Raja Rao

The princes of Kathiawar, like the Rajputs everywhere, claimed their descent from the sun and the moon, and belonged to the clan of Sri Rama (the Ikshavakus) or of Sri Krishna (the Yadavs). For, after all, Dwaraka was just around the corner of the peninsula, and that was where Sri Krishna ruled, so that most of the princes of course belonged to the Yadavas, or so they believed. Remember, Sri Krishna married his wife Rukmini at Madhavpur, and that is just down the curve of the bay, and the Yadavs once (and remember too) on that fateful day when they went gambolling to the sacred city of Prabhas Patan and got into a fist-fight, and then to a brawl and then to a real battle, they sent for Sri Krishna.


But he was not to be found -- he'd so planned as to go and lie under a tree (and had not Gandhari cursed that the race of Sri Krishna come to an end) -- and a hunter took Sri Krishna's heel for the head of a deer, and the greatest of Indians thus played the game of even death. He had planned it all, so everything happened according to his intent -- for, for him intent and action were just experience, there was none to act, as it were for he was in action as inaction and in inaction as action.

And so too was that other game of Sri Krishna. He had, while at school, a poor Brahmin companion, Sudhama was his name and he was raggedly miserable. Life had been most ungenerous to him (again it was Sri Krishna's game) -- Sudhama had a wife with a tongue of charcoal cinders and hair made as of rough hemp. When she called, even the street dogs would thgink it was one if their kind talking, and they started answering back. But Sudhama was so patient; he would always speak, when humiliated by his wife, of his friend Sri Krishna who ruled in Dwaraka. "Fine thing to have such a friend. But what does he do for you, you son of tamarind tree hag."

Hearing this morning after morning, Sudhama said one day to his wife: "I go and see Sri Krishna today, this very day."

"Oh yes you will go, you and your cononut-head and bursted-bleak belly, and seeing your tousled hair and bent back and bamboo limbs, even the guards will laugh at you."

"Perhaps you're right," said Sudhama, "but give me please a handful of puffed rice and a piece of molass."

"What for?" She asked.

"Well, when ones goes to see someone, you don't ever go empty handed, do you?"

"A beggar," declared Sudhama's wife, "has no manners," as the buffalo have no courtesies."

"Well, so be it. Give what puffed rice and molass you have to me."

Cursing her stars for such a rice miserable destiny -- that even the little puffed rice she had, she had perforce to give part of it away -- she threw a handful of it to Sudhama's dhoti-edge, broke a piece of molass from the pot, and said with wide-fingered threatening hands: "And don't you come back plain-handed, you cononut-head, after all this. I'll howl till the very spirits in the crematoriums will be frightened. Let's see what your great friend Sri Krishna is going to proclaim and perform for you. I take the name of God and tell you he will have chased out of city."

Sudhama was so accustomed to his wife's tongue and breath, it was as if he heard sound but not meaningful speech. He picked his stick from the corner, and taking the thought of Sri Krishna, he started towards holy Dwaraka. Long was the road to Dwaraka, but such sweetness of love, the trees seemed to open up and spread wide-limbed shade, and cool breezes blew from the northern mountains, and the sea seemed to churn in fervered joy. For when you take the thought of Lord true, all things that seek him rejoice in your rejoicement -- they also wave up and swell with your joy. Sudhama arrived at the city gates and these opened themeselves as if by vestal deities, and even the palace guards did not seem to mind his looks.

"Could you tell His Highness," he begged, "the Lord of Lords, his poor school fellow, the Brahmin Sudhama is at the door?"

"Be so kind as to wait here, Pundit Sir," said the guard politely, and "I'll go tell the Chamberlain."

It was all as if the gods were playing a trick, the Chamberlain seemed just waiting for this event at the door, and Sri Krishna himself, when he heard the news, he had such joy, people could see tears run down his lotus eyes. "Prepare," he commanded, "water to wash my guest's feet." And as Sri Krishna clad in silken blue, his eyes clear as sleep, his gait as if made by the curvature of form when he stood, the washing-stool before him, there he appeared across the courtyard, Sudhama, his boyhood friend. In their high niches the pigeons and parrots began to coo, and the ladies peered from the top apartments at this wondrous happening [You can see this in many a Rajput miniature today.]. Krishna bade Sudhama stand on the ivory stool, and rich with jugs of silver swan-shapen and filled with water, first warm and then cold, and when these were thrown on the Brahmin's withered and dusty feet -- then did Krishna taking the silken hand-cloth from Chamberlain's arm, wipe the friends feet himself. Now Sri Krishna took his friend to the marbled court and before all his noblemen, he said, "Friend, be seated," and showed Sudhama the throne. Could a poor Brahmin occupy such an august seat? No. He would not. Krishna himself sat on a couch beside his friend. They now spoke to each other of all that happened and passed by since their boyhood. They wept and laughed but all around there was as if a wall of luminescent silence. Everything moved in the palace as usual, the noblemen retired to their afternoon siesta, the servants moved with agility and calm, from corner to corner of the palace, the nine diurnal musics sounded, the elephants trumpeted after a good feed, the cows lowed for their returning calves, and when the evening fell a great banquet was laid for the poor Brahmin, Sudhama and he ate as if he was eating his own food, at his home. For, for the first time he was at home himself. Krishna enjoyed munching the puffed rice his friend had brought, and some of it was sent upstairs to Rukmini, and all the palace was given bits of it. It tasted, did this puffed rice, as nothing one has tasted before. It had the delicate saffron smell of Sri Krishna himself. And when the night drums sounded and the meal was over, the hands were washed, the betel leaves served, a carriage was made ready, and beneath the high chandeliers, flower and fruit hangings, as Sri Krishna said farewell to his friend, he embraced him again and again, and they wept. Yes, Sudhama had to go. He had now to go back home. But he'd forgot his promise to his wife. He had asked nothing of Sri Krishna., Pray what can you ask of a friend who is King?

Yet Sudhama was so happy the very mind-picture of Sri Krishna brought him to tears. And when he came nearer and nearer his home, he began however to have fits of fearfulness. What would it be like coming back plain handed? But when the town came it was all a different -- the very walls were shining as if white-washed and much repaired, and when his carriage stopped, his wife stood, a gold plate in her hands, flowers and Kumkum water floating in it, and lighting the lamp of auspiciousness, she welcomed her lord befittingly, and fell at his withered feet. And from that day onwards the city came to be called Sudhamaputi, or the city of Sudhama, but due to the crookedness of people's speech, it became Puri, and later someone added bunder[Bunder means harbour, haven] to it so that today it's called Porbandar, Haven-city.
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Mahavakyas of Advaita or Vedanta

The Mahavakyas -- or great sayings or principal statements -- from Upanishads are said to contain the essence of the teachings of Upanishads. It is said that meditating on the Mahavakyas would lead someone towards realization.

Pranava (Aum): Pranava is said to be the fundamental (true) Mahavakya of the Vedas as established in Chapter Twelve. (details?)

For a particular Vedantic system, some of the Mahavakyas are considered more important than the others. This is inspite of the fact that all schools in Vedanta accept Upanishads and all the Mahavakyas are from the Upanishads.

According to Arvind Sharma, Sri Adi Sankara stated that four Mahavakyas are fundamental for Advaitic learning. A web search also shows that Sri Adi Sankara set up the four schools, one for each Mahavakya. (Yet to find out which school stands for which Mahavakya.) They are the following:

1. Sarvam Khalvidam Brahma: meaning All this verily Brahman. This is from Chandogya Upanisad, 3.14.1.
   
   
2. Aham Brahmasmi: I am Brahman. This is from Brhad-aranyaka Upanisad, 1.4.10.
   
   
3. Ayam Atma Brahma The Self is Brahman or This Atman is Brahman. This is from ?
   
   
4. Tat tvam asi: That thou art. This Mahavakya is from Chandogya Upanisad, 6.8.7 (Tat tvam asi svetaketo, "O Svetaketo, you are that")
   
   

Additional Mahavakas from Vedanta. Some of them are:

Prajnanam Brahma: Brahman is Supreme Knowledge prajnanam brahma, This is from Aitareya Upanisad, 1.5.3.

So'ham: He am I. This is from ?? Read the rest of this entry >>

Story of Radha, Brahmachari-Krishna and Upavasi-Durvasa: Raja Rao

One day Radha had a very possessive thought of Krishna. "My Krishna", she said to herself, as though one could possess Krishna as one could possess a calf, a jewel. Krishna, the Absolute Itself, immediately knew her thought. And when the absolute knows, the knowing itself, as it were, is the action of the act; things do not happen according to his wish, but his wish itself is his own creation of his wish, as the action is the creation of his own action.



So, Durvasa the great Sage was announced.

"He is on the other side of the river, Lord", spake the messengers, and "and he sends his deep respects".

Then Krishna went into the deep chambers and said to Radha, "Radha, Durvasa the great Sage is come, my dear. We must feed him".

"Oh, then I will cook the food myself" said Radha, and Krishna was very happy at this thought. So he went back to the Hall of Audience, and not long after, Radha came in with all the cooked food. "Yes, the meal is ready my Lord. And I will take it myself to Sage Durvasa".

"Wonderful, wonderful!" exclaimed Sri Krishna, pleased with the devotion of his wife to the Sages.

"I'll go and come," said Radha, and hardly had she gone to the palace door when she remembered the Jumna was in flood. No ferryman would go across. She came back to Krishna and begged, "My Lord, how can I take the food? the river is in flood".

"Tell the river," answered Krishna, "Krishna the brahmachari [The celibate, or who has taken the vow of celibacy] wishes that the way be made for you to pass through."

And Radha went light of heart, but suddenly bethought herself it was a lie. Who better than she to know whether Krishna be brahmachari or not? "Ah the noble lie, the noble lie," she said to herself, and when she came to the river, she said, "Krishna, the Lord, the brahmachari, wishes that way be made for me to pass through".

And of course the river rose high and stood still, but suddenly opened out a blue lane, small as a village footpath, through which Radha walked to the other side. And coming to the opposite shore, she thanked the river, and saluting the great Sage Durvasa, in many a manner of courtesies and words of welcome, spread the leaf and laid him the food.

Durvasa was mighty hungry and he ate the food as though the palm of his hand went down his gullet. "Ah, ah," he said and belched and made himself happy, with curds and rice and many meats, perfumed and spiced with saffron, and when there was nothing left in leaf or vessel, he rose, went to the river and washed his hands. Radha took the vessels to the waters, too to wash, threw the leaf into the Jumna and stood there to leave. Then it was she who remembered, the river was in flood. Sri Krishna had told her what to say while going and not what to utter while coming back.

Durvasa understood her question before she asked - for the sages have this power too -- and he said, "Tell the river, Durvasa the eternal upavasi [He who fasts] says to the river, 'Open and let Radha pass through to the other shore.'"

Radha obeyed but she was more sorrowful. "I have seen him eat till his palm enter his gullet, and he has belched and passed his hand over his belly with satisfaction. It is a lie, a big lie," she said, but she went to the river thoughtful, very thoughtful. "River," she said, "Durvasa who is ever in upavasa says open and let me pass."

And the river opened a lane just as wide as a village pathway, and the waves held themselves over the head, and would not move. She came to the other shore and returned to the palace in heavy distress. "Yes, nature is a lie, nature believes and obeys lies. Lord, what a world," She said to herself and going into the Hall of Sorrowing, shut herself and began to sob. "Lord, what a lie the world is, what a lie."

Sri Krishna knew the cause and cadence of this all, and gently entered the Hall of Sorrowing. "Beloved, why might you be in sorrow?" he said.

"My Lord," she answered, "the river believes you are a brahmachari, and after all who should deny it better than me, your wife? and then I go to Durvasa and he eats with his palm going down his gullet, and he says, "Tell the river, Durvasa who is ever in upavasa asks you to open and let Radha pass." And the river opens herself, makes a way large as a village pathway, and I pass over to this side. The world is a fib, a misnomer, a lie."

"The world, my dear, is not a lie, it is an illusion. Besides, tell me, is my body your husband, Radha?"

"No, my Lord."

"Is my mind your husband, Radha?"

"No, my Lord."

"Then what is it when you say to yourself, 'Krishna, my husband?'"

"Assuredly something beyond the body and beyond the mind -- the Principle."

"And tell me, my love, can you possess that, can you possess it?"

"No, my lord, how can I possess the Absolute? The I is the Absolute." And she fell at the feet and understood, and lived ever after in the light of the Truth.
Read the rest of this entry >>

Illusion

Via Atanu, A very very cool illusion!! It is difficult to believe what is it that we see. Read the rest of this entry >>

Terms and Tennis Court Link

Some Useful Terms

A Polytope is different from a Polygon, which is a 2-dim figure. A Polytope is defined by as a Convex Hull of a set of vertices or a Cone of a set of halfspaces.

There are two main objects: a H-Polyhedron and V-Polyhedron. The bounded versions are the H-Polytope and the V-polytope.

Of course, there is the cone, which is the homogenized version of a polyhedron.

More about cones:

A Convex Cone can come in many varieties. A Blunt Cone does not contain the origin, while the origin belongs to a Pointed Cone. If the lineality space of a cone has dimension zero, is said to be pointed.

Direct sum

A theorem: The polar of a pointed cone is blunt and vice versa. Surprise, Surprise, this is a Farkas Lemma!!!!

A course in McMaster's that has amazingly beautiful slides on introduction to Polyhedral theory. It is obviously based on Schrijver's book.

---

Lattices and stuff:

Matrix decompositions, also the the polylib-lattices link.

A paper on Hermite Normal form. One of the authors wrote the lattices-and-cryptography-book with Goldwasser.

---

An Affine function is a x --> cx - z for c \in R^d*.
A trace (or trace)
is the weighted sum of the eigen values of the matrix. The weight of an eigen value
being its multiplicity. It is also the sum of the diagonal elements of the matrix. A
graph eigen value is the eigen value of a graph. This the defn. of graph spectrum .
This link also has some info about the eigen value of a graph.
The Caley hamilton theorem about the
characteristic equation.
Also look up the links from the Characteristic Polynomial
A Hadamard matrix is one that has elements from {-1,+1}.
A Hypergraph is a graph in which the edge set is any subset
from the powerset of the set of vertices (A usual graph is a 2-uniform hypergraph).
According to wolfram, a bigraph is the same as a bipartite graph, which
is possibly incorrect. A bigraph has signs on the edges. The proper definition is this?????? Google of bigraph returns bipartite graphs.
So, we need to check up on the definition.

This is a very good introduction to the Ellipsoid method (PDF link).

---

tennis court Read the rest of this entry >>

The Serpent and the Rope

Last updated: 11/05
=============
Due to running into rough weather (laptop disk crash and hand fracture), I was reasonably offline for most of the time. The nice part of that being that I had a chance to read the wonderful book The Serpent and the Rope by Raja Rao a second time. Also, I had a chance to read two comparative studies of works by Raja Rao. The one Shyamala Narayan titled Raja Rao: Man and his works and the one by Makarand Paranjape titled Best of Raja Rao.


begin side note


I have added an excellent biography of Shankara in the sidebar. It is this. It contains one sloka describing the success and glory of life of Adi Sankara:

“Ashta varshe chaturveedi
Dvaadasae sarva saastravit,
Shodasee kritavaan bhaashyam
Dvaatrimsee munirabhyagaat.”

meaning,

“At eight, a master of four Vedas
At twelve, a professor of all sastras,
At sixteen, an interpreter and commentator
At thirty two, a great sage.”

Also, the following biography by kamakoti peetham is good.


end side note.

Though the title and the beginning of the novel give hints that the plot may be Advaitic, no such hints would help the reader through the philosophical musings and metaphysical dilogues in which come as part of the plot. I myself was put off in the previous reading (in Bangalore?) by his sudden change of context, very slow movement of story and other things. Perhaps I was not slow enough then to appreciate such a great book!

The epigraph of the book is the deep Advaitic saying (from Astavakra Gita?):

Waves are nothing but water.
So is the sea
-- Sri Atmananda Guru

Sri Atmananda guru [Sri Krishna Menon] is the Guru of Sri Raja Rao.

Use of other languages in the book

The Serpent and the Rope is an amazing book and Raja Rao's scholarly writing puts me in his awe. He quotes from Sankara (Dakshninamurthy astakam, Kashi Panchakam, Kalabhairava astakam, Annapurna astakam, Manisha panchakam, Nirvana satakam) Bhavabhuthi's Uttararama Charita, Kalidasa's Raghuvamsam, French sources and others.

Purusha Suktam on page 272

In particular, this great (greatest?) author from India seems to have problems in expressing some feelings in English. At the places when words fail the author in English, he uses -- more often than not -- Sanskrit to express the emotion.

Beginning of the book

This is the often quoted beginning of the book:

I was born a Brahmin - that is devoted to Truth and all that. "Brahmin is he who knows Brahman," etc., etc., ... but how many of my ancestors since the excellent Yagnyavalkya, my legendary and Upanishadic ancestor, have really known the Truth excepting the sage Madhava, who founded an empire or, rather helped to build an empire, and wrote some of the most profound of Vedantic texts since Sri Sankara? There were otheres, so I'm told, who left hearth and riverside fields, and wandered to mountains distant and hermitages "to see god face to face." And some of them did see God face to face and built temples. But when they died -- for indeed did they "die" -- they too must have been burnt by tank or grove or meeting of two rivers, and they too must have known they did not die. I can feel them in me, and know they knew they did not die. Who is it that tells me they did not die? Who but me.

So, my ancestors went one by one and were burnt, and their ashes have gone down the rives.

When ever I stand in a river I remember how when young, on the day the monster ate the moon and the day fell into an eclipse, I used til and kusha grass to offer the manes my filial devotion. For withal I was a good Brahmin. I even knew grammar and the Brahma Sutras, read the Upanishads at the age of four, was given the holy thread at seven ...
...

Rama and the place his Advaitic thinking gives to Women in general and Man-Woman relationships in particular

There are a reasonable number of pointers in the book as to where the Advaitic thinking of Rama places Women in general. Also, where do the Man-Woman relationships figure in his thinking.

Some forward pointers of where his marriage is headed to given in one of the first pages (page 26?):

And Yagnavalkya had said to Maitreyi. "For whose sake, verily does a husband love his wife? Not for the sake of his wife, but verily for the sake of Self in her." Did little mother love the Self in my father? Did I love the Self in Madeline? I knew I did not ...

Also, Rama's description of what he feels about Savitri -- when he first meets her -- shows what he expects from a wife:

We had one thing in common: we both knew Sanskrit, and could entertain each other with Uttara Rama Charita or Raghuvamsa.

On pages 57, we get an insight as to what the Advaitic Rama expects from a wife.

It makes all the difference in the world whether the woman of your life is with you or not; she alone enables you to be in a world that is familiar and whole. If it is not his wife, then for an Indian it may be a sister in Mysore, or little Mother in Benares.

Love is a way of way of looking at things. If you love you forget yourself, and percieve the object not as you see it, but rather as seen. The woman therefore is the priestess of God.

The saying by Yagnyavalkya is again used when Rama goes to Cambridge to meet Savitri (page 171)

One cannot possess the world, one can become it: I could not possess Savitri -- I became I. Hence the famous saying of Yagnyavalkya to his wife. "The husband does not love the wife for the wife's sake, the husband loves the wife for the sake of the Self in her."

Second reference to the priestess of God??

(yet to add: stuff about Mother, little-mother, Madeline, Savitri, Lakshmi in Bombay, Lakshmi in Cambridge, Catherine)


Some parts are poetic

Rama and Madeline's pet Bull and the pet Elephant. Also their taking care of them by feeding the "pets". Madeline's saving of the Bull from the old man. Rama thinking that the signs on the Bull make it an auspicious Basavanna.

The symbolic marriage between Rama and Savitri.

Some parts are extremely poignant: The time when Rama takes leave of Madeline:

Devi, Devi ayam paccimas te Ramaciraca padapankajasparchah
Goddess, here for the last time
Does the head of Rama touch the lotus of your feet.

At the end of the novel, there are two upa-kadhas (pitta kadhas in Telugu):

The story of Rama-Deva and the

The story of Radha, Krishna and Durvasa.

Story of Radha, Brahmachari-Krishna, Upavasi-Durvasa

The story involving Radha, Krishna and Durvasa is a very interesting one, explaining a fundamental concept of Advaita: namely illusiory nature of the world. Here is the story:

"One day Radha had a very possessive thought of Krishna. 'My Krishna,' She said to herself, as though one could possess Krishna as one could possess a calf, a jewel. Krishna, the Absolute Itself, immediately knew her thought. And when the absolute knows, the knowing itself, as it were, is the action of the act; things do not happen according to his wish, but his wish itself is his own creation of his wish, as the action is the creation of his own action.

"So, Durvasa the great Sage was announced.
...

The full story is here.

The original story is also excerpted here.

The women of Vraja once asked Lord Krishna to name some Brahmin to whom they could offer food. Lord Krishna named “Durvasa”. The Gopis asked: “How can we approach him? There is the Yamuna which is in floods. How are we to cross it? Name some other Mahatma, please”. The Lord said “Request Yamuna in the name of Nitya Brahmachari Krishna to give way and it will instantly do so. The Gopis were amused, and though sceptic, did as requested and lo! the river at once gave way for the perplexed Gopis to cross it; Yamuna indeed knew the real Svarupa of Krishna, the spotless divine purity that He was! He was a Nitya Brahmacharin.

What amazes me -- besides the story itself -- is the story telling power of Raja Rao. Raja Rao added spice (and other ingredients) and explained the story so beautifully that the vivid images of the story and the moral stick permanently in the mind!

To continue the thoughts of Rama after the story:

To be free is to know one is free, beyond the body and beyond the mind; to love is to know one is love, to be pure is to know one is purity. Impurity is in action and reaction: what is born must die, what has form must vanish and stink. ....
....

Benares is everywhere where you are, says an old Vedantic text, and all waters are the Ganges. ...
...

Other than that, the concept of illusory nature of the world is experienced by Rama in the whole book. In fact, in pages 340?? Rama says to Madeline

"The world is either unreal or real -- the serpent or the rope. There is no in-between-the-two -- and all that's in between is poetry, is sainthood ... "

As Shyamala Narayan pointed out aptly in her book, The Serpent and the Rope begins and ends with the definition of a Brahmin. In the latter, some humor is added by giving another definition.

...
"Do you know what a Brahmin is, Catherine?"

"No, what is it?" She came back, having gone halfway to the kitchen.

"A Brahmin is he who knows Brahman. That is one definition," I said. "There is another rouguish definition. A Brahmin is he who loves a good banquet."

"You certainly do not belong to the second category, poor dear. Rama, what shall we do when you are gone? You have become so like one of us. We will be lost."

Georges looked at me. He looked so sad.

"We must have been brothers in a past life."
...

The roughish definition Raja Rao is hinting at is Brahmanah bhojana priyah. Rama can now afford to laugh at the two definitions as he has found a Guru and is on the path to realization.



Makarand Paranjape says the following about the book:

Published twenty two after Kanthapura, The Serpent and the Rope is the Rao's most appreciated work. If the former is modelled on a Upapurana, the latter is a kind of Mahapurana or epic: geographically, historically, philosophically and formally, its sweep is truly epical.

Raja Rao as a part of the Original Trinity of Indian English Authors

It is well known that the trinity of Raja Rao, R.K.Narayan and Mulk Raj Anand are the pioneers of Indian english writing in the form of novels.

Later, Makarand Paranjape goes into controversial area by comparing Raja Rao with R.K.Narayan and Mulk Raj Anand and (nearly) calling them trite. I agree however with his observation that Raja Rao is more scholarly when compared to others. I used to once agree with him on the former point. What happened was that, in the midst of a heated conversation I too called R.K Narayan names. This is surprising considering that I had read almost all his major works. After reading Raja Rao, I prefer Raja Rao to R.K.Narayan. I would not mind R.K.Narayan when asked to make a choice between him and other Indian authors. He is good. He is simple. He may be too simple.

The Cat and the Shakespeare is said to be the book where Raja Rao shows that the Absolute can be approached through play (as in The Cat and the Shakespeare) as effectively as through ascetic meditation (as in The Serpent and the Rope).

yet to add:

Rama and fear?

second place of reference woman priestess of the God???

Argument between Rama and Georges pages 108-112 about truth.

The two times when Rama feels he took a bath in Ganges: when he wins an argument with Georges and when Catherine tells him that she and Georges are going to have a baby(340-341?).

...
"I have news to give you," said Georges, pursuing his own thoughts, and playing with a knife on the table. He w as silent for a brief moment, looked at Catherine with adoration and announced: "Catherine will have a baby in five or six months. You are first person to know it, Rama."

To this day I cannot tell you why, but I felt somewhere I had been washed clean and whole by the Ganges, dipped again and again and made shining with Shravan Saturday sun. I must have looked very moved, for Catherine put the soup in front of me, touched my head as Saroja might have and said:
"You will look after him, when he grows up, and give him all your wisdom, won't you?"
...

I am reading The Great Indian Way, a biography of Mahatma Gandhi by Raja Rao. The beginning is very good and exceeds expectations.
Read the rest of this entry >>

Unbounded polyhedral domains and computability of SUREs

Problem: Give a single theorem that characterizes the theorems 1, corollaries 1/2 and skewed variations of corollaries 1/2. In other words, give a characterization of the computability of an SURE defined over an arbitrary polyhedral unbounded domain.

Explanation: The domain could be

F_n (as in theorem 1),

strip of the type Q_t* F_(n-t) (as in corollary 1/2)

a skew of the domain of collorary 1/2

a conical subset of F_n through the origin

a minkowski sum of a polytope and a cone (meaning, an arbitrary polyhedron which is a subset of F_n)


The statement of computability may be the following:

An SURE defined over an arbitrary unbounded polyhedral domain is incomputable iff

it has a cycle C of weight W(C)

-W(C) belongs to the characteristic cone of the domain.

Some assumptions

The SURE is defined as ... V_j(z-w_j)

The SURE is defined in all the integral (or rational points) in the domain

The RDG is strongly connected. We can do a greedy index splitting (ala Allan-Kennedy???)


We have two cones

C1: The dependence cone: constructed with the negative dependence vectors

C2: The characteristic (or recesssion) cone of the domain. This should have a vector other than {0} as the domain is assumed to be unbounded.

The convex cone of the (negative) dependence vectors strictly contains all the vectors which are the weight of cycles. Also, containing vectors just means of the same direction It could contain vectors which cannot be expressed by cycles of the weight vectors.

What has the separability of these two cones have to do with computability?: If C1 and C2 are separable, then of course the SURE is computable.

A sufficient condition for computability: If there exists a cycle whose weight is a vector that belongs to the two cones, then the SURE is incomputable. The weight vector of a cycle ofcourse belongs to a cycle. Does it belong to the char.cone of the domain?

How does the computability condition of SURE change if the domains are non-polyhedral, but still convex?
The concept of char.cone is from polyhedral domains. According to the book and lecture notes of Bertsekas et.al, the concept is perfectly valid in every convex domains. There are some caveats though. ????

Notes about importance of computability problem of S*RE's defined on unbounded Vs. bounded domains. According to QS, Unbounded domains are not that important while bounded domain are. This is because, they are more common. I disagree with this. More importantly, however, QS make the important observation that the computability problem of bounded domains is equivalent to the testing of the acyclicity of the EDG. From the outside, this seems like a condition which can be applied only to a non-parametrtized domain. However, thinking of the parameters as additional dimensions is a standard trick.
So, the question is given a bounded domain, and a computation defined over it, what is the computability problem?

Problem: An SURE defined over a bounded domain is computable iff, there exists no zero weight cycle.

Restatement of the problem: Given a partially integral polytope -- meaning some vertices are integral, others are rational -- find whether the polytope contains the vertex {0}

Algorithm:Use KMW decomposition. Find which edges donot participate in any cycle, remove them, recurse on the rest of the graph.


Question on finitizabilty of the domains

Later note:
I had originally defined the finitizability so the the computability condition of COR.1/2 of KMW can be applied to the case of SUREs defined over the skewed domains of COR.1/2.
Later note: With the precison we have for the condition for computability, we donot need this "finitizability".

Question: Given a computation defined over an unbounded polyhedral domain.
To find: A skew of the domain such that the application of which results in the maximization of for loops.

The number of finizable dimensions

the skewing matrix which can do this

A listing of the dimensions which are finitizable


Example1: if given a computation defined over Q_t * F_(n-t) (ala domain of cor.1/2 of KMW), we should give

a value of 2, meaning 1 out of the 2 domains is "finitizable"

The identity matrix: I_(2*2)

a listing of the domains which are finitizable: meaning j

Example2: If the domain of the computation is a skew of the domain of COR.1/2 of KMW, thr output should be

a value of 1, meaning 1 out of 2 domains are finitizable

a proper skew

a listing of the domains Read the rest of this entry >>

Cornuejols book: Combinatorial Optimization

I had been reading this book on and off. Very nice presentation, just like the other books -- like Tarjan's book -- in the series by CBMS-NSF.

The interesting part is the material in chapter 6, which talks about {0,+1,-1} matrices. The picture on page 50 gives the relationship between the classes of {0,+1,-1} matrices. The four classes used in the book are PERFECT, IDEAL, BALANCED and TOTALLY UNIMODULAR. A balanced matrix is both perfect and ideal. The set of matrices which are TUM is a (strict) subset of balanced matrices.

I had talked in a couple of posts about TUM matrices. The posts are here. In this post we will see the generalizations of TUM matrices.

Let n(A) be a function which takes a {0,+1,-1} matrix and returns a column vector whose ith component is the number of -1's in the ith column of A.

Now, compare Ax and 1 - n(A) for some x in {0,1}^n. If

the former wins, we have a ideal matrix --> COVERING

the latter wins, we have a perfect matrix --> PACKING (how is it related to perfect graphs?)

if they are tied, we have a balanced matrix.
--
Refer to the matrix (on page 61) which is called the 0,1 extension of a {0,+1,-1} matrix.
--
A bipartite graph is said to be a perfect graph. What about directed bipartite graphs?
The book uses the term bigraph. According to wolfram, a bigraph is the same as a bipartite graph, which is possibly incorrect. A bigraph has signs on the edges.
--

Partially integral Polytopes: Our incidence matrix induces a polytope. We donot know if the vertices of this polytope are integral or not. The dimensions of this polytope is d. However, this polytope is not as important as the polytope of (n+d) dimensions. The interesting things is, this (n+d) dim polytope is partially integral, as the n vertices always induce integral vertices (IS THIS RIGHT???) the question remaining, what about the d vertices? Read the rest of this entry >>