^{Last updated: 08/15/2008 07:42 AM}

**Infinite Cosmoses Of Infinite Algorithms For Infinite Transcendental Numbers**

The Book that is going to be released formally on the 2

^{nd}December 2007, at Nandhanam, Hotel Sarovaram, Eranakulam, Kerala, India, titled “Infinite Cosmoses Of Infinite Algorithms For Infinite Transcendental Numbers “, was essentially written between 1995 -1998 AD . It was in the chi-writer format and ten printed copies were hard-bound (in black calico, titled “Infinite Algorithms For Infinite Transcendental Numbers”) and circulated around the earth from around April 1999. The History of the random journey of the Manuscript(s) of the book is briefly described in the book itself.

This is Verily an Infinite Volume-d Book: The First Infinite Volume-d Seamless Book Ever Written on The Earth, The Set Of All Seeds of “A Book Of Sand “ or “A Library of Babel” as one may wish to call it, is contained here. Essentially, It is a book about the “Set Of All Transcendental Numbers” collected as infinite [infinitudes ] of formulas generating more formulas of convergent sequences, each transcendental number identified uniquely by a universal symbolic notation which is seamless ,infinite, eternally self-generating, for ever for ever all over all over .

Now We Know that there Exists Infinitudes of Transcendental Numbers. What Are They and Where Are They? This Seamless question Is Answered Eternally in This Seamless Library of Labyrinthine Babel. All Existing Transcendental Numbers including “e”, “pi”, Liouville’s transcendental number, Mahler’s decimal expansion etc. become ordinary examples of the infinite formulas enlisted endlessly in this strange divine work of Mathematics.

It can be self-evidently trivially proved that each power series defined here is a Transcendental Number, according to

**Liouville’s Criterion ~ Mahler’s Criterion etc. But the very Infinite Nature of the Infinite Book Requires developing the idea of Transcendentality defined to various Labyrinth levels.**

In order to truly exhaust the infinite seamless possibilities of these formulas, I re-cognized that I needed to generalize the idea of Polynomials, to CL(x) [Cosmic Labyrinth ] Functions and SD(x) [Seamless Divine] Functions. This generalization has phenomenal eternal repercussions on All Branches of Mathematics and Sciences where the idea of a polynomial function occurs.

There are also Infinite Possibilities of studying the idea of convergence (and divergence) of each series by its “velocity of convergence” and “infinite levels of accelerations of convergence (divergence)” . Thus, unique qualities and quantities of each convergent (divergent) series are defined, despite all of them approaching the same limit. This has deep and eternal consequences in the study of Analytic Functions.

The Book Leaves Endless Possibilities Open For Ever For All who want to pursue. This Infinite book has direct consequences to all the branches of Science ~ Mathematical Analysis, Algebra, Geometry, Arithmetic, Physics, Chemistry Computer Science, Artificial Intelligence, Various Branches of Engineering, Chaos theory to Catastrophe theory ad absurdum ad infinitum.

By chaining and by constructing towers and super-towers of labyrinth formulas we can create infinitudes more volumes of Transcendental Numbers for ever which are hinted and expanded in the book solemnly.

If you search Google for transcendentals {transcendental numbers} you will get very few, say, a paltry fifteen [including some spurious ones like i^i ]. All these examples are trivial examples of Our Library of Babel unfolding here. It is like the grains of sand on an infinite ocean-bed. Let the Transcendentals be the grains: Mathematicians knew about fifteen grains: I have named each grain for ever and given its eternal mathematical power-series ~identity-tag. It is impossible to add one more to this infinite listing of the seamless almanac of transcendental number stars.

In order that you understand this inaugural lecture a fair knowledge of the book, “Theory and Application of Infinite Series” by Konrad Knopp [Dover Publication] may help, But no essential preparation is really necessary, but an open seamless mind to comprehend the invincible infinite seamless Cosmoses of Transcendental numbers.

Let me humbly thank Prof. T.Thrivikraman who has consented to preside over this occasion.

I invite you personally dear friend, to attend and enhance this historic event of publishing which is perhaps comparable to the Sermon on the Mount.

NarayananRaghunathan ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I first encountered the summary of this work when I was in college in 2001. They were brought to me by Arun Parameswaran, the nephew of Narayanan Raghunathan, who was also my college mate. I was delighted by the vast and elemental discovery that mysteriously unfolded in these pages and felt eager to see more. Narayanan and I chatted about his work in the years that followed, and I slowly recognized more of the work's unspeakably infinite vastness and significance to mathematics and all the sciences. Here was the vast residue of the undiscovered real number line. The real numbers, which are widely (and falsely) believed to be un-denumerable, were literally enumerated here in vast ever wider steps. Every number that can be computed by any (even the final) computer is already either inscribed or defined inherently and classified in this work. Why? Almost All of It remains mysteriously beyond the memory of the last computer too. Each of these infinite numbers shares the special properties of the famous and ubiquitous "e". Hence an entire cosmos of mathematical analysis surrounds each number. Analysis is no longer an exotic esoteric pursuit anchored at a few special constants. Mathematics and all sciences are expanded infinitely forever and made seamless. All the real numbers are identifiable and symmetrically notated. It is surely the destiny of human knowledge, a discovery as inevitable and fatal as the miracle of counting.

I travelled alone by train to Thrissur, India in 2004 to meet Narayanan for the first time. He gifted me with the full manuscript as three fearsome black hardbound volumes. I was his sole audience there for 2 days in Thrissur. He was over-joyous that someone had come all the way to recognize the work and get involved. He spoke to me excitedly about the blissful and agonized discoveries that lead to the manuscript's writing. He made me see how it was the vast residue of the set of all transcendental numbers, almost all of which were literally unseen by mathematicians in history. He spoke about the formidable resistance weighed on him for trying to get this vast discovery seen by the academic circles. He was in tears as he rattled off the painful episodes of crude doubt and neglect that he faced trying to get the significance of the work seen by the responsible people. I saw how these people were concealing a jealousy at the work's vastness which they surely sensed as even I did. A mere amateur had done it all, their crowns were precariously slipping. All the same, since truth always wins, I had the faith that their transient noise would blow over. I felt a deep kinship to this cause of showing that there was a great and precious revelation being presented to the world.

I wondered if this book was truly infinite as it proclaimed there in gold embossed letters. (Narayanan would also call it "The Book of Sand" or the "Infinite Volume-d Book"). Yet what is an infinite book? Surely these pages are numbered and counted, but that is of course not what he meant. How else was this book infinite? Is it because it hints at infinitely long decimal expansions? Most mathematics books do that. These are not what make the book infinite. Little did I know how my question would get infinitely answered: Soon I recognized that it is truly an Infinite volume-d seamless treatise, seamlessly evolving in all possible directions. I, as everyone who has encountered him, know Narayanan's seamless concern with infinity, God(s), eternity and cosmoses from his other works, especially his aphorisms and poetry. If anyone knows the nature of the infinite and the last things, it is he. I knew there was no ambition in Narayanan to be a mathematician or make great public discoveries. Yet I wondered what could have motivated him to the tedious work of inscribing these formulas. He said it was the bliss of mantram-s and prayer that lead him through the fearsome labyrinths of this work. I believe it is only his seamless empty and ever infinite reaching mind that could accommodate and manifest these mathematical pathways into the undifferentiated infinite. It is this great humility before the infinite, that has made him the conduit on earth for this knowledge to reach mankind. This is surely a glorious and praiseworthy designation. Yet I have learned that such true glory is a terrible burden on the creator. The painful labour that these volumes have suffered to be where they are today cannot be overstated: that suffering truly borne by the one who desired and conceived and wrote them. The crude evasion by LMS, Annals and others was a painful blow to this work's pure and fierce hope for mankind. I felt a righteous anger at those people.

I took the books back with me to Chennai and then returned to my Master's studies in Oklahoma City University, USA. Narayanan suggested that I could base my Master's thesis on the CL(x) Functions and commence the design of software to compute and algebraically manipulate some of these formulas. I conceived a "Mathematica" like software which could model CL(x) [ complete labyrinth] functions and the general notation purveyed in this book. This would allow, when completed, computing with a relevant portion of Narayanan's infinite transcendental numbers just like existing computer algebra systems like Mathematica allow with a very few known transcendental constants. This system as was conceived would not have any bias towards the known transcendental constants such as "e" and represent them as hard-coded primitives: but would allow the entire real number line to be seamlessly represented using Narayanan's notation (of course, only as much as the given Computer’s finite memory could hold!) I thought this would help Narayanan demonstrate his work decisively. I tentatively completed this task over two semesters and presented my program and a paper describing it to the university. The software implementation was begun, but not at a state to be demonstrable. I felt often that I was working alone with blunt tools to complete an infinite task requiring an infinite meticulousness. Existing methods in computer algebra do not have enough generality to model these algorithms and I had to create my own, unassisted . Further, our goals were far more ambitious than these existing methods, requiring not approximate but exact results always. We deliberately shunned all stochastic and approximation methods. I was a little crestfallen for the lack of time and set down my tools to take up more mundane work. I graduated and took a job in Edison, New Jersey as a commercial software programmer.

Then followed a painful incident, which has bound me forever to the task of assisting Narayanan with his mission. I was required by my job to move to Wisconsin. Fearing the excess baggage of the volumes, I left them behind along with many other possessions in a box in the temporary housing provided by my employers. I was assured that they would keep it until I returned from the assignment. As a reminder of the computer software, I took Volume 2 of the three and the summary volume with me to Wisconsin. I needed them as I also considered taking the work up as a PhD dissertation. Upon returning from Wisconsin, I discovered that the house had been sublet to different tenants and my box had disappeared mysteriously. The people responsible claimed that it was too heavy to take along when they shifted. It was a very sad moment for both Narayanan and I. I regret and repent infinitely for my carelessness on that day when I left behind the precious unpublished manuscripts gifted to me with infinite love. I sought out the books desperately, even filed a complaint with Edison police, all to no avail. I even returned almost weeping to the same residence now occupied by complete strangers and inquired. The books seemed to be gone forever. I hoped secretly that no one would find them and that they may be destroyed, so that no one would steal them and cause shameful controversy over the authorship of the work. Narayanan and I have since feared the worst but hoped for the best. I am resting in the faith that there is a divine cosmic purpose to all happenings and some fruition will be borne out of all well-intended actions. My neglect and contribution to the burden on Narayanan's divine work has been not recognizing then, the preciousness of those manuscript volumes.

Hence I am relieved today that this book is finally appearing in print for mass distribution on earth. It has taken the utterly solitary path of not appearing in any journal or magazine, not anchored to any edifice of knowledge, not affiliated with any institution or school, but as these independent books. A start in any other direction has appeared to frustrate the deity that oversees its birth into the world. In our effort to have the books seen by mathematicians, there were surely some who were divinely favorable. Most eminent among them is Prof. D.D Nadkarni of Fort Myers, Florida. Professor Nadkarni, or Doc as he is fondly known, was formerly an engineer and applied mathematician at NASA after which he taught at various engineering and mathematics faculties around the USA. He was elated upon seeing the Summary of the transcendentals-volume and lauded the author as a future Ramanujan. Prof. Nadkarni kindly entertained me at his home in Florida in the summer of 2006. We chatted for long about the books and their content. Although his mathematical expertise does not include the area of transcendental numbers, he could appreciate the vastness and aesthetic perfection of the work. He noted that it was a work so unique and fundamentally important that it is unsurprising that it has taken many years to be recognized. He had made many unrequited efforts to publicize the work through his personal network of academic acquaintances. Professor Nadkarni was also very pleased that I was making an effort to computerize the algebra of the transcendentals. He supported my effort by reviewing my thesis and kindly providing his valuable referee letter for a PhD application to continue work in the same subject. Prof. Nadkarni has since been a continuous source of courage and support.

Is the journey over? No it has hardly begun. I hope that others can recognize the infinite potential that this discovery has gifted on all human pursuits. Mathematicians will delight and feel relieved that the whole spectrum of the real line is now notated in a seamless way and easily accessible to their fancy's exploration. The applications into sciences and computing will become known as the analytical properties of these numbers are elucidated. The black terrains of transcendental numbers are now brightly lit with a sun and there are paths laid for all to walk as they would like, taking off from a page of this infinite volume-d book.

**Infinite Algorithms for Infinite Transcendental Numbers**I first encountered the summary of this work when I was in college in 2001. They were brought to me by Arun Parameswaran, the nephew of Narayanan Raghunathan, who was also my college mate. I was delighted by the vast and elemental discovery that mysteriously unfolded in these pages and felt eager to see more. Narayanan and I chatted about his work in the years that followed, and I slowly recognized more of the work's unspeakably infinite vastness and significance to mathematics and all the sciences. Here was the vast residue of the undiscovered real number line. The real numbers, which are widely (and falsely) believed to be un-denumerable, were literally enumerated here in vast ever wider steps. Every number that can be computed by any (even the final) computer is already either inscribed or defined inherently and classified in this work. Why? Almost All of It remains mysteriously beyond the memory of the last computer too. Each of these infinite numbers shares the special properties of the famous and ubiquitous "e". Hence an entire cosmos of mathematical analysis surrounds each number. Analysis is no longer an exotic esoteric pursuit anchored at a few special constants. Mathematics and all sciences are expanded infinitely forever and made seamless. All the real numbers are identifiable and symmetrically notated. It is surely the destiny of human knowledge, a discovery as inevitable and fatal as the miracle of counting.

I travelled alone by train to Thrissur, India in 2004 to meet Narayanan for the first time. He gifted me with the full manuscript as three fearsome black hardbound volumes. I was his sole audience there for 2 days in Thrissur. He was over-joyous that someone had come all the way to recognize the work and get involved. He spoke to me excitedly about the blissful and agonized discoveries that lead to the manuscript's writing. He made me see how it was the vast residue of the set of all transcendental numbers, almost all of which were literally unseen by mathematicians in history. He spoke about the formidable resistance weighed on him for trying to get this vast discovery seen by the academic circles. He was in tears as he rattled off the painful episodes of crude doubt and neglect that he faced trying to get the significance of the work seen by the responsible people. I saw how these people were concealing a jealousy at the work's vastness which they surely sensed as even I did. A mere amateur had done it all, their crowns were precariously slipping. All the same, since truth always wins, I had the faith that their transient noise would blow over. I felt a deep kinship to this cause of showing that there was a great and precious revelation being presented to the world.

I wondered if this book was truly infinite as it proclaimed there in gold embossed letters. (Narayanan would also call it "The Book of Sand" or the "Infinite Volume-d Book"). Yet what is an infinite book? Surely these pages are numbered and counted, but that is of course not what he meant. How else was this book infinite? Is it because it hints at infinitely long decimal expansions? Most mathematics books do that. These are not what make the book infinite. Little did I know how my question would get infinitely answered: Soon I recognized that it is truly an Infinite volume-d seamless treatise, seamlessly evolving in all possible directions. I, as everyone who has encountered him, know Narayanan's seamless concern with infinity, God(s), eternity and cosmoses from his other works, especially his aphorisms and poetry. If anyone knows the nature of the infinite and the last things, it is he. I knew there was no ambition in Narayanan to be a mathematician or make great public discoveries. Yet I wondered what could have motivated him to the tedious work of inscribing these formulas. He said it was the bliss of mantram-s and prayer that lead him through the fearsome labyrinths of this work. I believe it is only his seamless empty and ever infinite reaching mind that could accommodate and manifest these mathematical pathways into the undifferentiated infinite. It is this great humility before the infinite, that has made him the conduit on earth for this knowledge to reach mankind. This is surely a glorious and praiseworthy designation. Yet I have learned that such true glory is a terrible burden on the creator. The painful labour that these volumes have suffered to be where they are today cannot be overstated: that suffering truly borne by the one who desired and conceived and wrote them. The crude evasion by LMS, Annals and others was a painful blow to this work's pure and fierce hope for mankind. I felt a righteous anger at those people.

I took the books back with me to Chennai and then returned to my Master's studies in Oklahoma City University, USA. Narayanan suggested that I could base my Master's thesis on the CL(x) Functions and commence the design of software to compute and algebraically manipulate some of these formulas. I conceived a "Mathematica" like software which could model CL(x) [ complete labyrinth] functions and the general notation purveyed in this book. This would allow, when completed, computing with a relevant portion of Narayanan's infinite transcendental numbers just like existing computer algebra systems like Mathematica allow with a very few known transcendental constants. This system as was conceived would not have any bias towards the known transcendental constants such as "e" and represent them as hard-coded primitives: but would allow the entire real number line to be seamlessly represented using Narayanan's notation (of course, only as much as the given Computer’s finite memory could hold!) I thought this would help Narayanan demonstrate his work decisively. I tentatively completed this task over two semesters and presented my program and a paper describing it to the university. The software implementation was begun, but not at a state to be demonstrable. I felt often that I was working alone with blunt tools to complete an infinite task requiring an infinite meticulousness. Existing methods in computer algebra do not have enough generality to model these algorithms and I had to create my own, unassisted . Further, our goals were far more ambitious than these existing methods, requiring not approximate but exact results always. We deliberately shunned all stochastic and approximation methods. I was a little crestfallen for the lack of time and set down my tools to take up more mundane work. I graduated and took a job in Edison, New Jersey as a commercial software programmer.

Then followed a painful incident, which has bound me forever to the task of assisting Narayanan with his mission. I was required by my job to move to Wisconsin. Fearing the excess baggage of the volumes, I left them behind along with many other possessions in a box in the temporary housing provided by my employers. I was assured that they would keep it until I returned from the assignment. As a reminder of the computer software, I took Volume 2 of the three and the summary volume with me to Wisconsin. I needed them as I also considered taking the work up as a PhD dissertation. Upon returning from Wisconsin, I discovered that the house had been sublet to different tenants and my box had disappeared mysteriously. The people responsible claimed that it was too heavy to take along when they shifted. It was a very sad moment for both Narayanan and I. I regret and repent infinitely for my carelessness on that day when I left behind the precious unpublished manuscripts gifted to me with infinite love. I sought out the books desperately, even filed a complaint with Edison police, all to no avail. I even returned almost weeping to the same residence now occupied by complete strangers and inquired. The books seemed to be gone forever. I hoped secretly that no one would find them and that they may be destroyed, so that no one would steal them and cause shameful controversy over the authorship of the work. Narayanan and I have since feared the worst but hoped for the best. I am resting in the faith that there is a divine cosmic purpose to all happenings and some fruition will be borne out of all well-intended actions. My neglect and contribution to the burden on Narayanan's divine work has been not recognizing then, the preciousness of those manuscript volumes.

Hence I am relieved today that this book is finally appearing in print for mass distribution on earth. It has taken the utterly solitary path of not appearing in any journal or magazine, not anchored to any edifice of knowledge, not affiliated with any institution or school, but as these independent books. A start in any other direction has appeared to frustrate the deity that oversees its birth into the world. In our effort to have the books seen by mathematicians, there were surely some who were divinely favorable. Most eminent among them is Prof. D.D Nadkarni of Fort Myers, Florida. Professor Nadkarni, or Doc as he is fondly known, was formerly an engineer and applied mathematician at NASA after which he taught at various engineering and mathematics faculties around the USA. He was elated upon seeing the Summary of the transcendentals-volume and lauded the author as a future Ramanujan. Prof. Nadkarni kindly entertained me at his home in Florida in the summer of 2006. We chatted for long about the books and their content. Although his mathematical expertise does not include the area of transcendental numbers, he could appreciate the vastness and aesthetic perfection of the work. He noted that it was a work so unique and fundamentally important that it is unsurprising that it has taken many years to be recognized. He had made many unrequited efforts to publicize the work through his personal network of academic acquaintances. Professor Nadkarni was also very pleased that I was making an effort to computerize the algebra of the transcendentals. He supported my effort by reviewing my thesis and kindly providing his valuable referee letter for a PhD application to continue work in the same subject. Prof. Nadkarni has since been a continuous source of courage and support.

Is the journey over? No it has hardly begun. I hope that others can recognize the infinite potential that this discovery has gifted on all human pursuits. Mathematicians will delight and feel relieved that the whole spectrum of the real line is now notated in a seamless way and easily accessible to their fancy's exploration. The applications into sciences and computing will become known as the analytical properties of these numbers are elucidated. The black terrains of transcendental numbers are now brightly lit with a sun and there are paths laid for all to walk as they would like, taking off from a page of this infinite volume-d book.

Shyam Santhanam

**~~~~~~~~~~~~~~~~~~~~**

**Narayanan- portrait of the artist as a mathematician**

Most of us put on a different cap when we enter a different domain. If it is science, I put on my “scientific” hat, and subject all propositions coming before me to the test of whether they stand up to scientific scrutiny. If the domain is art, then I don my “artistic” hat, and apply systems of aesthetics to the subject under consideration, or look at the subject from a strongly subjective perspective, coloured by my cultural and social influences. If the domain is philosophy, I put on my “philosopher’s” hat, and so on. I assume, of course, that most of us have a definite point of view, and do not suffer from an overpowering need to be politically correct in everything that we think and say.

Narayanan is that rare person who sees everything he looks at through the same set of lenses. Or it may even be that the light from things he looks at, directly burns into his consciousness, without the need for intervening lenses. It may therefore be said with a degree of truth that he approaches science as an aesthete, or philosophy as an artist, or art as a mathematician. Actually, I believe he approaches all subjects from the same standpoint, and this enables him to see things very differently from the rest of us.

In Narayanan, boundaries between disciplines get blurred and dissolve, as his mind seeks to get to the essence (or residue, as he would prefer to put it!) of the matter he is pursuing. An exercise of the intellect for him becomes a seamless movement, freely moving through the frontiers of different disciplines, in pursuit of his quarry.

I recall an aphorism of his, that emerged spontaneously when I exclaimed at what I felt was an extraordinary phenomenon. In 1984 I read in the newspapers that the AIDS virus had been identified, and its working understood. I was amazed when I recalled that Narayanan had, during a heated evening discussion at the Indian Coffee House in Trivandrum, observed that there are many ways in which disease and illness can be transmitted. He suggested that one way would be for the virus to undermine the body’s defences, so much so that one could die of a common cold! I remember the incredulity with which this proposition was greeted. Now, here was this newspaper item in 1984 stating something that was almost exactly what Narayanan had predicted in 1976! When I marveled at this, he replied: “Balu, if you can imagine it, it exists, because his imagination is greater than yours!” I remember thinking to myself that there you have a fine example of mathematical expression of a philosophical statement!

Narayanan’s private quest and solitary nature has prevented his ideas from getting the attention they deserve. I am glad that this book will bring his work in Mathematics to the attention of scholars everywhere.

C Balagopal

Trivandrum

November 16,2007

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Name of Other Books Available at the function ~

[ Published by Eternity Publishers , Thrissur ]

Kalki The Last Coming ~~~~~ Narayanan

[A Book Of Philosophical Aphorisms ~ 1]

Scrap Bits From the Note-Books Of A Lunatic ~~~~~ Narayanan [ Book Of Philiosophical Aphorisms ~2 ]

Solitary Infinity ~ Obituary to Transfinity ~~~~~ Narayanan Raghunathan [ R Narayanan ]

[ A Book on Philosphy Of Mathematics ]

Infinite Flame Silences ~~~~~

Narayanan Raghunathan [ R Narayanan ]

[ A Book Of Haiku Poems ]

Apocalyptic Rapture ~~~~~

Amanda Cazalet & Narayanan Raghunathan

[ A Book Of Haiku Poems ]