The Legacy of Alladi Ramakrishnan in the Mathematical Sciences

Professor Alladi Ramakrishnan, my father, belonged to a small eminent group of Indian scientists who made fundamental contributions to several fields of study and sustained a high level of productivity over a significant period of time. If among this select versatile group of researchers we seek those who also have made monumental contributions to the profession by creating leading institutions of advanced research, then we are down to a mere handful such as Professors Raman, Bhabha, Mahalanobis, Ramakrishnan, and a few more. In an illustrious scientific career that began in 1947, Professor Ramakrishnan has published about 150 influential research papers in leading journals on topics ranging over Stochastic Processes, Elementary Particle Physics, Matrix Algebra, and the Special Theory of Relativity, has guided 24 PhD students, lectured on his research at over 200 institutions of higher learning the world over and at numerous international conferences, and created MATSCIENCE, The Insitute of Mathematical Sciences in Madras. It is amazing that even after his retirement, and indeed until the very end, his passion for science and his spirit of enquiry remained unabated. Here I shall briefly describe some of his significant contributions including his most recent ones, and the circumstances that led to them.

After completing his PhD at the University of Manchester, my father returned to India and joined the physics department at the University of Madras as a Reader in 1952. He was later promoted as Professor. He was developing

the theory of product densities

that he had initiated in his PhD thesis and studying applications of it by himself and with his students. He availed every possible opportunity to invite eminent scientists to the University of Madras and to our family home

Ekamra Nivas

and encouraged his students to listen to their lectures and engage in discussions with them. When my father visited the Institute for Advanced Study in Princeton in 1957–1958 at the invitation of its Director Robert Oppenheimer, he had the opportunity to listen to over one hundred seminars on theoretical physics by the leading researchers of that generation. My father returned to India filled with a desire to expose students to the latest developments in modern physics. Not satisfied with the curriculum at the Madras University, he gave advanced lectures in theoretical physics to students at Ekamra Nivas. Eager students gathered at the seminar to hear his lectures, and this was formally called

The Theoretical Physics Seminar

. He invited eminent scientists to lecture in this seminar. My mother Mrs. Lalitha Ramakrishnan graciously hosted the foreign speakers and the students by arranging lavish South Indian dinners after the seminars. I was a very young boy, but I had the privilege of meeting the eminent visitors.

The inauguration of MATSCIENCE, The Institute of Mathematical Sciences, Madras, India, on 3 January 1962, was greeted with great enthusiasm by scientists from around the world. My father, Professor Alladi Ramakrishnan, in his inaugural speech as the Director of the new Institute, referred to the creation of MATSCIENCE as a miracle, because a series of unexpected pleasant circumstances came in rapid succession to bear fruit. About a week before the inauguration of MATSCIENCE, congratulatory telegrams and letters started pouring in from scientists around the world. I was just past my sixth birthday at that time, but I remember the sense of excitement at our family home

Ekamra Nivas

as my father was preparing for that sensational event. I remember a dinner at the roof garden of the Dasaprakash Hotel in Madras a few days before the inauguration at which my father’s students who attended his

Theoretical Physics Seminar

were present. At this dinner, my father asked each student to predict the number of congratulatory telegrams and messages that would be received by 3 January 1962. Such was the mood at that magic moment! My father had preserved these telegrams and had them photocopied and bound in two volumes. In the following pages I have presented photo copies of a selection of these telegrams and letters. I have made some observations about the person sending the message/telegram and my father’s association with that scientist.

My father Professor Alladi Ramakrishnan was a master of exposition, both in written and spoken form. He was a dynamic speaker, an orator in every sense. Right from my boyhood, I had the pleasure to listen to many of his scientific lectures and speeches, and was inspired by his manner of speaking and the power of his oratory. The finest speech he ever gave was perhaps at the inauguration of MATSCIENCE, the Institute of Mathematical Sciences, on January 3, 1962, when his thoughts came pouring out at that very exciting and memorable occasion. As a six year old boy, I was in the front row of the English Lecture Hall at the Presidency College in Madras when he delivered that speech extempore, as was his custom. The speech was later written up from a tape recording. This speech was printed in the appendix to The Alladi Diary, Vol I, East–West Books, Madras, (2000). It is reprinted here with the permission of East–West Books, Madras, Pvt. Ltd.

Professor Alladi Ramakrishnan believed that close interaction with the leading scientists around the world was essential for fundamental research. He traveled annually to present his work at conferences and universities worldwide. He has lectured at more than 30 international conferences, and given talks at about 200 centers of higher learning in North America, Europe, Asia, and Australia. In doing so, not only did he disemminate his research work and those of his group, but also used it as an opportunity to make new contacts and invite active researchers to Madras. Among his many trips abroad, we mention a few that were especially significant in terms of his career.

In my 1967 paper with almost the same title which appeared in volume 89 of the American Journal of Mathematics, I proved the invariance of the characteristic terms in the fractional power series expansion of a branch of an algebraic plane curve over fields of characteristic zero. Now I extend the results by a more generous interpretation of the characteristic terms, and by relaxing the characteristic zero hypothesis.

We obtain a series expansion for the product generating function of partitions in which the odd parts do not repeat. This is done by studying the 2-modular Ferrers graphs of such partitions via Durfee squares. This provides a unified approach to several fundamental identities in the theory of partitions and q-series such as those of Sylvester, Lebesgue, Gauss, and Rogers-Fine, and provides links with Göllnitz’s deep theorem.

q

-Analogs of the Catalan number identities of Touchard, Jonah, and Koshy are derived.

Brahmagupta extended Ptolemy’s theorem on cyclic quadrilaterals to find the lengths of the diagonals, the segments made when they are cut at the point of intersection of the diagonals, and the lengths of the sides of the needles, the figures formed when opposite sides of the quadrilateral are extended until they meet. Proofs of these results are given, and a derivation of the 19th century result of the length of the third diagonal is given. This “diagonal” is formed by connecting the tips of the needles with a line segment.

Two proofs are given for a series transformation formula involving the logarithmic derivative of the Gamma function found in Ramanujan’s lost notebook. The transformation formula is connected with a certain integral embodying the Riemann zeta function that is similar to integrals examined by Ramanujan in his one published paper on the zeta function.

We show that many of Ramanujan’s modular equations of degree 3 can be interpreted in terms of integral ternary quadratic forms. This way we establish that for any

n

∈

N

,

$$\begin{array}{rcl} & & \left \vert \left \{(x,y,z) \in {\mathbf{Z}}^{3} : \frac{x(x + 1)} {2} + {y}^{2} + {z}^{2} = n\right \}\right \vert \geq \\ & &\left \vert \left \{(x,y,z) \in {\mathbf{Z}}^{3} : \frac{x(x + 1)} {2} + 3{y}^{2} + 3{z}^{2} = n\right \}\right \vert, \\ \end{array}$$

just to name one among many similar “positivity” results of this type. In particular, we prove the recent conjecture of H. Yesilyurt and the first author, stating that for any

n

∈

N

,

$$\begin{array}{rcl} & & \left \vert \left \{(x,y,z) \in {\mathbf{Z}}^{3} : \frac{x(x + 1)} {2} + {y}^{2} + {z}^{2} = n\right \}\right \vert \geq \\ & &\left \vert \left \{(x,y,z) \in {\mathbf{Z}}^{3} : \frac{x(x + 1)} {2} + 7{y}^{2} + 7{z}^{2} = n\right \}\right \vert.\end{array}$$

We prove a number of identities for certain ternary forms with discriminants 144, 400, 784, or 3, 600 by converting every ternary identity into an identity for the appropriate η-quotients. In the process, we discover and prove a few new modular equations of degree 5 and 7. For any square free odd integer

S

with prime factorization

p

1

…p

r

, we define the

S

-genus as a union of 2

r

specially selected genera of ternary quadratic forms, all with discriminant 16

S

2

. This notion of

S

-genus arises naturally in the course of our investigation. It entails an interesting injection from genera of binary quadratic forms with discriminant − 8

S

to genera of ternary quadratic forms with discriminant 16

S

2

.

The positive integers

n

such that

n

! is a sum of three squares have a density which is equal to 7 ∕ 8. The key point for the proof of this result is to show that the above sequence is

automatic

and to study the matrix associated to the underlying automaton.

The polynomials commonly called “Eulerian” today have been introduced by Euler himself in his famous book “Institutiones calculi differentialis cum eius usu in analysi finitorum ac Doctrina serierum” [

5

, Chap. VII], back in 1755. They have been since thoroughly studied, extended, applied. The purpose of the present paper is to go back to Euler’s memoir, find out his motivation and reproduce his derivation, surprisingly partially forgotten. The rebirth of those polynomials in a

q

-environment is due to Carlitz two centuries after Euler. A brief overview of Carlitz’s method is given, as well as a short presentation of combinatorial works dealing with natural extensions of the classical Eulerian polynomials.

In this paper, we continue our previous investigations on applications of the Epstein zeta-functions. We shall mostly state the results for the lattice zeta-functions, which can be immediately translated into those for the corresponding Epstein zeta-functions. We shall take up the generalized Chowla–Selberg (integral) formula and state many concrete special cases of this formula.

An upper bound for the “positive homogeneous minimum” for

n

linearly independent linear forms in

n

real variables in terms of their determinant is obtained.

A representation of SL(2,

Z

) by integer matrices acting on the space of analytic ordinary Dirichlet series is constructed, in which the standard unipotent element acts as multiplication by the Riemann zeta function. It is then shown that the Dirichlet series in the orbit of the zeta function are related to it by algebraic equations.

In the course of studying a higher dimensional generalization of the Pythagorean equation and its connections to the Lorentz transformation, Alladi Ramakrishnan made a conjecture on a determinant of a certain circulant matrix and published it in his paper

Pythagoras to Lorentz via Fermat

. This conjecture was proved by the author in a letter to Alladi Ramakrishnan. That letter is reproduced here with a note by the Editor explaining the background.

A branching random walk is a branching tree such that with each line of descent a random walk is associated. This paper provides some results on the asymptotics of the point processes generated by the positions of the

n

th generation individuals. An application to the photon–electron energy cascade is also given.

The paper provides a series representation of the logistic probability density function in terms of differently scaled double exponential distributions with terms of the series alternating in signs. This representation is used to calculate moments, moment generating function, and characteristic function of a logistic distribution. The same representation is also used to derive the logistic distribution as the scale mixture of a normal distribution.

The paper provides an axiomatic setup for an entropy function as a measure of diversity. A general definition of cross entropy is given and its use in solving a variety of stochastic and nonstochastic optimization problems is mentioned. A method of deriving a cross entropy function associated with a given entropy function is given.

The problem of testing equality of survival distributions on the basis of paired censored survival data has received considerable attention in literature. Some of the important statistics used for such purposes can be expressed as linear combinations of two statistics, one based on uncensored pairs and the other based on the censored pairs. Raychaudhuri and Rao (Nonparametric Statistics, 1996, 6, 1–11) investigated properties of two classes of such statistics and derived expressions for the optimal coefficients (weights) for the linear combination that will maximize efficacy within each class. As the optimal weights depend upon the form of the underlying survival and censoring distributions, statistics with optimal weights can only be used with estimated weights. This article presents a method of estimating optimal weights on the basis of an assumed model that specifies the distribution of the difference between the observed survival times conditional on the censoring pattern. The model, in addition to dispensing with the usual assumption that the survival and censoring variables are independent, also permits a graphical check of its lack of fit on the basis of observed data. The performance of statistics with the estimated weights is evaluated by using two simulation studies – one with data generated under the assumed model and the other assuming independence of the survival and censoring times. Simulation results show that the optimal statistics with estimated weights have good power properties in all cases considered, and that they compare well with other commonly used tests for paired censored survival data. An advantage of the tests with optimal weights is that, unlike their competitors, these tests have demonstrated performance characteristics in some cases where the assumption of independent censoring may not be justified.

A sequence of random variables, each taking only two values “0” or “1,” is called a Bernoulli sequence. Consider the counts of occurrences of strings of the form {11}, {101}, {1001},

…

in Bernoulli sequences. Counts of such Bernoulli strings arise in the study of the cycle structure of random permutations, Bayesian nonparametrics, record values etc. The joint distribution of such counts is a problem worked on by several researchers. In this paper, we summarize the recent technique of using conditional marked Poisson processes which allows to treat all cases studied previously. We also give some related open problems.

We discuss a number of storage problems for a class of one-dimensional master equations with separable kernels. For this class of problems, the integral equation for the first overflow or first emptiness can be transformed exactly into ordinary differential equations. Analysis is done with a generalised separable kernel. Using imbedding method, closed form solutions are obtained for the first overflow without or with emptiness in a given time. The first passage time for emptiness without or with overflow in a given time is also obtained. The imbedding technique is also used to study the expected amount of overflow in a given time. Diffusion approximation for this model is also obtained using suitable statistical conditions.

This paper presents a novel variational model for inverse consistent deformable image registration. The proposed model deforms both source and target images simultaneously, and aligns the deformed images in the way that the forward and backward transformations are inverse consistent. To avoid the direct computation of the inverse transformation fields, our model estimates two more vector fields by minimizing their invertibility error using the deformation fields. Moreover, to improve the robustness of the model to the choice of parameters, the dissimilarity measure in the energy functional is derived using the likelihood estimation. The experimental results on clinical data indicate the efficiency of the proposed method with improved robustness, accuracy, and inverse consistency.

The deep inelastic scattering experiments reveal that the nucleon is a composite object consisting of quarks and gluons. Treating them as Fermi and Bose gases, statistical distribution functions are used to describe their momentum distributions in the rest frame. When transformed to the infinite momentum frame, they yield quark and gluon distribution functions. A thermodynamical bag model is proposed to obtain realistic distribution functions that yield correctly the nucleon structure functions. By including the spin degree of freedom in the Fermi statistical distribution functions, the quark spin distribution functions and the polarized nucleon structure functions are obtained.

Generalized Clifford algebras (GCAs) and their physical applications were extensively studied for about a decade from 1967 by Alladi Ramakrishnan and his collaborators under the name of

L

-matrix theory. Some aspects of GCAs and their physical applications are outlined here. The topics dealt with include: GCAs and projective representations of finite abelian groups, Alladi Ramakrishnan’s σ-operation approach to the representation theory of Clifford algebra and GCAs, Dirac’s positive energy relativistic wave equation, Weyl-Schwinger unitary basis for matrix algebra and Alladi Ramakrishnan’s matrix decomposition theorem, finite-dimensional Wigner function, finite-dimensional canonical transformations, magnetic Bloch functions, finite-dimensional quantum mechanics, and the relation between GCAs and quantum groups.

A (

p

,

q

)-analog of the classical Rogers-Szegö polynomial is defined by replacing the

q

-binomial coefficient in it by the (

p

,

q

)-binomial coefficient corresponding to the definition of (

p

,

q

)-number as

$${[n]}_{p,q} = ({p}^{n} - {q}^{n})/(p - q)$$

. Exactly like the Rogers-Szegö polynomial is associated with the

q

-oscillator algebra, the (

p

,

q

)-Rogers-Szegö polynomial is found to be associated with the (

p

,

q

)-oscillator algebra.

As applied to quantum theories, the program of renormalization is successful for ‘renormalizable models’ but fails for ‘nonrenormalizable models’. After some conceptual discussion and analysis, an enhanced program of renormalization is proposed that is designed to bring the ‘nonrenormalizable models’ under control as well. The new principles are developed by studying several, carefully chosen, soluble examples, and include a recognition of a ‘hard-core’ behavior of the interaction and, in special cases, an extremely elementary procedure to remove the source of all divergences. Our discussion provides the background for a recent proposal for a nontrivial quantization of nonrenormalizable scalar quantum field models, which is briefly summarized as well.

A number of features of living systems, reversible interactions and weak bonds underlying motor-dynamics; gel-sol transitions; cellular connected fractal organization; asymmetry in interactions and organization; quantum coherent phenomena; to name some, can have a natural accounting via

physical

interactions, which we therefore seek to incorporate by expanding the horizons of “chemistry-only” approaches to the origins of life. It is suggested that the magnetic “face” of the minerals from the inorganic world, recognized to have played a pivotal role in initiating Life, may throw light on some of these issues. A magnetic environment in the form of rocks in the Hadean Ocean could have enabled the accretion and therefore an ordered confinement of super-paramagnetic colloids within a structured phase. A moderate H-field can help magnetic nanoparticles to not only overcome thermal fluctuations but also harness them. Such controlled dynamics brings in the possibility of accessing quantum effects, which together with frustrations in magnetic ordering and hysteresis (a natural mechanism for a primitive memory) could throw light on the birth of biological information which, as Abel argues, requires a combination of order and complexity. This scenario gains strength from observations of scale-free framboidal forms of the greigite mineral, with a magnetic basis of assembly. And greigite’s metabolic potential plays a key role in the mound scenario of Russell and coworkers-an expansion of which is suggested for including magnetism.

The validity of the Ehrenfest theorem in Abelian and non-Abelian quantum field theories is examined. The gauge symmetries are taken to be unbroken. By suitably choosing the physical subspace, the above validity is proven in both the cases.