Cover page

Image Name: Professor Jayant Narlikar: Prof Jayant Narlikar needs no introduction. He was one of the greatest

visonaries India has produced.

For more information, visit: https://www.bbc.com/news/articles/cd62g8pn35yo

Managing Editor Chief Editor Editorial Board Correspondence

Ninan Sajeeth Philip Abraham Mulamoottil K Babu Joseph The Chief Editor

Ajit K Kembhavi airis4D

Geetha Paul Thelliyoor - 689544

Arun Kumar Aniyan India

Sindhu G

Journal Publisher Details

Publisher : airis4D, Thelliyoor 689544, India

Website : www.airis4d.com

Email : nsp@airis4d.com

Phone : +919497552476

Editorial

by Fr Dr Abraham Mulamoottil

airis4D, Vol.3, No.6, 2025

www.airis4d.com

This edition of the airis4D Journal is a Festschrift,

profoundly honoured to feature Professor Jayant

Narlikar on the cover page. Ajit Kembhavi’s Festschrift

article “Professor Jayant Narlikar: A Legacy of

Science and Inspiration (1938-2025)” was written not

just an acquaintance memoir, but as a close colleague,

collaborator, and successor in the institutions and

initiatives spearheaded by Jayant Narlikar, particularly

at IUCAA and in the realm of science outreach. As

a towering ﬁgure in Indian science, Dr. Narlikar

passed away on May 20, 2025, at the age of 86.

Renowned as a brilliant scientist, an astute institution

builder, and a compelling mass communicator, Narlikar

remained active until his last days, engaging in reading,

writing, and public discussions, including a popular

blog. He played a pivotal groundbreaking works in

cosmology, including the pioneering Hoyle-Narlikar

theory of gravity, and in establishing the esteemed

Inter-University Centre for Astronomy and Astrophysics

(IUCAA). Beyond his scientiﬁc contributions, Professor

Narlikar signiﬁcantly advanced the popularisation of

science in India through his numerous insightful

books and engaging programs. His impactful career

earned him prestigious accolades, including the Padma

Vibhushan and the Kalinga Prize.

In ”A Day with My Superhero - Prof. Jayant V

Narlikar,” Blesson George recounts a pivotal encounter

with his childhood scientiﬁc hero, Professor Jayant

V. Narlikar. The author, whose fascination began

with Narlikar’s captivating science ﬁction story ”The

Return of Vaman” during his undergraduate studies,

describes the surreal opportunity to organize a program

featuring the esteemed astrophysicist and his wife,

mathematician Dr. Mangala Narlikar. The event,

”A Day with Professor and Mrs. Narlikar,” held

at CMS College, Kottayam in February 2016, was

a successful outreach initiative for school students,

drawing overwhelming attendance. George shares a

personal highlight: a pre-event conversation with Prof.

Narlikar, who oﬀered profound insights on teaching.

During the program, Professor Narlikar captivated the

young audience with his talk on ”Searches for Micro Life

in Space,” discussing his experiments with ISRO, while

Dr. Mangala Narlikar charmed them with ”Mathematics

Through Puzzles.” For George, the day was a dream

realised, cementing Narlikar’s image as a ”superhero”

who not only contributed immensely to science but also

inspired countless individuals.

In ”The Legend — Remembering Prof. Jayant

V. Narlikar,” Joe Jacob reﬂects on his long-standing

admiration for Professor Jayant V. Narlikar, India’s

distinguished astrophysicist, and recounts signiﬁcant

encounters that shaped his own scientiﬁc journey.

Jacob’s fascination began in his undergraduate days,

inspired by Narlikar’s books. His ﬁrst direct

interaction was during a Visiting Students Research

Programme (VSRP) at TIFR in 1986, where he

attended Narlikar’s lecture and brieﬂy spoke with him

during a research scholar interview. Later, while

pursuing postgraduate studies, Jacob attended a crucial

”School on Astronomy” in Charalkunnu, where he

had the invaluable opportunity to interact closely

with Narlikar and other leading astronomers. This

experience deepened his understanding and solidiﬁed

his connection to the ﬁeld. His association with

Narlikar strengthened further through his role as a

Visiting Associate at the Inter-University Centre for

Astronomy and Astrophysics (IUCAA), an institution

Narlikar founded. Here, Jacob had numerous meetings

with Narlikar at various academic and social events,

noting Narlikar’s promptness in submitting an article

for a collection and his engaging style with students

during an astronomy popularisation program. Jacob

emphasises Narlikar’s humility, intellectual brilliance,

and unwavering dedication to science, highlighting how

each interaction left a lasting impact on his academic

and personal life. For Jacob and many others, Narlikar is

more than a scientist; he is a true legend who continues

to inspire generations.

The article, ”The Legacy of Jayant Narlikar” by

Moncy V John, eulogises Professor Jayant Narlikar as

a giant in cosmology, a great science popularizer, and a

remarkable institution builder. Despite the mainstream

shift away from his favored Steady State Cosmology

(later reﬁned into Quasi-Steady State Cosmology -

QSSC), Narlikar’s extensive research and contributions

remain signiﬁcant. The author notes India’s historical

strength in quantum mechanics but a lag in general

relativity, a ﬁeld where Narlikar’s father, V.V. Narlikar,

was a pioneer. Jayant Narlikar’s early life showed a

natural inclination towards mathematics. The article

details Narlikar’s journey from Cambridge, where

he was inspired by Fred Hoyle’s institution-building

eﬀorts, to his crucial role in establishing the Inter-

University Centre for Astronomy and Astrophysics

(IUCAA) in Pune. IUCAA, designed by Charles

Correa, became a world-class research hub, reﬂecting

Narlikar’s vision for fostering academic excellence

and interaction. While Narlikar expressed frustration

with the Big Bang model’s dominance and perceived

lack of open inquiry in modern cosmology, his

contributions to alternative theories and his insistence

on rigorous scientiﬁc methodology were invaluable.

The author concludes by sharing a personal anecdote of

Narlikar’s collaboration on a new cosmological model,

highlighting Narlikar’s genuine love for physics and his

compassionate mentorship. Narlikar’s legacy is one of

unwavering scientiﬁc inquiry and profound inspiration

The article “In Memory of Prof. Jayant

V. Narlikar: A Personal Tribute” by Tharanath

R recounts three signiﬁcant encounters with Prof.

Narlikar at diﬀerent stages of his academic journey.

Initially, as a postgraduate student at a summer

school, he was too naive to fully appreciate Narlikar’s

presence. Later, during his research days at CUSAT, the

IUCAA Resource Centre (IRC) provided a platform for

interaction. Tharanath, deeply engaged with Narlikar’s

book ”An Introduction to Relativity,” still felt too

inhibited to speak to him personally during a visit

to CUSAT. The most memorable meeting was the

third, when Tharanath, along with colleagues, assisted

Prof. Narlikar during his visit to Kochi. By then,

Tharanath had shed his hesitation and openly discussed

his research struggles with Narlikar, who listened

patiently. Narlikar’s encouraging remark about working

together left a lasting impression. Tharanath concludes

by remembering Prof. Narlikar not just as a brilliant

scientist, but as a humble and gentle soul who listened

intently. He cherishes these encounters as ”gemstones”

of his research life, highlighting Narlikar’s lasting legacy

on those he inspired.

The article, ”Reinforcement Learning for

LLMs” by Arun Aniyan, explores how Reinforcement

Learning (RL) enhances Large Language Models

(LLMs) by enabling them to learn from interactions

and feedback. It deﬁnes core RL concepts (agent,

environment, actions, state, reward, policy) in the

context of LLMs and details the iterative learning

loop. A key methodology discussed is Reinforcement

Learning from Human Feedback (RLHF), involving

pre-training, supervised ﬁne-tuning, reward model

training (based on human comparisons), and RL ﬁne-

tuning (e.g., using PPO). The article also mentions

other RL algorithms like A2C. Advanced applications

include improving response quality, reducing harmful

outputs, aligning with human values, and personalising

responses. Beneﬁts highlighted are enhanced

performance, improved control, and better alignment.

Challenges include data collection for human feedback,

reward hacking, training instability, and interpretability.

The author concludes that RL for LLMs is a crucial and

evolving ﬁeld, promising more nuanced and ethically

aligned AI interactions.

Blesson George introduces Group Equivariant

iii

Convolutional Networks (G-CNNs), an extension of

traditional CNNs that incorporates broader symmetries

beyond simple translations, such as rotations and

reﬂections. The core idea is based on group theory,

where transformations like translations, rotations (e.g.,

90 degrees), and reﬂections form symmetry groups

(e.g., Z2, p4, p4m). G-CNNs perform G-convolution,

which is a convolution operation over these symmetry

groups, allowing the network to recognise features

regardless of their orientation or position. The concept

involves structured feature maps representing group

elements (poses), where transformations lead to both

local data changes and permutations of data across

these poses. This ensures equivariance: transformed

input results in a predictably transformed output. The

beneﬁts of G-CNNs include data eﬃciency, parameter

sharing, and better generalisation due to their inherent

understanding of symmetries. They have applications

in image classiﬁcation, medical imaging, molecular

modelling, and astronomy. The article also mentions

extensions like Steerable CNNs and 3D G-CNNs,

concluding that G-CNNs represent a signiﬁcant step

towards symmetry-aware deep learning.

The article, ”Type II and Type III Supernovae:

A Detailed Exploration of Stellar Cataclysms” by

Sindhu G, discusses two categories of stellar explosions.

Type II supernovae are well-understood core-collapse

events of massive stars

M > 8 × M

solar

that retain their

hydrogen envelopes, leading to prominent hydrogen

lines in their spectra. The explosion mechanism

involves a core collapse, rebound shock, and neutrino

energy transfer. Subtypes like IIP, IIL, IIn, and

IIb are based on light curve characteristics. These

supernovae are crucial for chemical enrichment, neutron

star/black hole formation, galactic dynamics, and

cosmic distance scaling. In contrast, the term

”Type III supernova” is not recognised in modern

astronomical classiﬁcation. Historically, it appeared

in speculative contexts for hypothetical or unusual

explosions. Sometimes informally associated with pair-

instability supernovae (PISNe)—theoretical explosions

of extremely massive stars 140 to 260 M that leave

no remnant—Type III ultimately remains an obsolete

designation. Modern understanding categorises unusual

supernovae as subtypes of Type II or as superluminous

supernovae. The article concludes that while Type II

supernovae are fundamental to understanding stellar

death, the defunct Type III classiﬁcation highlights

the evolving nature of scientiﬁc discovery, suggesting

that future observations might reveal new stellar death

phenomena.

This article, ”Invasive Alien Species in India: A

Growing Threat to India’s Biodiversity, Ecosystems

and Economy” by Geetha Paul, highlights the severe

impact of Invasive Alien Species (IAS) on India’s

biodiversity, ecosystems, and economy. IAS, deﬁned

as non-native organisms that aggressively establish and

spread, are a major threat, facilitated by anthropogenic

pathways like trade and aquaculture. Examples include:

Lantana camara (terrestrial plant): Forms dense

thickets, outcompetes native plants, is toxic to livestock,

and alters ﬁre regimes. Parthenium hysterophorus

(Congress Grass): Rapidly colonises disturbed lands,

reduces crop yields, and causes health risks (allergies).

Eichhornia crassipes (Water Hyacinth): Forms dense

mats on water bodies, depletes oxygen, reduces

biodiversity, and promotes disease vectors. Clarias

gariepinus (African Catﬁsh): A generalist predator that

outcompetes and preys on native ﬁsh. Phenacoccus

solenopsis (Cotton Mealybug): A polyphagous pest

causing signiﬁcant agricultural losses. Aedes albopictus

(Asian Tiger Mosquito): Transmits diseases like dengue

and chikungunya. The article emphasises the need

for a One Health approach, integrating ecological,

agricultural, and public health perspectives. It

proposes management strategies including: Preventive

measures: Stricter biosecurity and early detection.

Control methods: Mechanical, chemical, biological

control (e.g., Zygogramma bicolorata for Parthenium),

and ecological restoration. Policy recommendations:

National Invasive Species Strategy, research funding for

biocontrol, and integration with climate adaptation

plans. In conclusion, the article stresses that a

science-driven, multi-stakeholder approach is crucial

for sustainable IAS management in India, focusing on

prevention, eradication, and restoration, along with

future research into genomic tools and climate-resilient

biocontrol agents.

This article, ”Entropy in Physics and

Information: One Concept, Many Realms” by

Jinsu Ann Mathew, explores the unifying concept of

entropy across thermodynamics and information theory.

Initially, entropy in physics, introduced by Rudolf

Clausius, quantiﬁes disorder and the tendency of energy

to spread in a closed system, explaining irreversible

processes like melting ice. Boltzmann’s formula

S = k × ln(W )

deﬁnes it as a measure of microstates

corresponding to a macrostate, illustrating how systems

naturally move towards higher disorder (e.g., solid to

liquid to gas). Later, Claude Shannon redeﬁned entropy

in information theory as a measure of uncertainty or

information content in a message. Shannon’s formula

quantiﬁes the unpredictability of a random variable,

where higher entropy means less predictability and

more information per symbol. This concept is crucial

for data compression and communication limits. The

article highlights the bridge between these two worlds:

both use entropy to quantify unpredictability, whether in

molecular motion or symbol variability. Beyond these

core ﬁelds, entropy has expanded to diverse disciplines

like neuroscience, linguistics, ecology, and machine

learning, serving as a fundamental principle to quantify

the unknown, manage uncertainty, and understand

system evolution and information processing.

The article, ”Aﬃne transformations in

Convolutional Neural Networks” by Linn Abraham,

explores the relationship between aﬃne transformations

and convolutional neural networks (CNNs). The article

explains that aﬃne transformations, which preserve

dimensions and ratios of parallel line segments, are

fundamental to understanding how CNNs process

information. While neural networks generally perform

aﬃne transformations on input data, CNNs leverage this

concept in two key ways: Neural Network Operations:

Any neural network transforms input by multiplying it

with a weight matrix and adding a bias, which is an

aﬃne transformation. Image Transformations: Image

manipulations like rotation, ﬂipping, and scaling can

be seen as aﬃne transformations. CNNs, however, are

not invariant to such transforms, which is why data

augmentation is used. The convolution operation in

CNNs, which utilises ﬁlters to extract features, can

also be represented as a linear transformation that

preserves ordering. The article provides a mathematical

background on aﬃne spaces and related concepts to

provide a rigorous understanding of these relationships.

News Desk

Figure 1: Prof. Jayant Narlikar and Prof. V C Kuriakose

Figure 2: A teacher with a passion for the subject

Figure 3: With students of Prof V C Kuriakose

Figure 4: Prof Jayant Narlikar at Marthoma College, Thiruvalla

vii

Contents

Editorial ii

1 Professor Jayant Vishnu Narlikar

19 July 1938 – 20 May 2025 1

1.1 Narlikar the Scientist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Narlikar, the Institution Builder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Narlikar and Kerala . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Narlikar and Public Outreach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 A Day with My Superhero - Prof. Jayant V Narlikar 5

3 The Legend — Remembering Prof. Jayant V. Narlikar 8

4 The Legacy of Jayant Narlikar 10

5 In Memory of Prof. Jayant V. Narlikar: A Personal Tribute 14

I Artiﬁcial Intelligence and Machine Learning 16

1 Reinforcement Learning for LLMs 17

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 Introduction to Reinforcement Learning in LLMs . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3 Core Concepts of Reinforcement Learning for LLMs . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Reinforcement Learning Methodologies for LLMs . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.5 Reinforcement Learning Algorithms for LLMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6 Advanced Applications of Reinforcement Learning for LLMs . . . . . . . . . . . . . . . . . . . . 19

1.7 Beneﬁts of Reinforcement Learning for LLMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.8 Challenges and Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Introduction to Group Equivariant CNN

Part - II 21

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Group Convolutions and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Structured Feature Maps and Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Transformation of Structured Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Two Vital Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6 Why Equivariance Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7 Applications of G-CNNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.8 Extensions and Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

CONTENTS

II Astronomy and Astrophysics 24

1 Type II and Type III Supernovae: A Detailed Exploration of Stellar Cataclysms 25

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.2 Supernova Classiﬁcation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3 Type II Supernovae: Core-Collapse of Massive Stars . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.4 Astrophysical Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.5 The Mystery of Type III Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.6 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

III Biosciences 29

1 Invasive Alien Species in India: A Growing Threat to India’s Biodiversity, Ecosystems

and Economy

(A tribute to International Day for Biological Diversity- May 22, 2025) 30

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.2 Major Invasive Species in India: Ecological and Economic Impacts . . . . . . . . . . . . . . . . 31

1.3 Mechanisms of Invasion Success . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.4 Management Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

IV General 36

1 Entropy in Physics and Information: One Concept, Many Realms 37

1.1 Entropy in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.2 Entropy in Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.3 Bridging the Two Worlds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

V Computer Programming 40

1 Aﬃne transformations in Convolutional Neural Networks 41

1.1 Aﬃne transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.3 Linear algebra: Aﬃne Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Professor Jayant Vishnu Narlikar

19 July 1938 – 20 May 2025

by Ajit Kembhavi

airis4D, Vol.3, No.6, 2025

www.airis4d.com

Professor Jayant Narlikar has for long been one

of the best known scientists of the country. He passed

away in his sleep in the early hours of May 20, following

a brief illness. He would have been 87 years of age in

July. He was active right to the end, reading, writing,

discussing and occasionally interacting with the public.

Since January 2024, he has been writing a blog, covering

diﬀerent stages of his life, which surely will be sorely

missed by the tens of thousands of people who read it.

Narlikar was a great scientist, institution builder and

mass communicator.

After graduating from Banaras Hindu University,

where his father was a Professor of Mathematics,

Narlikar left for Cambridge University, where he

completed the mathematical tripos as Senior Wrangler

(in the ﬁrst position) in 1959. He then worked as a

research student under the guidance of the great British

astronomer, Professor Fred Hoyle. Over the next few

years, he did an astonishing amount of work, got coveted

awards including the Adams Prize, and held his own

in ﬁery meetings of the Royal Astronomical Society.

On a visit to India in the mid-sixties, he thoroughly

impressed the scientiﬁc community, and captivated the

public imagination through his ever smiling face, a

gentle way of talking and a great ability to convey to

non-expert people his research on the Universe. He

continued to interact with the public in many ways to

the end of his life. He was awarded a Padma Bhushan

at the young age of 26 years and a Padma Vibhushan in

2013.

1.1 Narlikar the Scientist

Narlikar’s ﬁrst work was on the distribution of

cosmic radio sources as a function of their measured

brightness, which is known as the log N-Log S

distribution. The shape of the distribution depends

on the geometry of the Universe, the distribution of

the sources in space, and whether or not the source

population evolves as a function of cosmic time. The

radio data was from the Cambridge radio telescopes

built by astronomers from the Cavendish Laboratory in

Cambridge, led by Professor Martin Ryle. Ryle favoured

the Big Bang theory of the Universe in which the radio

source population could be evolving, while Fred Hoyle,

one of the creators of the steady state theory, believed

that the distribution should be constant in time. A clear

resolution of the problem was not possible because of

the very limited data then available. And yet the debates

had far reaching consequences, not only for cosmology,

but also for the career paths of the people involved in

the work. Martin Ryle won the Nobel Prize in Physics

in 1974, jointly with Professor Antony Hewish, who

discovered radio pulsars with his student Jocelyn Bell.

Along with the astronomical data analysis, Narlikar

also worked on diﬃcult theoretical problems, including

Newtonian cosmological models with rotation and shear,

the age of galaxies and the avoidance of singularities

in the steady state cosmology, Mach’s Principle

and the Creation of Matter, and Time Symmetric

Electrodynamics and Arrow of Time in Cosmology.

In 1966, Hoyle and Narlikar published work on a new

1.2 Narlikar, the Institution Builder

theory of gravitation, which is invariant under conformal

transformations. This output was remarkable by any

standards; it had amazing variety, depth and novelty,

and went against many cherished conventional ideas.

Narlikar spent much eﬀort, ﬁrst with Fred

Hoyle, and then with other distinguished astronomers,

including Professor Geoﬀrey Burbidge, on the steady

state theory of the Universe, working out its

astrophysical implications. In this theory, the Universe

forever remains the same, in spite of its expansion,

because of the continuous creation of matter. In this

model, there is no creation of the Universe in a Big Bang,

and so there is no early phase in which the Universe

was very hot. The steady state theory lost much of

its allure after the discovery in 1964 of the cosmic

microwave background radiation. Over the years, it

was established that the radiation had the spectrum of

black body radiation at 2.73 K with a near-perfect ﬁt to

Planck’s law. The background also had a remarkably

isotropic distribution in the sky. It, therefore, could only

have arisen in a hot phase of the Universe. That required

a modiﬁcation of the original steady-state theory to a

quasi-steady state theory, which was made by Narlikar

and Burbidge in 1993. In this theory, hot phases would

be possible with mini-bangs, but there would be no

singularity, making the Universe eternal as in the pure

steady state theory. A possibility here is that galaxies

from an earlier phase of the Universe could survive

to appear as seemingly prematurely evolved galaxies

in early epochs of our phase. Narlikar carried out

observations with collaborators to ﬁnd such galaxies,

and it is intriguing that the JWST is ﬁnding just such

objects in the very early epochs. With Burbidge and

Halton Arp, Narlikar also worked on the possible

anomalous redshift of quasars.

In later years, Narlikar pursued the idea, originally

due to Fred Hoyle, that microorganisms could have

entered the Earth’s atmosphere from outer space. The

idea seemed very fanciful when Hoyle ﬁrst proposed

it. He was denied publication of his theory in scientiﬁc

journals, and he had to publish it as a science ﬁction

novel. Narlikar proposed experiments that could be

carried out to detect organisms in the upper atmosphere,

which could not have got there from the surface of

the Earth, and which could possibly have a nature

distinct from their terrestrial counterparts. While much

planning was done in collaboration with people from

ISRO and other organisations, the experiment was never

carried out. That was possibly a great lost opportunity,

especially given the growing realisation that living

organisms could exist in several locations in the Solar

system, and the ubiquity of habitable extrasolar planets

in our Galaxy, even though those planets are too distant

to contribute organisms to our atmosphere.

In 1972, Narlikar moved from Cambridge to the

Tata Institute of Fundamental Research (TIFR). There,

he continued his work on various fronts in gravitation

and cosmology. He mainly worked with a number

of talented graduate students, with some working on

problems of his interest, while others devoted their eﬀort

to areas of their own choice. He was very democratic

in the matter, as he was in all his interactions at every

level, and that attitude seems to have worked very well.

Many of his students and other young researchers who

worked under his guidance have done excellently in

their professions, and others who have worked for him

in various capacities have always contributed their best.

1.2 Narlikar, the Institution Builder

In 1987, Professor Yash Pal, who was then the

Chairman of the University Grants Commission, invited

Narlikar to set up a new institution, which would be

unique in addressing the diﬃculties of the universities

in carrying out research in astronomy and astrophysics.

It was decided that the new institution would be located

in Pune and was eventually named the Inter-University

Centre for Astronomy and Astrophysics (IUCAA).

Narlikar moved to Pune for the purpose on June 1,

1989, and with the help of Professor Naresh Dadhich

and the author of this article, began the process of setting

up IUCAA. Within a few years, the unique buildings,

But even before the facilities became available, the

scientiﬁc work and all related activities had started, and

soon IUCAA became known as a place where good

astronomy was done.

The unique feature of IUCAA, of course, was the

1.3 Narlikar and Kerala

Figure 1: A photograph taken during the late evening

public meeting held during the 1997 workshop at Charal

Mount in Kerala. From left, Rt. Rev Mar Chisostom,

Prof. Jayant Narlikar, Prof. Babu Joseph (CUSAT),

Prof. Ajit Kembhavi and Prof. Naresh Dhadich are

seen.

tens of visitors from the universities and colleges who

came all the way from distant parts of the country,

even though there were hardly any facilities. They

worked in collaboration and brought their students, and

soon there was a thriving astronomical community

in the universities. Narlikar helped by interacting

personally with the visitors, who soon increased greatly

in number. He often visited departments all over

the country lecturing and introducing teachers and

students to IUCAA, and providing basic email and

other then emerging facilities at IUCAA’s cost. The

development of the university community is Jayant’s

greatest contribution to astronomy in India.

1.3 Narlikar and Kerala

In the early years of IUCAA, Narlikar and a few

of his colleagues from IUCAA travelled to Cochin,

Thiruvananthapuram and other centres in Kerala. The

purpose was to inform faculty and research students

working in the universities and colleges in the state about

IUCAA, and the facilities available there for pursuing

research in astronomy, gravitation theory, cosmology

and allied areas. Following that, several faculty and

young researchers started visiting IUCAA regularly,

and soon some novel collaborative programmes were

set up. But the activities did not remain conﬁned to

IUCAA.

A number of workshops and other events were

organised at diﬀerent levels in Kerala, so that a large

Figure 2: During the International Astronomy Year

2009, a sequence of programs and competitions for the

students in the Idukki district of Kerala under the label

“Sasthragramam” was carried out by Joe Jacob and Ravi

Pillai (Newman College, Thodupuzha). The ﬁnal round

of twenty students who topped in these programs were

brought to IUCAA, where they interacted with faculty

members and research students and visited various

facilities like the IGO, GMRT, etc.

number of local students could learn about the subject.

Narlikar visited Kerala on several occasions, gave many

talks and as always, endeared himself to everyone there.

He and his colleagues developed an excellent rapport

with Mar Philiopose Chrysostom, Metropolitan of the

Mar Thoma Church, who enthusiastically facilitated

IUCAA activities in the region. Narlikar’s initiative and

the work of his colleagues there have led to the rapid

spread of world-class astronomy in Kerala. The visitors

from Kerala now coming to IUCAA are third and fourth

generation academic descendants of the people who

ﬁrst came from there, while some of the ﬁrst generation

visitors are still into active research collaborations.

1.4 Narlikar and Public Outreach

Narlikar was a great communicator. He was able

to convey ideas about astronomy and cosmology to

the general public in a very simple yet interesting and

lucid manner. He did that through his many talks,

articles, books and also science ﬁction stories. The

public came in great numbers whenever and wherever

he lectured, and were mystiﬁed that the great person

they had heard so much about was, after all, one of their

own. When they asked questions, he enthusiastically

provided answers, but when they wanted his autograph,

1.4 Narlikar and Public Outreach

he politely refused. But as consolation, he oﬀered to

send a signed reply if the concerned person sent him a

question on a postcard.

Narlikar made public outreach an integral part

of IUCAA. The activities began with hundreds of

school children coming to the campus, still under

construction, for Saturday lectures. Some of those

young children, now in their middle age, still fondly

recount the inspiration that they received from Narlikar

to do well, and better, in whatever they were doing.

That was a simple message, but it has produced many

stars over the decades.

Truly, Jayant Narlikar was a great man in all ways.

About the Author

Professor Ajit Kembhavi is an emeritus

Professor at Inter University Centre for Astronomy

and Astrophysics and is also the Principal Investigator

of the Pune Knowledge Cluster. He was the former

director of Inter University Centre for Astronomy and

Astrophysics (IUCAA), Pune, and the International

Astronomical Union vice president. In collaboration

with IUCAA, he pioneered astronomy outreach

activities from the late 80s to promote astronomy

research in Indian universities.

A Day with My Superhero - Prof. Jayant V

Narlikar

by .Blesson George

airis4D, Vol.3, No.6, 2025

www.airis4d.com

It all began back in our B.Sc. second year. Like

many students of physics, I was swept up by the textbook

chapter titled The Return of Vaman. Authored by none

other than Prof. Jayant V. Narlikar, this story wasn’t

just another chapter—it was a portal into the universe.

It was my ﬁrst encounter with his genius, and it left an

imprint that has stayed with me ever since. I remember

being utterly fascinated, captivated by how beautifully

science could be told as a story.

(a)

(b)

Figure 1: Prof. Jayant V. Narlikar

Years later, on 2016, February 8

, life gave me

a surreal opportunity: to organize a program with

Prof. Jayant V. Narlikar himself. It felt like hosting a

superhero I had admired since my student days.

(a)

(b)

Figure 2: Mrs. Narlikar addressing the students.

The event, titled A Day with Professor and Mrs.

Narlikar, was organized by the Department of Physics,

CMS College, Kottayam, in association with the

IUCAA Resource Centre, Kochi. The program was

designed as a special outreach initiative for school

students of classes VII to IX.

The IUCAA team, including Dr. Ninan Sajith

Philip and Dr. Moncy V. John, lent great support

in shaping the academic direction of the sessions.

CMS College Principal, Dr. Roy Sam Daniel, and

Dr. P. Rajagopal (Head, Department of Physics,

CMS College) were equally enthusiastic and excited to

welcome such a distinguished guest to our campus.

(a)

(b)

Figure 3: Students actively participating in the session.

Not only did we have Prof. Narlikar with us,

but his wife, Dr. Mangala Narlikar, a renowned

mathematician, joined us too. Their presence was more

than academic—it was inspiring, human, and deeply

humbling.

We were overwhelmed by the response from

schools. Registrations ﬂooded in, and to our

astonishment, we had to stop accepting more due to

space limitations. It was a rare moment where we felt

both elated and guilty—for having to turn away eager

young minds.

One of the moments etched in my memory is the

privilege of receiving Prof. Narlikar at his hotel. I spent

nearly an hour with him before Prof. V. C. Kuriakose

arrived. That hour was a blessing beyond words. He

spoke warmly, thoughtfully, and gently. We spoke about

teaching, science, students—and he shared his thoughts

on what a teacher ought to be: curious, patient, and

above all, inspiring. It felt like a masterclass in humility

and wisdom.

(a)

(b)

Figure 4: Prof. Narlikar with the author.

His session, titled Searches for Micro Life in

Space, was a fascinating window into a domain that

fused space science with biology. He spoke about

his experiments—initiated with ISRO—to search for

microbial life in the upper atmosphere at 41 km altitude.

His storytelling made even the youngest students listen

with rapt attention.

Dr. Mangala Narlikar’s talk on Mathematics

Through Puzzles was equally delightful. She showed

how puzzles aren’t just games, but gateways into deeper

understanding—how they have even led to the creation

of new mathematical ﬁelds.

That day, held at the E-Learning Centre of CMS

College, wasn’t just an event. It was a culmination

of dreams. It brought together young minds, science

enthusiasts, and two extraordinary educators whose

legacy is not just in their academic contributions but in

how they inspire and uplift others.

For me, Prof. Narlikar is not just a name in a

textbook—he is the superhero who stepped out of the

pages and spent a day with us. And I will forever be

grateful for having been a small part of that moment.

About the Author

Dr. Blesson George presently serves as

an Assistant Professor of Physics at CMS College

Kottayam, Kerala. His research pursuits encompass

the development of machine learning algorithms, along

with the utilization of machine learning techniques

across diverse domains.

The Legend — Remembering Prof. Jayant V.

Narlikar

by Joe Jacob

airis4D, Vol.3, No.6, 2025

www.airis4d.com

My introduction to Professor Jayant V. Narlikar,

one of India’s most distinguished astrophysicists, dates

back to my undergraduate days. His name ﬁrst came

up during a lecture on cosmology, and soon after, I

discovered some of his popular science books in our

college library. With a deep-rooted fascination for

astronomy since childhood, I felt an intense desire to

someday meet this iconic ﬁgure in Indian science.

During my postgraduate studies, a friend owned a

copy of Introduction to Cosmology by Prof. Narlikar.

Eager to delve into the subject, I oﬀered my two-volume

set of Resnick and Halliday in exchange — a trade I

never regretted. That book still occupies a prominent

place on my bookshelf.

Though I had always hoped to meet him, I did

not expect the opportunity to come so early. In the

summer of 1986, I was selected for the Visiting Students

Research Programme (VSRP) at the Tata Institute of

Fundamental Research (TIFR), Mumbai — a major

milestone in my academic journey. It was also the

longest train journey I had taken up to that point, with

my brother accompanying me to the bustling city.

While working on my VSRP project under the

guidance of Prof. Alfred Stephens, I had the opportunity

to attend a special event held at the Homi Bhabha

Auditorium in honour of Prof. B.V. Sreekantan, on

his birthday. I still vividly remember Prof. Narlikar’s

engaging lecture, in which he traced the evolution of

Prof. Sreekantan’s research ﬁeld vis-

a-vis his age from

childhood, indicating how the stage was all set for his

launch as a pioneer in the ﬁeld. With characteristic

clarity and wit, he described how the scientiﬁc landscape

was primed for Prof. Sreekantan’s groundbreaking

contributions.

I had hoped for a personal meeting with Prof.

Narlikar during the program, but our paths crossed only

at the ﬁnal selection interview for research scholars

from among the VSRP students. He chaired the ﬁnal

interview panel. I clearly remember his question about

the motion of an electron in mutually perpendicular

electric and magnetic ﬁelds — a moment etched in

my memory. Though I wasn’t selected, I had the

privilege of speaking with him brieﬂy during the closing

dinner. That brief interaction made the entire experience

worthwhile.

After completing my postgraduation, I aspired to

pursue doctoral research in astronomy or cosmology.

I approached Prof. Babu Joseph at Cochin University

of Science and Technology, but unfortunately, there

were no available openings at the time. On his advice,

I joined a research program in microwave antenna

development under the guidance of Prof. K.T. Mathew

at the School of Pure and Applied Physics, Mahatma

Gandhi University, Kottayam.

Meanwhile, Dr. Ninan Sajeeth organized a School

on Astronomy at Charalkunnu, near Tiruvalla — a

landmark event in my scientiﬁc journey. When I learned

that Prof. Narlikar and other leading astronomers would

be participating, I applied and was thrilled to be selected.

The school provided not only a wealth of knowledge but

also a rare opportunity to interact closely with scientists

like Prof. Narlikar, Prof. A.K. Kembhavi (AKK),

Figure 1: Prof. Jayant Narlikar and the author with his

students

and Prof. A.K. Pandey, among others. I still recall

the extended panel discussions in the auditorium —

enlightening sessions that deepened my understanding

of cosmology and astronomy and allowed me to ask

questions. The encouragement I received there gave me

a sense of belonging in the ﬁeld.

Years later, in 2006, seeking to reconnect with my

passion for astronomy, I attended a refresher course

for college teachers at the Inter-University Centre for

Astronomy and Astrophysics (IUCAA). There, I met

Prof. Joydeep Bagchi, a radio astronomer who was

then in the process of establishing the Radio Astronomy

Lab at IUCAA. Upon learning about my background in

microwave antennas, he encouraged me to apply for an

associateship at IUCAA. I was selected the following

year, marking the beginning of my academic journey

in radio astronomy under Prof. Joydeep’s mentorship.

Over the years, during my visits to IUCAA, I had

the privilege of meeting Prof. Narlikar on numerous

occasions — at events, academic gatherings, and the

dinners that often followed them. I fondly recall a time

when I requested an article from Prof. Narlikar for a

collection being put together by IUCAA faculty, under

the guidance of Prof. Ajit Kembhavi. Prof. Narlikar

was the ﬁrst to submit his piece — well ahead of the

deadline!

Another cherished memory involves a visit to

IUCAA with twenty selected school students from

Idukki district, Kerala, during the International Year

of Astronomy. This visit marked the culmination of

a year-long astronomy popularisation programme in

village schools, named Sasthragramam. Prof. Narlikar

enjoyed engaging with students, explaining the wonders

of the universe in his inimitable, accessible style.

In 2013, I had the honour of attending a meeting

during his visit to the IUCAA Resource Centre

at the Department of Physics, Cochin University.

When he visited again in 2018, Prof. Kuriakose

entrusted me with organizing student interactions at

Nirmala College, Muvattupuzha, and CMS College,

Kottayam. Facilitating these sessions and witnessing

young students engage with such a towering ﬁgure in

science was an immensely rewarding experience.

Looking back, every interaction with Prof.

Narlikar — however brief — left a lasting impact

on my academic and personal life. His humility,

intellectual brilliance, and unwavering dedication to

science continue to inspire generations of students,

educators, and researchers. For many of us, he is more

than a scientist — he is a legend.

About the Author

Dr. Joe Jacob retired as an Associate

Professor in the Department of Physics at Newman

College, Thodupuzha, Kerala, which is aﬃliated

with Mahatma Gandhi University, Kottayam. Since

2006, he has been a Visiting Associate at the Inter-

University Centre for Astronomy and Astrophysics

(IUCAA), Pune, and remains actively engaged in

astronomical research, as well as science education

and popularization in the region.

Dr. Joe was part of the team that discovered the galaxy

supercluster named Saraswati in 2017—a ﬁnding that

received wide coverage in both Indian and international

media. In recognition of his contributions to education,

he received the Prof. Sivaprasad Memorial Best

Teacher Award in 2018. He is also the co-recipient of

the New Discovery Award and the Zubin Kembhavi

Award for Science Popularization by the Astronomical

Society of India.

The Legacy of Jayant Narlikar

by Moncy V John

airis4D, Vol.3, No.6, 2025

www.airis4d.com

Professor Jayant Narlikar, who passed away

recently, was literally a giant in the ﬁeld of astronomy,

astrophysics and cosmology. His contributions to these

ﬁelds will always be remembered, especially those

he made in cosmology. Though the ‘Steady State

Cosmology’, which was developed in the 1940s by

his mentor Fred Hoyle, along with Geoﬀrey Burbidge

and Chandra Wikramasinghe, fell out of favour among

mainstream cosmologists lately, Narlikar is considered

a towering ﬁgure in Indian academic circles. He did

extensive research and made signiﬁcant contributions

to steady state cosmology, and later developed it into

the Quasi-Steady State Cosmology (QSSC). In addition.

he was a great science populariser and an ardent

stalwart of proper scientiﬁc temper, a fast receding

ideal in most parts of the world. This padmavibhushan

recipient cosmologist was a great institution builder too,

showing India how a research institution should work,

by founding the Inter-University Centre for Astronomy

and Astrophysics (IUCAA) at Pune.

Looking backward, it may seem astonishing that

during the evolution of quantum mechanics in the

ﬁrst half of the twentieth century, there were major

contributions from scientists in India, which was only a

British colony at that time. Here are some examples.

All the fundamental particles in nature can be

divided into two categories, namely bosons and

fermions. The former, which includes photons, the

quanta of light, is named after the Bengali physicist

Satyendra Nath Bose. It was he who discovered

their collective quantum behavior. Similarly, Sir C.V.

Raman, Meghnath Saha, etc. have made signiﬁcant

contributions to the development of quantum mechanics.

But contrary to this, India has no names to project in the

case of general theory of relativity during this period.

Even S. Chandrasekhar, who wrote the ‘horoscope’ of

stars, showed interest in general relativity only very

lately, in the 1960’s. The physics research in India

shows this ‘quantum leaning’, in general. The ﬁrst

theorist who paved the foundations of general relativity

in India was Prof. V.V. Narlikar, then a professor of

mathematics at the Banaras Hindu University. Most of

the general relativists in this country belong to the clan

of Prof. V.V. Narlikar. His son, Prof. Jayant Narlikar,

later shot to world wide fame for his contributions to

the ‘Steady State cosmological Model’.

Jayant Narlikar has revealed that his desire to

become a mathematician was not deliberately cultivated

in him by his father. Here is an incident that took place

while he was a student in standard three: The teacher

asked each student what his/her parent is doing. Most of

them were children of staﬀ members of Banaras Hindu

university. ”My father is a professor” was Jayant’s reply.

”Professor of what?” the teacher again asked, but the

child could not answer it. ”Your father is a professor

of mathematics” the teacher said. Narlikar remembers

that the feeling of inadequacy at not knowing the full

answer soon gave way to one of elation, as his father

is a professor of his best liked subject, which was

mathematics.

Even then, he never forgot to acknowledge the ideal

conditions he could enjoy in his pursuits. This humble

professor attributed his success to the right people he

had around him to support him in every matters. When

he says that at t = 0 he was fortunate to have the right

kind of parents, we recognise that the gentle humour in

it is aimed at the big bang model!

In the 1960’s, when Narlikar joins Fred Hoyle for

research in cosmology, the big bang and the steady

state models were almost equals. But now in the

midst of those who believe that the cosmic microwave

background radiation predicted by Milne in 1935 and

discovered by Penzias and Wilson in 1965 has falsiﬁed

the steady state model, there were only a few senior

cosmologists, including Narlikar, who did not accept

defeat.

Among the criticisms they raise against big bang

model, the most important one is that this model does not

provide a deep insight or revelation that triggers thought.

The big bang simply follows an empiricist epistemology.

The former students of Hoyle, namely Narlikar, Geoﬀrey

Burbidge, Chandra Wickramasinghe etc. used to say

that even young researchers in cosmology do not hesitate

to join the big bang ﬂock, without evaluating the

situation objectively. The witty Burbidge had once

qualiﬁed themselves as ‘old revolutionaries’ and the

opponents as ‘young conservatives’ in cosmology!

After obtaining his Ph.D. in 1963, Narlikar started

his career as a researcher and a professor in Cambridge

and later in some of its allied institutions. At Cambridge,

in order to cope with the fast changing situation on the

observational and computational front in astronomy,

Hoyle was feeling the need to set up an institution

where visitors from active centres in the world would

visit and discuss their work and thereby positively

and constructively inﬂuence the working of academics

there. When the response from the university and

the government was not very forth-coming, private

organisations such as Wolfson Foundation, Nuﬀeld

Foundation etc. came to support him. Finally when

Cambridge University donated the necessary land for

construction, Hoyle’s dream project named ‘Institute

of Theoretical Astronomy’ materialised. To what will

happen to the institute when the Nuﬀeld grant runs out,

Hoyle replied that if the institute does not grow to a

world class institute by that time, he for one would shed

no tears at its abolition!

Narlikar was among the founding faculty of this

institute. He got inspiration to start such an institution

in India from this experiment. Narlikar opines that

whereas institutions are created to boost egos of certain

individuals, and continue long past their usefulness

because no one had the courage to abolish them, the

success of the institute justiﬁes Hoyle’s vision that such

an institution was needed.

While returning to India in 1972, even though the

steady state picture was fading, Narlikar was considered

a national hero. Visiting on an invitation from the

President of India, he toured to make a series of lectures

in his sharp and transparent style, attracting students

and researchers to this new ﬁeld. From 1972 to 1988 he

worked as the Head of Theoretical Astrophysics at the

Tata Institute of Fundamental Research (TIFR), Mumbai.

This institute has by that time become a world renowned

research institute under the able leadership of Homi J.

Bhabha. Narlikar has disclosed that Bhabha’s insights

as to how to run a research institute has helped him a

lot. In 1988, the then University Grants Commission

(UGC) Chairman Prof. Yash Pal entrusted Narlikar

with the task of establishing a world class institution

for astronomy, astrophysics and allied subjects. On the

outskirts of the Pune University Campus, by the side

of the old Mumbai-Pune highway, the space for this

was made available. Thus started the beautiful ‘Inter-

University Centre for Astronomy and Astrophysics’

(IUCAA), designed by the world famous architect

Charles Corrhea. This land was formerly grasslands

and small woods where cattle used to graze, at hardly

two kilometers from the Khadki railway station in Pune.

Narlikar was its founder Director. Around hundred

researchers, stay and do research here. Many students

and teachers from various Indian universities come to

visit IUCAA quite often for interactions and references.

After being at the helm of action as Director for

ﬁfteen years, Narlikar served as Professor Emiratus at

IUCAA. The most curious thing is that by this time the

paradigm of steady state model is almost wiped out.

In 1994, Hoyle, Narlikar and Burbidge have together

proposed the quasi-steady state cosmology (QSSC), a

modiﬁed version of steady state model. In this new

model, it is conceived that the universe oscillates, i.e.,

cycles of expansion and contraction repeats, even when

it is in a steady state. We are now in an expanding

phase of it. The model will have a hot past, just as in

the standard big bang model. Thus it can explain the

microwave background and other phenomena, without

much diﬀerence from that of big bang. Many people

consider it as not much diﬀerent from standard big

bang model, even when they are not willing to test

any diﬀerence with it at the observational front. After

Shyamal Banerjee and Ram Gopal Vishwakarma, who

helped Prof. Narlikar in his research in QSSC left for

teaching assignments elsewhere, there were no students

working in this ﬁeld at IUCAA.

Narlikar and co., who were very much conﬁdent

with QSSC, have expressed their annoyance that theories

of science are not defeated; instead, they disappear

with the death or aging of their proponents. That the

steady state model now can be used for studies on

the methodology of science is really an irony. That

Narlikar is disgusted by the plight of this branch of

science is evident from his words. In an interview

given to Frontline after his retirement, he said: ”When

I entered the ﬁeld of cosmology as a research student in

1960, the subject was open and there were observational

possibilities of checking theories. Today one relies on N-

body simulations based on speculative initial conditions

to assert what is the correct model of the universe. If

I were a research student today, cosmology would not

attract me.”

He regrets that many people now use the theories

developed by Hoyle and himself in the 1960s, such as

negative energy scalar ﬁelds, black holes in galactic

nuclei, super clusters and voids, oscillating universe

which has no singularity, etc., without bothering to

acknowledge. Most are simply believers in big bang

cosmology, though it is inconsistent with ground

realities - even the measured value of the basic Hubble

constant remains controversial. It is opposed to the

spirit of science, which asks for repeatable experiments

to check a theory.

However, the role Narlikar and coworkers had in

keeping cosmology a science is beyond mention. Prof.

Richard Ellis, from Caltech in USA, says: ”.. the reason

why most astronomers believe in the big bang model

is that it is the simplest picture that is consistent with

the data. But it is very important that there are people

who are constantly pushing to be provocative to make

us question in more detail, whether this is the right

picture or not”. Echoing similar views, E.P.J. van den

Heuvel of the University of Amsterdam says: ”It is very

important that you have people like Narlikar who are

exploring other possibilities. There is a lot that people

do not basically understand. And it is now being told

that with WMAP there are only a few details to be ﬁlled

in and then we know everything. It is not like that. I do

not believe that.”

Lastly, let me script my personal acquaintances

with Professor Narlikar and how he helped me to test a

new cosmological model, which I have developed with

my supervisor Prof. K. Babu Joseph, at the Cochin

University of Science and Technology. This model,

which is named the ‘Eternal Coasting Cosmological

Model’, has an evolution of the universe, which is

neither accelerating nor decelerating. During my ﬁrst

visit to IUCAA in 1997, I could discuss this model

with Professor Narlikar. He was greatly enthused, for it

arose from some fundamental revision of cosmological

assumptions. In 1999, when I paid another visit

to IUCAA, he suggested me to analyse the then

released sensational Supernova data, which suspected

an accelerated of the universe, to see how our model

copes with it. He kindly agreed to collaborate with me

in this endeavor, which was a great recognition for a

novice like me. Professor Narlikar is not known to have

worked in such extensive manner in any cosmological

model, other than the steady state or related theories. In

2002, the results of our research were published in the

prestigious Physical Review journal. It was during these

long period of close association with him that I came

across the genuine lover of physics, the hard and fast

adherent of scientiﬁc methodology and compassionate

teacher in him. I pay respects to the great son of India,

who would lead us light, for generations to come.

About the Author

Dr. Moncy V. John is currenly a Visiting

Professor at the School of Pure and Applied Physics,

Mahatma Gandhi University, Kottayam. Formerly he was

Associate Professor and HoD, Department of Physics,

St. Thomas College, Kozhencherry and a Visiting

Associate at IUCAA, Pune. His research interests include

Theoretical Cosmology and Foundations of Quantum

Mechanics.

In Memory of Prof. Jayant V. Narlikar: A

Personal Tribute

by Tharanath R

airis4D, Vol.3, No.6, 2025

www.airis4d.com

Prof. Jayant V. Narlikar was not just a towering

ﬁgure in the world of astrophysics; to many of us, he

was an inspiration whose quiet intellect and gracious

demeanour left a lasting impression. I had the rare

fortune of meeting him three times over diﬀerent phases

of my academic life, and each encounter added a unique

layer to my admiration for him.

The ﬁrst time I saw Prof. Narlikar was during

my postgraduate days, when I was selected to attend

a summer school at IUCAA, Pune. Back then, my

understanding of astrophysics was still in its infancy,

and I hardly grasped the magnitude of the stalwarts

around me. I was too na

ıve to fully appreciate what it

meant to be in the same space as someone like Prof.

Narlikar. He walked the corridors with quiet grace, and

though I didn’t have a personal interaction with him

during that time, his presence lingered in my memory

like a silent inspiration waiting to bloom.

Our paths crossed again during my research days

at CUSAT, especially at the IUCAA Resource Centre

(IRC) established on campus. The IRC played a

signiﬁcant role in bridging aspiring researchers like

me with the premier Astrophysics’ community in India.

It created a valuable platform for interactions with

legends like Prof. Narlikar, and also opened doors to

visit IUCAA itself — an institution synonymous with

excellence in theoretical astrophysics. For many of

us, the IRC was not just a space; it was a conduit of

mentorship, academic growth, and lasting connections.

During one such opportunity, Prof. Narlikar

visited CUSAT. I had the privilege of welcoming him

along with my guide, the late Dr. V. C. Kuriakose,

and my fellow researchers. By then, I had started

studying General Theory of Relativity, and on my

guide’s suggestion, I picked up An Introduction to

Relativity by Prof. Narlikar — the one with the

iconic yellow cover. That book became my constant

companion and guide. Although I couldn’t muster

the courage to speak to him personally during that

visit—my inhibitions got the better of me—I listened

intently to his lectures. His clarity and poise made

complex ideas feel accessible and exciting. It was one

of those deﬁning moments when a book and its author

both became pillars in my learning journey.

The third meeting remains the most memorable.

By then, I was deep into my research work. Prof.

Narlikar had come to Kochi along with his wife,

intending to spend a few quiet days and then travel

to Andaman. Our guide entrusted me, along with Nijo

Varghese and Jishnu Suresh, with the responsibility of

assisting him during his stay. We were thrilled. This

time, I had ﬁnally shed the layers of hesitation that had

held me back during earlier encounters. I opened up

to him—not just about science, but about my research

struggles, my doubts, even the occasional blunders. He

listened with incredible patience and calm, never once

interrupting or dismissing my concerns. At the end

of our conversations, he gently remarked, “When you

visit IUCAA next time, we can think about some work

together.” That moment ﬁlled me with both pride and

a quiet regret. Perhaps I wasn’t conﬁdent enough yet

to follow through on that opportunity, but his words

still echo in my mind as an encouragement that I carry

forward.

I had hoped to get his signature in my now heavily

worn copy of his book, but by then, the cover had faded

and frayed from overuse — a testament to how deeply

I had studied it. To this day, I consider that book my

Bible. More than a scientist, I remember Prof. Narlikar

as a gentle soul — a man who listened more than he

spoke, whose humility was as profound as his intellect.

If there had been a fourth meeting, I might have even

gathered the courage to share my poems with him — a

thought that still brings a smile.

These encounters, brief yet meaningful, remain

the gemstones of my research life — moments of grace

and learning that I will cherish forever. Prof. Jayant

V. Narlikar’s legacy lives on not just in the cosmos he

studied, but in the minds and hearts of those he touched,

however brieﬂy.

About the Author

Dr. Tharanath R presently serves as an

Assistant Professor of Physics at Aquinas College,

Edacochin, Kerala.

Part I

Artiﬁcial Intelligence and Machine Learning

Reinforcement Learning for LLMs

by Arun Aniyan

airis4D, Vol.3, No.6, 2025

www.airis4d.com

1.1 Introduction

Reinforcement Learning (RL) has emerged as a

pivotal methodology for enhancing the capabilities

and reﬁning the behaviors of Large Language Models

(LLMs). By enabling LLMs to learn from interactions

and feedback within an environment, RL provides a

robust framework for optimizing their performance

across a diverse range of tasks. This article provides

a very detailed exploration of reinforcement learning

techniques speciﬁcally applied to LLMs, delving into

core concepts, methodologies, advanced applications,

and future directions.

1.2 Introduction to Reinforcement

Learning in LLMs

Reinforcement Learning is a branch of machine

learning where an agent learns to make decisions by

interacting with an environment. The agent receives

feedback in the form of rewards or penalties for

its actions. In the context of LLMs, the agent is

the language model, the environment is the task or

interaction scenario, and the actions are the generated

textual outputs. The objective of applying RL to

LLMs is to ﬁne-tune the models to perform tasks

more eﬀectively, align their responses with human

preferences, and minimize undesirable outputs.

RL enables LLMs to adapt and improve over time

through trial and error. Unlike traditional supervised

learning, where models learn from labeled data, RL

allows models to learn from the consequences of their

actions. This is particularly valuable for LLMs, which

need to handle a wide variety of open-ended tasks and

interactions.

1.3 Core Concepts of Reinforcement

Learning for LLMs

Reinforcement Learning (RL) applied to Large

Language Models (LLMs) leverages a framework

where the LLM, acting as an agent, operates within

an environment. This environment can manifest in

various forms, such as direct user interaction through

an interface, accessing and manipulating data within

a database, or navigating within a simulated scenario

designed for training.

The actions taken by the LLM are its generated

textual responses or outputs. These actions are

contingent on the perceived state of the environment,

which is the LLM’s understanding or representation

of the current situation or context. This state could

encompass the preceding dialogue, the content of a

database entry, or the parameters of a simulation.

A crucial element of the RL framework is the

reward, a scalar value that provides feedback on the

quality or desirability of the LLM’s action in a given

state. The design of the reward function is critical, as it

directly shapes the learning process, guiding the LLM

towards preferred behaviors and away from undesirable

ones. Rewards can be based on factors like coherence,

relevance, factual accuracy, user satisfaction, or task

completion.

The policy represents the LLM’s learned strategy

or set of rules for selecting actions based on the current

state. Initially, this policy might be random, but through

1.4 Reinforcement Learning Methodologies for LLMs

the iterative process of interaction, reward feedback, and

policy updates, the LLM learns to reﬁne its policy to

consistently choose actions that maximize its expected

future rewards.

The core of RL lies in the iterative interaction loop

between the LLM (agent) and the environment. In each

step of this loop:

The LLM observes the current state of the

environment.

Based on its current policy, the LLM takes an

action.

The environment transitions to a new state

and provides the LLM with a reward signal,

indicating the consequences of its action.

This cycle repeats continuously. The LLM uses the

received rewards to evaluate the eﬀectiveness of its

actions and update its policy accordingly. The goal is to

learn an optimal policy that maximizes the cumulative

reward received over time. This process often involves

exploring diﬀerent actions to discover which yield the

highest rewards and exploiting the knowledge gained

to consistently choose those beneﬁcial actions. The

challenges in applying RL to LLMs often revolve around

deﬁning eﬀective state representations, designing

appropriate reward functions that align with desired

outcomes, and eﬃciently exploring the vast action space

of text generation.

1.4 Reinforcement Learning

Methodologies for LLMs

Several methodologies have been developed for

applying RL to LLMs, with one of the most prominent

being Reinforcement Learning from Human Feedback

(RLHF).

1.4.1 Reinforcement Learning from Human

Feedback (RLHF)

RLHF is a sophisticated approach that leverages

human preferences to guide the learning process. It

involves multiple stages:

Pre-training: An initial LLM is trained on a

vast corpus of text data to learn general language

patterns.

Supervised Fine-tuning: The pre-trained LLM

is ﬁne-tuned on a smaller dataset of labeled

examples to improve its performance on speciﬁc

tasks.

Reward Model Training: A reward model

is trained to predict the quality of the

LLM’s responses based on human judgments.

Human annotators provide comparisons between

diﬀerent LLM outputs, and the reward model

learns to assign higher scores to preferred

responses.

Reinforcement Learning Fine-tuning: The

reward model is used to guide the ﬁne-tuning

of the LLM using a reinforcement learning

algorithm, such as Proximal Policy Optimization

(PPO). The LLM’s policy is updated to maximize

the reward predicted by the reward model.

1.4.2 Reward Modeling in RLHF

The reward model serves as a crucial link

in Reinforcement Learning from Human Feedback

(RLHF), connecting the subtleties of human preferences

with the algorithmic nature of reinforcement learning.

Its primary function is to quantify the desirability of

Language Learning Model (LLM) outputs, thereby

providing a scalar reward signal that the RL agent can

optimize. This reward signal guides the LLM towards

generating responses that align with human expectations

and values.

The training of the reward model relies on carefully

curated datasets consisting of human comparisons. In

this process, human annotators are presented with

multiple responses generated by the LLM for a given

prompt. Their task is to rank or compare these responses

based on various criteria, such as helpfulness, accuracy,

coherence, and harmlessness. These comparisons

provide the reward model with valuable information

about the relative quality of diﬀerent LLM outputs.

Through exposure to a substantial number of

these human comparisons, the reward model learns to

discern the subtle nuances that distinguish a preferred

response from a less desirable one. It develops an

1.5 Reinforcement Learning Algorithms for LLMs

internal representation of what constitutes a high-quality

response according to human judgment. This learned

understanding enables the reward model to assign a

numerical reward to any given LLM output, reﬂecting its

estimated quality based on the patterns observed in the

human preference data. The accuracy and robustness

of the reward model are paramount to the success of

RLHF, as it directly inﬂuences the learning direction of

the LLM. A well-trained reward model can eﬀectively

guide the LLM towards generating more human-aligned

and valuable outputs.

1.5 Reinforcement Learning

Algorithms for LLMs

Various RL algorithms can be used to ﬁne-tune

LLMs, each with its own strengths and weaknesses.

1.5.1 Proximal Policy Optimization (PPO)

PPO is a popular policy gradient algorithm that

strikes a balance between exploration and exploitation.

It restricts policy updates to a certain range, which helps

to stabilize the training process and prevent drastic

changes in behavior. PPO has been widely used in

RLHF due to its robustness and eﬃciency.

1.5.2 Advantage Actor-Critic (A2C)

A2C is another commonly used algorithm that

combines actor-critic methods. The actor learns the

policy, while the critic estimates the value of diﬀerent

states or actions. A2C helps to reduce variance in the

training process, leading to more stable and reliable

learning.

1.6 Advanced Applications of

Reinforcement Learning for

LLMs

RL applied to LLMs enables several advanced

applications.

Improving Response Quality: RL can optimize

LLMs to produce more coherent, engaging,

and contextually appropriate responses. By

rewarding responses that are perceived as high-

quality by humans, LLMs can learn to generate

better outputs.

Reducing Harmful Outputs: RL can be used

to penalize harmful, biased, or toxic outputs. By

training LLMs to avoid generating such content,

it can enhance the safety and ethical behavior.

Aligning with Human Values: RLHF enables

LLMs to align with human preferences and values.

This is particularly important for ensuring that

AI systems are helpful and harmless.

Personalized Responses: RL can be used

to tailor LLM responses to individual users’

preferences and needs. By learning from user

feedback and interactions, LLMs can provide

more personalized and relevant experiences.

1.7 Beneﬁts of Reinforcement

Learning for LLMs

Enhanced Performance: RL allows ﬁne-tuning

LLMs for speciﬁc tasks, leading to improved

performance and eﬀectiveness.

Improved Control: RL provides greater control

over the LLM’s behavior, enabling developers to

steer its responses in desired directions.

Better Alignment: RLHF ensures that LLMs

are better aligned with human preferences and

values, leading to safer and more responsible AI

systems.

1.8 Challenges and Considerations

Despite its many beneﬁts, RL for LLMs also

presents several challenges.

Data Collection: Gathering high-quality human

feedback can be expensive and time-consuming.

The quality and diversity of the feedback data are

crucial for the success of RLHF.

Reward Hacking: LLMs may ﬁnd ways to

exploit the reward model, leading to unintended

behaviors.

1.9 Conclusion

Stability: RL training can be unstable and require

careful tuning. Poorly tuned RL can lead to

erratic behavior and poor performance.

Interpretability: Understanding why an LLM

makes certain decisions after RL training can

be challenging, which can hinder debugging and

improvement eﬀorts.

1.9 Conclusion

The burgeoning ﬁeld of Reinforcement Learning

for Large Language Models (LLMs) represents a critical

frontier in the advancement of artiﬁcial intelligence.

This interdisciplinary area harnesses the principles

of reinforcement learning to ﬁne-tune and optimize

the capabilities of increasingly sophisticated LLMs.

Given the rapid pace of research and development in

both natural language processing and reinforcement

learning, the methodologies employed to train and

evaluate these models are in a state of constant ﬂux

and improvement. As researchers delve deeper into

the complexities of language understanding, generation,

and interaction, we can anticipate the emergence of

novel and signiﬁcantly more eﬀective techniques for

aligning LLMs with intricate human preferences and

expectations. These forthcoming advancements hold

the promise of dramatically enhancing the capabilities

of AI models to engage in nuanced and meaningful

dialogues with human users across a wide spectrum

of applications. Furthermore, the application of

reinforcement learning methodologies oﬀers a powerful

avenue for ensuring that AI systems are not only

capable of performing complex tasks but also that

their behavior is intrinsically aligned with fundamental

human values, ethical considerations, and societal

norms. This alignment is crucial for fostering trust and

enabling the widespread adoption of AI technologies

in sensitive and impactful domains.

References

Christiano, P., et al. ”Deep reinforcement

learning from human preferences.” Advances in

neural information processing systems 30 (2017).

Ouyang, Long, et al. ”Training language models

to follow instructions with human feedback.”

Advances in Neural Information Processing

Systems 35 (2022).

Schulman, John, et al. ”Proximal policy

optimization algorithms.” arXiv preprint

arXiv:1707.06347 (2017).

About the Author

Dr.Arun Aniyan is leading the R&D for

Artiﬁcial intelligence at DeepAlert Ltd,UK. He comes

from an academic background and has experience

in designing machine learning products for diﬀerent

domains. His major interest is knowledge representation

and computer vision.

Introduction to Group Equivariant CNN

Part - II

by Blesson George

airis4D, Vol.3, No.6, 2025

www.airis4d.com

2.1 Introduction

Convolutional Neural Networks (CNNs) have

revolutionized computer vision by exploiting spatial

structure via translation equivariance—the ability to

detect features regardless of their spatial position.

However, standard CNNs are limited to translations

and do not inherently support other transformations

such as rotations and reﬂections.

In 2016, T. S. Cohen and M. Welling proposed

Group Equivariant Convolutional Networks (G-

CNNs). Their work introduced the concept of

incorporating broader symmetry groups into the

convolution operation, leading to improved performance

on datasets such as CIFAR-10.

2.2 Group Convolutions and

Symmetry

2.2.1 Symmetry Groups

The core idea of G-CNNs is grounded in group

theory. A group is a set of transformations (called

group elements) that includes an identity transformation,

has a deﬁned inverse for every element, and supports

composition. In the context of G-CNNs, relevant

transformations include:

Translations: Shifting images in space.

Rotations: Typically in multiples of 90

○

Reﬂections: Horizontal or vertical mirroring.

Common groups used in practice include:

: Translations on a 2D grid.

p4: Translations + 90

○

rotations.

p4m

: Translations +

○

rotations + mirror

reﬂections.

2.2.2 G-Convolution: Convolution Over a

Group

In standard CNNs, the convolution operation is

translation-equivariant. G-CNNs generalize this idea

by performing convolution over a group

. The G-

convolution is deﬁned as:

[f ∗ ψ](g) =

∑

h∈G

f(h)ψ(g

−1

where

is the input feature map deﬁned on the

group

, and

is the convolutional kernel, also deﬁned

over

. The result

[f ∗ ψ](g)

is a function on

—

that is, a feature map indexed by elements of the group.

2.3 Structured Feature Maps and

Transformation

2.3.1 Graph Representation of Group

Elements

To visualize this, think of a graph:

Nodes represent possible poses (e.g., positions

and orientations).

Edges represent group actions that transform one

pose into another.

2.4 Transformation of Structured Objects

The connections between these poses reﬂect the

structure of the group. For instance, rotating a node by

○

might move it to a neighboring node in the graph.

2.4 Transformation of Structured

Objects

When a transformation is applied to a structured

object (i.e., a feature map on G), two things happen:

Local Transformation: The data at each node

(e.g., a rotated image patch) is individually

transformed.

Permutation: The data is moved to another node

in the graph according to the transformation. This

is often described as a permutation of nodes.

2.4.1 Understanding Through Rotation

Consider a 90

○

rotation:

Each image patch (data at the node) is rotated by

○

The rotated patch is reassigned to a new node that

corresponds to the new orientation, following a

pre-deﬁned arrow (transformation rule) in the

graph.

This mechanism ensures that the network’s

response is equivariant: rotating the input results

in a corresponding rotation (permutation) of the output.

2.5 Two Vital Operations

To fully understand G-convolutions, two core

operations are essential:

Transformation of a structured object:

As discussed, this involves both local data

transformation and permutation across nodes.

Dot-product over the group: The G-

convolution can be seen as computing a group-

level dot-product between the input feature map

and the ﬁlter, integrating over all group elements.

2.6 Why Equivariance Matters

Equivariance ensures that transformations in the

input lead to predictable transformations in the output.

This has several beneﬁts:

Data Eﬃciency: The network can generalize

from fewer examples, as it no longer needs to see

every transformation explicitly.

Parameter Sharing: Fewer parameters are

needed to handle transformed inputs.

Better Generalization: The model generalizes

better to unseen orientations and poses.

2.7 Applications of G-CNNs

G-CNNs have shown promise in several domains:

Image classiﬁcation (e.g., CIFAR-10, rotated

MNIST)

Medical imaging (e.g., histopathology, radiology

where orientation varies)

Molecular modeling (where rotational

symmetry is key)

Astronomy (e.g., galaxy shape classiﬁcation)

2.8 Extensions and Related Work

Steerable CNNs: Generalize G-CNNs by

allowing continuous groups and steerable ﬁlters.

Gauge Equivariant CNNs: Incorporate local

symmetry and gauge theory into CNNs.

3D G-CNNs: Extensions to 3D data for

applications in robotics and volumetric analysis.

Lie Group CNNs: Handle continuous symmetry

groups using Lie algebra techniques.

2.9 Conclusion

Group Equivariant Convolutional Networks extend

the power of CNNs by incorporating symmetry

beyond translation. By modeling feature maps on

structured domains (groups) and applying group

actions, G-CNNs maintain equivariance under broader

transformations. This leads to more robust, data-

2.9 Conclusion

eﬃcient, and generalizable models — a step closer

to truly symmetry-aware deep learning.

About the Author

Dr. Blesson George presently serves as

an Assistant Professor of Physics at CMS College

Kottayam, Kerala. His research pursuits encompass

the development of machine learning algorithms, along

with the utilization of machine learning techniques

across diverse domains.

Part II

Astronomy and Astrophysics

Type II and Type III Supernovae: A Detailed

Exploration of Stellar Cataclysms

by Sindhu G

airis4D, Vol.3, No.6, 2025

www.airis4d.com

1.1 Introduction

Supernovae are among the most violent and

luminous events in the cosmos, signaling the end of a

star’s life and profoundly inﬂuencing galactic evolution.

They are responsible for dispersing heavy elements

into the interstellar medium, altering the chemical

composition of galaxies, and triggering the formation

of new stars. Supernovae are classiﬁed based on

their spectral properties and the physical mechanisms

driving the explosion. Among these, Type II supernovae

represent a well-established category associated with

the core-collapse of massive stars. In contrast, Type III

supernovae remain a subject of ambiguity and historical

curiosity, having no place in the current astrophysical

classiﬁcation scheme.

This article delves into the known physics of

Type II supernovae and examines the origins and

interpretations of the enigmatic Type III designation,

providing a comprehensive view of these stellar

explosions.

1.2 Supernova Classiﬁcation

Overview

The classiﬁcation of supernovae hinges primarily

on their spectral features, especially the presence or

absence of hydrogen lines:

Type I Supernovae: Lack hydrogen in their

spectra.

Figure 1: The expanding remnant of SN 1987A, a

peculiar Type II supernova in the Large Magellanic

Cloud. (Image Credit: NASA)

1.3 Type II Supernovae: Core-Collapse of Massive Stars

Type II Supernovae: Show prominent hydrogen

lines.

Further subdivisions are made based on other

spectral characteristics and light curve behaviors. While

Type Ia supernovae originate from white dwarf systems

undergoing thermonuclear runaway, Types II, Ib, and

Ic are all core-collapse supernovae associated with the

deaths of massive stars.

Historically, supernovae were initially identiﬁed

based on photographic observations and crude

spectroscopic data. As instrumentation improved, so

too did the reﬁnement of classiﬁcation. Within this

evolving landscape, the term ”Type III” occasionally

surfaced but never solidiﬁed into an accepted category.

1.3 Type II Supernovae:

Core-Collapse of Massive Stars

1.3.1 Progenitor Stars

Type II supernovae originate from stars with

initial masses greater than about 8 solar masses

(

⊙

). These stars evolve through successive nuclear

burning stages—hydrogen, helium, carbon, and heavier

elements—until iron forms in the core. Since iron

fusion is endothermic, it cannot support the core against

gravitational collapse.

The core rapidly collapses into a neutron star

or black hole, and the outer layers are expelled in a

cataclysmic explosion. The deﬁning feature of Type

II supernovae is the presence of hydrogen lines in

their optical spectra, indicating that the progenitor star

retained a signiﬁcant hydrogen envelope.

1.3.2 Explosion Mechanism

When the core collapses, it reaches nuclear

densities, and a rebound shock is created. Neutrinos,

released in vast numbers, play a key role in transferring

energy to the outer layers, driving the explosion. The

result is a bright and energetic display, with luminosities

reaching 10

–10

times that of the Sun.

1.3.3 Subtypes of Type II Supernovae

Type II supernovae are further classiﬁed into

subtypes based on the shape of their light curves:

Type IIP (Plateau): Characterized by a relatively

ﬂat light curve for several weeks due to hydrogen

recombination in the ejected envelope.

Type IIL (Linear): Show a more steady, linear

decline in brightness.

Type IIn (Narrow): Exhibits narrow emission

lines resulting from interaction with dense

circumstellar material (CSM), often indicative of

signiﬁcant mass loss before the explosion.

Type IIb: Begin with hydrogen lines that fade over

time, suggesting the progenitor lost most—but

not all—of its hydrogen envelope. These can

serve as a bridge between Type II and Type Ib.

1.4 Astrophysical Importance

Type II supernovae are critical for:

Chemical enrichment: They synthesize and

disperse elements like oxygen, silicon, and

calcium.

Neutron star and black hole formation.

Galactic dynamics: Supernova remnants inject

energy into the interstellar medium, inﬂuencing

star formation.

Cosmic distance scaling: Though not standard

candles like Type Ia, their plateau phase and other

features can be used for distance estimation.

1.5 The Mystery of Type III

Supernovae

1.5.1 Historical Mentions and Theoretical

Origins

The term ”Type III supernova” is not recognized

in the modern classiﬁcation system. However, it has

appeared sporadically in older literature or speculative

contexts. Its usage has typically referred to:

Hypothetical new classes of supernovae that did

not ﬁt into the Type I or II categories.

1.6 Comparative Analysis

Theoretical models involving unusual progenitor

stars or explosion mechanisms.

Rare phenomena not yet well understood

observationally.

One interpretation associates ”Type III” with an

early attempt to classify supernovae that show

extremely slow light curve evolution or unique

spectral signatures.

1.5.2 Association with Pair-Instability

Supernovae

In some discussions, ”Type III” has been

informally equated with pair-instability supernovae

(PISNe), an exotic and rare type of explosion theorized

to occur in stars with initial masses between 140–260

⊙

Pair-Instability Mechanism:

At very high core temperatures, energetic gamma

rays can produce electron-positron pairs, reducing

radiation pressure and causing the core to contract.

This leads to explosive oxygen and silicon burning that

completely disrupts the star—leaving no remnant.

While PISNe exhibit extremely luminous and long-

lasting light curves, they are not classiﬁed as ”Type

III” in standard terminology. Instead, they are often

grouped under Type I (lacking hydrogen) or considered

a separate theoretical class.

1.5.3 Current Status of the Term

Modern supernova taxonomy, based on spectral

features and light curves, has rendered the term ”Type

III” obsolete. No observational evidence has ﬁrmly

established a distinct supernova class that necessitates

its resurrection. Most unusual or borderline cases are

now better understood as subtypes of Type II or as

superluminous supernovae (SLSNe), which may span

both core-collapse and thermonuclear origins.

1.6 Comparative Analysis

Feature

Type II

Supernova

Type III

Supernova

Recognition

Oﬃcial and

well-studied

classiﬁcation

Not recognized

in modern

taxonomy

Progenitor

Massive stars

(

> 8 M

⊙

)

with hydrogen

envelopes

Hypothetical

or extremely

massive stars

(PISNe?)

Spectrum

Prominent

hydrogen lines

Ambiguous;

possibly unusual

spectra or light

curves

Mechanism Core-collapse

Theorized: pair-

instability or

exotic collapse

Remnant

Neutron star or

black hole

None (in case of

pair-instability)

Observational

Status

Numerous

observations

No conﬁrmed

observations as

a distinct class

Table 1.1: Comparison between Type II and Type III

supernovae.

1.7 Conclusion

The study of supernovae remains a vital area

of astrophysics, oﬀering insights into the life cycles

of stars, the formation of compact objects, and the

chemical evolution of galaxies. Type II supernovae are

a cornerstone of our understanding of stellar death

in massive stars. Their clear classiﬁcation, well-

understood progenitors, and rich observational data

make them a crucial subject of ongoing research.

In contrast, Type III supernovae, as an undeﬁned

or obsolete classiﬁcation, highlight the evolving nature

of scientiﬁc understanding. While they may no longer

hold a formal place in supernova taxonomy, the concept

underscores the need to remain open to new discoveries

and theoretical possibilities.

As telescope technology advances and surveys like

1.7 Conclusion

the Vera C. Rubin Observatory’s LSST come online, we

may uncover new forms of stellar death that challenge

current classiﬁcation systems. Whether or not a true

”Type III” supernova emerges, the universe continues

to surprise and inspire, one explosion at a time.

References:

Type II supernova

The diﬀerent types of supernovae explained

Discovery of new type of supernova explains

ancient mystery

Supernova

Photometric and spectroscopic diversity of Type

II supernovae

Astronomers conﬁrm there’s a third type of

supernova explosion

Supernovae Ib and Ic from the explosion of

helium stars

Evolutionary Models for Type Ib/c Supernova

Progenitors

Type Ib and Ic Supernovae: Models and Spectra

Near-infrared Spectral Properties of Type Ib/Ic

Supernova Progenitors and Implications for

JWST and NGRST Observations

About the Author

Sindhu G is a research scholar in Physics

doing research in Astronomy & Astrophysics. Her

research mainly focuses on classiﬁcation of variable

stars using diﬀerent machine learning algorithms. She

is also doing the period prediction of diﬀerent types

of variable stars, especially eclipsing binaries and on

the study of optical counterparts of X-ray binaries.

Part III

Biosciences

Invasive Alien Species in India: A Growing

Threat to India’s Biodiversity, Ecosystems and

Economy

(A tribute to International Day for Biological Diversity- May 22, 2025)

by Geetha Paul

airis4D, Vol.3, No.6, 2025

www.airis4d.com

1.1 Introduction

Invasive alien species (IAS) represent one

of the most pervasive and accelerating threats

to global biodiversity, ecosystem stability, and

socio-economic systems. As non-native organisms

that establish, proliferate, and spread beyond their

natural biogeographical ranges, IAS disrupt ecological

networks, alter habitat structures, and drive native

species toward extinction through competitive exclusion,

predation, and habitat modiﬁcation. In India, a mega-

diverse country harbouring four global biodiversity

hotspots, the unchecked proliferation of IAS has

precipitated severe ecological consequences, including

the degradation of forest ecosystems, eutrophication of

freshwater systems, and the collapse of native species

assemblages.

The introduction and spread of IAS in India

are facilitated by multiple anthropogenic pathways,

including international trade, horticultural exchange,

aquaculture expansion, and climate-mediated range

shifts. Key invasive taxa such as Lantana camara,

Parthenium hysterophorus, and Eichhornia crassipes

have demonstrated remarkable adaptive plasticity,

enabling them to outcompete indigenous ﬂora and fauna

while disrupting critical ecosystem services. These

invasions impose substantial economic burdens, with

agricultural losses exceeding 10,000 crores annually due

to crop infestations by pests like the cotton mealybug

(Phenacoccus solenopsis) and the fall armyworm

(Spodoptera frugiperda). Furthermore, public health

systems are strained by IAS-linked zoonoses and vector-

borne diseases, exempliﬁed by the role of Aedes

albopictus in arboviral transmission.

This review synthesises current knowledge

on the invasion ecology of high-impact IAS in

India, examining their 1) introduction pathways,

2) physiological and reproductive adaptations

driving invasiveness, 3) trophic-level impacts on

native biodiversity, and 4) cascading eﬀects on

ecosystem functioning (e.g., soil nutrient cycling,

hydrology). We evaluate evidence-based management

strategies, emphasising Integrated Pest Management

(IPM) frameworks that combine biological control

(e.g., Zygogramma bicolorata for Parthenium),

mechano-chemical interventions, and restoration

ecology principles. Policy gaps in India’s

biosecurity infrastructure are critically analysed, with

recommendations for 1) strengthening early detection

systems, 2) enforcing stricter quarantine protocols,

and 3) fostering cross-border collaborations under

international agreements like the CBD and IPBES.

1.2 Major Invasive Species in India: Ecological and Economic Impacts

By elucidating the mechanistic drivers of IAS

success and highlighting scalable mitigation approaches,

this article underscores the urgency of adopting a One

Health approach, integrating ecological, agricultural,

and public health perspectives, to curb the escalating

IAS crisis in India.

Deﬁnition

Invasive alien species (IAS) are non-indigenous

organisms that establish, proliferate, and spread

aggressively in new environments, often outcompeting

native species. According to the Convention on

Biological Diversity (CBD), IAS are the second-

largest threat to global biodiversity after habitat

destruction. IAS entered India through multiple

anthropogenic pathways. Intentional introductions

(e.g., Lantana camara as ornamental plants, Nile

tilapia for aquaculture). Unintentional introductions

(e.g., Parthenium hysterophorus via contaminated grain

imports). Climate change and habitat fragmentation

facilitate IAS spread.

1.2 Major Invasive Species in India:

Ecological and Economic Impacts

1.2.1 Terrestrial Plant Invasions

Lantana, Origin: Neotropics (Central and

South America)

Lantana camara is considered one of the world’s

worst invasive species due to its rapid spread,

adaptability, and severe ecological impact. It forms

dense thickets that outcompete native plants, reduce

biodiversity, and disrupt natural ecosystems. The plant

produces toxic chemicals that inhibit the growth of

other species and are harmful to livestock and wildlife.

Its seeds are easily dispersed by birds and animals,

enabling quick colonisation of new areas. Additionally,

Lantana alters soil conditions and increases ﬁre risks,

further damaging habitats. Its widespread invasion

across forests, grasslands, and agricultural lands in many

countries makes it diﬃcult to control and a signiﬁcant

threat to both natural ecosystems and human livelihoods.

Allelopathic chemicals inhibit native plant germination.

Figure 1: Lantana camara (lantana); habitat, showing

ﬂowers and foliage, as well as the thorny stems Image courtesy:

https://www.cabidigitallibrary.org/doi/full/10.1079/cabicompendium.29771

Rapid vegetative propagation by stem fragments. Forms

monocultures, reducing native plant diversity by 30–

60% in invaded forests (Kohli et al., 2006).

Ecological Impact:

Alter ﬁre regimes due to high ﬂammability. 500–

1000 crores/year spent on removal in protected areas

(Forest Survey of India, 2020).

Parthenium hysterophorus (Congress Grass),

Origin: Tropical Americas

Parthenium hysterophorus, commonly known as

Congress grass, is one of the world’s most aggressive

and harmful invasive plant species. It rapidly colonised

disturbed lands such as roadsides, pastures, farmlands,

and wastelands, often causing signiﬁcant reductions

in crop yields and biodiversity. Its success as an

invader is due to proliﬁc seed production—up to around

20,000 seeds per plant—combined with its ability to

germinate across a wide temperature range and persist

in soil seed banks for up to a decade. The plant’s

allelopathic nature releases chemicals like parthenin

that inhibit the growth of native plants and crops, further

exacerbating its impact on agriculture and ecosystems.

1.2 Major Invasive Species in India: Ecological and Economic Impacts

Figure 2: Parthenium hysterophorus on open land

Image courtesy: https://en.wikipedia.org/wiki/Parthenium hysterophorus

Additionally, it poses serious health risks to humans and

animals, causing dermatitis, respiratory allergies, and

toxic eﬀects in livestock. Parthenium’s seeds disperse

easily via wind, water, vehicles, and animals, enabling

its spread across more than 50 countries worldwide.

Its adaptability to diverse environmental conditions,

lack of natural enemies in invaded regions, and high

competitive ability make it a formidable invasive weed

requiring integrated management strategies to control

its spread and mitigate its negative eﬀects. High seed

production (˜15,000 seeds/plant) with prolonged soil

viability. Allelopathic suppression of crops like wheat

and pulses.

Ecological Impact:

Reduces agricultural yields by 40–90% in heavily

infested regions (GISD, 2021). Causes severe allergic

contact dermatitis in humans.

1.2.2 Aquatic Invasions

Figure 3: Water hyacinth single (A), and in the form

of a dense mat (B).

Image courtesy: https://pmc.ncbi.nlm.nih.gov/articles/PMC3218481/ﬁgure/F1/

Eichhornia crassipes (Water Hyacinth), Origin:

Amazon Basin

Eichhornia crassipes, commonly known as water

hyacinth, is a free-ﬂoating aquatic plant native to South

America that has become one of the world’s most

notorious invasive species. It rapidly forms dense,

monospeciﬁc mats on the surface of lakes, rivers,

ponds, and wetlands, often doubling its population

within two weeks under ideal conditions. These

thick mats outcompete native aquatic plants for light,

nutrients, and oxygen, severely reducing biodiversity

and altering aquatic ecosystems. The mats also

deplete dissolved oxygen levels, disrupt ﬁsh spawning

areas, degrade waterfowl habitats, and can lead to

ﬁsh kills due to hypoxic conditions. Additionally, as

the plant decays, it contributes to water pollution and

can create breeding grounds for disease vectors such

as mosquitoes. Its extraordinary adaptability, rapid

vegetative reproduction, and lack of natural predators

outside its native range make Eichhornia crassipes

extremely diﬃcult to control and a major threat to

freshwater habitats worldwide.

Doubles biomass in 5–15 days under optimal

conditions. Forms dense mats, blocking sunlight and

depleting dissolved oxygen (<2 mg/L).

Ecological Impact:

80% decline in native ﬁsh populations in infested

lakes (e.g., Dal Lake, Kashmir). Promotes mosquito

breeding (e.g., Anopheles spp.), increasing malaria risk.

1.2 Major Invasive Species in India: Ecological and Economic Impacts

Figure 4: Invasive catﬁsh rule reservoirs. Image courtesy:

https://www.google.com/search?client=ﬁrefox-b-d&sca esvInvasion Mechanism

Clarias gariepinus (African Catﬁsh), Origin:

Africa

Escaped from aquaculture farms into the Ganga

and Yamuna rivers. Generalist predator with high

fecundity. Clarias gariepinus (African sharptooth

catﬁsh) is considered invasive in many non-native

regions due to its aggressive predation, rapid growth,

and high reproductive capacity. Its ability to survive in

low-oxygen and polluted waters allows it to outcompete

native ﬁsh species, disrupt local ecosystems, and reduce

biodiversity. Additionally, it can hybridise with native

catﬁsh, threatening genetic diversity. Because of these

ecological impacts, it is often regulated or banned in

countries where it poses a threat to native aquatic life.

Preys on native ﬁsh (e.g., Tor putitora) and amphibian

eggs. Competes with Indian catﬁsh (Clarias batrachus).

Ecological Impact:

Clarias gariepinus, the African sharptooth catﬁsh,

negatively impacts ecosystems outside its native range

by preying on and outcompeting native species,

disrupting food webs, and reducing aquatic biodiversity.

1.2.3 Invasive Arthropods and Pathogens

Phenacoccus solenopsis (Cotton Mealybug),

Origin: North America

Phenacoccus solenopsis, commonly known as

the cotton mealybug, is a polyphagous, sap-sucking

insect pest in the family Pseudococcidae. Native

to North America, it has become a highly invasive

species and now occurs worldwide, causing severe

damage to economically important crops, especially

cotton. The adult female is oval, about 2–5 mm

Figure 5: Colony of the cotton

mealybug, Phenacoccus solenopsis.Image courtesy:

https://apps.lucidcentral.org/pppw v10/text/web full/entities/cotton mealybug 373.htm

1.3 Mechanisms of Invasion Success

Figure 6: Invasive Aedes aegypti and Aedes albopictus

mosquitoes Image courtesy: https://www.sciencedirect.com/topics/agricultural-and-

biological-sciences/aedes-aegypt

long, covered in white wax with dark stripes on the

body, and is yellowish-green underneath. The pest

multiplies rapidly, infesting over 200 plant genera,

and feeds by extracting plant sap, which leads to

yellowing, stunted growth, and sometimes plant death. P.

solenopsis is particularly problematic in warm climates

and has developed resistance to several insecticides,

making its management challenging. They have a rapid

parthenogenetic reproduction (30–45 generations/year),

and they are resistant to conventional pesticides.

Economic Impact:

8,000–10,000 crores/year losses in cotton

production (ICAR, 2019).

Aedes albopictus (Asian Tiger Mosquito) and

Aedes albopictus mosquitoes, Origin:

Southeast Asia

Aedes aegypti and Aedes albopictus can transmit

viruses to people when they bite, including dengue,

chikungunya, and Zika. Currently, only the dengue

virus has been transmitted by mosquitoes in California,

and this happens very rarely. The A. aegypti and

albopictus mosquitoes have white stripes on their legs

and a lyre-shaped (musical harp-like) pattern on their

thorax (the upper body). This white marking on a black

body is a strong visual cue for distinguishing them from

other mosquitoes, which tend to be more uniformly

coloured. A. albopictus mosquitoes can also live in a

broader temperature range and at cooler temperatures

than Ae. aegypti.

Public Health Impact:

Invasive Aedes aegypti and Aedes albopictus

mosquitoes transmit viruses such as dengue,

chikungunya and Zika, posing a huge public health

burden as well as having a less well-understood

economic impact.

Competes with Aedes aegypti, amplifying

dengue/chikungunya transmission.

1.3 Mechanisms of Invasion Success

1.3.1 Ecological Traits of Successful IAS

Selected life history (rapid reproduction, short

generation time), phenotypic plasticity (adaptability to

varied climates), the enemy release hypothesis (absence

of natural predators/parasites).

1.3.2 Human-Induced Drivers

Land-use changes (deforestation aids Lantana

spread), climate change (warming favours tropical IAS

like Parthenium).

1.4 Management Strategies

1.4.1 Preventive Measures

Stricter biosecurity under the Plant Quarantine

Order (2003).

Early Detection and Rapid Response (EDRR)

using GIS and citizen science.

1.4.2 Control Methods

1.5 Conclusion

Method Example Limitations

Mechanical

Manual removal

of water hyacinth

Labour-intensive,

high recurrence

Chemical

Glyphosate for

Parthenium

Non-target

toxicity,

resistance

Biological

Control

Zygogramma

bicolorata beetle

Slow

establishment,

host speciﬁcity

Ecological

Restoration

Replanting

native species

Requires long-

term monitoring

1.4.3 Policy Recommendations

National Invasive Species Strategy (modelled after

Australia’s framework), research funding for biocontrol

agents (e.g., mycoherbicides for Lantana).

1.5 Conclusion

IAS pose a multidimensional threat to India’s

ecosystems, food security, and public health.

A science-driven, multi-stakeholder approach—

combining prevention, eradication, and restoration is

critical for sustainable management. Future research

should focus on:

Genomic tools to predict invasiveness, Climate-

resilient biocontrol agents, and Policy enforcement to

curb illegal species trade.

Key Takeaways for Policymakers

Strengthen biosecurity laws to prevent new

introductions.

Invest in biocontrol research for sustainable

management.

Integrate IAS management with climate

adaptation plans.

References

Kohli, R.K. et al. (2006). Lantana invasion: An

ecological threat. Biodiversity & Conservation.

ICAR (2019). Economic impact of invasive pests

on Indian agriculture.

GISD (2021). Global Invasive Species Database.

https://www.sciencedirect.com/topics/agricultura

l-and-biological-sciences/aedes-aegypti

https://apps.lucidcentral.org/pppw v10/text/we

b full/entities/cotton mealybug 373.htm

https://www.google.com/search?client=firefox-b

-d&sca esvInvasionMechanism

About the Author

Geetha Paul is one of the directors of

airis4D. She leads the Biosciences Division. Her

research interests extends from Cell & Molecular

Biology to Environmental Sciences, Odonatology, and

Aquatic Biology.

Part IV

General

Entropy in Physics and Information: One

Concept, Many Realms

by Jinsu Ann Mathew

airis4D, Vol.3, No.6, 2025

www.airis4d.com

Entropy is a powerful and far-reaching concept that

lies at the heart of both physical science and information

theory. First introduced in the 19th century by Rudolf

Clausius in the context of thermodynamics, entropy

originally described the tendency of energy to spread out

or become disordered within a closed system. It helped

explain why certain processes—such as ice melting

or heat ﬂowing from a hot body to a cold one—occur

naturally and irreversibly. This early formulation of

entropy was deeply rooted in the physical world, oﬀering

a mathematical framework to capture the arrow of time

and the unavoidable march toward disorder.

However, in the mid-20th century, entropy took

on a new and profoundly abstract meaning. Claude

Shannon, a pioneer in electrical engineering and

mathematics, reimagined entropy in the realm of

communication. In his 1948 landmark paper, he deﬁned

entropy as a measure of uncertainty or information

content in a message. Shannon entropy became the

cornerstone of modern digital communication, enabling

the development of data compression, error correction,

and cryptography.

At ﬁrst glance, it may seem that thermodynamics

and data transmission have little in common.

Yet, both domains use entropy to quantify

unpredictability—whether it is the random motion of

molecules in a gas or the variability of symbols in a

text. In both cases, entropy serves as a bridge between

the known and the unknown, a metric for quantifying

disorder or surprise.

(image courtesy:https://medium.com/udacity/shannon-entropy-information-gain-and-picking-balls-

from-buckets-5810d35d54b4)

Figure 1: Water in its three states, and their respective

entropies

1.1 Entropy in Physics

Entropy in physics quantiﬁes the degree of

disorder or randomness in a system. It helps explain

why natural processes tend to evolve in a particular

direction—typically from order to disorder. In

thermodynamics, entropy is often associated with the

dispersal of energy: the more ways energy can be

arranged at the microscopic level, the higher the entropy.

A simple way to understand entropy is by

comparing the three states of matter: solids, liquids,

and gases. In a solid, particles are tightly packed in an

ordered structure and can only vibrate in place. This

limited movement means fewer possible arrangements,

so the entropy is low. In a liquid, particles are less

ordered and can slide past each other, increasing the

number of possible conﬁgurations—and hence, entropy.

In a gas, particles move freely and occupy any part of

the container, leading to a highly disordered system

with the highest entropy among the three (Fig 1).

For example, when ice melts into water, the

structured arrangement of molecules in the solid breaks

down, allowing them to move more freely. This

transition increases the system’s entropy. Similarly,

1.2 Entropy in Information Theory

when water evaporates into steam, the molecules

become even more dispersed and disordered, resulting

in an even greater increase in entropy.

From a statistical perspective, Boltzmann deﬁned

entropy using the equation:

S = k ⋅ ln W (1.1)

where S is the entropy, k is the Boltzmann constant,

and W is the number of microstates corresponding to a

macrostate. For example, when ice melts into water, the

ordered molecular structure breaks down, increasing W,

and thus S. Similarly, when water vaporizes into gas, W

increases further, reﬂecting the higher entropy of the

gaseous state.

Entropy not only explains changes in states of

matter but also underlies the irreversible nature of many

physical processes. It provides a fundamental arrow of

time, showing why systems naturally progress toward

equilibrium and why some processes—like unmixing

gases or refreezing melted ice without energy input—do

not occur spontaneously.

1.2 Entropy in Information Theory

In information theory, entropy quantiﬁes the

uncertainty or unpredictability associated with a random

variable or information source. The concept was

introduced by Claude Shannon in his seminal 1948

paper ”A Mathematical Theory of Communication”,

where he sought to understand the fundamental limits

of compressing and transmitting data. Just as entropy in

thermodynamics measures disorder in physical systems,

Shannon entropy measures the amount of ”disorder” or

uncertainty in an information source. The greater the

unpredictability of the source, the higher its entropy.

The mathematical formulation of Shannon entropy

is both elegant and intuitive. For a discrete random

variable X with possible outcomes x1, x2, x3.....xn and

corresponding probabilities P(x1), P(x2)...... P(xn), the

entropy H(X) is given by:

H(X) = −

∑

i=1

P (x

) log

P (x

) (1.2)

This formula computes the expected value of the

information content of each possible outcome. The

logarithmic term

log

P (x

)

captures the ”surprise”

or information gained when a particular event

occurs. Events with lower probability convey more

information when they occur, which aligns with our

intuitive understanding: rare events are more surprising

and hence more informative. To understand this better,

consider the simple example of a fair coin toss. A fair

coin has two outcomes—heads and tails—each with a

probability of 0.5. Plugging into Shannon’s formula,

we get:

H = −(0.5 log

0.5 + 0.5 log

0.5)

= −(0.5 × (−1) + 0.5 × (−1)) = 1 bit

(1.3)

This result implies that each coin toss yields one bit

of information. Now, imagine a biased coin where the

probability of heads is 0.9 and tails is 0.1. The entropy

becomes:

H = −(0.5 log

0.9 + 0.5 log

0.1) = 0.47bit (1.4)

Here, entropy is lower because the outcome is more

predictable—thus, each toss carries less information.

Entropy also determines the theoretical limit of

data compression. If the entropy of a message source

is 2 bits per character, then no lossless compression

algorithm can, on average, represent each character

in fewer than 2 bits. Huﬀman coding and arithmetic

coding are examples of algorithms that approach this

theoretical limit by assigning shorter codes to more

probable events. Thus, Shannon entropy provides a

baseline for what is achievable in eﬃcient data encoding.

1.3 Bridging the Two Worlds

Entropy stands as a profound and unifying concept

that transcends disciplinary boundaries, oﬀering a

shared language to describe uncertainty, complexity,

and transformation across both the physical and

informational realms. In physics, entropy helps us

understand why systems evolve toward equilibrium,

why time appears to move forward, and why certain

processes—like mixing gases or spreading heat—are

irreversible. It quantiﬁes the hidden micro-level

complexity underlying macroscopic order, revealing

how nature itself leans toward disorder not out of chaos,

1.3 Bridging the Two Worlds

but out of statistical inevitability.

In information theory, entropy captures the essence

of unpredictability in communication. It allows us to

measure how much information a message carries, how

eﬃciently it can be encoded, and how resilient it is

to noise and errors. Just as high entropy in a gas

implies a vast number of molecular arrangements, high

entropy in a message source implies greater variability

and less predictability—more “surprise” per symbol.

This analogy is not coincidental but foundational: both

systems are governed by the mathematics of probability

distributions and logarithmic scales.

Moreover, the concept of entropy has expanded

far beyond its origins. Today, it is central to

disciplines as diverse as neuroscience (to measure brain

signal variability), linguistics (to analyze language

complexity), ecology (to quantify biodiversity), and

machine learning (to regularize models and measure

uncertainty). In each case, entropy enables us to

quantify the unknown, manage uncertainty, and make

informed decisions in complex systems.

In conclusion, entropy is more than a measure of

disorder or uncertainty—it is a deep, mathematical

principle that describes how systems evolve, how

information is processed, and how knowledge itself

is structured. Whether we are probing the laws of

thermodynamics or transmitting messages across a

noisy channel, entropy guides our understanding of the

limits and possibilities of the universe. Its elegance lies

in its simplicity, and its power lies in its universality.

References

Axiomatic Relation between Thermodynamic

and Information-Theoretic Entropies

Entropy in Information Theory

Shannon Entropy, Information Gain, and Picking

Balls from Buckets

A Gentle Introduction to Information Entropy

What entropy really is : the contribution of

information theory

About the Author

Jinsu Ann Mathew is a research scholar

in Natural Language Processing and Chemical

Informatics. Her interests include applying basic

scientiﬁc research on computational linguistics,

practical applications of human language technology,

and interdisciplinary work in computational physics.

Part V

Computer Programming

Aﬃne Transformations in Convolutional

Neural Networks

by Linn Abraham

airis4D, Vol.3, No.6, 2025

www.airis4d.com

In this article we try to explore the connection

between the convolution operation that is used

in Convolutional Neural Networks (CNNs) and

mathematical operations called aﬃne transformations.

We bring out the main ideas to motivate both these

concepts and also start with some of the basic math

needed for a more rigourous understanding. Since aﬃne

transformations are a special case of the more general

concept of aﬃne maps, which belong to aﬃne spaces,

the end goal of the article is also to have a technical

understanding of an aﬃne space.

1.1 Aﬃne Transformation

The deﬁnition of an aﬃne transformation taken

from Wikipedia is as follows: “An aﬃne transformation

is an automorphism of an aﬃne space (Euclidean spaces

are speciﬁc aﬃne spaces) which preserves both the

dimensions of any aﬃne subspaces (meaning that it

sends points to points, lines to lines, planes to planes,

etc.) and the ratios of the lengths of parallel line

segments.” Figure 1 gives an intuitive idea of what

aﬃne transformations are. An aﬃne transformation

does not necessarily preserve angles between lines or

distances between points, though it does preserve ratios

of distances between points lying on a straight line.

If X is the point set of an aﬃne space, then

every aﬃne transformation on X can be represented

as the composition of a linear transformation on X

and a translation of X. From the deﬁnition of linear

transformations it can be shown that the zero vector

of the ﬁrst space is always mapped to the zero vector

of the second space. Hence translation operations

are not linear transformations because the origin of

the space gets mapped to a non-zero vector. Thus

every linear transformation is aﬃne, but not every

aﬃne transformation is linear. Examples of aﬃne

transformations include translation, scaling, similarity,

reﬂection, rotation, shear mapping and the composition

of these in any combination and sequence.

The set of all linear transformations from a space

of dimension m to another of dimension n is isomorphic

to the set of all

m × n

matrices. Whereas a non-linear

tranformation like the translation operation could be

represented by a matrix of a diﬀerent size. However

aﬃne transformations can be still be represented as

a special case of the general matrix multipication

operation with certain constraints on the elements

[Halmos(1995)].

1.2 Convolution

Aﬃne transformation enter into the discussion of

CNNs in two diﬀerent ways. Any neural network can be

seen as performing an aﬃne transformation on the input

data (vector) by multiplying it with a matrix (weight) and

translating it (adding a bias) before applying a non-linear

function (activation) to it [

Dumoulin and Visin(2016)

The second way being that many image transformations

like rotation, ﬂipping, scaling and so on can be seen

in general as aﬃne transformations of the input image.

Incidently the learning of most neural networks are not

1.3 Linear Algebra: Aﬃne Spaces

Figure 1: A visual depiction of an aﬃne transformation.

The image shows a fern-like fractal. Each of the leaves

of the fern is related to each other leaf by an aﬃne

transformation. The red leaf can be taken to the dark

blue leaf by a comination of reﬂection, rotation, scaling

and translation.

invariant to such aﬃne transforms and that is the reason

data augmentations involving such transformations are

often applied to make the network learn better.

The usual aﬃne transformations that we are

familiar with are still lacking in some sense. Natural

sources of data like images, sound recordings etc. has

some intrinsinc structure which the aﬃne transformation

do not take adavantage of. For example the data is stored

as multi-dimensional arrays where ordering of elements

along certain dimensions matter (think of the height

and width dimensions in an image or the time axis in

a sound recording). Also certain channels of data are

used to represent diﬀerent views of the same underlying

object(e.g. the red, green and blue channles of a color

image, or the left and right channles of a stereo sound

clip). This fact can be seen as the motivation for

the (discrete) convolution operation. The process of

convolution which usually involves muliplying a small

patch of the whole image with a kernel of the same size,

element-wise and adding up the values and doing it for

the whole image by using translations in a certain way,

can be represented using a single matrix. Such a matrix

is sparse in nature (only a few input units contributing

to a given output unit) and reuses parameters (the same

weights are applied to multiple locations in the input).

Thus the convolution operation can be seen as a linear

transformation, and one which additionally preserves

the notion of ordering.

1.3 Linear Algebra: Aﬃne Spaces

Ordered Pair

An ordered pair

(a, b)

is the set

{a, {a, b}}

. By

this deﬁnition, we can then show that the ordered pair

(a, b)

and

(b, a)

are not the same by just interchanging

a and b. Whereas the set

{a, b}

and the set

{b, a}

are

the same.

Cartesian Product

The cartesian product between two sets A and B

denoted

A × B

is the set of all ordered pairs (a,b) where

a belongs to A and b belongs to B.

Relation

A relation between the sets A and B is a subset of

A × B.

Map (Functional Relation or Function)

A relation

between A and B is called a map if

and only if for every a in A, there exists a unique b in B

such that (a,b) belongs to

. We say then that,

maps

a to b.

Bijective Map (Isomorphism)

A map from A to B which is both injective (one to

one) and surjective (onto) is called a bijective map or

an isomorphism. Then A and B are called isomophic

to each other.

Injective Map (one to one)

A map from A to B is called injective if for every

element in the range, there exists a unique a in A.

REFERENCES

Surjective Map (onto)

A map from A to B is called surjective if the range

is equal to the set B.

Group

A group is a set together with a map that satisiﬁes

three properties - Associativity, the presence of an

identity element (for this map), presence of an inverse

element (for this map). [Artin(1991)]

Homomorphism

If G and

′

are groups, a homomorphism

ϕ ∶ G →

′

is a map from G to

′

such that for all a and b in G,

φ(a ⋅

b) = φ(a) ⋅

′

φ(b)

Note that the product on the left side takes place in

G, while the product on the right hand side takes place

′

. Thus the above equation gives a relation between

these two binary operations and hence between the two

group structures. [Fraleigh and Katz(2003)]

Tuple

A tuple is an ordered list of objects.

Permutation

A bijection from a set to itself is also called a

permutation. [Berger(2009)]

Symmetric Group

The set of all permutations of a set X under the

composition of maps:

fg = f o g

is a group called the

symmetric group of X denoted S

Group Action

Let G be a group and X a set. A group action on

X is a homomorphism

ϕ ∶ G → S

. If

is a G-action

on X, we say that G acts or operates on X (by ϕ).

Faithful Action

The action

(G, X, ϕ)

is called faithful if

injective (in other words, if only

e ∈ G

maps to the

identity Id

Transitive Action

The action

(G, X, ϕ)

is called transitive if for every

x, y in X there exists g ∈ G such that g(x) = y.

Aﬃne Space

An aﬃne space over a ﬁeld K is a faithful and

transitive group action

(X, X, ϕ)

, where X is the

underlying set of points,

is a vector space over

K considered with its additive group structure and

a group action. The vector space

is said to underlie

the aﬃne space X. [

Schuller(2017)

] [

Halmos(1974)

]

[Krohn(2020)]

References

[Halmos(1995)]

Paul R. Halmos. Linear Algebra

Problem Book. Number 16 in The Dolciani

Mathematical Expositions. Mathematical

Association of America, Washington, DC, 1995.

ISBN 978-0-88385-322-1.

[Dumoulin and Visin(2016)]

Vincent Dumoulin and

Francesco Visin. A guide to convolution arithmetic

for deep learning. arXiv:1603.07285 [cs, stat],

March 2016.

[Artin(1991)]

Michael Artin. Algebra. Prentice Hall,

Upper Saddle River, N.J, 1991. ISBN 978-0-13-

004763-2.

[Fraleigh and Katz(2003)]

John B. Fraleigh and

Victor J. Katz. A First Course in Abstract Algebra.

Addison-Wesley, Boston, 7th ed edition, 2003.

ISBN 978-0-201-76390-4.

[Berger(2009)]

Marcel Berger. Geometry. 1.

Universitext. Springer, Berlin Heidelberg, corrected

4. print edition, 2009. ISBN 978-3-540-11658-5.

REFERENCES

[Schuller(2017)]

Frederik Schuller. Lectures

on geometrical anatomy of theoretical physics.

https://www.youtube.com/playlist?list=PLPH7f 7ZlzxTi6kS4vCmv4ZKm9u8g5yic,

2017.

[Halmos(1974)]

P. R. Halmos. Finite-Dimensional

Vector Spaces. Undergraduate Texts in

Mathematics. Springer-Verlag, 1974. ISBN 0-387-

90093-4 978-0-387-90093-3.

[Krohn(2020)]

Jon Krohn. Aﬃne transformations

— topic 27 of machine learning foundations.

https://www.youtube.com/watch?v=6H-fSbV-Jzw,

2020.

About the Author

Linn Abraham is a researcher in Physics,

specializing in A.I. applications to astronomy. He is

currently involved in the development of CNN based

Computer Vision tools for prediction of solar ﬂares

from images of the Sun, morphological classiﬁcations

of galaxies from optical images surveys and radio

galaxy source extraction from radio observations.

About airis4D

Artiﬁcial Intelligence Research and Intelligent Systems (airis4D) is an AI and Bio-sciences Research Centre.

The Centre aims to create new knowledge in the ﬁeld of Space Science, Astronomy, Robotics, Agri Science,

Industry, and Biodiversity to bring Progress and Plenitude to the People and the Planet.

Vision

Humanity is in the 4th Industrial Revolution era, which operates on a cyber-physical production system. Cutting-

edge research and development in science and technology to create new knowledge and skills become the key to

the new world economy. Most of the resources for this goal can be harnessed by integrating biological systems

with intelligent computing systems oﬀered by AI. The future survival of humans, animals, and the ecosystem

depends on how eﬃciently the realities and resources are responsibly used for abundance and wellness. Artiﬁcial

intelligence Research and Intelligent Systems pursue this vision and look for the best actions that ensure an

abundant environment and ecosystem for the planet and the people.

Mission Statement

The 4D in airis4D represents the mission to Dream, Design, Develop, and Deploy Knowledge with the ﬁre of

commitment and dedication towards humanity and the ecosystem.

Dream

To promote the unlimited human potential to dream the impossible.

Design

To nurture the human capacity to articulate a dream and logically realise it.

Develop

To assist the talents to materialise a design into a product, a service, a knowledge that beneﬁts the community

and the planet.

Deploy

To realise and educate humanity that a knowledge that is not deployed makes no diﬀerence by its absence.

Campus

Situated in a lush green village campus in Thelliyoor, Kerala, India, airis4D was established under the auspicious

of SEED Foundation (Susthiratha, Environment, Education Development Foundation) a not-for-proﬁt company

for promoting Education, Research. Engineering, Biology, Development, etc.

The whole campus is powered by Solar power and has a rain harvesting facility to provide suﬃcient water supply

for up to three months of drought. The computing facility in the campus is accessible from anywhere through a

dedicated optical ﬁbre internet connectivity 24×7.

There is a freshwater stream that originates from the nearby hills and ﬂows through the middle of the campus.

The campus is a noted habitat for the biodiversity of tropical Fauna and Flora. airis4D carry out periodic and

systematic water quality and species diversity surveys in the region to ensure its richness. It is our pride that the

site has consistently been environment-friendly and rich in biodiversity. airis4D is also growing fruit plants that

can feed birds and provide water bodies to survive the drought.