**13.
Mr. Raj Kumar : STUDIES ON CODES AND GRAPHS WITH ROSEBLOOM-TSFASMAN METRIC **

The thesis titled
"Studies on codes and graphs with Rosenbloom-Tsfasman Metric" submitted by Mr.R.Rajkumar makes a
significant contribution to the theory of error correcting codes and distance graphs. The thesis
mainly focuses on the role played by the RT metric-introduced by Rosenbloom and Tsfasman in diverse
areas, namely coding theory, graph theory and interconnection networks. The RT metric is known for
its usefulness in case of interference in several consecutive channels in coding theory literature.
And there has been a considerable amount of study going on in this direction in recent years.
The work presented by Mr. Rajkumar proceeds in making a good contribution to the advancement of
knowledge in this field.
* Overall, the thesis is well-structured, easy to read, and the results are correct and clearly
stated. In summary, this thesis is a significant contribution to coding and graph theory that opens
up possible applications to interconnection networks. The candidate exhibits skill and made substantial
advancement of knowledge in this field. Hence without hesitation I strongly commend the thesis for the
award of the Ph.D. degree.
*.

**--Dr.R.S.Selvaraj, Assoc. Prof, NIT Warangal, Andhra Pradesh, India.**

The thesis is dedicated to the theoretical properties and applications of the Rosenbloom
and Tsfasman metric in several very important areas algebraic coding theory, metric graph
theory and interconnection networks. A number of novel concepts are introduced and explored.
For example, a new class of circulant RT distance codes and the notion of fuzzy RT distance
codes are defined and investigated. *
In my opinion, this is the very best doctoral thesis among the many theses that I have evaluated.
The results obtained in the thesis have been published in eight high-quality journal articles and
have been presented at seven conferences. This work belongs to the intersection of several research
directions. All of these directions are very important and the results obtained in the thesis are
excellent. They clearly demonstrate that R. Rajkumar has acheived high level of expertise in all of
these branches and is well qualified for productive work in these areas. The text in the present
form is well-written and constitutes enjoyable and enlightening reading. This is why the thesis in
its present form is highly commended for the award of the PhD degree. *

**--Prof. Andrei Kelarev, Dept. of Computing and Maths., Univ. of Tasmania,
Tasmania, Australia.**

**12.
Mr. K. Paramasivam: COLORING OF ZERO-DIVISOR GRAPH OF SOME SPECIAL ALGEBRAIC
STRUCTURES**

The
researcher has tried to discuss the zerodivisor graphs of some special class
of loopring using Z2 and Z and their chromatic numbers are determined in chapter
4. Fifth chapter deals with the concept of generalized k-strongly edgemagic
labeling. For this, necessary and sufficient conditions have been applied
to get the desired results. On the whole the thesis provides sufficient evidence
of the candidate’s capacity for originality and capability of carrying
out research work independently. *The thesis is satisfactory in point of
language and presentation of the subject matter. In my view, the thesis can
be approved for the award of Ph.D. degree in mathematics in the present form*.

**--Prof. Bijendra Singh, S.S. in Mathematics, Vikram University, Ujjain,
India.**

*The thesis
contains a detailed exposition of the research work undertaken by the candidate.*
It is devoted to the important notion of a zero-divisor graph, which falls
neatly into the intersection of graph theory and universal algebra. The thesis
begins with a thorough and comprehensive overview of the required preliminaries
and previous results obtained by scientists who have contributed to this research
direction.* Subsequent chapters contain new facts and techniques discovered
by the candidates. Complete proofs are given to new theorems obtained by the
author of the thesis. The Thesis contains many new interesting results and
is strongly recommended for the award of Ph.D. degree.*

**--Prof. Andrei Kelarev, Dept. of Computing and Maths., Univ. of Tasmania,
Tasmania, Australia.**

**11.
Mr. T. Johnson: SOME NEW FUZZY DATA ANALYSIS MODELS AND THEIR APPLICATIONS**

This
thesis is a considerable work covering ideas of the Data analysis models using
fuzzy techniques. The author has taken ideas from the existing fuzzy models
and generalized to some interesting new fuzzy models. . . The theis provides
appropriate illustrations emphasizing on the social and real world problems
as a whole. The formulation of the thesis is correct and well planned. I recommend
the thesis for the award of the degree of doctor of philsosophy. **Dr
Kuncham Syam Prasad, Dept. of Maths, Manipal Institute of Technology, Manipal
Academy of Higher Education, Manipal, India **

*The thesis
is devoted to novel applications of important concepts of fuzzy set theory
and is focusing on data processing. The kind of data considered in the thesis
is particularly difficult to analyze using previous traditional methods. It
is remarkable that the choice of fuzzy set theory has turned out more meaningful
and appropriate for tackling broad classes of problems including social and
psychological problems. The classes considered are very broad and can be also
applied in engineering and to handle scientific q**uestions. The thesis
introduces several new models, all of these are novel and esstenial contributions
seriously advancing this direction. In my opinion the thesis is very well
written and should be commened for the award of the Ph.D degree. *

**--Prof. Andrei Kelarev, Dept. of Computing and Maths., Univ. of Tasmania,
Tasmania, Australia.**

**10.
Mr. R.S.Selvaraj: METRIC PROPERTIES OF RANK DISTANCE CODES**

The thesis is
self content and well presented. It contains many new ideas and orginal results.
The coding theory is becoming more and more important in communication. This
way he as developed the thesis shows that the author reveals a complete mastery
over the subject and the author is familiar with contemporary research on
RD codes. Hence, I strongly feel that the one must congratulate the supervisor
and the student for this valuable work.

I have great pleasure in recommending that R.S.Selvaraj may be awarded the
Ph.D degree in Mathematics and the thesis may be regarded as Highly commended
**--Prof. Kirangi, Dept. of Math., Univ. of Mysore, Mysore, India. **

*This thesis
provides several new results on rank distance codes. The work presented by
Selvaraj proceeds in good tradition with previous work conducted by Dr(Mrs)
Vasantha and her collaborators. The thesis is well-written, well-structured,
and the results are clearly stated. The candidate exhibits skill to solve
substantial mathematical problems (in particular, Chapter 2, 4 and 5). Based
on this thesis, the candidate deserves to receive the Ph.D in Mathematics.***-
Prof. Karl-Heinz Zimmermann, Technical Univ. of Hamburg-Harburg, Dept. of
Comp. Elec. Engineering, Hamburg, Germany.
**

I
must congratulate the supervisor and the candidate for one of the good theses
in Algebra. They have introduced new definitions, proved many results in detail
and suggested interesting research problems in their thesis. The thesis not
only studies the special elements of both associative rings and non-associative
rings, it also introduces several new Smarandache special elements and using
these new Smarandache algebraic structures have been introduced. To summarize,
the thesis is self-contained, theorems are proved with all details. Further,
it contains many examples and counter examples, which clarifies the theory.
Contains many new ideas and opened the channel of S-rings. The author is familiar
with contemporary research on related topics. The thesis makes a valuable
contribution to the field of Algebra. **--Prof. Kirangi, Dept. of Math.,
Univ. of Mysore, Mysore, India. **

*In
this thesis, Moon Kumar Chetry introduces and studies about Smarandache special
elements in rings. It is rare to see a thesis, which carries out stufy in
both associative and non-associative rings. He uses number theoretic techniques
in solving the problems. The concepts are very clearly illustrated by examples,
which happens to be a positive point, as these examples not only help in understanding
the theorems but also makes one concretely understand the definitions and
results. This thesis is well presented... Prof.Dr.Florentin Smarandache,
Department of Mathematics, University of New Mexico, Gallup, USA. *

**8.
Mr. R. S. Rajadurai: ON LINEAR CODES WITH RANK METRIC CONSTRUCTIONS, PROPERTIES,
AND APPLICATIONS**

One
just can't think of the world without information. Today it is mainly digital,
but it is prone to error. Any new method which detects and corrects errors,
maintains secrecy is appreciated. This thesis is on those lines. The Introduction
and preliminaries are well presented and it makes the thesis self-contained.
The Chapter II introduces combined error-erasure decoding technique for the
first time to the class of [n,k,d] and MRD codes. Most of the results are
worth mentioning and in particular are theorem 2.4.11 (p.52) and example 2.2.1
(pp. 31-35) really deserve to be appreciated. Good contribution to the field.
The chapter V deserves to be appreciated as it presents some of the possible
applications and further scope for research. The author has mastered the works
of many eminent scholars. Most of the examples add clarity of the theme. For
this the thesis supervisor and the candidate deserve to be congratulated.
Without any hesitation, I strongly recommend for the award of the Ph.D. degree
to R.S. Raja Durai in Mathematics. **--Prof. Kirangi, Dept. of Math.,
Univ. of Mysore, Mysore, India. **

A well-written thesis […] The last section on applications is a persuasive
summary of the importance of some of the results in the thesis. In sum, the
number and the significance of the results appear to be suitable for the award
of Ph.D. **-– Prof. Hema Srinivasan, Dept. of Mathematics, University
of Missouri, Colombia, MO 65211, USA.**

**7.
Ms. S. Ramathilagam: MATHEMATICAL APPROACH TO THE CEMENT INDUSTRY PROBLEMS
USING FUZZY THEORY**

Keeping
in mind the problems faced by the Cement Industry in the country and recalling
the previous works done by various authors, S. Ramathilagam made good attempt
to solve some of the problems faced by the Cement Industry by using new techniques
and infact these were tested. [...] Over all the candidate made substantial
survey and colleced enough data related to the work and analyzed the data
by using new techniques to suggest the cement industry to mitigate its hardships
in running the industry more effectively. As the work suggests various benefits
like increase in production, reduction in power consumption, consistent quality,
reduction in mill stoppages, etc. this can be considered as one of the most
useful works either directly or indirectly to the cement industry. -- **Prof.
G. L. Reddy, Dept. of Maths. and Computer Science, Univ. of Hyderabed, AP,
India.**

*She
has researched the earlier models and have incorporated them in her work using
Fuzzy matrices. The thesis gives a nice introduction to Fuzzy Theory in teh
beginning which would be useful to an outsider. From the rest of the thesis
it is clear that Ms. Ramathilagam has obtained an expert knowledge in Fuzzy
logic and techniques from Fuzzy theory. Techniques from Fuzzy logic have already
become the norm dealing with many difficult or imprecise problems of the industry.
Ms. Ramathilagam's research appears to be a very good addition to our existing
knowledge in that field. Much of the results in the thesis has been presented
in professional conferences as well as published or submitted for publishing
in journals. The amount of work is certainly quite substanial. -- ***Prof.
Hema Srinivasan, Dept. of Maths., Univ. of Missouri, Colombia, USA.**

**6.
Mr. S. R. Kannan: MATHEMATICAL APPROACH TO THE CHEMICAL INDUSTRY PROBLEMS
USING FUZZY THEORY**

The
thesis is well written. The candidate has taken up the problems which are
very relevant to the present day situation. -- **Prof.
P.Dheena, Dept. of Maths, Annamalai Univ. TN, India.**

*The
thesis contains significant new results devoted to the development of new
valuable practical mathematics algorithms using the novel methods of fuzzy
theory and neural networks. The results have been published in several good
publications, and the overall amount of work is significant by all international
standards. I am sure that the methods developed by the applicant will find
applications also in other branches of industry. The is very well written,
and I strongly recommend awarding the degree without making any changes to
the text of the thesis. -- ***Prof.
Andrei Kelarev, Dept. of Maths., Univ. of Tasmania, Tasmania, Australia.**

**5.
Ms. R. Sujatha: CONSTRUCTION OF CONCATENATION AND ERASURE TECHNIQUES FOR THE
CLASS OF MRD CODES AND THEIR APPLICATIONS**

I
have gone through the thesis with great interest. The thesis is well-written
and the exposition is good. I recommend the thesis for the award of the Ph.D.
degree. -- **Prof. Loganathan, Dept. of Maths.,
RIASM, Univ. of Madras, India.**

*The
study of Rank Distance codes (RD codes) is a newer branch of algebraic coding.
RD codes have been invented by E.M.Gabidulin. He has studied so called MRD
codes, a subclass of RD codes which attain the Singleton bound. Another major
contribution of RD codes has been given by N.Suresh Babu, a former student
of IIT, Madras. The thesis of R. Sujatha is a major contribution to RD codes...
The results may have some practical relevance in the future. In particular,
the guessing technique for MRD codes provides a very nice example of the strength
of RD codes... These codes may be useful in byte-organized memories. This
sort of construction is very original. *--
**Prof. Karl-Heinz Zimmermann, Techical Univ.
of Hamburg-Harburg, Dept. of Comp. Elec. Engineering, Hamburg, Germany. **

**4.
Ms. Indra Venkatbabu: MATHEMATICAL APPROACH TO THE PASSENGER TRANSPORTATION
PROBLEM USING FUZZY THEORY**

Indra's
thesis contributes decisively to the study of the passenger transportation
problem using Fuzzy Theory... On the other hand, good analysis is made to
present the transportation problem solely for the operator's benefit in such
a way that perfect balance was obtained both from the operator's as well as
from the passenger's point of view... The entire study of the transportation
problem was presented lucidly and satisfactorily. -- **Prof.
G. Lakshma Reddy, Dept. of Comp. Sci and Math., Univ. of Hyderabad, Hyderabad,
India. **

*This
is a very good thesis at a very high degree difficulty particularly because
it combines mathematical modeling and analysis with empirical work, which
can be found rather solemnly in a Ph.D. thesis. The work is undoubtedly of
such a quality that it makes significant contributions to the advancement
of knowledge and I recommend it by all means to accept this thesis for the
award of the Ph.D. degree. The combination of computer programming, emperical
work (working on a real problem as well as acquiring real data) and mathematical
modeling also normally requires much more work than in either purely mathematical
or purely emperical research work. I have also checked the programs in the
world wide web and I am impressed by the work...* -- **Prof.
Dr. Dr. h.c.H-J. Zimmermann, (Author: Fuzzy Sets and Systems), RWTH,
Aachen, Germany. **

**3.
Mr. D. Meiyappan: STUDIES ON FUZZY SUBGROUPS**

The
author and the student deserve to be congratulated for their detailed study
of the concept and beautiful presentation of the results... the thesis makes
an valuable contributiin to the theory of fuzzy subgroups.-- **Prof.
Kirangi, Dept. of Math., Univ. of Mysore, Mysore, India. **

*Fuzzy
sets and groups is an interesting and important topic of current research
in algebra. It is a topic that is perhaps at a relatively early age of deverlopment
and this makes it especially suitable for a thesis. I highly commend Professor
Vasantha for her ingenuity in guiding her students in this interesting and
fruitful direction. In summary this is a very well written thesis which contains
a number of interesting results on an important topic.*-- **Prof.
Hema Srinivasan, Dept. of Math., Univ. of Missouri-Columbia, USA. **

**2.
Mr. N. Suresh Babu: STUDIES OF RANK DISTANCE CODES**

This
thesis gives a detailed analysis of error correcting codes in rank metric
which is a very effiecient metric in the sense that it can be utilized for
error correcting codes defined over any higher dimensional Galois field wherein
usual Hamming metric is inappropriate. In conclusion, I would like to say
that this thesis is an excellent piece of work in coding theory. Hence, this
thesis is commended for the award of the Ph.D. degree. -- **Prof.
M. Vijayalakshmi, Dept. of Math., RIASM, Univ. of Madras, India.**

*This
thesis makes a significant contribution to the theory of error correcting
codes. Indeed the thesis contains several very important results: a) the non-existence
theorem of perfect rank distance codes over GF(2 ^{N}) which is certainly
one of the truimphs of the theory...--*

**1.
Mr. Satya Veer Singh : ON A NEW CLASS OF LOOPS AND LOOP RINGS**

The
result seems to be new and interesting... Finally I may say that this is a
reasonably good thesis. *--***
Prof. R.K.Sharma, Dept. of Math., IIT Kharagpur, WB, India.**

*This
thesis studies a special type of loops and motivates the reader by providing
an application...--
***Prof.
M. Satyanarayana, Dept. of Math., Univ. of California, USA. **